Osiris Wavelets in Three Dimensions

Annals of Physics 286, 23107 (2000) doi:10.1006aphy.2000.6081, available online at http:www.idealibrary.com on

Osiris Wavelets in Three Dimensions Guy Battle Department of Mathematics, Texas A 6 M University, College Station, Texas 77843-3368 Received March 20, 2000

We extend to three dimensions an unusual wavelet construction that we first introduced in a two-dimensional context (in ``Wavelet Transforms and TimeFrequency Signal Analysis'' (L. Debnath, Ed.), Birkhauser, Basel, in press). This is part of an on-going program to analyze critical behavior in classical equilibrium statistical mechanics. Following Golner's general idea of using an incomplete multiscale set of functions (1973, Phys. Rev. B 8, 339) to obtain more realistic modeling that is still hierarchical, we introduce a wavelet set whose mother wavelets are continuous, piecewise-linear functions supported in the unit cube. Such a wavelet set is necessarily incomplete, but in three dimensions we have packed seven mother wavelets into the unit cubefour based on the 8 sub-cubes, and three based on the 12 octahedra that intersect adjacent sub-cubes. The generated wavelet set is not Sobolevorthogonal, but we derive a positive lower bound on the multi-scale Sobolev overlap matrix.  2000 Academic Press

1. INTRODUCTION In a previous paper we raised the possibility of using an incomplete set of wavelets as a new hierarchical approximation for the study of critical behavior in classical statistical mechanics [1]. Since the energy norm in this context is a Sobolev norm, the hierarchical approximations in the past have been based on modifications of the lattice Laplacian. The standard idea is to throw away the nearest-neighbor couplings in the fluctuations when the renormalization group transformation is applied [24], and this approach has been successful up to a point. The fundamental problem has been that the critical exponent ' automatically vanishes for such modeling. An alternate way to understand this is to apply Wilson's point of view [5]i.e., to regard the hierarchical approximation as a modification of the fluctuations instead. While his formulation was not intended to be mathematically rigorous, it provided more insight than the other point of view. It suggested other modifications of the fluctuations that would admit a non-zero value for ' and yet preserve the hierarchical nature of the approximation, which was so essential to the tractability of the computational approach. This program was pioneered by Golner [6] with some degree of success. Our mathematical work is inspired by his achievement. The GolnerWilson work in the study of critical phenomena was done over 25 years ago, and one fascinating aspect of their approach is that it anticipates wavelet analysis. The modification of the fluctuations in a renormalization group transformation is implemented by a multi-scale expansion of continuum configurations, 23 0003-491600 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.

24

GUY BATTLE

where the expansion functions are assumed to have rather drastic properties. Both the Wilson expansion functions and the Golner expansion functions are necessarily fictitious, but the latter set of expansion functionsor waveletscannot possibly be complete, even from a qualitative point of view. Simple multi-scale counting of the degrees of freedom involved in the Golner fluctuations shows that the wavelets do not cover all degrees of freedom, as we have pointed out in [1]. Yet Golner obtained reasonable values for the critical exponents. From a field-theoretic point of view, the Golner wavelet set can be regarded as a cutoff on the continuum configurations that is neither infrared nor ultraviolet, but multi-scale. We are currently pursuing a mathematically precise version of this modeling. The name of the game is to construct wavelets that are sharply localized in the blocks to which they are associated. Since the Hilbert space inner product is given by

|

( f, g) = { f } {g,

(1.1)

it is clear that the simplest wavelets possible must be piecewise linear and continuous. This means that the multi-scale rectangular lattice structure on which the wavelets are defined must be enhanced by multi-scale triangulation. We constructed such a wavelet set in two dimensions [1]. There were four mother wavelets 9 1 , 9 2 , 9 3 , and 9 4 i.e., four wavelets associated with the unit square [0, 1] 2. The wavelet set is generated by the mother wavelets as (

(

( ), 9 rm( @(! )=9 @(2 &r! &m

( m # Z 2,

r # Z,

(1.2)

where the normalization

| ({9 ) =1 2

@

(1.3)

is obviously preserved. In general, the amplitude in (1.2) would require a scale factor for the normalization condition, but for this Sobolev norm, the scale factor is unity in two dimensions. Since the mother wavelets 9 @ are sharply localized in ( . Note that our [0, 1] 2, the wavelets 9 rm( @ are sharply localized in [0, 2 r ] 2 +2 rm convention is to assign the positive values of r to large length scales and the negative values of r to small length scales. Thus, 9 rm( @ is an r-level wavelet or, alternatively, a 2 r-scale wavelet. With this degree of coherence, the properties of the wavelet set obviously depend very much on the construction of the mother wavelets. We have already stipulated the piecewise-linear continuity property, and in [1] we based 9 1 and 9 2 on the first and second triangulations shown in Fig. 1, respectively. In each case, the function is zero on the two sub-squares that are not triangulated, and has a constant gradient on each triangle in the other two squares, where the directions of the gradients are given by the arrows. On the other hand, 9 3 and 9 4 are based on the first and second triangulations shown in Fig. 2, respectively. In each of these cases, the function is zero on the large triangles and on the two diamonds that are not

OSIRIS WAVELETS IN THREE DIMENSIONS

25

FIGURE 1

triangulated. On the triangulated diamonds, the arrows have the same meaning as in Fig. 1. For obvious reasons, we named these functions Osiris wavelets. The generated set of wavelets is not complete, and the hierarchical approximation introduced in [1] is defined as the Sobolev-orthogonal projection of continuum configurations onto the span of this wavelet set. On the other hand, the wavelet set is not orthonormal, either, and the issue here is that the expansion of a hierarchical configuration in the wavelets must be unique and stable. However, this condition is satisfied. Indeed, we found a positive lower bound on the overlap matrix when we investigated the partial orthogonality of this wavelet set in our previous paper [1]. The purpose of this paper is to construct Osiris wavelets in three dimensions and establish a positive lower bound on the multi-scale overlap matrix. Naturally, our construction will not be as simple as it was in two dimensions. For example, the analog of sub-dividing a square into four isosceles triangles is to sub-divide a cube into six pyramids. On the other hand, the analog of sub-dividing a diamond (which is similar to a square) into four isosceles triangles is to sub-divide an octahedron (which is not similar to a cube) into eight simplexes. Accordingly, we base one kind of wavelet on continuous, piecewise-linear functions described by the diagram shown in Fig. 3, where the arrows represent the constant gradients in each pyramid, and we have feathered the arrows in the front and back pyramids in the interests of three-dimensional perspective. All six of these gradients are normal to their respective cubic faces, where the function vanishes. As long as these six gradients have the same magnitude, this function is obviously continuous across the pyramidal faces inside the cube. We base the other kind of wavelet on functions described by the diagram shown in Fig. 4, where the arrows have the same meaning as in Fig. 3, and we have feathered those arrows that are normal to the faces of the back pyramid. The octahedron consists of two pyramids with a common base, just as the

FIGURE 2

26

GUY BATTLE

FIGURE 3

cube consists of six pyramids with a common apex. Each pyramid of the octahedron is sub-divided into four simplexes. If the constant gradients on these simplexes have the same magnitude, then our piecewise linear function is continuous across the simplicial faces inside the octahedron. Consider the unit cube [0, 1] 3 and partition it into the eight sub-cubes [0, 12 ] 3 + 12 =( with (= # [0, 1] 3. A sub-cube with the associated sub-division into pyramids is a basic cube. The continuous, piecewise-linear function on a basic cube defined by Fig. 3 is a basic C-function. Four of our mother wavelets are linear combinations of such functions. These wavelets are schematically given by the diagrams shown in Figs. 5.0.05.1.1, where the \ signs indicate whether the basic C-function is added or subtracted. These four mother wavelets are mutually orthogonal because their supports are mutually disjoint. Now consider the square face common to any pair of adjacent basic cubes. An octahedron of the type we have discussed is certainly obtained by joining the two pyramids having that square face as baseone from each basic cube. Such an octahedron with the associated sub-division into simplexes is a basic octahedron. Clearly, there are 12 basic octahedra in the unit cube, and they are mutually disjoint. The continuous, piecewise-linear function on a basic octahedron defined by Fig. 4 is a basic O-function. We define three mother wavelets as linear combinations of such functionsschematically given by the diagrams shown in Figs. 6.16.3, where each basic octahedron is represented only by the square face common to the two basic cubes that

FIGURE 4

OSIRIS WAVELETS IN THREE DIMENSIONS

FIGURE 5.0.0

FIGURE 5.0.1

FIGURE 5.1.0

FIGURE 5.1.1

27

28

GUY BATTLE

FIGURE 6.1

contribute pyramids to the octahedron. The \ signs indicate whether the basic O-function is added or subtracted. While these three wavelets are not mutually orthogonal, it is visually obvious that each one is orthogonal to all four of the wavelets defined by Fig. 5. The unit cube together with all of the interior structure we have introduced is a basic block. Let 9 +& denote the mother wavelet given by Fig. 5.+.&, where we have chosen the normalization

| ({9

+&

) 2 =1.

(1.4)

Let ' @ denote the mother wavelet given by Fig. 6.@, where we have chosen the normalization

| ({' ) =3.

(1.5)

} {9 +$&$ =$ ++$ $ &&$ ,

(1.6)

2

@

Then

| {9

+&

| {9

+&

} {' @ =0,

|

(1.7)

| {'

1

} {' 2 = {' 2 } {' 3 =1,

(1.8)

| {'

1

} {' 3 =&1.

(1.9)

FIGURE 6.2

OSIRIS WAVELETS IN THREE DIMENSIONS

29

FIGURE 6.3

It is easy enough to calculate the eigenvalues of the 7_7 overlap matrix for the mother wavelets. The matrix has eigenvalues 1 and 4 with multiplicities 5 and 2, respectively. The wavelet set is generated by the three-dimensional version of (1.2). For each r # Z, the r-level wavelets are given by (

(

( 9 rm( ; +&(! )=2 &r29 +&(2 &r! &m ), (

' rm( @(! )=2

&r2

' @(2

&r (

(

! &m ),

(+, &) # [0, 1] 2,

(1.10)

@=1, 2, 3,

(1.11)

( # Z 3, where the scale factor for the amplitudes preserves the normalization with m conditions (1.4) and (1.5). As in the two-dimensional case, our wavelet set is incomplete, and orthogonality properties are the focus of our investigation. Our analysis in three dimensions is much more complicated, however. The two-dimensional wavelet system has a couple of nice properties that simplified our analysis in [1]. The most important property was that a given wavelet is orthogonal to any other wavelet on a different level, provided the two levels (length scales) are not adjacent. Our three-dimensional wavelet system simply does not have this property. We shall have to deal with non-zero inner products between wavelets that are arbitrarily separated with respect to their length scales. On the other hand, we shall prove that

| {9

( ; +& rm

} {' sn(@ =0,

s
(1.12)

This is not obvious until all cases are checked, but it motivates our definition of ' 1 , ' 2 , ' 3 , which may have seemed too elaborate at first glance. Another property of the two-dimensional wavelet set is that the subset generated by the mother wavelets defined by Fig. 1 is actually orthogonal. The mother wavelets defined by Fig. 5 are the three-dimensional analogs, but unfortunately, the generated wavelets given by (1.10) are not mutually orthogonal at all. Indeed, we shall see that the inner products among these wavelets are the most significant offdiagonal contributions to the overlap matrix. It is ironic thatas our analysis showsthese wavelets are orthogonal in the special case where they lie on adjacent levels. Thus, the only non-zero inner products between adjacent levels must involve

30

GUY BATTLE

an ' @ -wavelet on the larger length scale and a 9 +& -wavelet on the smaller length scale. Actually, the adjacent-level inner products will have to be analyzed separately because the associated case structure differs from that of the more general scenario. As far as the more general scenario is concerned, the calculation of Sobolev inner products will involve a few geometric cases. One must consider the ways in which the inclined planes associated with a larger-scale wavelet can divide the basic block associated with a smaller-scale wavelet. The cases can be illustrated as shown in Figs. 7.17.4. When we formally introduce these cases, we shall refer to the divider in Fig. 7.1 as a simple divider, the divider in Fig. 7.2 as a Type C compound divider, and the divider in Fig. 7.3 as a Type O compound divider. The divider in Fig. 7.4 consists of two Type O compound dividers that have a common face. A simple divider can arise in three different ways: it may be part of a face of a larger-scale octahedron, it may be part of one of the inclined planes dividing a larger-scale octahedron into simplexes, or it may be part of one of the faces common to two pyramids in a larger-scale cube. A Type C compound divider, on the other hand, is induced only by a larger-scale basic C-function, while a Type O compound divider is induced only by a larger-scale basic O-function. Figure 7.4 covers the case where two basic O-functions contributing to the same larger-scale ' @ -wavelet induce dividers on the same smaller-scale basic block. For convenient representation of the overlap matrix, we order our seven mother wavelets with the notation 9 0 , 9 1 , ..., 9 6 by setting 9 +& =9 2++& ,

(1.13)

' @ =9 3+@ .

(1.14)

The overlap matrix for the Sobolev inner product is given by

|

S rr$; m(m($; }}$ = {9 rm(} } {9 r$m($}$ .

(1.15)

The systematic calculation of these matrix elements is at the heart of this paper. Our goal is the operator inequality 1 4 S1& 18 - 2& 16 - 3& 24 - 6(182)

r0.159677.

(1.16)

The operator-theoretic estimation that supplements our calculations is based on a couple of standard lemmas [7]. Suppose X is a bounded operator on a Hilbert space such that &X&; and &X k&: k&1;,

k=1, 2, 3, ...

(1.17)

for some :<;. Then &X+X*&:+;.

(1.18)

OSIRIS WAVELETS IN THREE DIMENSIONS

FIGURE 7.1

FIGURE 7.2

FIGURE 7.3

FIGURE 7.4

31

32

GUY BATTLE

We cover the proof here for convenience. Since X+X* is a bounded, self-adjoint operator, we have &X+X*& N =&(X+X*) N &

(1.19)

for every positive integer N. On the other hand (with X 1 =X ) &(X+X*) N &

&X *1 X *2 } } } X *N &,

:

(1.20)

* # [V, 1] N

and there is precisely a two-to-one correspondence between [V, 1] N and the subsets B/[1, ..., N&1]. Indeed, we may define B * as the set of all k such that * k+1 {* k .

(1.21)

Clearly, a given subset B arises from two *'sone for which * 1 =V and one for which * 1 =1. If we isolate the intervals of X-factors and the intervals of X*-factors with &PQ&&P& &Q&

(1.22)

&(X*) k&=&X k&,

(1.23)

and then apply

we can reduce the bound to a sum over the subsets B. Thus N&1

n

&(X+X*) N &2 : n=0

B

B

` &X k i+1 &k i &,

:

(1.24)

B/[1, ..., N&1] i=0 card B=n

where k Bi denotes the i th integer in B in order of size. This holds if we have the understanding that k Bn+1 =N and k B0 =0 when we sum over the B with fixed cardinality n. Now by (1.17) we have n

B

n

B

B

B

` &X k i+1 &k i & ` (: k i+1 &k i &1 ;) i=0

i=0

=; n+1: N&n&1.

(1.25)

Since the inner sum is now only counting the n-cardinality subsets B, it follows that N&1

&(X+X*) N &2 : n=0

\

N&1 ; n+1: N&n&1 n

+

=2;(:+;) N&1.

(1.26)

OSIRIS WAVELETS IN THREE DIMENSIONS

33

Combining this with (1.19), we see that N &X+X*& 2; (:+;) 1&1N,

(1.27)

and so (1.18) is obtained in the limit as N Ä . The other lemma that we shall use is a special variation on the first. Suppose X is a bounded operator and X 2 =0. Then &X+X*&&X&.

(1.28)

The proof is a simplification of the argument just given.

2. CUBES AND OCTAHEDRA Recall that when we introduced Osiris wavelets in the two-dimensional context [1], the most fundamental geometric objects used in our construction were inscriptions of the type shown in Figs. 1 and 2 in the Introduction. We called each inscription in Fig. 1 a basic square and each inscription in Fig. 2 a basic diamond. In two dimensions the only difference is in scaling and orientation, but in three dimensions the two analogous objects are essentially different. The three-dimensional version of the basic diamond is an octahedron, while a cube now replaces the basic square. Another difference lies in the associated subdivisions, where in two dimensions the basic square and the basic diamond are each partitioned into 4 right isosceles triangles. In three dimensions the basic cube is partitioned into 6 pyramids with common apex at the center and the basic octahedron is partitioned into 8 simplexes comprising 2 pyramids with a common base. Our wavelet construction will be more difficult to describe in three dimensions, sinceas we have already seen in the Introductionit is not as easy to illustrate the piecewise-constant gradient fields of continuous, piecewise-linear functions. Recall that in the two-dimensional case [1] such illustrations were useful visual aids in the calculation of Sobolev inner products. Here our descriptions will be more verbal, but we hope that the figures in the Introduction may enhance the reader's three-dimensional perception. We now define a basic simplex in the following way. Consider the unique bounded region 2 enclosed by the 4 planes y=x,

z=0,

y=z,

x+ y= 12 .

(2.1)

More explicitly, 2=[(x, y, z): 0z 14 , z y 14 , yx 12 & y].

(2.2)

34

GUY BATTLE

This subset of the cube [0, 12 ] 3 is a right simplex whose vertices are (0, 0, 0), ( 12 , 0, 0), ( 14 , 14 , 0), and ( 14 , 14 , 14 ). The basic simplexes are certain congruent copies of this right simplex. Indeed, there are 24 orientations of interest to usrealized in the partitioning of the cube [0, 12 ] 3 by the 6 planes y=x,

x+ y= 12 ,

z=y,

y+z= 12 ,

x=z,

z+x= 12 .

(2.3)

Half-integer coordinate translations of these right simplexes are also basic simplexes. We define a basic pyramid, which we first exemplify by P L =[(x, y, z): 0z 14 and zx, y 12 &z].

(2.4.L)

The possible orientations of interest to us are given by P U =[(x, y, z): 14 z 12 and 12 &zx, yz],

(2.4.U)

P W =[(x, y, z): 0x 14 and x y, z 12 &x],

(2.4.W)

P E =[(x, y, z): 14 x 12 and 12 &x y, zx],

(2.4.E)

1 4

1 2

P S =[(x, y, z): 0 y and yz, x & y],

(2.4.S)

P N =[(x, y, z): 14  y 12 and 12 & yz, x y].

(2.4.N)

Obviously, these 6 pyramids form another partition of the cube [0, 12 ] 3, where the center is the common apex. We refer to P L (resp. P U , P W , P E , P S , P N ) as a lower (resp. upper, west, east, south, north) basic pyramid. Half-integer-coordinate translates of these pyramids are also basic pyramids. Actually, the basic simplicial partition of [0, 12 ] 3 is a refinement of the basic pyramidal partition, where each basic pyramid is partitioned into 4 basic simplexes. This suggests a simple way to label the basic simplexesnamely according to the pyramid and the side it contributes to that pyramid. For example, the basic simplex 2 defined above can be denoted by 2 LS because it contributes the south side to the lower basic pyramid. With this notation we have the decompositions P L =2 LN _ 2 LS _ 2 LE _ 2 LW ,

(2.5.L)

P U =2 UN _ 2 US _ 2 UE _ 2 UW ,

(2.5.U)

P W =2 WU _ 2 WL _ 2 WN _ 2 WS ,

(2.5.W)

P E =2 EU _ 2 EL _ 2 EN _ 2 ES ,

(2.5.E)

P S =2 SE _ 2 SW _ 2 SU _ 2 SL ,

(2.5.S)

P N =2 NE _ 2 NW _ 2 NU _ 2 NL .

(2.5.N)

OSIRIS WAVELETS IN THREE DIMENSIONS

35

Any half-integer-coordinate translate of 2 LS is a south lower basic simplex, while such a translate of 2 SL is a lower south basic simplex. We define a basic cube as any half-integer-coordinate translate of [0, 12 ] 3 with the associated pyramidal partition. We also define a basic octahedron as the union of two basic pyramids with a common base, where both pyramids lie in the same integer-coordinate translate of the unit cube [0, 1] 3. The basic simplicial partition involving the 8 simplexes that constitute the two pyramids is understood to be associated with a given basic octahedron. Clearly, a basic octahedron intersects with two adjacent basic cubes lying in the same integer-coordinate translate of the unit cube. Indeed, there are as many basic octahedra in the unit cube as there are faces common to adjacent basic cubesi.e., as many as there are 12_ 12 distinct faces of cubic subdivision interior to the unit cube. This means that there are 12 basic octahedra in the unit cube, and that they are mutually disjoint. We are ready to introduce the two building blocks for our wavelet construction. Let .(x, y, z) be supported by the basic cube [0, 12 ] 3 and defined by

.(x, y, z)=

{

x,

(x, y, z) # P W ,

y,

(x, y, z) # P S ,

z,

(x, y, z) # P L ,

1 2 1 2

&x,

(x, y, z) # P E ,

& y,

(x, y, z) # P N ,

1 2

&z,

(x, y, z) # P U ,

(2.6)

where the basic pyramids are given by (2.4). It is easy to check that this piecewise linear function is continuous. For example, the common face of P W and P U is part of the plane x+z= 12 . We shall refer to any half-integer translate of .(x, y, z) as a basic C-function. The other building block is not quite a single type of function, but rather three orientations of a single type. Let .$(x, y, z) be supported by the basic octahedron h 1 =P E _ (P W +( 12 , 0, 0))

(2.7)

and defined by

.$(x, y, z)=

x& y, x&z,

(x, y, z) # 2 EN , (x, y, z) # 2 EU ,

x+ y& 12 , x+z& 12 , 1&x& y,

(x, y, z) # 2 ES , (x, y, z) # 2 EL , (x, y, z) # 2 WN +( 12 , 0, 0),

1&x&z, y&x+ 12 ,

(x, y, z) # 2 WU +( 12 , 0, 0, ), (x, y, z) # 2 WS +( 12 , 0, 0),

z&x+ 12 ,

(x, y, z) # 2 WL +( 12 , 0, 0),

(2.8)

36

GUY BATTLE

where we are using the simplicial decompositions (2.4.E) and (2.4.W) of the east and west pyramids, respectively. It is straightforward to check that this piecewise linear function is continuous as well. For example, the common face of 2 EN and 2 EU is part of the plane y=z, while the common face of 2 EN and 2 WN =( 12 , 0, 0) is part of the plane x= 12 . We shall refer to any discrete translate of the form .$(x+n 1 , y+ 12 n 2 , z+ 12 n 3 ) with n 1 , n 2 , n 3 # Z as an x-oriented basic O-function. Now introduce ."(x, y, z) supported by the basic octahedron h2 =P N _ (P S +(0, 12 , 0))

(2.9)

and defined in the same piecewise-linear manner on h2 as .$(x, y, z) was defined on h1 . This function is really just a rotation and translation of .$(x, y, z), but we will need the explicit definition, y+x& 12 , 1 2

."(x, y, z)=

(x, y, z) # 2 NW ,

y+z& , y&z,

(x, y, z) # 2 NL , (x, y, z) # 2 NE ,

y&z, x& y+ 12 , z& y+ 12 ,

(x, y, z) # 2 NU , (x, y, z) # 2 SW +(0, 12 , 0), (x, y, z) # 2 SL +(0, 12 , 0),

1& y&x, 1& y&z,

(x, y, z) # 2 SE +(0, 12 , 0), (x, y, z) # 2 SU +(0, 12 , 0),

(2.10)

where we are now using the simplicial decompositions (2.5.N) and (2.5.S) of the respective pyramids involved in (2.9). Any discrete translate of the form ."(x+ 12 n 1 , y+n 2 , z+ 12 n 3 ) with n 1 , n 2 , n 3 # Z is a y-oriented basic O-function. Finally, we introduce the function .$$$(x, y, z) supported by the basic octahedron h3 =P U _ (P L +(0, 0, 12 ))

(2.11)

and defined by

.$$$(x, y, z)=

z+x& 12 , z+ y& 12 ,

(x, y, z) # 2 UW , (x, y, z) # 2 US ,

z&x, x& y, x&z+ 12 ,

(x, y, z) # 2 UE , (x, y, z) # 2 UN , (x, y, z) # 2 LW +(0, 0, 12 ),

y&z+ 12 , 1&z&x,

(x, y, z) # 2 LS +(0, 0, 12 ), (x, y, z) # 2 LE +(0, 0, 12 ),

1&z& y,

(x, y, z) # 2 LN +(0, 0, 12 ),

(2.12)

OSIRIS WAVELETS IN THREE DIMENSIONS

37

where the simplicial decompositions (2.5.U) and (2.5.L) are now relevant. Any discrete translate of the form .$$$(x+ 12 n 1 , y+ 12 n 2 , z+n 3 ) with n 1 , n 2 , n 3 # Z is a z-oriented basic O-function. Like ."(x, y, z), the function .$$$(x, y, z) is just a rotation and translation of .$(x, y, z).

3. UNIT-SCALE INNER PRODUCTS AND MULTI-SCALE OBSERVATIONS Our goal is to construct a multi-scale hierarchy of continuous, piecewise-linear wavelets having orthogonality properties with respect to the Sobolev inner product. Since we propose to build the mother wavelets with the basic C-functions and basic O-functions, one major task is to calculate Sobolev inner products for translations and scalings of these more fundamental functions. Their gradients are piecewiseconstant vector functions. We have

{.(x, y, z)=

{

{.$(x, y, z)=

{."(x, y, z)=

i^, j^,

(x, y, z) # P S ,

k,

(x, y, z) # P L ,

&i^, &j^,

(x, y, z) # P E ,

&k,

(x, y, z) # P U ,

i^ & j^, i^ &k, i^ + j^, i^ +k, &i^ & j^,

(x, y, z) # 2 EN , (x, y, z) # 2 EU ,

(x, y, z) # P W ,

(3.1)

(x, y, z) # P N ,

(x, y, z) # 2 ES , (x, y, z) # 2 EL , (x, y, z) # 2 WN +( 12 , 0, 0),

&i^ &k, j^ &i^, k &i^,

(x, y, z) # 2 WU +( 12 , 0, 0), (x, y, z) # 2 WS +( 12 , 0, 0),

j^ +i^, j^ +k,

(x, y, z) # 2 NW , (x, y, z) # 2 NL ,

j^ &i^, j^ &k, i^ & j^, k & j^, & j^ &i^,

(x, y, z) # 2 NE , (x, y, z) # 2 NU , (x, y, z) # 2 SW +(0, 12, 0), (x, y, z) # 2 SL +(0, 12 , 0), (x, y, z) # 2 SE +(0, 12 , 0),

&j^ &k,

(x, y, z) # 2 SU +(0, 12 , 0),

(3.2)

(x, y, z) # 2 WL +( 12 , 0, 0),

(3.3)

38

GUY BATTLE

k +i^, k + j^, k &i^, {.$$$(x, y, z)=

(x, y, z) # 2 UW , (x, y, z) # 2 US , (x, y, z) # 2 UE , (x, y, z) # 2 UN ,

k & j^, i^ &k, j^ &k,

(x, y, z) # 2 LW +(0, 0, 12 ), (x, y, z) # 2 LS +(0, 0, 12 ),

&k &i^, &k & j^,

(x, y, z) # 2 LE +(0, 0, 12 ), (x, y, z) # 2 LN +(0, 0, 12 ).

(3.4)

On the other hand, we shall realize the mother wavelets as linear combinations of the functions . *+&(x, y, z)= .(x& 12 *, y& 12 +, z& 12 &), 1 2

1 2

(3.5)

.$+&(x, y, z)= .$(x, y& +, z& &),

(3.6)

."*&(x, y, z)= ."(x& 12 *, y, z& 12 &),

(3.7)

1 2

1 2

.$$$*+(x, y, z)= .$$$(x& *, y& +, z), (*, +, &) # [0, 1] 3.

(3.8) (3.9)

The eight functions given by (3.5) are all of the basic C-functions whose supports lie in the unit cube; all other basic C-functions are generated by integer-component translations of these. The other twelve functions given by (3.6)(3.8) are all of the basic O-functions whose supports lie in the unit cube; integer-component translations generate the rest of the basic O-functions as well. It is straightforward to calculate the Sobolev inner products of basic O-functions with basic C-functions in the cases where their supports intersect. For example,

| {. } {.$=&

1 48

(3.10)

because supp . & supp .$=P E , {.(x, y, z) } {.$(x, y, z)=1,

(3.11) (x, y, z) # 2 E* ,

and the volume of every basic pyramid is

1 48

*=N, U, S, L,

(3.12)

. Similarly,

| {. } {."=&

1 48

,

(3.13)

| {. } {.$$$=&

1 48

.

(3.14)

OSIRIS WAVELETS IN THREE DIMENSIONS

39

More generally,

| {.

*+&

1 } {.$+& =& 48 $ ++ $ && ,

(3.15)

| {.

*+&

1 } {."*& =& 48 $ ** $ && ,

(3.16)

*+&

1 } {.$$$ *+ =& 48 $ ** $ ++ .

(3.17)

| {.

The Kronecker deltas arise from the observation that for a basic cube intersecting a basic octahedron, those basic cubes given by translations in directions perpendicular to the orientation of the basic octahedron are disjoint from the latter. It is also important that all twelve basic octahedra in the unit cube are clearly disjoint from one another. Thus

| {.$

1 } {.$+& = 12 $ ++ $ && ,

(3.18)

| {.$

} {."*& =0,

(3.19)

| {.$

} {.$$$ *+ =0,

(3.20)

| {."

1 } {."*& = 12 $ ** $ && ,

(3.21)

| {."

} {.$$$ *+ =0,

(3.22)

+&

+&

+&

*&

*&

| {.$$$ } {.$$$ = *+

*+

1 12

$ ** $ ++

(3.23)

because in the case of coincident basic octahedra the dot product is equal to the 1 constant 2 on that common support; the volume of every basic octahedron is 24 . Similarly, all eight basic cubes in the unit cube are mutually disjoint. The volume of each of these is 18 , so

| {.

*+&

} {. *+& = 18 $ ** $ ++ $ &&

because the dot product is equal to unity on coincident basic cubes.

(3.24)

40

GUY BATTLE

Having calculated inner products for functions on just one length scale, we must now consider inner products over the multi-scale set of functions generated by the basic C-functions and basic O-functions. An r-level basic C-function is any function of the form (

(

( . rm( ; *+&(! )=2 &r2 . *+&(2 &r! &m ),

( m # Z 3,

(3.25)

(

with ! =(x, y, z). The purpose of the factor 2 &r2 is to preserve the Sobolev norm under scaling in three dimensions. An r-level x-oriented basic O-function has the form (

(

( .$rm( ; +&(! )=2 &r2.$+&(2 &r! &m ),

( m # Z 3.

(3.26)

Similarly, an r-level y-oriented basic O-function and an r-level z-oriented basic O-function have the forms (

(

(

&r (

( ."rm( ; *&(! )=2 &r2."*&(2 &r ! &m ), &r2 ( ; *+(! )=2 .$$$ .$$$ rm *+(2

( ! &m ),

(3.27) (3.28)

respectively. The calculation of inner products will be greatly simplified by some rules based on general observations. First, it is straightforward to check that for an arbitrary but fixed vector (v,

| v } {.=0, (

| v } {. (

*

=0,

(3.29) *=$, ", $$$,

(3.30)

and this property obviously extends to all r-level basic functions. This implies that every such function is Sobolev-orthogonal to every piecewise-linear function that is linear on the support of the former. In particular, the given function is Sobolevorthogonal to every higher-level basic C-function for which the support of the former is contained in a basic pyramid of the latteror disjoint from the support of the latter. The given function is also Sobolev-orthogonal to every higher-level basic O-function for which the support of the former is contained in a basic simplex of the latteror disjoint from its support. As far as interscale inner products are concerned, the nonzero values to be calculated arise in the case where the support of the lower-level basic function is divided by a plane involved in the triangulation for the higher-level basic function. Second, the cubic structure of our multi-scale hierarchy implies that the support of the lower-level basic function is not divided by any constant-coordinate plane involved in the larger-scale triangulation. The planes to be considered in the largerscale triangulation are the xy-inclined planes, the yz-inclined planes, and the zx-inclined

41

OSIRIS WAVELETS IN THREE DIMENSIONS

planes. An r-level xy-included plane, for example, is any plane whose equation has the form y=\x+2 r&1l,

l # Z,

(3.31)

and such planes can divide the supports of basic functions at arbitrarily lower levels (smaller scales). Third, if an r-level inclined plane does indeed divide the support of a lower-level basic O-function, then that plane is also part of the triangulation for the function. Actually, this property is not quite shared by basic C-functions, as an r-level inclined plane dividing the support of a lower-level basic C-function also divides two of the pyramids in the triangulation for the lower-level basic function. Fourth, an r-level xy-inclined plane cannot divide any lower-level basic octahedron that is either x-oriented or y-oriented. Similarly, an r-level yz-inclined (resp. zxinclined) plane cannot divide any lower-level basic octahedron that is either y-oriented or z-oriented (resp. z-oriented or x-oriented). This restriction is not as great an advantage as it may sound, since every higher-level basic function is defined by planes of all three inclinations.

4. INTERSCALE INNER PRODUCTSOCTAHEDRAL DIVIDERS We now calculate the Sobolev inner products between our basic functions when they have different length scales. First, consider an r-level basic octahedron, and since our geometric hierarchy is isotropic, assume without loss of generality that this octahedron is x-oriented. Now a divider of an octahedron is defined to be the intersection of the octahedron with a plane separating four of its basic simplexes from the other four. Obviously, there are only three dividers of an octahedrona constant-coordinate divider and two inclined dividers. For the x-oriented case, the inclined dividers are yz-inclined, while the faces of the octahedron are either xy-inclined or zx-inclined. Second, consider an s-level basic octahedron where s
42

GUY BATTLE

LS supp .$rm(; +& into upper-north and lower-south subsets A UN (; +& and A rm (; +& , respecrm tively. Then the included smaller-scale divider partitions supp .$sn(; +& into A UN (; +& and sn , and each of these subsets is contained in one of the r-level basic simplexes A LS ( sn; +& LS comprising A UN (; +& and A rm (; +& , respectivelyagain, by multi-scale compatibility. It rm follows that

| {.$

( ; +& rm

} {.$sn( ; +& =

|

( UN

v 1 } {.$sn( ; +& +

A ( sn ; +&

|

( LS

v 2 } {.$sn( ; +& ,

(4.1)

A ( sn ; +&

where (v 1 and (v 2 are the constant values of {.$rm(; +& on the respective r-level basic simplexes. By (3.2), the possibilities are (

v 1 =2 &3r2(i^ & j^ )

and

(

v 1 =2 &3r2(i^ &k )

and

(

(

v 2 =2 &3r2(i^ +k ),

(4.2.1)

v 2 =2 &3r2(i^ + j^ ),

(4.2.2)

(k &i^ ),

(4.2.3)

v 2 =2 &3r2( j^ &i^ ),

(4.2.4)

(&i^ & j^ )

and

(

v 1 =2 &3r2(&i^ &k )

and

(

(

v 1 =2

&3r2

(

v 2 =2

&3r2

where the case for (v 2 is linked to the case for (v 1 by the constraint that the r-level basic simplex on one side of the divider would have to share a face with the r-level basic simplex on the other side. To calculate each integral in (4.1), we need only observe that we are integrating over all four possibilities at the s-level. As far as the first integral is concerned, the domain A UN (; +& is composed of four s-level basic simplexes, each with volume sn 1 3(s&1) , such that 24 2 {.$sn(; +& =2 &3s2(i^ & j^ ),

(4.3.E.N )

(i^ &k ),

(4.3.E.U )

{.$sn(; +(& =2

&3s2

{.$sn(; +& =2 &3s2(&i^ & j^ ),

(4.3.W.N)

(&i^ &k )

(4.3.W.U)

{.$sn(; +& =2

&3s2

are the constant values of the gradient on these four sub-domains. Hence

|

1 v 1 } {.$sn(; +& =& 96 2 3s2 (v 1 } ( j^ +k ).

( UN

A sn(; +&

(4.4)

On the other hand, v 1 } ( j^ +k )=&2 &3r2

(

(4.5)

in all four of the cases for (v 1 , so

|

1 v 1 } {.$sn(; +& = 96 2 3(s&r)2

( UN

A ( sn ; +&

(4.6)

43

OSIRIS WAVELETS IN THREE DIMENSIONS

in any case. From this we can infer

|

1 v 2 } {.$sn(; +& = 96 2 3(s&r)2

( LS

A ( sn ; +&

(4.7)

in each case, because {.$rm(; +& } {.$sn(; +& is invariant with respect to reflection through the yz-inclined plane dividing both octahedra. Thus

| {.$

(; +& rm

1 } {.$sn(; +& = 48 2 3(s&r)2

(4.8)

in all of these cases where an r-level divider includes an s-level divider. In the case where the s-level basic octahedron is y-oriented, the situation changes. The Sobolev inner product of the associated function ."sn(; *& with .$rm(; +& can still be non-zero, but only a face of supp .$rm(; +& can include one of the dividers of supp ."sn(; *& indeed, only a zx-inclined face (again, by Observation 4 in the preceding section). Assume without loss of generality that the pertinent zx-inclined face of supp .$rm(; +& LE partitions supp ."sn(; *& into A UW (; *& and A sn (; *& . One set is contained in one of the sn simplexes comprising supp .$rm(; +& while the other is disjoint from supp .$rm(; +& . Assume further that the upper-west set is the one disjoint from this r-level basic octahedron. Then it is easy to see that the lower-east set is contained in the upper simplex of the east pyramid of the r-level basic octahedron. (Bear in mind that an east pyramid is actually the western half of an x-oriented octahedron.) Therefore

| {.$

(; +& rm

} {."sn(; *& =

|

( LE

v } {."sn(; *& ,

(4.9)

A (  sn ; *&

v =2 &3r2(i^ &k ),

(

(4.10)

where the value of (v can be inferred from (3.2). On the other hand, the domain 1 3(s&1) A LE , such (; *& is composed of four s-level basic simplexes, each with volume 24 2 sn that {."sn(; *& =2 &3s2( j^ +k ),

(4.11.N.L)

{. "sn(; *& =2 &3s2( j^ &i^ ),

(4.11.N.E)

{."sn(; *& =2 &3s2(k & j^ ),

(4.11.S.L)

{."sn(; *& =2

&3s2

(&i^ & j^ )

(4.11.S.E)

are the constant values of the gradient on these four sub-domains. Hence

|

1 v } {."sn(; *& = 96 2 3s2 (v } (k &i ),

( LE

A (  sn ; *&

(4.12)

44

GUY BATTLE

and so

| {.$

(; +& rm

1 } {."sn(; *& =& 48 2 3(s&r)2.

(4.13)

The analysis of the s-level z-oriented case is identical, and we have

| {.$

(; +& rm

1 } {.$$$sn(; *+ =& 48 2 3(s&r)2,

(4.14)

in all non-zero cases. For our next collection of cases, we consider an r-level basic cube together with an s-level basic octahedron with s
| {.

(; +&* rm

} {.$sn(; +& =

|

( UN

v 1 } {.$sn(; +& +

A sn(; +&

|

( LS

v 2 } {.$sn(; +& ,

(4.15)

A sn(; +&

where (v 1 and (v 2 are the constant values of {. rm(; +&* on the respective r-level basic pyramids. Now the r-level divider can be either the face common to the north pyramid and lower pyramid in the block, or the face common to the south pyramid and upper pyramid. By (3.1), these two possibilities translate into (

v 1 =&2 &3r2 j^

and

(

v 2 =2 &3r2k,

(4.16.1)

v 1 =&2 &3r2k

and

(

v 2 =2 &3r2j^.

(4.16.2)

(

To calculate the first integral in (4.15), we recall the remark that A UN (; +& is composed sn 1 2 3(s&1) with the constant values of four s-level basic simplexes, each with volume 24 of the gradient on these sub-domains given by (4.3). Hence, the integral is still given by (4.3) with (v 1 now given by (4.16), and so we have

|

1 v 1 } {.$sn(; +& = 96 2 3(s&r)2

( UN

A ( sn ; +&

(4.17)

OSIRIS WAVELETS IN THREE DIMENSIONS

45

in both cases. Since {. rm(; +& } {.$sn(; +& is invariant with respect to reflection through the yz-plane containing both dividers, the second integral in (4.15) has the same value. Thus

| {.

(; +&* rm

1 } {.$sn(; +& = 48 2 3(s&r)2.

(4.18)

This essentially completes our analysis of an r-level basic cube versus an s-level basic octahedron with s
5. INTERSCALE INNER PRODUCTSCUBIC DIVIDERS Continuing our calculation of Sobolev inner products between an r-level basic function and an s-level basic function with s
46

GUY BATTLE

LS divider partitions the r-level basic octahedron into A UN ( ; +& and A rm (; +& . This divider rm also partitions the s-level basic cube into the subset U N WU WN EU EN B UN (; *+& =P sn (; *+& _ P sn (; *+& _ 2 sn (; *+& _ 2 sn (; *+& _ 2 sn (; *+& _ 2 sn (; *+& sn

(5.1)

and the relative complement L S WL WS EL ES B LS (; *+& =P sn (; *+& _ P sn (; *+& _ 2 sn (; *+& _ 2 sn (; *+& _ 2 sn (; *+& _ 2 sn (; *+& . sn

(5.2)

For convenience, we adopt the multi-index notation q=(s, (n; *, +, & ). The inner product is given by

| {.$

(; +& rm

} {. q =

|

( B UN q

v 1 } {. q +

|

( B LS q

v 2 } {. q ,

(5.3)

whereas in (4.1)v( 1 and (v 2 are the constant values of {.$rm(; +& on the respective r-level simplexes. The possibilities are already given by (4.2). As for the upper-north part of the s-level basic cube, we have &2 &3s2k, &2 &3s2 j^, {. q(x, y, z)= &2 &3s2 i^, 2 &3s2 i^,

{

Since each s-level basic pyramid has volume

|

(x, y, z) # P U q , (x, y, z) # P N q , (x, y, z) # 2 WU _ 2 WN , q q EU . (x, y, z) # 2 q _ 2 EN q 1 6

(5.4)

2 3(s&1), it follows that

1 v 1 } {. q =& 48 2 3s2( j^ +k ) } (v 1

( UN

Bq

1 = 48 2 3(s&r)2,

(5.5)

where we have applied (4.5) as well as the i^-cancellation. By a reflection-symmetry argument, this implies

|

1 v 2 } {. q = 48 2 3(s&r)2

( LS

Bq

(5.6)

as well, and therefore

| {.$

(; +& rm

1 } {. q = 24 2 3(s&r)2

(5.7)

in this first case of a simple divider. Let us next examine the case where only a face of the r-level, x-oriented basic octahedron partitions the s-level basic cube. We may assume without loss of generality that the relevant face is zx-inclined. Indeed, we may assume that this simple divider

OSIRIS WAVELETS IN THREE DIMENSIONS

47

partitions the s-level basic cube into the subsets B UW and B LE q q with the former disjoint from the r-level basic octahedron and the latter interior to it. In this event,

| {.$

(; +& rm

} {. q =

|

( LE

v } {. q

(5.8)

y, z) # P Lq , y, z) # P Eq , _ 2 NL y, z) # 2 NE q q , SE SL y, z) # 2 q _ 2 q ,

(5.9)

Bq

with (v given by (4.10). On the other hand, 2 &3s2k, &2 &3s2 i^, {. q(x, y, z)= &2 &3s2 j^, 2 &3s2 j^,

(x, (x, (x, (x,

{

so we have

|

1 v } {. q = 48 2 3s2(k &i^ ) } (v

( LE

Bq

1 =& 24 2 3(s&r)2

(5.10)

as the value of the inner product in this second case. The third case to be considered involves a Type O compound divider. It is the case where a divider and two faces of the r-level basic octahedron are involved in a partitioning of the s-level basic cube. One face is zx-inclined and the other is xy-inclined, but neither face partitions the s-level basic cube by itself. If we assume that the former is an upper-east face of the r-level basic octahedron and that the latter is a north-east face, then their common boundary inside the s-level cube is either the diagonal line from the lower northwest vertex to the upper southeast vertex, or the diagonal line from the lower southeast vertex to the upper northwest vertex. If we assume the latter scenario, it is easy to see that these two faces together partition the s-level basic cube into the subset SW LW B SWL =P W _ 2 SL _ 2 LS q q _ 2a q _ 2q q

(5.11)

and the relative complement U SE SU LE LN C SWL =P Eq _ P N q q _ Pq _ 2q _ 2q _ 2q _ 2q .

(5.12)

On the other hand, a yz-inclined divider of the r-level basic diamond meets the common boundary of the two faces. Indeed, it partitions the smaller subset B SWL q into =2 WU _ 2 WS _ 2 SW _ 2 SL B SWL q q q q q , B8

SWL q

=2

WL q

_2

WN q

_2

LW q

_2

LS q

.

(5.13) (5.14)

48

GUY BATTLE

Since C SWL is disjoint from the r-level basic octahedron, q

| {.$

(; +& rm

} {. q =

|

( SWL

B q

v 1 } {. q +

|

( SWL

v 2 } {. q ,

(5.15)

B8 q

where (v 1 and (v 2 are the constant values of {.$rm(; +& on the r-level simplexes containand B8 SWL , respectively. In this case, ing B SWL q q (

v 1 =&2 &3r2(k +i^ ),

(5.16)

v 2 =&2 &3r2(i^ + j^ ),

(5.17)

(

while 2 &3s2i^, 2 &3s2 j^, {. q(x, y, z)= &3s2 ^ i, 2 &3s2 2 k,

{

_ 2 WS (x, y, z) # 2 WU q q , SW SL (x, y, z) # 2 q _ 2 q , _ 2 WN , (x, y, z) # 2 WL q q LW LS (x, y, z) # 2 q _ 2 q .

(5.18)

Hence

|

1 v 1 } {. q = 96 2 3s2(i^ + j^ ) } (v 1

( SWL

B q

1 =& 96 2 3(s&r)2,

|

(5.19)

1 v 2 } {. q = 96 2 3s2(k +i^ ) } (v 2

( SWL

B8 q

1 =& 96 2 3(s&r)2,

(5.20)

1 } {. q =& 48 2 3(s&r)2

(5.21)

and therefore

| {.$

(; +& rm

in this last case for an r-level basic octahedron paired with an s-level basic cube. We now turn to the last general situation to be consideredpairing an s-level basic cube with an r-level basic cube. By Observation 2 at the end of Section 3, only the dividers of the r-level basic cube can partition the s-level basic cube, as the faces ( of the r-level basic cube are constant-coordinate planes. Let m and (*, +, &) be the position indices for the r-level basic cube. We first consider the case where a simple divider is involved, and we assume without loss of generality it is a yz-inclined

49

OSIRIS WAVELETS IN THREE DIMENSIONS

divider that partitions the s-level basic cube into B UN and B LS q q . The inner product is given by

| {.

(; *+& rm

} {. q =

|

( UN

v 1 } {. q +

Bq

|

( LS

v 2 } {. q ,

(5.22)

Bq

where (v 1 and (v 2 are the constant values of {. rm(; *+& on the r-level basic pyramids containing B UN and B LS q q , respectively. The cases are v 1 =&2 &3r2j^

( (

v 1 =&2

&3r2

k

v 2 =2 &3r2k,

(5.23.1)

j^.

(5.23.2)

and

(

and

(

v 2 =2

&3r2

The former case corresponds to the interface between the lower r-level basic pyramid and the north r-level basic pyramid, while the latter case corresponds to the interface between the upper pyramid and the south pyramid. On the other hand, (5.4) implies

|

1 v 1 } {. q =& 48 2 3s2( j^ +k ) } (v 1 .

(5.24)

|

(5.25)

( UN

Bq

We obtain 1 v 1 } {. q = 48 2 3(s&r)2

( UN

Bq

in both cases. As for the lower-south part of the s-level basic cube, we have 2 &3s2 k, 2 &3s2 j^, {. q(x, y, z)= &2 &3s2 i^, 2 &3s2 i^,

{

(x, (x, (x, (x,

y, z) # P Lq , y, z) # P Sq , ES y, z) # 2 EL q _ 2q , y, z) # 2 WL _ 2 WS q q ,

(5.26)

and so

|

1 v 2 } {. 2 = 48 2 3s2( j^ +k ) } (v 2

(

B LS q

1 = 48 2 3s2

(5.27)

1 } {. q = 24 2 3(s&r)2,

(5.28)

in both cases. Thus

| {.

(; *+& rm

in this situation where a simple divider is involved.

50

GUY BATTLE

The cases that remain to be considered involve three r-level inclined half-planes whose common boundary is a diagonal line passing through opposite vertices of the s-level basic cubei.e., a Type C compound divider. Assume without loss of generality that the diagonal line passes from the lower southeast vertex to the upper northwest vertex. As far as the r-level basic cube is concerned, this could be either the edge common to the upper, north, and west pyramids or the edge common to the lower, south, and east pyramids. Consider the former case first. Then the s-level basic cube is partitioned into the subsets SE SU ES EU B SUE =P U q q _ 2q _ 2q _ 2q _ 2q ,

(5.29)

SW LW =P W _ 2 SL _ 2 LS B SWL q q _ 2q q _ 2q q ,

(5.30)

LN EN =P N _ 2 LE _ 2 EL B LNE q q _ 2q q _ 2q q ,

(5.31)

and we have the decomposition

| {.

( ; *+& rm

} {. q =

|

( SUE

v 1 } {. q +

Bq

|

( SWL

v 2 } {. q +

Bq

|

( LNE

v 3 } {. q ,

(5.32)

Bq

where (v 1 , (v 2 , and (v 3 are the constant values of {. rm(; *+& on the r-level basic pyramids , B SWL , and B LNE , respectively. These pyramids are the upper, containing B SUE q q q west, and north pyramids, so the vectors are given by (

v 1 =&2 &3r2 k,

(5.33.1)

(

v 2 =2 3r2 i^,

(5.33.2)

v 3 =&2 &3r2j^,

(5.33.3)

(

while (3.1) yields &2 &3s2k, {. q(x, y, z)= 2 &3s2j^, &2 &3s2 i^,

{ { {

(x, y, z) # P U q SU (x, y, z) # 2 SE q _ 2q , (x, y, z) # 2

ES q

_2

EU q

,

2 &3s2i^, {. q(x, y, z)= 2 &3s2j^, 2 &3s2k,

(x, y, z) # P W q , (x, y, z) # 2 SW _ 2 SL q q ,

&2 &3s2j^, {. q(x, y, z)= 2 &3s2k, &2 &3s2i^,

(x, y, z) # P N q , LN (x, y, z) # 2 q _ 2 LE q ,

(x, y, z) # 2

LW q

_2

(5.34)

LS q

(5.35)

,

(x, y, z) # 2 EN _ 2 EL q q .

(5.36)

51

OSIRIS WAVELETS IN THREE DIMENSIONS

Hence

|

1 v 1 } {. q = 96 2 3s2(&i^ + j^ &2k ) } (v 1

( SUE

Bq

1 = 48 2 3(s&r)2,

|

(5.37.1)

1 v 2 } {. q = 96 2 3s2(2i^ + j^ +k ) } (v 2

( SWL

Bq

1 = 48 2 3(s&r)2,

|

(5.37.2)

1 v 3 } {. q = 96 2 3s2(&i^ &2j^ +k ) } (v 3

( LNE

Bq

1 = 48 2 3(s&r)2,

(5.37.3)

and therefore

| {.

(; *+& rm

1 } {. q = 16 2 3(s&r)2

(5.38)

in this case. Now consider the case where the diagonal line is the edge common to the r-level lower, east, and south basic pyramids. Then the s-level basic cube is partitioned into B WLN =P Lq _ 2 NL _ 2 NW _ 2 WL _ 2 WN , q q q q q

(5.39)

=P Eq _ 2 NU _ 2 NE _ 2 UE _ 2 UN B NEU q q q q q ,

(5.40)

B

USW q

S q

S q

=P _ 2 _ 2 q _ 2

US q

_2

UW q

,

(5.41)

respectively. The decomposition of the inner product is given by

| {.

(; *+& rm

} {. q =

|

( WLN

v 1 } {. q +

Bq

|

( NEU

Bq

v 2 } {. q +

|

( USW

v 3 } {. q ,

(5.42)

Bq

(

v 1 =2 &3r2k,

(5.43.1)

(

v 2 =&2 &3r2 i^,

(5.43.2)

v 3 =2 &3r2j^.

(5.43.3)

(

As far as the s-level basic function is concerned, we break down (3.1) as 2 &3s2k, {. q(x, y, z)= &2 &3s2j^, 2 &3s2i^,

{

(x, y, z) # P Lq , NW (x, y, z) # 2 NL , q _ 2q (x, y, z) # 2 WL _ 2 WN , q q

(5.44)

52

GUY BATTLE

&2 &3s2 i^, {. q(x, y, z)= &2 &3s2j^, &2 &3s2k,

{ {

2 &3s2j^,

(x, y, z) # P Eq (x, y, z) # 2 NU _ 2 NE q q , UE (x, y, z) # 2 q _ 2 UN q ,

(5.45)

(x, y, z) # P Sq

{. q(x, y, z)= &2 &3s2k, 2 &3s2i^,

UW (x, y, z) # 2 US , q _ 2q WS (x, y, z) # 2 q _ 2 WU . q

(5.46)

Actually, the formulas (5.43)(5.46) differ from their counterparts (5.33)(5.36) by sign-reversal only. Therefore, the value of our inner product in this case does not differ at all from the value in that other casei.e., we have

| {.

(; *+& rm

1 } {.= 16 2 3(s&r)2

(5.47)

in this last case.

6. THE MOTHER WAVELETS In this section we finally define the system of wavelets arising from the basic functions we have already associated with our cubes and octahedra. We return to the unit cube [0, 1] 3, on which our mother wavelets are to be defined. For (+, &) # [0, 1] 2 we introduce 9 +& =2. 0+& &2. 1, 1&+, 1&&

(6.1)

as the type of mother wavelet associated with the basic cubes [0, 12 ] 3 + 12 =(. All four of theses functions are supported in the unit cube and have mutually disjoint supports. This means that, not only are they mutually orthogonal, but their integer-coordinate translates comprise an orthogonal set. The factor of 2 is a normalizing factor, since

| ({.

0+&

|

|

&{. 1, 1&+, 1&& ) 2 = ({. 0+& ) 2 + ({. 1, 1&+, 1&& ) 2 = |supp . 0+& | + |supp . 1, 1&+, 1&& |

(6.2)

and each volume is 18 . With regard to mother wavelets associated with basic cubes, this is the obvious extension of the definition of two-dimensional mother wavelets associated with basic squares. The definition of mother wavelets associated with the basic octahedra is more involved. With an eye to interscale orthogonality properties, we already see that the coefficient of .$+& (resp. ."*& , .$$$ *+ ) in a linear combination of basic O-functions must be minus the coefficient of .$1&+, 1&& (resp. ."1&*, 1&& , .$$$ 1&*, 1&+ ). At the same time, we require orthogonality of these mother wavelets to the mother wavelets given

OSIRIS WAVELETS IN THREE DIMENSIONS

53

by (6.1). These two constraints can be met. For example, if .$+& has a non-zero coefficient, then for the basic O-functions ."*& and .$$$ *+ , the coefficient of one can be zero, while the coefficient of the other can be minus the coefficient of .$+& . On the other hand, there are only three linearly independent combinations of basic O-functions meeting the two constraints. Our prescription yields the functions ' 1 =- 6 ( .$00 & ."10 + .$$$ 11 & .$11 + ." 01 &.$$$ 00 ),

(6.3.1)

' 2 =- 6 ( .$00 & ."00 +.$$$ 01 &.$11 +." 11 &.$$$ 10 ),

(6.3.2)

' 3 =- 6 ( .$10 &."00 +.$$$ 00 & .$01 +." 11 &.$$$ 11 ),

(6.3.3)

where - 6 is a normalizing factor that we explain below (as it does not normalize ' 1 , ' 2 , ' 3 ). We have already given a pictorial scheme for these functions in the Introduction, and that description provides the best way to understand them. In any case, the orthogonality properties

| {'

} {9 +& =0

@

(6.4)

follow from the orthogonality properties

| {'

@

} {. +&* =0,

(6.5)

which, in turn, follow from the inner product relations (3.15)(3.17). These functions ' @ are not mutually orthogonal. By the disjointness of the octahedral supports of the basic O-functions, we can easily calculate inner products among the ' @ by using the relations (3.18)(3.23). We have

| {'

1

} {' 2 = {' 2 } {' 3 =1,

|

(6.6)

| {'

1

} {' 3 =&1,

(6.7)

| ({' ) =3, 2

@

@=1, 2, 3.

(6.8)

Following the notation adopted in the Introduction, we set 9 +& =9 2++& , ' @ =9 @+3

(+,&) # [0,1] 2, @=1, 2, 3,

(6.9) (6.10)

54

GUY BATTLE

for conveniently indexing elements of the overlap matrix. Let E be the overlap matrix for the seven mother waveletsi.e.,

|

E }}$ = {9 } } {9 }$ .

(6.11)

Then

E=

_

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0 3 0 1 0 &1

0 0 0 0 0 0 0 0 1 &1 3 1 1 3

&

.

(6.12)

This positive matrix has the eigenvalues 1 and 4, with the multiplicities 5 and 2, respectively. We introduced the factor - 6 in the definition of ' @ because we have chosen the convention that the minimum eigenvalue of E be normalized. The most natural orthonormal basis of eigenvectors for this matrix is the set 1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 &1 1

0 0 0 0 0 1 1

0 0 0 0 2 1 &1

{_ & _ & _ & _ & _ & _ & _ &= ,

,

,

,

1

-3

,

1

-2

,

1

-6

,

where the first 5 have eigenvalue 1 and the last 2 have eigenvalue 4. Naturally, we can use this orthonormal basis to diagonalize E and apply the functional calculus. We obtain

E 12 = 13

_

3 0 0 0 0 0 0

0 3 0 0 0 0 0

0 0 3 0 0 0 0

0 0 0 0 0 0 3 0 0 5 0 1 0 &1

0 0 0 0 0 0 0 0 1 &1 5 1 1 5

&

(6.13)

55

OSIRIS WAVELETS IN THREE DIMENSIONS

E &12 = 16

_

6 0 0 0 0 0 0

0 6 0 0 0 0 0

0 0 6 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 4 &1 1 0 &1 4 &1 0 1 &1 4

&

.

(6.14)

7. SOBOLEV ORTHOGONALITY PROPERTIES, I We shall refer to the set of all functions generated by the seven defined on the unit cube as Osiris wavelets. The r-level Osiris wavelets have either of the forms (

(

( ), 9 rm(; +&(! )=2 &r29 +&(2 &r! &m (

' rm(@(! )=2 &r2' @(2

&r (

( ! &m ),

(+, &) # [0, 1] 2,

(7.1)

@=1, 2, 3,

(7.2)

( ranges over all vectors with integer coordinates. Since the mother wavelets where m

[9 00 , 9 01 , 9 10 , 9 11 , ' 1 , ' 2 , ' 3 ] are supported in the unit cube, the set generated by integer-coordinate translations of these seven functions is translationally orthogonal. These wavelets are the 0-level Osiris wavelets (or unit-scale Osiris wavelets). It immediately follows from the scaling of inner product integrals that the set of r-level Osiris wavelets has the translational orthogonality in the scale-commensurate sense for every integer r. Our task is to investigate interscale orthogonality propertiesi.e., to determine how much Sobolev orthogonality there is between the r-level wavelets and the s-level wavelets for all s
(

9 &1, =(; +&(! )=- 2 9 +&(2! &=( ), (

(

' &1, =(@(! )=- 2 ' @(2! &=( ),

(+, &) # [0, 1] 2,

(7.3)

@=1, 2, 3.

(7.4)

This property follows from the observation that the intersection ( ) [0, 1] 3 & ([0, 12 ] 3 + 12 m ( Â [0, 1] 3. is either null or two-dimensional for m

56

GUY BATTLE

First consider the event where both wavelets are of the kind composed of basic C-functions (the 9 +& -kind of wavelets). With 9 +& =2. 0+& &2. 1, 1&+, 1&& , 9 &1, =(; +& =2. &1, =(; 0+& &2. &1, =(; 1, 1&+, 1&& ,

(7.5) (7.6)

we can easily calculate the Sobolev inner product. We observe that

| {.

0+&

|

} {. &1, =(; 0+& = {. 0+& } {. &1, =(; 1, 1&+, 1&& =0, =( {+j^ +&k,

| {.

1, 1&+, 1&&

(7.7)

|

} {. &1, =(; 0+& = {. 1, 1&+, 1&& } {. &1, =(; 1&+, 1&& =0, =( {i^ ++j^ +&k,

(7.8)

as a consequence of disjoint supports in each case. Thus

| {9

+&

} {9 &1, =(; +& =0,

=( {+j^ +&k, i^ ++j^ +&k,

(7.9)

so assume (= =+j^ +&k. In this case, (7.8) still holds, and so

| {9

+&

|

|

} {9 &1, =(; +& =4 {. 0+& } {. &1, =(; 0+& &4 {. 0+& } {. &1, =(; 1, 1&+, 1&& . (7.10)

On the other hand, . 0+& is reflection-invariant through the point ( 14 , 12 ++ 14 , 12 &+ 14 ). Since the same reflection transforms . &1, =(; 0+& into . &1, =(; 1, 1&+, 1&& when =( = +j^ +&k, it follows that

| {9

+&

} {9 &1, =(; +& =0,

=( =+j^ +&k.

(7.11)

In the case =( =i^ ++j^ +&k, it is (7.7) that still holds, and it is now reflection through the point ( 34 , 12 ++ 14 , 12 &+ 14 ) that transforms . &1, =(; 0+& into . &1, =(; 1, 1&+, 1&& . It follows from the reflection-invariance of . 1, 1&+, 1&& through this point that

| {9

+&

} {9 &1, =(; +& =0,

=( =i^ ++j^ +&k.

(7.12)

In summary, the unit-scale wavelets of the kind composed of basic C-functions are Sobolev-orthogonal to the half-scale wavelets of the same kind. Next, consider the event where the unit-scale wavelet is still of the kind composed of basic C-functions, but the half-scale wavelet is now of the kind composed of basic

OSIRIS WAVELETS IN THREE DIMENSIONS

57

O-functions. Equation (4.18) is relevant in this event. With 9 +& as the unit-scale wavelet, we first observe that 1

| {9

+&

} {. * &1, =(; +& =

| {9

+&

} {. * &1, =(; +& =&

| {9

+&

} {. * &1, =(; +& =0,

48 - 2

,

1 48 - 2

,

(

= =+j^ +&k,

(7.13)

=( =i^ ++j^ +&k,

(7.14)

otherwise.

(7.15)

Now combine this observation with the half-scale version of (6.3) to infer

| {9

+&

@ =1, 2, 3,

} {' &1, =(@ =0,

(7.16)

in all three cases. Thus 9 +& is Sobolev-orthogonal to the half-scale wavelets of the kind composed of basic O-functions as well. Now consider the event where the unit-scale wavelet is of the kind composed of basic O-functions and the half-scale wavelet is of the kind composed of basic C-functions. With 9 &1, =(; +& =2. &1, =(; 0+& &2. &1, =(; 1, 1&+, 1&& ,

(7.17)

it is clear from (6.8) that we need to calculate the inner products of ' 1 , ' 2 , and ' 3 with . &1, =(; 0+& and . &1, =(; 1, 1&+, 1&& for every (= and for every (+, & ) # [0, 1] 2. Equations (6.3) reduce these calculations to the calculation of the inner products of the basic O-functions .$+& , ."*& , and .$$$ *+ with the . &1, =(; *+& . Now for adjacent levels, only compound dividers are possible for the basic (&1)-level cubes, and they are Type O compound dividers in this scenario, so Eq. (5.21) is relevant here. Visual inspection of disjoint supports shows that

| | |

{. &1, =(; *+& } {.$+& =

{. &1, =(; *+ & } {."*& =

{. &1, =(; *+ & } {.$$$ *+ =

{ { {

&

1 96 - 2

,

otherwise,

0, &

1 96 - 2

,

&

1 96 - 2

,

(7.18.1)

=( =*i^ +(1&+ ) j^ +&k, otherwise,

0,

0,

= =(1&* ) i^ ++j +&k,

(

= =*i ++j^ +(1&& ) k,

(7.18.2)

(

otherwise.

(7.18.3)

58

GUY BATTLE

Hence

| {9

&1, =(; +&

| {9

| {9

&1, =(; + &

&1, =(; + &

} {.$+& =

} {."*& =

} {.$$$ *+ =

{ { {

&

1

48 - 2 1 , 48 - 2 0, &

,

=( =+j^ +&k,

1

1

48 - 2 1 , 48 - 2 0,

(7.19.1)

otherwise, =( =*i^ +(1&+ ) j^ +&k,

48 - 2 1 , 48 - 2 0, &

= =i^ ++j^ +&k,

(

=( =*i^ ++j^ +&k,

(7.19.2)

otherwise ,

= =*i^ ++j^ +(1&& ) k,

(

=( =*i^ ++j^ +&k,

(7.19.3)

otherwise.

These evaluations enable us to calculate inner products of the wavelets 9 &1, =(; +& with ' 1 , ' 2 , and ' 3 , but visual inspection of disjoint supports reveals that ' 1 = 9 &1,

,

' 1 =9 &1, i^ +k; +& ,

(7.20.1)

' 2 = 9 &1, k; +& ,

' 2 = 9 &1, i^ + j^; +& ,

(7.20.2)

' 3 = 9 &1, i^; +& ,

' 3 = 9 &1, j^ +k; +& ,

(7.20.3)

j^; +&

where we use the perpendicularity symbol to denote Sobolev orthogonality. To further reduce the number of calculations, we exploit rotational symmetries. Let R 1 be the 90% yz-counter-clockwise rotation about the axis y=z= 12 and R 2 the 90% xy-counter-clockwise rotation about the axis x= y= 12 . Then ' 1 b R &1 1 =' 3 ,

(7.21)

' 3 b R &1 2 =&' 2 ,

(7.22)

and since these rotations only permute the set of 9 &1, =(; +& up to some sign-reversals, it is enough to calculate the inner products for ' 1 . Eight of the inner products are accounted for by (7.20.1), and these are not the only inner products that vanish. There are inner products where the supports overlap but the non-zero contributions cancel out. An accounting of this case yields the further Sobolev-orthogonality relations

OSIRIS WAVELETS IN THREE DIMENSIONS

59

' 1 = 9 &1, 0(; +0 ,

' 1 =9 &1, i^ + j^ +k; +0 ,

(7.23.1)

' 1 = 9 &1, i^; 1& ,

' 1 =9 &1, j^ +k; 1& ,

(7.23.2)

' 1 = 9 &1, k; +, 1&+ ,

' 1 = 9 &1, i^ + j^; +, 1&+ .

(7.23.3)

This leaves us with 12 non-zero inner products. We calculate

| {9

( &1, 0; +1

| {9

&1, i^; 0&

|

(7.24)

|

(7.25)

|

(7.26)

|

(7.27)

1 } {' 1 = {9 &1, i^ + j^ +k; +1 } {' 1 =& 24 - 3,

1 } {' 1 = {9 &1, j^ +k; 0& } {' 1 = 24 - 3,

| {9

&1, k; 00

1 } {' 1 = {9 &1, i^ + j^; 00 } {' 1 =& 24 - 3,

| {9

&1, k; 11

1 } {' 1 = {9 &1, i^ + j^; 11 } {' 1 = 24 - 3.

In contrast to the situation where the unit-scale wavelets were of the kind composed of basic C-functions, this situation does not admit complete orthogonality between the scales, but the non-zero inner products have now been computed implicitly, given the rotational symmetries. Finally, consider the event where both wavelets are of the kind composed of basic O-functions. Equations (4.8), (4.13), and (4.14) are relevant in this event, but the possibilities are a little special for adjacent length scales. If a half-scale octahedron is oriented in the same direction as a unit-scale octahedron, then there is no octahedral divider of the former provided by the latter. This observation, in turn, implies that the inner product of the corresponding basic O-functions is zero in that case. In the case where a half-scale octahedron and a unit-scale octahedron do not have the same orientation, an octahedral divider of the former can be given only by a face of the latter, and the inner product is the same negative number given by (4.13) and (4.14) in every case where there is such a divider. With this observation in mind, one can show that

| {.

* +&

} {' &1, =(@ =0

(7.28)

in all cases. In some cases this inner product vanishes by disjoint supports, while in other cases it vanishes because the non-zero contributions cancel. This can be verified by visual inspection. Thus

| {'

@

} {' &1, =(@ =0,

(7.29)

60

GUY BATTLE

and so the unit-scale wavelets of the kind composed of basic O-functions are orthogonal to all of the half-scale wavelets of the same kind.

8. SOBOLEV ORTHOGONALITY PROPERTIES, II Our investigation of interscale orthogonality now turns to the case of nonadjacent length scalesi.e., the case s<&1, since we have already set r=0 without loss of generality. As before, it is sufficient to consider those unit-scale (0-level) wavelets that are supported in the unit cube. In this case, however, there are many of the smallerscale wavelets supported in the unit cube. On the basis of support intersection properties we can say that the mother wavelets are orthogonal to all of the 2 s-scale (s-level) wavelets except the translates (

(

(

9 sl(; +&(! )=2 &s29 +&(2 &s! &l ), (

(

(

' sl(@(! )=2 &s2' @(2 &s! &l ),

(+, &) # [0, 1] 2,

(8.1)

@=1, 2, 3,

(8.2)

where (

l # [0, 1, 2, ..., 2 &s &1] 3.

(8.3)

As in the previous section, there are four situations to investigate because there are two kinds of wavelets. First consider the situation where both the mother wavelet and the s-level wavelet are of the kind composed of basic C-functions. With 9 +& =2. 0+& &2. 1, 1&+, 1&& , 9 sl(; +& =2. sl(; 0+& &2. sl(; 1, 1&+, 1&& ,

(8.4) (8.5)

we know that the Sobolev inner product is zero unless the support of the s-level wavelet is contained in one of the basic cubes comprising the support of the mother wavelet. We may assume with no essential loss of generality that supp 9 sl(; +& /supp . 0+& ,

(8.6)

l # [0, 1, 2, ..., 2 &s&1 &1] 3 +2 &s&1+j^ +2 &s&1&k.

(8.7)

or equivalently, (

61

OSIRIS WAVELETS IN THREE DIMENSIONS

Thus

| {9

( sl ; +&

|

} {9 +& =4 {. sl(; 0+& } {. 0+& &4

| {.

( sl ; 1, 1&+, 1&&

} {. 0+& ,

(8.8)

so we compare the two integrals in each of the cases already discussed for basic C-functions in previous sections. As we observed in Section 3, the inner product of a basic C-function with a lower-level basic C-function vanishes unless there is a divider of the latter basic cube induced by the former. Actually, the cases can be examined more naturally by considering dividers of the s-level basic block containing the support of the s-level wavelet. Following our terminology in the Introduction, we ( define the s-level basic block to be the 2 s-scale cube [0, 2 s ] 3 +2 sl together with the seven s-level wavelets supported by it. Now in the case of a simple divider of the s-level basic block, it is obvious that either a separation of supp . sl(; 0+& from supp . sl(; 1, 1&+, 1&& is induced or a simple divider is induced on both s-level basic cubes. Equation (5.28) applies to the latter occurrence, but it is more to the point to observe that

| {.

( sl ; 0+&

|

} {. 0+& = {. sl(; 1, 1&+, 1&& } {. 0+& ,

(8.9)

so the inner product (8.8) vanishes in that case. We turn to a consideration of a Type C compound divider of the s-level basic block and note that both simple dividers and Type C compound dividers are possible for the s-level basic cubes. Only for one (+, & ) is a Type C compound divider induced on one of the s-level basic cubes, and in that case, such a divider is induced on the other basic cube as well. Equation (5.38) applies to this case, but the point is that (8.9) still holds, and so the inner product (8.8) still vanishes. In the case where a simple divider is induced on one s-level basic cube by the Type C compound divider of the s-level basic block, there is no divider for the other s-level basic cube. Equation (5.28) applies once again, but (8.9) no longer holds. One integral is zero, while the other is not. For example,

| {.

} {. 0+& =0

(8.10)

1 } {. 0+& = 24 2 3s2

(8.11)

} {9 +& =\ 16 2 3s2,

(8.12)

( sl ; 0+&

if and only if

| {.

( sl ; 0+&

in this case. Thus

| {9

( sl ; +&

62

GUY BATTLE

in this case. Actually, three of the four (+, & ) are covered by this case; we have already pointed out that the remaining pair indexes the two s-level basic cubes with Type C compound dividers induced on them. Next, consider the situation where the mother wavelet is still of the kind composed of basic C-functions, but the s-level wavelet is now of the kind composed of basic O-functions. Again, we assume that (8.6) holds for the sake of definiteness. We must analyze

| {'

( sl @

} {9 +& =2

| {'

( sl @

} {. 0+& ,

(8.13)

and we again find it convenient to examine the cases by considering dividers of the s-level basic block. Now in the case of a simple divider, assume without loss of generality that it is yz-inclined. By Observation 4 at the end of Section 3, it can divide those s-level basic octahedra that are x-oriented. By Observation 1, it follows that

| {'

( sl 1

} {. 0+& =- 6

| {.$

} {. 0+& &- 6

| {.$

} {. 0+& ,

(8.14.1)

| {'

( sl 2

} {. 0+& =- 6

| {.$

} {. 0+& &- 6

| {.$

} {. 0+& ,

(8.14.2)

| {'

( sl 3

} {. 0+& =- 6

| {.$

} {. 0+& &- 6

| {.$

} {. 0+&

(8.14.3)

( sl ; 00

( sl ; 00

( sl ; 10

( sl ; 11

( sl ; 11

( sl ; 01

in this case. Equation (4.18) applies to these integrals, but the point is that the yz-inclined plane divides supp .$sl(; 00 (resp. supp .$sl(; 10 ) if and only if it divides supp .$sl(; 11 (resp. supp .$sl(; 10 ). Therefore, the integrals in each difference are equal, and so the inner product (8.13) vanishes. Now suppose . 0+& induces a Type C compound divider of the s-level block, and suppose further that the common line in that divider is part of where the east, north, and upper basic pyramids of supp .0+& meet. Visual inspection of which octahedra are divided yields the reduction

| {'

( sl 1

( } {. 0+& =- 6 {.$$$ sl ; 11 } {. 0+& &- 6

|

| {.$

| {'

( sl 2

} {. 0+& =&- 6 {.$sl(; 11 } {. 0+& +- 6 {."sl(; 11 } {. 0+& ,

| {'

( sl 3

( } {. 0+& =- 6 {."sl(; 11 } {. 0+& &- 6 {.$$$ sl ; 11 } {. 0+& .

( sl ; 11

|

|

} {. 0+& ,

|

|

(8.15.1)

(8.15.2)

(8.15.3)

OSIRIS WAVELETS IN THREE DIMENSIONS

63

Equation (4.18) applies to all of these more basic inner products, so all three differences are zero. If the common line in the divider is part of where the east, south, and lower basic pyramids of supp . 0+& meet, then we have the reduction

| {'

( sl 1

} {. 0+& =- 6 {.$sl(; 00 } {. 0+& &- 6 {."sl(; 10 } {. 0+& ,

|

|

(8.16.1)

| {'

( sl 2

( } {. 0+& =- 6 {.$sl(; 00 } {. 0+& &- 6 {.$$$ sl ; 10 } {. 0+& ,

|

|

(8.16.2)

| {'

( sl 3

} {. 0+& =0,

(8.16.3)

and the more basic inner products are all equal, as before. If we check the other six possibilities for the common line in the divider, we find that the inner product of . 0+& with ' sl(@ is either manifestly zero or equal to a difference that is zero. Thus (8.13) vanishes in every case, and so we have complete orthogonality in the situation where the unit-scale wavelet is of the kind composed of basic C-functions and the s-level wavelet is of the kind composed of basic O-functions. Now consider the reverse situationi.e., where the mother wavelet is ' 1 , ' 2 , or ' 3 and the s-level wavelet is the 9 +& -kind of wavelet. Once again, the inner product is zero in the case of a simple divider of the s-level basic block. It does not matter that the simple divider is induced by some basic O-function instead. As in the first situation investigated above, it either separates supp . sl(; 0+& from supp . sl(; 1, 1&+, 1&+ or it divides both basic cubes. While (5.10) now applies instead of (5.28), the contributions to the inner product still cancel out. The remaining possibility is that one or two Type O compound dividers are induced on the s-level basic block. To understand why there can be two Type O compound dividers induced on the same s-level basic block by the unit-scale wavelet ' @ , it is necessary only to inspect the schematic representation of ' @ in the Introduction. For example, suppose @=1 and the s-level basic block lies somewhere on the line segment from ( 34 , 34 , 14 ) to (1, 12 , 12 ). By Fig. 6.1 in the Introduction, it is clear that the basic O-functions ."10 and .$$$ 11 contribute to ' 1 , that their supports have a common face, and that the line segment we have picked is a boundary of the common face. This means that ."10 and .$$$ 11 both induce Type O compound dividers on the s-level basic block. On the other hand, the two compound dividers share the same antipodal line segment for the basic block in this case. Before we examine inner products in the case of two Type O compound dividers, we first examine the case of a single Type O compound dividerexemplified by an s-level basic block somewhere on, say, the line segment from ( 12 , 0, 0) to ( 34 , 14 , 14 ) in Fig. 6.1. (Only the basic O-function .$00 induces a Type O compound divider in that scenario.) As we pointed out in the Type C situation, a compound divider is induced on the s-level basic cubes of only one of the (+, & ). The only difference is in the type of compound divider. Equation (5.21) is now relevant, but the Type O

64

GUY BATTLE

compound divider is induced for both basic cubes, so the integrals cancel out. For each of the other pairs (+, & ) of indices, the Type O compound divider of the s-level block induces a simple divider for one basic cube, but no divider for the otherjust as the Type C compound divider did in the first situation. However, this situation is a little different, in that the simple dividers for two of the (+, & ) are induced by faces of a unit-scale basic octahedron in supp ' @ , while the simple divider for the remaining (+, & ) is part of a divider of that octahedron. In summary,

| {'

@

} {9 sl(; +& =0

(8.17)

in every case except these two. In the case where the induced simple divider is part of an octahedral face, Eq. (5.10) applies, so

| {'

@

- 6 3s2 } {9 sl(; +& =\ 2 . 12

(8.18)

In the case where the simple divider is induced by an octahedral divider, (5.7) applies, so the inner product still has one of the values given by (8.18). We now shift our attention back to the possibility we have already raisedthat two Type O compound dividers are induced. As before, a compound divider is induced on the s-level basic cubes of only one of the (+, & ), and it is still true that the integrals cancel out in the inner product of that wavelet with the unit-scale waveletwhatever the values of the integrals may be. For each of the other pairs (+, & ) of indices, only simple dividers can be induced on the s-level basic cubes by our pair of Type O compound dividers of the s-level basic block. The similarity to the preceding case ends there, but actually, this case is more favorable in the sense that the inner products are zero for these other (+, & ) as well. For one of them, one cube has no simple divider, while the other cube is divided by the face common to the supports of the two basic O-functions. The integral associated with the first cube is zero because that cube lies in a unit-scale basic simplex, while the integral associated with the other cube is zero because it is the cancellation of the inner product of the basic C-function with one basic O-function against the inner product with the other. The point here is that the two basic O-functions must have opposite signs, as can be seen from the definition of ' @ (see Fig. 6 in the Introduction). For each of the remaining two (+, & ), one cube is divided by a support face of only one basic O-function, while the other cube is divided by the support divider of the other basic O-function. Now the inner product of a basic C-function with a basic O-func1 1 tion is &( 12 - 6) 2 3s2 in the first case and ( 12 - 6) 2 3s2 in the latter case, so we have cancellation for the remaining two (+, & ) also. Finally, consider the situation where both wavelets are of the kind composed of basic O-functions. In the case where a simple divider of the s-level block is induced, assume that it is yz-inclined without loss of generality. We have the reduction

65

OSIRIS WAVELETS IN THREE DIMENSIONS

| {'

( sl 1

} {' @ =\- 6

| {.$

} {. * +& Ã- 6

| {.$

} {. * +& ,

(8.19.1)

| {'

( sl 2

} {' @ =\- 6

| {.$

} {. * +& Ã- 6

| {.$

} {. * +& ,

(8.19.2)

| {'

( sl 3

} {' @ =\- 6

| {.$

} {. * +& Ã- 6

| {.$

} {. * +& ,

(8.19.3)

( sl ; 00

( sl ; 00

( sl ; 10

( sl ; 11

( sl ; 11

( sl ; 01

where . * +& is the unit-scale basic O-function associated with the octahedron that induces the simple divider. Now if this involves a divider of supp . * +& , then *=$ and Eq. (4.8) applies. On the other hand, this yz-inclined plane divides supp .$sl(; 00 (resp. supp .$sl(; 10 ) if and only if it divides supp .$sl(; 11 (resp. supp .$sl(; 01 ), so the integrals cancel or vanish in each of the three differences. If a face of supp . * +& induces the simple divider, then (4.14) applies instead, but the observation and the conclusion do not change. Now suppose . * +& induces a Type O compound divider of the s-level block, and suppose further that *=$ and that the upper east and north east basic simplexes in supp .$+& are involved. Visual inspection of which of the s-level block octahedra are divided yields the reduction

| {'

( sl 1

} {' @ =\- 6

| {.$

} {.$+& \- 6 {.$sl(; 01 } {.$+& ,

| {'

( sl 2

} {' @ =\- 6

| {.$

( } {.$+& \- 6 {.$$$ sl ; 01 } {.$+& ,

| {'

( sl 3

} {' @ =0.

( sl ; 00

( sl ; 00

|

(8.20.1)

|

(8.20.2) (8.20.3)

In each of the first two expressions, Eq. (4.8) applies to the first integral, while (4.14) applies to the second. Notice that these values cancel out, so all three inner products vanish. If *=" instead, and if the lower north and east north basic simplexes are now involved, then we have the reduction

| {'

( sl 1

} {' @ =\- 6

| {.$

} {."+& \- 6 {."sl(; 01 } {."+& ,

| {'

( sl 2

} {' @ =\- 6

| {.$

( } {."+& Ã- 6 {.$$$ sl ; 10 } {." +& ,

| {'

( sl 3

} {' @ =0.

( sl ; 00

( sl ; 00

|

(8.21.1)

|

(8.21.2) (8.21.3)

In the first expression, the first integral is given by (4.14), while the second integral is given by (4.8), so they cancel out. In the second expression, both integrals are given by (4.14), so we still have cancellation. If we check all of the other

66

GUY BATTLE

possibilities, we find that all of the inner products vanish for one or another of these reasons. We have complete orthogonality in the situation where both unit-scale and s-level wavelets are of the kind composed of basic O-functions.

9. OVERLAP MATRIX ELEMENTS FOR ADJACENT LENGTH SCALES Following the notation adopted in the Introduction, we index our set of wavelets with a dyadic scaling parameter r # Z, a discrete translation parameter n( # Z 3, and a mother wavelet parameter }=0, 1, ..., 6, where we have already set 9 +& =9 2++& ,

(9.1)

' @ =9 @+3 .

(9.2)

The overlap matrix is given by

|

S rr$; n(n($; }}$ = {9 rn(} } {9 r$n($}$ , (

(

9 rn(}(! )=2 &r2 9 }(2 &r! &n( ).

(9.3) (9.4)

We obviously have the condition S r+q, r$+q; n(n($; }}$ =S rr$; n(n($; }}$ ,

(9.5)

S rr$; n(n($; }}$ =T r$&r; n(n($; }}$ .

(9.6)

so we have the form

We propose to calculate T s; n(n($; }}$ for each s # Z. Now the s=0 elements comprise the overlap matrix for the unit-scale wavelets, which are obviously supported in unit blocks. Wavelets associated with different unit blocks have mutually disjoint supports, so T 0; n(n($; }}$ =E }}$ $ n(n($ ,

(9.7)

where E }}$ is the overlap matrix for the seven wavelets associated with one unit block. We have already examined this positive matrix and found that it has eigenvalues 1 and 4, with multiplicities 5 and 2, respectively. Thus, the trivially infinite matrix T 0 has only these eigenvalues. On the other hand, T &s =T s* ,

(9.8)

so it suffices to calculate the elements T s; n(n($; }}$ for, say all s<0. The infinite matrix T &1 is significantly different from T s for s<&1, so we calculate the elements T &1; n(n($; }}$ first.

OSIRIS WAVELETS IN THREE DIMENSIONS

67

The structure of T &1 is a little more involved than that of T 0 , as in this case of adjacent length scales, the support properties of the wavelets imply T &1; n(n($; }}$ =0,

( n($&2n  [0, 1] 3.

(9.9)

Indeed, T &1; n(, 2m( +=(; }}$ =T &1; 0(=(; }}$ $ n(m( ,

=( # [0, 1] 3,

(9.10)

so we set (

T &1; 0(=(; }}$ =E =}}$ .

(9.11)

(

E = is the matrix of inner products between the unit-scale wavelets supported in [0, 1] 3 (i.e., the mother wavelets) and the half-scale wavelets supported in [0, 12 ] 3 + 12 =(. Our task is to calculate these eight 7_7 matrices. We found in Section 7 that the mother wavelets 9 +& =9 2++& composed of basic C-functions are Sobolev-orthogonal to the half-scale wavelets of both kinds. Since (

|

E =}}$ = {9 } } {9 &1, =(}$ ,

(9.12)

it follows that (

E =}}$ =0,

}=0, 1, 2, 3, }$=0, 1, ..., 6,

(9.13)

for every =( # [0, 1] 3. We also found that the mother wavelets ' @ =9 @+3 composed of basic O-functions are Sobolev-orthogonal to the half-scale wavelets of the same kind. Hence (

E =}}$ =0,

}=4, 5, 6, }$=4, 5, 6,

(9.14)

for every =( # [0, 1] 3 as well. The other cases where matrix elements must vanish are =-dependent. For example, the orthogonality relations (7.20) imply (

^

^

j +k E 4}$ =E i4}$ =0, ^

(9.15.1)

^

 +j =E i5}$ =0, E k5}$ ^

^

j +k E i6}$ =E 6}$ =0,

(9.15.2) }$=0, 1, 2, 3.

(9.15.3)

68

GUY BATTLE

On the other hand, the orthogonality relations (7.23) imply (

(

^

i^ 42

i^ 43

j^ +k 42

k 41

k 42

i^ + j^ 41

^

^

^

E 040 =E 042 =E i40+ j +k =E i42+ j +k =0, E =E =E E =E =E

=E

=E

j^ +k 43

i^ + j^ 42

=0,

(9.16.1) (9.16.2)

=0.

(9.16.3)

With our focus on the }=4 entries of all 8 matrices, we see that only 12 such matrix elements remain to be evaluated, but the evaluation has already been done in Section 7. Indeed, (7.24)(7.27) imply (

(

^

i^ 40

i^ 41

j^ +k 40

k 40

i^ + j^ 40

k 43

i^ + j^ 43

^

^

^

1 - 3, E 041 =E 043 =E i41+ j +k =E i43+ j +k =& 24

E =E =E E =E E =E

=E

j^ +k 41

=

1 24

- 3,

(9.18)

1 =& 24 - 3,

=

1 24

(9.17)

(9.19)

- 3.

(9.20)

To derive the }=6 entries of our matrices, we need only apply the rotation R 1 introduced in Section 7. Since 9 &1, 0(; 00 b R &1 1 =9 &1, j^ ; 10 , 9 &1, 0(; 10 b R

&1 1

9 &1, 0(; 01 b R &1 1 =9 &1, j^ ; 00 ,

(9.21.0)

9 &1, 0(; 11 =9 &1, j^ ; 01 ,

(9.21.1)

=9 &1, j^ ; 11 ,

9 &1, i^ ; 00 b R &1 1 =9 &1, j^ +i^ ; 10 ,

9 &1, i^ ; 01 b R &1 1 =9 &1, j^ +i^ ; 00 ,

(9.22.0)

9 &1, i^ ; 10 b R

9 &1, i^ ; 11 b R

(9.22.1)

&1 1

=9 &1, j^ +i^ ; 11 ,

&1 1

=9 &1, j^ +i^ ; 01

( 9 &1, k; 00 b R &1 1 =9 &1, 0; 10 ,

( 9 &1, k; 01 b R &1 1 =9 &1, 0; 00 ,

(9.23.0)

( 9 &1, k; 10 b R &1 1 =9 &1, 0; 11 ,

( 9 &1, k; 11 b R &1 1 =9 &1, 0; 01 ,

(9.23.1)

9 &1, i^ +j^ ; 00 b R

&1 1

9 &1, i^ +j^ ; 01 b R

=9 &1, i^ +j^ +k; 10 ,

&1 1

=9 &1, i^ +j^ +k; 00 , (9.24.0)

9 &1, i^ +j^ ; 10 b R &1 1 =9 &1, i^ +j^ +k; 11 ,

9 &1, i^ +j^ ; 11 b R &1 (9.24.1) 1 =9 &1, i^ +j^ +k; 01 ,

9 &1, j^ +k; 00 b R &1 1 =9 &1, k; 10 ,

9 &1, j^ +k; 01 b R &1 1 =9 &1, k; 00 ,

(9.25.0)

9 &1, j^ +k; 10 b R &1 1 =9 &1, k; 11 ,

9 &1, j^ +k; 11 b R &1 1 =9 &1, k; 01 ,

(9.25.1)

9 &1, i^ +j^ +k; 00 b R &1 1 =9 &1, i^ +k; 10 ,

9 &1, i^ +j^ +k; 01 b R &1 1 =9 &1, i^ +k; 00 ,

(9.26.0)

9 &1, i^ +j^ +k; 00 b R

9 &1, i^ +j^ +k; 11 b R

(9.26.1)

&1 1

=9 &1, i^ +k; 11 ,

&1 1

=9 &1, i^ +k; 01 ,

it follows from (7.21) and (9.16) that ^

^

^

^

j j =E 63 =E i62+k =E i63+k =0, E 62

E

i^ + j^ 63 ( 0 60

=E

i^ + j^ 61 ( 0 63

(9.27.1)

  =E k63 =E k61 =0,

E =E =E

i^ + j^ +k 60

=E

i^ + j^ +k 63

(9.27.2) =0.

(9.27.3)

69

OSIRIS WAVELETS IN THREE DIMENSIONS

If we combine the same rotational relations with (9.17)(9.20), we obtain ^

^

^

^

j j 1 =E 61 =E i60+k =E i61+k =& 24 - 3, E 60 ^

^

^

^

  1 E i62+ j =E i60+ j =E k62 =E k60 = 24 - 3, ( 0 62 ( 0 61

E =E

i^ + j^ +k 62

E =E

i^ + j^ +k 61

1 =& 24 - 3,

=

1 24

(9.28) (9.29) (9.30)

- 3.

(9.31)

The effect of R 1 is really more simple than our laborious tabulation may suggest. The correspondence (

0 Ä j^,

(

i^ Ä i^ +j^,

j^ +k Ä k,

k Ä 0,

i^ +j^ Ä i^ +j^ +k,

i^ +j^ +k Ä i^ +k,

is consistent with the 90%-revolution of the sub-cubes labeled by these vectors about the axis y=z= 12 . The correspondence (0, 0) Ä (1, 0),

(0, 1) Ä (0, 0),

(1, 0) Ä (1, 1),

(1, 1) Ä (0, 1),

with no sign-reversals of the 9 &1, =(; +& is consistent with the associated 90% changes in the orientations of these sub-cubes supporting these half-scale wavelets. The }=5 entries are now derived by applying the rotation R 2 introduced in Section 7. The relevant correspondences are (

( j^ Ä 0,

0 Ä i^,

k Ä i^ +k,

i^ +k Ä i^ +j^ +k,

i^ +j^ Ä j^,

i^ +j^ +k Ä j^ +k

and (0, 0) Ä (1, 0)

(sign reversal)

(0, 1) Ä (1, 0)

(sign reversal)

(1, 0) Ä (0, 0)

(no sign reversal)

(1, 1) Ä (0, 1)

(no sign reversal)

in this case. Combining the induced relations with (7.22) and (9.27), we obtain (

(

^

j^ 51

j^ 52

i^ +k 51

i^ 53

i^ 51

j^ +k 53

^

^

^

E 050 =E 051 =E i50+ j +k =E i51+ j +k =0, E =E =E E =E =E

=E =E

i^ +k 52

=0,

j^ +k 51

=0.

(9.32.1) (9.32.2) (9.32.3)

70

GUY BATTLE

The same relations together with (9.28)(9.31) imply (

(

^

^

^

^

^

^

1 E 053 =E 052 =E i53+ j +k =E i52+ j +k =& 24 - 3, ^

^

j j 1 &E 50 =E 53 =&E i50+k =E i53+k = 24 - 3, ^

(9.34)

^

j +k 1 &E i50 =&E 50 =& 24 - 3, ^

(9.33)

(9.35)

^

j +k 1 E i52 =E 52 = 24 - 3.

(9.36)

We have finally calculated all matrix elements. Now if we combine (9.13) with (9.14), we see that the only non-zero elements of ( E = lie in both the first 4 columns and the last 3 rows. It is clearly convenient to ( record only this 3_4 sub-matrix, which we denote by E = . Thus (

1 E 0 = 24 -3

i^

E =

j^

E =

k

E =

^

1 24

-3

1 24

-3

1 24

-3

_ _ _ _

0 &1

0 &1

0 &1

&1 ,

0

1 &1

0

1 1 0

0

1 0 1

0 ,

0 0 0

0

&

0

0 0

&1

0

0 1 ,

&1

&1

&1

0 0 1

0 0

&

0 0 0 ,

1

0 1 0

^

E j +k =E i, ^

^

i^ + j^ +k

( 0

E i +k =E j, E

&

0

^

=E .

(9.37.1)

(9.37.2)

0

E i + j =E k, ^

&

0

(9.37.3)

(9.37.4)

(9.38.1) (9.38.2) (9.38.3) (9.38.4)

This completes our calculation of the infinite matrix T &1 . T &1 T &12 which will be important in what Actually, it is the infinite matrix T &12 0 0 is to come. However, this is easy enough to calculate. Since (T &12 ) n(n($; }}$ =(E &12 ) }}$ $ n(n( $ , 0

(9.39)

OSIRIS WAVELETS IN THREE DIMENSIONS

71

it follows that (

T &1 T &12 ) n(, 2m( +=(; }}$ =(E &12E = E &12 ) }}$ $ n(m( . (T &12 0 0

(9.40)

(

Since the last 3 columns of E = have only zero elements, it is clear from (6.14) that (

(

E = E &12 =E = ,

(9.41)

so we have (

(

1 E &12 E = E &12 = 144 A=,

(9.42)

(

(

A = =144E &12 E = . (

(9.43)

(

As in the case of E = , the only non-zero entries of A = lie in both the first 4 columns ( and the last 3 rows, and we shall denote this 3_4 sub-matrix by A = . We have

(

A 0 =3 - 3

_ _

i^

_

0 &1

0 &1

&

0

0 &1

&1 ,

0

1 &1

0

3

4 &1

A =- 3 3 &1 0

0

&

4 0 ,

1 &1

0

0 &1 0 &1 ^ A j =- 3 &3 1 0 4 ,

A k =- 3

^

^

A i + j =A k, ^

^

A j +k =A i, ^

^

A i +k =A j, ^

^

(

A i + j +k =A 0.

_

&3

&4

0 &1

&3

0

1

0 0 &1 3 0

4

4

& &

&1 , 1

(9.44.1)

(9.44.2)

(9.44.3)

(9.44.4)

(9.45.1) (9.45.2) (9.45.3) (9.45.4)

72

GUY BATTLE (

To obtain the norms of the A = , we calculate 0 0 0 0 2 &1 A * A =27 0 &1 2 0 1 1

0 1 , 1 2

(9.46.1)

2 1 1 1 2 &1 ^ ^ A i * A i =27 1 &1 2 0 0 0

0 0 , 0 0

(9.46.2)

( 0

_ _ _ _

( 0

& & & &

2 1 A *A j =27 0 &1

1 2 0 1

0 0 0 0

&1 1 , 0 2

(9.46.3)

2 0 A k *A k =27 1 &1

0 0 0 0

1 0 2 1

&1 0 . 1 2

(9.46.4)

j^

Each of these matrices has only the eigenvalues 0 and 81. Hence, (

(

&A = * A = &=81,

=( # [0, 1] 3,

(9.47)

and so (

&A = &=9,

=( # [0, 1] 3.

(9.48)

10. OVERLAP MATRIX ELEMENTS FOR NON-ADJACENT LENGTH SCALES We now turn to calculating the elements of the infinite matrix T s for arbitrary s<&1. In this general case, the support properties of the wavelets imply T s; n(n($; }}$ =0,

n($&2 &s n( Â [0, 1, ..., 2 &s &1] 3,

(10.1)

with &s as a positive integer. Indeed, T s; n(, 2 &s m( +l( ; }}$ =T s; 0(l( ; }}$ $ n(m( ,

(

l # [0, 1, ..., 2 &s &1] 3,

(10.2)

OSIRIS WAVELETS IN THREE DIMENSIONS

73

so we set ((

; =| T s; 0(l( ; }}$ =F N }}$ ,

(10.3)

(

( l =2 N (= +| ,

=( # [0, 1] 3 ( | # [0, 1,..., 2 N &1] 3,

s=&N&1.

(10.4) (10.5)

((

F N; = | is the matrix of inner products between the unit-scale wavelets supported in [0, 1] 3 and the 2 &N&1-scale wavelets supported in ( [0, 2 &N&1 ] 3 + 12 =( +2 &N&1| .

The calculation of these 7_7 matrices is determined by the rules discussed in Section 8, as these inner products were implicitly evaluated there. Actually, our first task is to count the non-zero matrices, and to this end, we recall that an inner product is zero unless the larger-scale wavelet induces a compound divider of the s-level basic block supporting the smaller-scale wavelet. In every case, we saw that the inner product vanishes if the induced divider is simple. On the other hand, every compound divider induced on some s-level basic block involves a line segment connecting a vertex of the relevant unit-scale basic cube [0, 12 ] 3 + 12 (= to its center. Moreover, that line segment connects two opposite vertices of that s-level basic block. Such s-level basic blocks are easy to count. Clearly, there are 2 N&1 of them on each of these 8 line segments in [0, 12 ] 3 + 12 =(. This observation (( reduces the number of matrices F N; = | by two orders of magnitudei.e., the number N of non-zero matrices is O(2 ) instead of O(2 3N ). We may also make the general observation that ((

((

F N; = |$ =F N; = |

(10.6)

( ( if | and | $ label s-level blocks on the same line segment. We index these 8 line segments with [0, 1] 3 such that (=$ labels the line segment connecting the center of (( [0, 12 ] 3 + 12 =( to the vertex whose position vector is 12 (=( +=($). Let E N; = = $ denote the ( matrix (10.6) common to all of the s-level basic blocks on the =$-segment in [0, 12 ] 3 + 12 =(. Our counting and identifications have reduced our task to the calculation of 64 matrices. Another general observation derived from the analysis in Section 8 is that the inner product vanishes when the s-level wavelet is of the kind composed of s-level basic O-functions. Thus ((

==$ E N; }}$ =0,

}=0, 1, ..., 6, }$=4, 5, 6.

(10.7)

74

GUY BATTLE

Now pick a representative s-level basic block on the =($-segment in [0, 12 ] 3 + 12 (= say, the one touching the center of that basic cubeand denote the associated (=($ for }$=0, 1, 2, 3. Then s-level 9 }$ -wavelet by 9 =s}$ ((

((

|

==$ ==$ E N; }}$ = {9 } } {9 s}$ ,

}=0, 1, ..., 6, }$=0, 1, 2, 3,

(10.8)

and since [0, 12 ] 3 + 12 =( lies in supp 9 +& for only one (+, &) # [0, 1] 2, we know that (=($ for the other three (+, &). Let the wavelet  2++& =9 +& is Sobolev-orthogonal to 9 =s}$ ( }(= )=2++& for the distinguished (+, &). Then the condition (10.7) is complemented by ((

== $ E N; }}$ =0,

}{}(=( ), 4, 5, 6 }$=0, 1, 2, 3.

(10.9)

( }(0 )=}(i^ + j^ +k )=0,

(10.10.0)

}( j^ )=}(i^ +k )=1,

(10.10.1)

}(i^ )=}( j^ +k )=2,

(10.10.2)

We tabulate

}(i^ + j^ )=}(k )=3

(10.10.3)

for future reference. With 64 matrices to reckon with, we need to exploit the rotational symmetries in a more systematic way than we did in the previous section. We have already defined R 1 (resp. R 2 ) as the yz-counter-clockwise (resp. xy-counter-clockwise) 90%-rotation about the axis y=z= 12 (resp. x= y= 12 ). We now define R 3 as the zx-counter-clockwise 90%-rotation about the axis z=x= 12 . In the previous section we discussed the permutations of the basic cubes [0, 12 ] 3 + 12 =( induced by R 1 and R 2 . Let P 1 , P 2 , and P 3 be the permutations induced by R 1 , R 2 , and R 3 , respectively. We tabulate these permutations in Table I. It is easy enough to tabulate the action of these rotations on the four wavelets (=($ 9 2++& =9 +& , shown in Table II. The action of these rotations on the wavelets 9 =s}$ has a simple decomposition into an action on =((=$ and an action on 9 }$ . By tracking the change of orientation of a basic cube associated with its revolution, we see that R @ induces the change =( =($ [ P @(=( ) P @(=($). We have the relations shown in Table III. Having listed the rotational relations for all 9 +& -wavelets at both levels, we still need to tabulate them for the unit-scale ' @ -wavelets. Recalling (7.22) and (7.23), we now tabulate all such equations, shown in Table IV. If we insert our notation ' @ =9 3+@ , Table IV becomes Table V. Combining Tables II, III, and V with (10.8), we obtain the matrix product relations

75

OSIRIS WAVELETS IN THREE DIMENSIONS TABLE I (

(

(

P 1( 0 )= j^ P 1(i^ )=i^ + j^ P 1( j^ )= j^ +k

P 2( 0 )=i^ P 2(i^ )=i^ + j^ ( P 2( j^ )= 0

P3( 0 )=k ( P 3(i^ )= 0 P 3( j^ )= j^ +k

P 1(k )= 0 P 1(i^ + j^ )=i^ + j^ +k P 1(i^ +k )=i^

P 2(k )=i^ +k P 2(i^ + j^ )= j^ P 2(i^ +k )=i^ + j^ +k

P 3(k )=i^ +k P 3(i^ + j^ )= j^ P 3(i^ +k )=i^

(

P 1( j^ +k )=k

P 2( j^ +k )=k

P 3( j^ +k )=i^ + j^ +k

P 1(i^ + j^ +k )=i^ +k

P 2(i^ + j^ +k )= j^ +k

P 3(i^ + j^ +k )=i^ + j^

TABLE II 9 0 b R &1 1 =9 2 9 1 b R &1 1 =9 0

9 0 b R &1 2 =&9 3 9 1 b R &1 2 =&9 2

9 0 b R &1 3 =9 1 9 1 b R &1 3 =&9 2

9 2 b R &1 1 =9 3 9 3 b R &1 1 =9 1

9 2 b R &1 2 =9 0 9 3 b R &1 2 =9 1

9 2 b R &1 3 =9 3 9 3 b R &1 3 =&9 0

TABLE III ((

((

(

(

P2 (= ) P2 (= $) 9 =s0= $ b R &1 2 =&9 s3

P1 (= ) P1 (= $) 9 =s1= $ b R &1 1 =9 s0

(

(

P2 (= ) P2 (= $) 9 =s1= $ b R &1 2 =&9 s2

(( 9 =s2=

(( 9 =s2= $

b R &1 2 =9

=(=( $ 9 s3

2 b R &1 2 =9 s1

(( 9 =s2= $

P3 (=( ) P3 (=( $) b R &1 3 =9 s3

P1 (= ) P1 (= $) 9 =s0= $ b R &1 1 =9 s2 ((

b R &1 1 =9

(( 9 =s3= $

1 b R &1 1 =9 s1

=(=( $ 9 s0

P3 (=( ) P3 (=($) b R &1 3 =9 s1

((

((

P1 (=( ) P1 (=($) s3 P (=( ) P (=($) 1

(

(

==$ P3 (= ) P3 (= $) 9 s1 b R &1 3 =&9 s2

((

(

(

(

(

P2 (=( ) P2 (=($) s0 P (=( ) P (=($) 2

(

(

==$ P3 (= ) P3 (= $) 9 s3 b R &1 3 =&9 s0

TABLE IV ' 1 b R &1 1 =' 3 ' 2 b R &1 1 =' 1 &' 2 +' 3

' 1 b R &1 2 =&' 1 +' 2 &' 3 ' 2 b R &1 2 =&' 1

' 1 b R &1 3 =' 3 ' 2 b R &1 3 =&' 1

' 3 b R &1 1 =&' 2

' 3 b R &1 2 =&' 2

' 3 b R &1 3 =&' 1 +' 2 &' 3

TABLE V 9 4 b R &1 1 =9 6

9 4 b R &1 2 =&9 4 +9 5 &9 6

9 4 b R &1 3 =9 6

9 5 b R &1 1 =9 4 &9 5 +9 6 9 6 b R &1 1 =&9 5

9 6 b R &1 2 =&9 4 9 6 b R &1 2 =&9 5

9 5 b R &1 3 =&9 4 9 6 b R &1 3 =&9 4 +9 5 &9 6

76

GUY BATTLE ((

(

(

E N; = = $ =U _ E N; P_ (= ) P_ (= $) U _*,

U1=

U2=

U3=

_ _ _

0 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 1 0 0 0 0

0 0 &1 0 0 0 0 &1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 &1 1 &1 0 0 0 &1 0 0 0 0 0 0 &1 0

0 0 0 &1 0 0 0

1 0 0 0 0 0 0

0 0 1 0 0 0 0

_=1, 2, 3,

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 &1 1 0 &1 0

1 0 0 &1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 &1 0 &1

&

(10.11)

,

(10.12.1)

0 0 0 0 0 0 0 0 0 1 0 0 1 &1

& &

,

.

(10.12.2)

(10.12.3)

Combinations of (10.11) generate our 64 matrices from just 8 of them. We now (( focus on the matrices E N; 0= $, and we can reduce the number of calculations further with more symmetries. Let % 1 be reflection through the plane y=z, % 2 reflection through the plane x= y, and % 3 reflection through the plane z=x. These transformations preserve the unit cube [0, 1] 3, and their action on the relevant mother wavelets 9 0 , 9 4 , 9 5 , 9 6 is tabulated in Table VI. The action of these reflections on the s-level 9 +& -wavelets is given in Table VII. TABLE VI 9 4 b % &1 1 =&9 5

9 4 b % &1 2 =&9 6

9 4 b % &1 3 =&9 4

9 5 b % &1 1 =&9 4 9 6 b % &1 1 =&9 6 9 0 b % &1 1 =9 0

9 5 b % &1 2 =&9 5 9 6 b % &1 2 =&9 4 9 0 b % &1 2 =9 0

9 5 b % &1 3 =&9 6 9 6 b % &1 3 =&9 5 9 0 b % &1 3 =9 0

77

OSIRIS WAVELETS IN THREE DIMENSIONS TABLE VII ((

(

((

(

(

((

(

(

(

0 %1 (= $) 9 0s0= $ b % &1 1 =9 s0

0 %2 (= ) 9 0s0= $ b % &1 2 =9 s0

0 %3 (= $) 9 0s0= $ b % &1 3 =9 s0

(( 9 0s1= $ (( 9 0s2= $ (( 9 0s3= $

(( 9 0s1= $ (( 9 0s2= $ ( 0 =($ 9 s3

(( 9 0s1= $ (( 9 0s2= $ ( 0 =( $ 9 s3

( 0 %1 (=($) b % &1 1 =9 s2 ( 0 %1 (=($) b % &1 1 =9 s1 ( 0 %1 (=($) b % &1 1 =9 s3

( 0 %2 (=($) b % &1 2 =9 s1 ( 0 %2 (=($) b % &1 2 =&9 s3 ( 0 %2 (=( $) b % &1 2 =&9 s2

( 0 %3 (=($) b % &1 3 =&9 s3 ( 0 %3 (=($) b % &1 3 =9 s2 ( 0 %3 (=( $) b % &1 3 =&9 s1

Combining these tables, we obtain the matrix product relations ((

(

(

1 0 0 0 0 0 0

0 0 1 0 0 0 0

0 1 0 0 0 0 0

1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 &1 0 0 0 0 &1 0 0 0 0 0 0 0 0 0 &1 0 0 0 0 &1 0 0 0 0 &1 0 0

1 0 0 0 0 0 0

0 0 0 &1 0 0 0

E N; 0= $ =V _ E N; 0%_ (= $) V _*,

V1 =

V2 =

V3 =

_ _ _

_=1, 2, 3,

(10.13)

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 &1 0 0 &1 0 0 0 0 0 &1

&

,

(10.14.1)

0 0 0 0 0 0 &1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 &1 0 0 0 0 0 0 &1 0 0 0 &1 0 ((

& &

,

(10.14.2)

.

(10.14.3)

In this way, we can generate the eight matrices E N; 0= from just four of them. We choose to calculate ((

E N; 00, directly.

(^

E N; 0i,

( ^

^

E N; 0, i + j,

( ^

^

E N; 0, i + j +k

78

GUY BATTLE ((

For E N; 00 the s-level basic block on the associated line segment is disjoint from supp 9 } when }=4, 5, 6. This means that ((

00 E N; }}$ =0,

}=4, 5, 6, }$=0, 1, ..., 6,

(10.15)

so by (10.7) and (10.9) the only possible non-zero entries are ((

((

00 E N; , 00

((

00 E N; , 01

00 E N; , 02

((

00 E N; . 03

In this case, each inner product involves a Type C compound divider of the s-level basic block, and according to our discussion in Section 8, it induces a Type C compound divider of the s-level basic cubes comprising the support of only one of these s-level( ( wavelets. Moreover, the corresponding inner product vanishes. In this context, 9 0s00 is that distinguished wavelet, so ((

00 E N; =0. 00

(10.16.0)

Applying (8.12) to this context as well, we obtain (( 1 1 00 E N; 2 &3N2, = 2 3s2 = 01 6 12 - 2

(10.16.1)

(( 1 1 00 E N; = 2 3s2 = 2 &3N2, 02 6 12 - 2

(10.16.2)

(( 1 1 00 E N; =& 2 3s2 =& 2 &3N2. 03 6 12 - 2

(10.16.3)

This matrix( is the simplest of the four matrices. ^ For E N; 0i the s-level basic block on the associated line segment is disjoint from supp 9 6 but intersects both supp 9 4 and supp 9 5 . Thus (^

0i E N; 6}$ =0,

}$=0, 1, ..., 6,

(10.17) (^

(^

0i N; 0i by the argument following (8.18). On the other hand, E N; 4}$ and E 5}$ do not vanish for all }$. In this case, each inner product involves a Type O compound divider of the s-level basic block. Combining the discussion preceding (8.17) and (8.18) with visual inspection of basic supports, we see that

(^

1 2 3s2 =& 2 &3N2, 2 -6 8 -3

(^

1

0i E N; 50 =&

0i E N; 51 =

1

2 -6

2 3s2 =

1 8 -3

2 &3N2,

(10.18.0) (10.18.1)

OSIRIS WAVELETS IN THREE DIMENSIONS

1

(^

0i E N; 52 =

2 -6

2 3s2 =

1 8 -3

2 &3N2,

(^

0i E N, 53 =0,

79 (10.18.2) (10.18.3)

and that (^

(^

0i N; 0i E N; 4}$ =E 5}$ .

(10.19)

Inspection also shows that the inner products involving 9 0 are ( 1 3s2 1 0i^ 2 &3N2, E N; = 00 = 2 6 12 - 2

(10.20.0)

( 1 3s2 1 0i^ E N; =& 2 &3N2, 01 =& 2 6 12 - 2

(10.21.1)

( 1 3s2 1 0i^ 2 &3N2, E N; =& 02 =& 2 6 12 - 2

(10.20.2)

(^

0i E N; 03 =0.

(10.20.3)

(^

All other elements of E N; 0i vanish. ( N; 0, i^ + j^ the s-level basic block on the associated line segment intersects For E supp 9 4 , supp 9 5 , and supp 9 6 . Indeed, with regard to 9 5 , the block intersects with both supp .$00 and supp ."00 two basic octahedra with a common face. This is another case where visual inspection together with our geometric analysis following (8.18) applies. We have ( ^

1

^

0, i + j E N; =& 60 ( ^

^

( ^

^

2 -6

2 3s2 =&

1 8 -3

2 &3N2,

0, i + j E N; =0, 61

0, i + j E N; = 62

( ^

(10.21.1) 1

2 -6

( ^

( ^

^

2 -6

1

^

2 -6

0, i + j E N; =0, 41

2 3s2 =

1

^

0, i + j E N; =& 63

0, i + j E N; = 40

(10.21.0)

1 8 -3

2 3s2 =&

2 3s2 =

2 &3N2, 1

8 -3

1 8 -3

2 &3N2,

2 &3N2,

(10.21.2)

(10.21.3)

(10.22.0) (10.22.1)

80

GUY BATTLE ( ^

1

^

0, i + j E N; =& 42

( ^

1

^

0, i + j E N; = 43

2 -6

2 -6

2 3s2 =&

2 3s2 =

1 8 -3

1 8 -3

2 &3N2,

2 &3N2

(10.22.2) (10.22.3)

for the inner products involving either supp .$00 or supp ."00 exclusively. The inner products involving both supp .$00 and supp ."00 actually vanish: ( ^

^

0, i + j =0. E N; 5}$

(10.23)

We have ( 1 1 0, i^ + j^ = & 2 3s2 =& 2 &3N2, E N; 00 6 12 - 2 ( ^

^

(10.24.0)

0, i + j E N; =0, 01

(10.24.1)

( 1 1 0, i^ + j^ E N; = 2 3s2 = 2 &3N2, 02 6 12 - 2

(10.24.2)

( 1 1 0, i^ + j^ 2 &3N2 E N; =& 2 2s2 =& 03 6 12 - 2

(10.24.3)

for the inner products involving 9 0 . ( ^ ^ For E N; 0, i + j +k the s-level basic block on the associated line segment not only intersects supp 9 4 , supp 9 5 , and supp 9 6 , but also intersects two basic octahedra with a common fact in each of these supports. Our geometric analysis following (8.18) implies that all of the inner products vanish for 9 4 , 9 5 , and 9 6 in this case. Thus ( ^

^

0, i + j +k E N; =0, }}$

}=4, 5, 6.

(10.25)

For inner products involving 9 0 we have ( ^

^

0, i + j +k E N; =0, 00

(10.26.0)

( 1 1 0, i^ + j^ +k =& 2 3s2 =& 2 &3N2, E N; 01 6 12 - 2

(10.26.1)

( 1 1 0, i^ + j^ +k E N; =& 2 3s2 =& 2 &3N2, 02 6 12 - 2

(10.26.2)

( 1 1 0, i^ + j^ +k 2 &3N2. E N; = 2 3s2 = 03 6 12 - 2

(10.26.3)

This completes our calculation of the infinite matrix T s .

OSIRIS WAVELETS IN THREE DIMENSIONS

81

11. BASIC ESTIMATION Our goal is to obtain a positive lower bound on the overlap matrix associated with the set of Osiris wavelets. First observe that to obtain such a result for S, we need only find a positive, {-independent lower bound on the matrix-valued function T ({) given by the Fourier series T ({) n(n($; }}$ = : T s; n(n($; }}$ e is{.

(11.1)

s#Z

This reduction is an immediate consequence of (9.6). We may further write T ({)=T 0 + : (T s e is{ +T s*e &is{ )

(11.2)

s<0

as a consequence of (9.8). Clearly, it suffices to prove that

" : (T

&12 0

"

T s T &12 e is{ +T &12 T s* T &12 e &is{ ) c 0 0 0

s<0

(11.3)

for some constant c<1. The first constant we obtain will be greater than unity, but the estimation is worthwhile. In this section we derive basic bounds on the norms of the T &12 T s T &12 . We pursue more delicate estimation in the next two sections. 0 0 We consider the special case s=&1 first. By (9.40), the operator norm of T &1 T &12 is just the norm of the 56_7 matrix with columns indexed by T &12 0 0 ( ( (=, }$) and with the } th entry in the (=(, }$)th column just (E &12E = E &12 ) }}$ . If we break up this matrix into 7_7 matrices with the obvious estimation, we obtain &T &12 T &1 T &12 &2 0 0

:

(

&E &12E = E &12& 2.

(11.4)

=( # [0, 1] 3

On the other hand, (9.42) and (9.48) imply (

1 &E &12 E = E &12&= 16 .

(11.5)

Hence &T &12 T &1 T &12 & 0 0

1 4 -2

.

(11.6)

Now consider the general case s=&N&1, N1. Our estimation now depends on the analysis in the previous section together with the calculations in Appendix B. By (10.2)(10.5) we have ((

(T &12 T &N&1 T &12 ) n(, 2 N+1 m( +2 N =( +|( ; }}$ =(E &12F N; = | E &12 ) }}$ $ n(m( , 0 0

(11.7)

82

GUY BATTLE

where we have applied (9.39) once again. In this case, the operator norm of T &12 T &N&1 T &12 is equal to the norm of a matrix with columns indexed by 0 0 ( ( (=, |, }). This is a 2 3N+37_7 matrix, but as we observed in Section 10, most of the 7_7 sub-matrices are zero, while many others are equal to one another. Indeed, the non-zero part consists of just 2 N&1 copies of a 448_7 matrix with columns indexed by (=(, (=$, }), where (=$ labels the line segments discussed in Section 10. As we did there, we apply (10.6) to set ((

((

F N; = | =E N; = = $

(11.8)

( lying on the =($-segment. If we now break up our long rectangular matrix for all | into 7_7 matrices with the standard estimation, we get

T &N&1 T &12 & 2 2 N&1 &T &12 0 0

:

((

&E &12E N; = = $E &12 & 2.

(11.9)

=(, =($ # [0, 1] 3

On the other hand, we have calculated all sixty-four of these matrices. By (B.4) we have ((

((

1 (2 &3N2 ) &A = = $&, &E &12 E N; = = $ E &12 &= 144

(11.10)

((

and the matrices A = = $ are given in Appendix B. Since all of these matrices have only rank one, the operator norms are just HilbertSchmidt norms, so it is easy to see that 16 of them each have norm 6 - 6 while the remaining 48 matrices each have norm 3 - 42. Hence 1 2 &3N &T &12 T &N&1 T &12 & 2 2 N&1( 144 ) 2 0 0

((

&A = = $& 2

: =(, =($ # [0, 1] 3

1 2 =2 &2N&1( 144 ) (16 } 216+48 } 378) &2N , = 25 48 2

(11.11)

and therefore 

T &N&1 T &12 & : &T &12 0 0 N=1

5



: 2 &N =

4 - 3 N=1

5 4 -3

.

(11.12)

12. POWER ESTIMATION We now supplement the estimates derived up to now with more delicate estimation, where our strategy is based on the operator lemma discussed in the Introduction. First we observe that (T &12 T &1 T &12 ) 2 =0, 0 0

(12.1)

83

OSIRIS WAVELETS IN THREE DIMENSIONS

which follows from (9.40) and (9.42) together with the property (

(

A = A = $ =0,

=(, =($ # [0, 1] 3.

(12.2)

The last property is a consequence of the fact that the only non-zero elements of ( each A = are in the 3_4 lower left sub-matrix. By the special operator lemma, (12.1) implies T &1 T &12 e &i{ +T &12 T*&1 T &12 e i{&&T &12 T &1 T &12 &. &T &12 0 0 0 0 0 0

(12.3)

If we combine this with (11.6), we obtain T &1 T &12 e &i{ +T &12 T*&1 T &12 e i{ & &T &12 0 0 0 0

1 4 -2

.

(12.4)

Our goal is to find the best constant c in (11.3) that we can. If we set 

T &N&1 T &12 e &i(N+1) {, X({)= : T &12 0 0

(12.5)

N=1

we have

" : (T

&12 0

"

T s T &12 e is{ +T &12 T s* T &12 e &is{ )  0 0 0

s<0

1 4 -2

+&X({)+X({)*&. (12.6)

We need to analyze powers of the operator X({), so we expand accordingly:

\

J

X({) J = `



:

j=1 Nj =1

+ \

J

exp &i{ : (N j +1) j=1

+

_(T &12 T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 ). 0 0 0 0

(12.7)

Our purpose is to control this operator expansion, and we propose to do so by estimating each operator product. For N1, (10.2)(10.5) together with (9.39) yields the 7_7 matrix equation ((

T &N&1 T &12 ) n(, 2 N+1 m( +2 N =( +|( =$ n(m( E &12 F N; = |, (T &12 0 0 ((

( | # [0, 1, ..., 2 N &1] 3, (12.8)

( where F N; = | actually vanishes for all but O(2 N ) of the | , as we pointed out in (( ( Section 10. Indeed, we saw that the set of | for which F N; = | {0 is the union of eight sets, each corresponding to a line segment labeled by =($ # [0, 1] 3. Recall that the =($-segment in the sub-cube [0, 12 ] 3 + 12 (= is the line segment from the center of that sub-cube to the vertex 12 (=( +=($). The non-zero matrices correspond to the (&N&1)-level basic blocks through which these line segments run. Let 4(N; =(, =($)

84

GUY BATTLE

( denote the set of | labeling these basic blocks for the =($-segment in [0, 12 ] 3 + 12 =(. Then by the property (10.6) and the notation based on it, we have ((

F N; = | =

((

E N; = = $

: =($ # [0, 1] 3

:

( l # 4(N; =(, =($)

$ l(|( .

(12.9)

Combining this with (B.4) and inserting it in (12.9), we obtain T &N&1 T &12 ) n(, 2 N+1 m( +2 N =( +|( (T &12 0 0 1 = 144 (2 &3N2 ) $ n(m(

((

A==$

: =($ # [0, 1] 3

:

( l # 4(N; =(, =($)

$ l(|( .

(12.10)

This is the 7_7 matrix equation for N1. To estimate each operator product in (12.7), we must write it in terms of the 7_7 building blocks. Clearly, T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 )] n(, 2 NJ +1 m(J +2 N J =(J +|(J [(T &12 0 0 0 0 J 1 J =( 144 ) ` (2 &3Nj 2 ) j=1

\

J&1

` : j=1

( m j

+\

J&1

+\ ` :+ \ ` :+ $ +.

j=1

J&1

( | j

j=1

=(j

J

_$ n(m(1 ` $ 2 Nj+1 m(j +2 N j =(j +|(j , m(j+1 j=1 ( (

J&1

` :

j=1

( (

J

_A = 1 = $1 } } } A = J = $J ` j=1

\

:

=($j

(( l| j

( l # 4(Nj ; =(j =($j )

(12.11)

The product of sums of Kronecker deltas obviously restricts each | j -summation to the set 4(N j ; (= j (=$j ) for 1 jJ&1. The other Kronecker deltas restrict the multiple summation as ( ( m 1 =n,

(12.12.1)

( N1 +1 ( ( m 1 +2 N1 (= 1 +| m 2 =2 1,

(12.12.2)

b

b

b

b

( NJ&2 +1 ( ( m J&2 +2 NJ =( J&2 +| m J&1 =2 J&2 .

(12.12.J-1)

( This replaces all of the m j -summations with the identification

( m J&1 =

\

J&2

+

J&2

` 2 Nj +1 (n + : j=1

k=1

\

J&2

` j=k+1

+

( 2 Nj +1 (2 Nk =( k +| k ),

and since ( Nk +1 2 Nk (= k +| &1] 3, k # [0, 1, ..., 2

(12.13)

85

OSIRIS WAVELETS IN THREE DIMENSIONS

( ( ( ( ( ( it is obvious that m J&1 is uniquely represented by n, = 1 , ..., = J&2 , | 1 , ..., | J&2 . Combining these observations, we see that

[(T &12 T &N1 &1 T &12 ) } } } T &12 T &NJ &1 T &12 )] n(, 2 NJ +1 m(J +2 N J =(J +|(J 0 0 0 0 J

\

1 J =( 144 ) ` (2 &3Nj 2 ) j=1

\

_

J&1

` : =($j

j=1

+\

J

+

( (

=(j$

j=1

J&1

:

( l # 4(NJ ; =(J =($J

++ $ +\ ` (( l| J

( (

` : A = 1 = $1 } } } A = J = $J :

( # 4(N ; =( =($ ) j=1 | j j j j

+$

( N ( ( ( 2 N J&1 +1 m J&1 +2 J&1 = J&1 +|J&1 , mJ

(12.14)

( with m J&1 given by its unique representation (12.13). Thus, for an arbitrary squaresummable 7-vector-valued function ( NJ ( ( 1(n($)=1(2 NJ +1m = J +| J +2 J)

on the lattice Z 3, ( T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 ) 1 ](n ) [(T &12 0 0 0 0 J

J

j=1 ( (

J

` :

j=1

( (

_A = 1 = $1 } } } A = J = $J

J

\ +\ +\ 1 \\ ` 2 + n+ : \

1 J =( 144 ) ` (2 &3Nj 2 )

` :

=(j

`

=($j

j=1

J

:

( # 4(N ; =( =( $ ) j=1 | j j j j

J

Nj +1

(

j=1

k=1

J

2 Nj +1

` j=k+1

+ + (2

+

Nk (

( = k +| k) .

(12.15) Since J

J

: \ +\ + _ 1 " \\ ` 2 + n + : \ ` 2 + (2

: &1(n($)& 2 =:

`

($ n

j=1 =(j # [0, 1] 3

( n

:

`

( # [0, 1, ..., 2 Nj &1] 3 j=1 | j

J

J

Nj +1

J

Nj +1

(

j=1

k=1

2

Nk (

( = k +| k)

j=k+1

+" , (12.16)

we can apply the Schwarz inequality to obtain &(T &12 T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 )& 2 0 0 0 0 J 1 2J ( 144 ) ` (2 &3Nj ) j=1

\

J

` : j=1

=(j

+\

J

` : j=1

=($j

+

J

( (

( (

` card 4(N j ; =( j (=$j ) &A = 1 = $1 } } } A = J = $J & 2. j=1 (12.17)

On the other hand, we have already pointed out in Section 10 that card 4(N j ; =( j (=$j )=2 Nj &1,

(12.18)

86

GUY BATTLE

so we have the further reduction T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 )& 2 &(T &12 0 0 0 0 J 1 2J ( 12 ) J ( 144 ) ` (2 &2Nj ) j=1 ( (

\

J

` : j=1

=(j

+\

J

` : j=1

=($j

+

( (

_&A = 1 = $1 } } } A = j = $J & 2.

(12.19)

We have reduced our estimation of the norm of the operator product to estimation of the norm of a matrix product. (( Now the structure of the 7_7 matrices A = = $ enables us to reduce an arbitrary product rather drastically. Superficial inspection of all 64 matrices in Appendix B shows that for a given (=( J&1 , =($J &1 ), (

(

( (

A = J&1 = $J&1 A = J = $J =0

(12.20)

for 16 of the (=( J , (=$J ). For each of the other 48 (=( J , =($J ), (

(

( (

(

(

A = J&1 = $J&1 A = J = $J =A = J&1 = $J&1 C

(12.21)

for a large affine space of matrices C. In particular, we may choose C such that (a)

every non-zero entry is \6 - 2.

(b) every non-zero entry is the only non-zero entry in both the row and the column of that entry. These properties imply that the norm of C is 6 - 2, and so ( (

(

(

( (

( (

(

(

$ &A = 1 = $1 } } } A = J&1 = J&1 A = J = $J &6 - 2 &A = 1 = 1$ } } } A = J&1 = $J&1 &

(12.22)

for each of the (=( J , =($J ) where (12.20) does not hold. Hence ( (

(

(

( (

( (

(

(

: : &A = 1 = $1 } } } A = J&1 = $J&1 A = J = $J & 2 48(6 - 2) 2 &A = 1 = $1 } } } A = J&1 = $J&1 & 2. =(J

(12.23)

=($J

If we now iterate the inequality, we reduce (12.19) to &(T &12 T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 )& 2 0 0 0 0 1 2J ( 12 ) J ( 144 ) 48 J&1(6 - 2) 2J&2

\

J

+

( (

` 2 &2Nj : : &A = 1 = $1 & 2. j=1

=(1

(12.24)

=($1

As we have already observed in the previous section, ( (

: : &A = 1 = $1 & 2 =16 } 216+48 } 378, =(1

=($1

(12.25)

87

OSIRIS WAVELETS IN THREE DIMENSIONS

so we finally have the estimate &(T &12 T &N1 &1 T &12 ) } } } (T &12 T &NJ &1 T &12 )& 0 0 0 0

J

1 5 2 2 -3

\ +\

J

+

` 2 &Nj . j=1

(12.26) This controls an arbitrary operator product in the expansion. If we now insert this bound in the multiple summand of (12.7), we have the multiple geometric series estimation

\

J



&X({) J & `

:

j=1 Nj =1

1 5 2 2 -3

+\

J

` 2 &Nj j=1

+ \ +

J

J

\ + 1 5 = 4 - 3 \ 2 - 3+ =

5 1 2 2 -3

J&1

.

(12.27)

On the other hand, (11.12) implies &X({)&

5 4 -3

.

(12.28)

It follows from the main operator lemma that &X({)+X({)*& =

5 4 -3 7 4 -3

+

1 2 -3

,

(12.29)

so by (11.6) we finally have

" : (T

&12 0

"

T s T &12 e is{ +T &12 T s*T &12 e &is{ )  0 0 0

s<0

1 4 -2

+

7 4 -3

.

(12.30)

13. THE RECKONING In the previous section we obtained a concrete operator norm inequality of the form (11.3), where c=

1 4 -2

+

7 4 -3

r1.187395.

(13.1)

88

GUY BATTLE

With this constant larger than unity, we do not yet have a positive lower bound on the overlap matrix S. However, we now have a great deal of information about S, and it was instructive to carry out this estimation. It demonstrated the application of the operator lemmas and the effect of contributions of the matrix elements. We now proceed with the more delicate estimation. The key is to obtain a sharper bound on the operator norm of X({). While the bound (12.27) was generalized to the more subtle power bound (12.26), the derivation of the former was based on applying the estimation 

T &N&1 T &12 & &X({)& : &T &12 0 0

(13.2)

N=1

and then controlling each term with HilbertSchmidt estimation on each rectangular matrix in a diagonal system of rectangles. We now propose to use the refinement &X({)& 2 =&X({)* X({)& 



: &(T &12 T*&N&1 T &12 )(T &12 T &M&1 T &12 )&, 0 0 0 0

 :

(13.3)

N=1 M=1

where we control each term in the same basic way, except the rectangles are larger and there are matrix products to calculate. Our first step is to recall that 1 (T &12 T &N&1 T &12 ) n(, 2 N+1 m( +2 N =( +|( = 144 (2 &3N2 ) $ n(m( 0 0

((

A== $

: =($ # [0, 1] 3

:

$ |(|($ .

($ # 4(N; =(, =($) |

(13.4) Obviously the adjoint is given by 1 T*&N&1 T &12 ) 2 N+1 m( +2 N =( +|(, n( = 144 (2 &3N2 ) $ n(m( (T &12 0 0

((

A = = $*

: =($ # [0, 1] 3

:

$ |(|($ .

($ # 4(N; =(, =( ) |

(13.5) Hence [(T &12 T*&N&1 T &12 )(T &12 T &M&1 T &12 )] 2 N+1 m( +2 N =( +|(, 2 M+1 l( +2 M _( +:( 0 0 0 0 1 2 =( 144 ) (2 &3N2 )(2 &3M2 ) $ m(l(

_

: ($ # 4(N; =(, =($) |

$ |(|( $

:

((

((

A = = $ * A __$

($ # [0, 1] 3 =($_

: ($ # 4(M; _ (, _ ($) :

$ :(:($ .

(13.6)

89

OSIRIS WAVELETS IN THREE DIMENSIONS

By standard operator estimation, we have &(T &12 T*&N&1 T &12 )(T &12 T &M&1 T &12 )& 2 0 0 0 0 1 4 ( 144 ) (2 &3N )(2 &3M )

_

"

((

:

:

2

((

A = = $ * A __$

:

:

( # [0, 1] 3 | ( # [0, 1, ..., 2 N &1] 3 : ( # [0, 1, ..., 2 M &1] 3 =(, _

($ # [0, 1] 3 =($, _

:

$ |(|($

($ # 4(N; =(, =($) |

$ :(:($

: ($ # 4(M; _, _ ($) :

".

(13.7)

Clearly, the matrix expression is non-zero only for ( | #

4(N; =(, =("),

. =(" # [0, 1] 3

:( #

( ( 4(M; _ , _").

. (" # [0, 1] 3 _

( ( Moreover, the subsets 4(N; =(, =(") (resp. 4(M; _ , _")) are mutually disjoint, so the ( ( ( ") contribution when sums of Kronecker deltas pick out the (=$, _$)=(=(", _ ( ( ( ( (| , : ) # 4(N; =(, =(")_4(M; _ , _").

We have the reduction T*&N&1 T &12 )(T &12 T &M&1 T &12 )& 2 &(T &12 0 0 0 0 1 4 ( 144 ) (2 &3N ) (2 &3M )

:

:

card 4(N; =(, =(")

( # [0, 1] 3 =(", _ (" # [0, 1] 3 =(, _ ((

((

( ( _card 4(M; _ , _") &A = = " * A __ "& 2.

(13.8)

On the other hand, we have previously had occasion to observe that card 4(N; =(, =(")=2 N&1,

(13.9)

( ( , _")=2 M&1, card 4(M; _

(13.10)

so we may write T*&N&1 T &12 )(T &12 T &N&1 T &12 )& 2 &(T &12 0 0 0 0 1 4 ( 144 ) (2 &2N&1 )(2 &2M&1 )

:

:

((

((

&A = = " *A __"& 2.

(13.11)

( # [0, 1] 3 =(", _ (" # [0, 1] 3 =(, _

Since the multiple summation accounts for 2 12 terms, the calculation may appear daunting at first glance. However, we have also had occasion to observe previously (( that each matrix A = = " has rank one.

90

GUY BATTLE ((

As an operator on 7-vectors, A = = " is a one-dimensional, non-orthogonal projec(( (( tion. Specifically, there are two 7-vectors ; = = " and u = = " such that ((

((

((

A = = "#=( #, ; = = " ) u = = "

(13.12)

for all 7-vectors #. The adjoint is given by ((

((

((

A = = " *#$=(#$, u = = " ) ; = = ",

(13.13)

so the composition is given by ((

((

((

((

((

((

((

((

A = = " * A __"#=( #, ; __" )( u __", u = = " ) ; = = ".

(13.14)

Thus ((

((

((

((

&A = = " * A __"&= |( u __", u = = " ) | & ; __"& & ; = = "&.

(13.15)

=(=("

=(=($

Now if we inspect the matrices A in Appendix B, we see that each ; has exactly 3 non-zero entries and that each entry is \1. Obviously the norm of each such vector is - 3, so we have the reduction ((

((

((

((

&A = = " * A __"&=3 |( u __", u = = " ) |.

(13.16)

In the calculation of these dot products, it will be easier to track the relative signs ( of the vector components if we have a sign convention. Since each A = " is invariant under the joint sign reversal ((

((

((

(

(; = = ", u = = " ) [ (&; = = ", &u = =" ), ((

we adopt the convention that the first non-zero component of ; = = " be +1. This (( (( choice identifies u = = " with the first non-zero column of A = = ", so we may easily refer to Appendix B. (( Our next step is to decompose each vector. Let w = = " be the vector contributed by ( ( the component of u = = " involving - 2 and set ((

((

((

u = = " &w = = " #u~ = = ".

(13.17)

Since the first four components of every u~-vector vanish and the last three components of every w-vector vanish, the former are orthogonal to the latter, and therefore ((

((

((

((

((

((

( u = = ", u __" ) =( u~ = = ", u~ __ " ) +( w = = ", w __" ).

(13.18)

On the other hand, inspection of the matrices in Appendix B yield the symmetries ( ^

^

(

((

u~ = , i + j +k &= " =u~ = = " w

=(, i^ + j^ +k &=( "

=&w

(13.19) =(=( "

.

(13.20)

91

OSIRIS WAVELETS IN THREE DIMENSIONS

Now we define ( 4 0 =[0, i^, j^, k ]

(13.21)

to introduce the set decomposition [0, 1] 3 =4 0 _ (i^ + j^ +k &4 0 ).

(13.22)

We may write ((

((

( u = = ", u __" ) 2 =

: ( " # [0, 1] 3 =( ", _

((

((

((

((

((

((

((

((

(( u = = ", u~ __" ) +( w = = ", w __" ) ) 2

: ( " # [0, 1] 3 =( ", _

=2

[(( u~ = = ", u~ __" ) +( w = = ", w __" ) ) 2

: (" # 4 =( ", _ 0 ((

((

((

((

+(( u~ = = ", u~ __" ) &( w = = ", w __" ) ) 2 ] =4

((

((

((

((

[( u~ = = ", u~ __" ) 2 +( w = = ", w __" ) 2 ].

:

(13.23)

(" # 4 =( ", _ 0

Combining this with (13.16), we reduce (13.11) to &(T &12 T*&N&1 T &12 )(T &12 T &M&1 T &12 )& 2 0 0 0 0 1 4 9 } 4( 144 ) (2 &2N&1 )(2 &2M&1 )

_

:

((

((

((

((

[( u~ = = ", u~ __" ) 2 +( w = = ", w __" ) 2 ].

:

(13.24)

( # [0, 1] 3 =( ", _ (" # 4 =(, _ 0

Consider the w-inner products first. Clearly, only 32 of the w-vectors are involved, since the second index is confined to 4 0 . Moreover, inspection of the matrices in Appendix B shows that the w-vectors have the symmetry (13.20) in the first index as well: ^

^

( (

((

w i + j +k &= , = " =&w = = ".

(13.25)

We also observe that ((

w 0= " =6 - 2 e( 0 , ( k =( "

(13.26.0)

=6 - 2 e( 1 ,

(13.26.1)

w j = " =6 - 2 e( 2 ,

(13.26.2)

w

^( ^(

w i = " =&6 - 2 e( 3 ,

(13.26.3)

92

GUY BATTLE

where e( @ denotes the unit 7-vector in the @ th-coordinate direction. Thus ((

((

( w = = ", w __" ) 2 =2 } 4 } 4 2(72) 2,

:

:

(13.27)

( # [0, 1] 3 =( ", _ (" # 4 =(, _ 0

where we have counted the non-zero inner productsall of which have the value \72. We now turn our attention to the u~-inner products, and our first remark is that the u~-vectors have the symmetry (13.19) in the first index as well. Thus ^

^

( (

((

u~ i + j +k &= , = " =u~ = = ",

(13.28)

and so :

:

((

((

( u~ = = ", u~ __" ) 2 =2

( # [0, 1] 3 =( ", _ (" # 4 =(, _ 0

:

:

((

((

( u~ = = ", u~ __" ) 2.

(13.29)

( # 4 =( ", _ (" # 4 =(, _ 0 0

For the 16 u~-vectors now involved, inspection of the matrices in Appendix B yields: (a)

4 vectors that are zero,

(b)

2 of the vectors &3 - 3 (e( 4 +e( 5 ),

(c)

2 of the vectors 3 - 3 (e( 5 +e( 6 ),

(d)

2 of the vectors 3 - 3 (e( 4 &e( 6 ),

(e)

1 of each of the vectors \- 3 (e( 4 &4e( 5 +e( 6 ),

(f)

1 of each of the vectors \- 3 (4e( 4 &e( 5 +e( 6 ),

(g)

1 of each of the vectors \- 3 (e( 4 &e( 5 +4e( 6 ).

Now the inner product of two vectors of the same type is \54, while the inner product of a vector of the type (g) (resp. type (f), type (e)) with the vector (b) (resp. (c), (d)) vanishes. The remaining inner products are \27. Hence :

:

((

((

( u~ = = ", u~ __" ) 2 =2 2[6(54) 2 +24(27) 2 ]

( # 4 =( ", _ (" # 4 =(, _ 0 0

=2 2 } 48(27) 2,

(13.30)

and so the value of (13.29) is twice this number. Pulling everything together, we can now evaluate the bound in (13.24). We have shown that &(T &12 T*&N&1 T &12 )(T &12 T &M&1 T &12 )& 2 0 0 0 0 1 4 9 } 4( 144 ) (2 &2N&1 )(2 &2M&1 )[2 } 4 } 4 2(72) 2 +2 } 2 2 } 48(27) 2 ] 1 4 1 ) ( 4 )(2 &2N )(2 &2M )[3 4 } 2 7(64+27)] =9 } 4( 144 91 = 4608 (2 &2N )(2 &2M ).

(13.31)

OSIRIS WAVELETS IN THREE DIMENSIONS

93

Thus 1 T*&N&1 T &12 )(T &12 T &M&1 T &12 )& 96 - 182 (2 &N )(2 &M ), &(T &12 0 0 0 0

(13.32)

and so it follows from (13.3) that 



1 - 182 : &X({)& 2  96

: (2 &N )(2 &M )

N=1 M=1

=

1 96

- 182.

(13.33)

Obviously, the estimate &X({)&

1 4 -6

4 182

(13.34)

is much sharper than the estimate (12.27). The corresponding improvement on the power estimation is obtained by combining the approach of this section with the analysis of the previous section. We write &X({) J & 2 =&(X({)*) J X({) J &,

(13.35)

insert (12.5), and estimate the multiple sum term by term. With no subtleties beyond those already covered, we obtain &X({) J &

1 4 -6

4 182

1

\ 2 - 3+

J&1

.

(13.36)

Combining this with (13.34) and the main operator lemma in the Introduction, we infer &X({)+X({)*&

1 2 -3

+

1 4 -6

4 182.

(13.37)

Finally, if we combine this, in turn, with (12.4), we see that a bound of the form (11.3) holds with c=

1 4 -2

+

1 2 -3

+

1 4 -6

4 182.

(13.38)

This constant is less than unity. Indeed, cr0.840323.

(13.39)

94

GUY BATTLE

APPENDIX A ((

In this section we explicitly give all 64 matrices E N; = = $ for arbitrary N1. It is clear by now that each matrix has the form ((

((

1 E N; = = $ = 24 (2 &3N2 ) D = = $, 1 where the extraction of the scalar factor 24 is just a bookkeeping convention. Since the last 3 columns of each matrix has zero entries only, it is obviously convenient (( to record only the sub-matrix D = = $ consisting of the first 4 columns. Now if we apply (10.13) and (10.14) to the matrices already calculated, we have all eight of the (( matrices D 0= $, which we tabulate as

- 2 - 2 &- 2

0

((

D 00=

(

^ D 0j =

(

^ ^ D 0, i +j =

(

^ ^ D 0, i +j =

_ _ _ _

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 &- 2 0

& _ & & _ & & _ & & _ & ((

D 0i =

,

&- 2 0 - 2

0

-2

,

0

0

0

0

0

0

0

0

0 &- 3

0 -3

0 -3

0 0

&- 3

-3

-3

0

0

0

0

0

-2

0

&- 2

-2

0

0

0

0

0

0

0

0

0

0

0

0

(

D 0k =

-3

-3

&- 3 0 - 3

-3

&- 3 0 - 3

-3

0 &- 3

0

0

&- 3 0

0

0

-3

&- 3

-2

&- 2

&- 2

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 0

-3 0

0 &- 3 0

&- 3 0

0

-3 0

-3

&- 3 0

-3

&- 3

-3

&- 3 0

-3

0 -3

&- 2 - 2 - 2

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3 - 3 - 3 &- 3 - 3 - 3 0

0

0

(

^ D 0, i +k =

,

0 &- 2

0 0 0

0 &- 2 &- 2 - 2

,

(

^ ^ D 0, i +j +k =

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.

95

OSIRIS WAVELETS IN THREE DIMENSIONS

We apply (10.11) and (10.12) to these eight matrices to obtain

^(

D i 0=

^ ^

D i j =

^ ^

^

D i, i +j =

^ ^

D i, j +k =

^(

D j 0=

_ _ _ _ _

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 &- 2

0

0

0

0

& _ & & _ & & _ & & _ & & _ &

-2

^ ^

D i i =

,

&- 2 - 2 - 2 0

0 &- 3

&- 3 - 3

0

0

0

0 &- 3

&- 3 - 3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 - 2 0 &- 2 0

0

0

0

^

D i k =

,

&- 2 0 - 2 &- 2

0

0

0

0 -3

&- 3

&- 3 - 3 0 &- 3 0

0

0

0

0

0

0

0

0

0

-2

0

&- 2

-2

0

&- 3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

^ ^

D i, i +k =

,

-2

&- 2 0

-3

&- 3

0

0

0

0

0

0

&- 3

-3

0

0

0

0

0

0

&- 3 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 -2

- 2 &- 2

0

0

0

0

0

0

0

0

0 -3

- 3 &- 3

0 -3

- 3 &- 3

0

0 &- 3 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-2

-2

&- 2

- 2 &- 2 &- 2 0 0

0

0

-2

,

^ ^

^

D i, i +j +k =

0

0 &- 3

&- 3

-3

0

0 &- 3

&- 3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

- 2 &- 2 &- 2 0

,

^ ^

D j i =

0

0

0

0

0

0

0

0

- 3 &- 3 &- 3 0 0

0

0

0

96

GUY BATTLE 0

0

0

0

0

0

0

0

0

- 2 &- 2 0 - 2

^ ^

D j j =

^ ^

^

^ ^

D j, j +k =

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-2

&- 2

0

0

0

0

0

0

0

0

0

0

0

0

- 3 0 &- 3

0

0

0

0

0

0

0

0

-3

0 0

&- 2 - 2 0

,

^ ^

D j, i +k =

&- 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

0

&- 3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2

-2

-2

0

0 &- 2 &- 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3 &- 3 0

0

-3

-3

&- 3

0

-3

-3

&- 3

0

0

,

^ ^

^

D j, i +j +k =

0

0

0

0

0

-2

-2

&- 2

-2

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

-3

0

0

0

0

0

0

0

0

&- 3

-3

0

0

0

0

0

0

0

&- 2 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

0 &- 3

0

&- 2 &- 2

0

0

0

0

0

0

0

&- 3 &- 3

0

0

0

- 2 0 &- 2 - 2

,

D k k =

0

0

0

-2

0

&- 3

^

D k i =

0

0

0

,

0

0

0

-2

^

^

D j k =

,

0

&- 2 - 2

0

0 &- 3

D k j =

-2 0

0

0

(

0

0

0

-3

D k 0=

0

0

0

&- 2 0

D j, i +j =

0

_ & _ & _ & _ & _ & _ & _ & _ & _ & _ & 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-2

97

OSIRIS WAVELETS IN THREE DIMENSIONS 0

0

0

^

^

D k, i +j =

^

D k, j +k =

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

^

D k, i +k =

,

^

0

0

0

0

0

0

0

0

0

0

0

0

0

0 &- 2 &- 2

0

0

0

0

0

0

0

0

^

^

D k, i +j +k =

,

&- 3 &- 3 0

0

0

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

0

0

0

0

0

0

0

0

0

0

0 &- 3 &- 3

0

0

-3

0

0

0

0

0

0

&- 2

-2

&- 2

-2

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

0

0

^

^ ^

D i +j, i =

,

-3

&- 3 &- 3 0

0

0

0

0

0

0

&- 3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-2

0

0

0

0

^ ^

0

0

-2 0

^

0

0

-2

0

0

0

0

0

- 2 &- 2

&- 2 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

^

^

D i +j, k =

,

&- 3 - 3 0 &- 3

D i +j, i +j =

0

0

0

0

^ ^

0

&- 3 - 3 0 &- 3

&- 2 - 2 0 &- 2

^

0

-2

0 &- 3

D i +j, j =

0

0

0 &- 2

^ (

0

&- 2 - 2 0 &- 2

&- 2

0

^

0

_ & _ & _ & _ & _ & _ & _ & _ & _ & _ & 0

-3

D i +j, 0=

0

- 2 &- 2

&- 2 0

0

0

&- 2 - 2

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3

-3

^

^ ^

D i +j, i +k =

0

&- 2 0

0

,

0

0

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 &- 3

98

GUY BATTLE 0

^

^ ^

D i +j, j +k =

(

^

D i +k, 0=

^

^

D i +k, j =

^

^

^

D i +k, i +j =

^

^

D i +k, k +k =

_ _ _ _ _

0

0

0

0

& & & & &

- 2 &- 2 &- 2 0 0

0

0

0

0

0

0 0

^

^ ^

^

- 3 &- 3 &- 3 0 0

0

0

0

0

0

0

0

0

0

0

0

0 &- 2

0 0

0

-2 0

0

0

0

0

0

0

0

-3

-3

&- 3

0

-3

-3

&- 3

0

0

0

0

0

0

0

^

^

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-2 0

&- 2 - 2

0

0

0

0

0

0

0

0

0

0

0

0

-3 0

&- 3 - 3

0

0

0

0

0

0

^

D i +k, k =

0

0

^

^

D i +k, i +k =

0

0

0

0

0

- 3 &- 3 &- 3 0 0

0

0

0

,

0

0

0

-3

-3

0

0

0

0

-3

0

0

0

0

&- 2

-2

-2

0

0

0

0

0

0

^

^

^

0 0

0

0

0

&- 3 &- 3 0

0

0

0

0

0

0

0

0

0

0

0 0

-2

&- 2

0

0

0

0

0

0

0

0

0

0

0

-3

0 &- 3

0

0

0

0

0

0

0

0

-3

0 -2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

D i +k, i +j +k =

& & & 0

&- 3

0

- 2 &- 2

0

0

0

0

0

- 2 &- 2 &- 2 0 0

0

0

&- 2 0

,

,

0

-3

&- 2 - 2 0 &- 2 0

&- 2

0

D i +k, i =

,

0

0

0

-2

0 &- 3 &- 3

0

&- 2

0

-2

_ _ _ _ _

D i +j, i +j +k =

,

0

0

0

0

0

0

0

0

0 -2 -2

& 0 0

&- 2

0

0

0

0

0

0

0

0

0 -3 -3 0 -3

&- 3

- 3 &- 3

&

99

OSIRIS WAVELETS IN THREE DIMENSIONS

(

^

D j +k, 0=

^

^

D j+k, j =

^

^

^

D j +k, i +j =

^

^

D j +k, j +k =

^

^

(

D i +j +k, 0=

_ _ _ _ _

0

0

0

0

0

0

0

0

0

0

-2

-2

&- 2

0 &- 3

&- 3

-3

0 &- 3

&- 3

-3

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

-2

&- 2 0

0

0

&- 3

-3

0

0

^

0 0

-2

0

^

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 0 - 2

&- 2

,

^

^

D j +k, i +k =

- 3 &- 3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 - 2 - 2

0

0

0

0

0

0

0

0

0

0

0

0

0

,

^

^

^

D j +k, i +j +k =

&- 2 - 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

,

_ _ _ _ _

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-2

^

^

^

& & & & & 0 0

0 0

0

0 0 0

-2

0 &- 2 -3

&- 3

0

0

0

0

0

0

0

&- 3 0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 - 2 0 &- 2 0

0

0

0

&- 3 - 3 0 &- 3 0

0

0

0

0

0

0

0

0

0

0

0

0 &- 2

0

0 0 0

&- 2 - 2

0

&- 3 &- 3

-3

0

&- 3 &- 3

-3

0

0

0

&- 2 - 2 - 2

D i +j+k, i =

0

- 2 &- 2 &- 2 0

D j +k, k =

,

0 &- 3

0

0 &- 2

^

D j +k, i =

,

0

0

&- 3 0

& & & & & 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3 - 3 - 3 0 &- 3 - 3 - 3 0 0

0

0

0

100

GUY BATTLE

^

^

^

D i +j +k, j =

^

^

^

^

D i +j +k, i +j =

^

^

^

D i +j +k, j +k =

_ _ _

&- 2

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 &- 2

-3

&- 3 0

-3

-3

&- 3 0

-3

-2

0

&- 2

-2

0

0

0

0

0

0

0

0

0

0

0

0

-3

0

&- 3

-3

0

0

&- 3 0 -2

&- 2

-2

0

& _ & & _ & & _ & ^

^

D i +j +k, k =

,

0

0

0

0

0

0

0

0

0

0

-3

0

0 &- 3

0

0

0

0

0

-3

&- 3

^

^

^

D i +j +k, i +k =

,

0 &- 3

&- 2 &- 2 0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3

-3

-3

0

&- 3

-3

-3

0

0

0

0

0

0 -2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

0

&- 3 0 - 3

- 3 &- 3 0

^

^

^

^

D i +j +k, i +j +k =

,

-3

&- 3

- 2 &- 2

-3

&- 2

0

0 -3

- 2 - 2 &- 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

APPENDIX B ((

Having calculated all 64 matrices E N; = = $ for arbitrary N1, we now need to calculate the corresponding matrices ((

((

1 (2 &3N2 ) E &12 D = = $ E &12. E &12 E N; = = $ E &12 = 24

(B.1)

Recall that E is the overlap matrix for the seven mother wavelets. In Section 6 we calculated this matrix and found its eigenvalues and eigenvectors. The spectral analysis yielded

E &12 =

_

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 2 0 3 0 & 16 1 0 6

0 0 0 0 & 16

0 0 0 0

2 3

& 16

&

1 6

1 6

2 3

&

.

(B.2)

101

OSIRIS WAVELETS IN THREE DIMENSIONS ((

Since the last 3 columns of D = = $ have zero entries only, it follows that ((

((

D = = $ E &12 =D = = $,

(B.3)

so (B.1) reduces to ((

((

1 (2 &3N2 ) A = = $, E &12 E N; = = $ E &12 = 144 =(=( $

(B.4)

=(=( $

A =6E &12 D .

(B.5)

((

Since the last 3 columns of A = = $ also have zero entries only, we record only the sub(( matrix A = = $ consisting of the first 4 columns. Calculating the matrix products (B.5), (( with the matrices D = = $ obtained in Appendix A, we obtain 0 -2

((

A 00=6

(^

( ^

^

A 0, i +j =3

_ & _ & _ & _ & _ & _ & _ & _ & 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3

-3

-3

0

0

0

0

0

&- 3

-3

-3

0

0

0

0

0

0

0

0

0

,

(^

A 0i =3

2 -2

0

&2 - 2

2 -2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

0

&- 3

-3

-3

&- 3

0

-3

0

0

-3

&- 3

0

-3

&2 - 2 0 2 - 2

(

A 0k =3

,

&- 3 0

0

0

-3

&- 3

&2 - 2

0

2 -2

&2 - 2

&2 - 2

2 -2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

0

0

0

0 &- 3

,

( ^

A 0, i +k =3

0 &2 - 2

0

0

0

0

0

-3

&- 3 0

-3

&- 3

0

-3

&- 3

-3

&- 3 0

-3

&2 - 2

( ^

&2 - 2 &2 - 2 0

0

-3

A 0, j +k =3

2 -2

0

2 -2

A 0j =3

- 2 &- 2

0

2 -2 2 -2 0

0 &- 2

&- 2 - 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3

-3

-3

0

0

0

0

0

&- 3

-3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

,

( ^

^

A 0, i +j +k =6

102

GUY BATTLE 0

^(

A i 0=3

^ ^

A i j =

0

0

0

0

0

0

0

_ & _ _ & _ _ & _ _ & _ _ & _

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 &2 - 2 &2 - 2

^ ^

A i, j +k =6

^(

-2 0

&- 3

-3

0

0

0

0

&- 3

&- 3

-3

0

0

0

0

0

0

0

0

0

0

0

0

&

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&6 - 2

6 -2

&6 - 2 0

6 -2

&6 - 2

&4 - 3 0

4 -3

&4 - 3

0 &6 - 2

4 -3

^

A i k =

,

-3

&- 3 0

0 &4 - 3

-3

-3

&- 3 0

-3

0 &- 3

&- 3

0

-3

&- 3

&

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6 -2

&6 - 2

0

6 -2 -3

6 -2

0 &6 - 2

6 -2

^ ^

A i, i +k =

,

&4 - 3

0

4 -3

&4 - 3

-3

&- 3

0

-3

0

&- 3

-3

&4 - 3

4 -3

0 &4 - 3

&- 3

0

-3

&- 3

-3

&- 3

0

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2 -2

&2 - 2

- 2 &- 2 &- 2 0

^ ^

^

A i, i +j +k =3

,

0 2 -2

0

0

0

0

0 &- 3 &- 3

-3

0

0

0

0

0 &- 3 &- 3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&2 - 2

6 -2

0 2 -2 2 -2

A j 0=3

&- 2 - 2

&- 3

-3

^

^ ^

A i i =6

,

0

-3

^ ^

2 -2

0

&4 - 3

A i, i +j =

0

0

0

0

0

0

,

^ ^

A j i =

0

0

0

0

0

&- 3

0

-3

-3

&- 3

4 -3

0

-3

-3

&- 3

&- 3

&6 - 2 &6 - 2

0

0

0

0

-3

-3

0

&4 - 3 &4 - 3 0 -3

-3

0

& &

&

103

OSIRIS WAVELETS IN THREE DIMENSIONS 0

0

0

0

0

0

0

0

0

- 2 &- 2 0 - 2

^ ^

A j j =6

^ ^

^

A j, i +j =

^ ^

A j, j +k =

^

A j k =

0

0

0

0

0

0

0

0

0

&- 3 0

0

0

0

0

4 -3

,

0

6 -2

0

0

0

0

-3

0

&- 3

-3

-3

&- 3

0 &4 - 3

4 -3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&6 - 2

0

6 -2

&6 - 2

&- 2

0

0

0

0

0

0

0

-3

0

&- 3

-3

0

0

0

0

&- 3

0

-3

&- 3

0

0

0

0

4 -3

0 &4 - 3

4 -3

0

0

0

0

^ ^

A j, i +k =6

,

0 0

-2 0

&- 2 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&6 - 2

6 -2

6 -2

0

0 &2 - 2

&2 - 2

2 -2

0

0

0

0

&- 3

-3

-3

0

^ ^

^

A j, i +j +k =3

,

&4 - 3 &4 - 3 0 -3

-3

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

0

-3

-3

&- 3

0

0

0

0

0

2 -2

2 -2

&2 - 2

6 -2

0

0

0

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

4 -3

&4 - 3

0

0

0

0

&- 3

-3

-3

0

-3

-3

&- 3

&- 3

0

0

0

0

&6 - 2 0

^

A k i =

,

0 0

0

0

-2

0

0

0

0

0

0

0

0

&- 3

-3

0

-3

&- 3

0

&4 - 3

4 -3

0 &4 - 3

0

0

&6 - 2 &6 - 2

6 -2

0

0

0

0

&4 - 3 0

0

0

0 &- 2 - 2

0

0

0

0

0

0

&- 3

0

0

0

0

-3

0

0

0

0

0

0

0

0

A k k =6

0

0

0

,

0

0

0

6 -2

^

0

0

0 &6 - 2

0

0 &- 3 &- 3

A k j =

6 -2

0

4 -3

(

0

0

0

&- 3

A k 0=3

0

_ & _ & _ & _ & _ & _ & _ & _ & _ & _ & 0

0

104

GUY BATTLE 0

0

0

0

0

^

^

A k, i +j =6

^

A k, j +k =

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

^

^ ^

A i +j, j =

0

&- 3

-3

0

&- 3

-3

0

-3

&- 3 0

&4 - 3 4 - 3 0

&4 - 3

0

0

0

0 &2 - 2 &2 - 2

0

0

0

0

0

0

0

0

,

^

^

A k, i +j +k =3

&4 - 3 &4 - 3 0

0

0

0

2 -2

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

&- 3

-3

-3

0

0

0

0

0

-3

&- 3

&- 3

0

0

&- 3

&- 3

-3

0

0

0

0

2 -2

&6 - 2

6 -2

6 -2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

4 -3

&4 - 3

^

^ ^

A i +j, i =

,

0

&4 - 3 0

0

0

0

0

&- 3

-3

-3

0

0

&- 3

&- 3

-3

-3

&- 3

&- 3

0

0

0

&6 - 2

6 -2

0

0

0

0

0

0

0

0

&- 3

-3

0

0

^

0

0

0

0

4 -3

0

0

0

0 &6 - 2

&- 3 0

-2 0

^ ^

0

0

6 -2

-3

^

0

0

0

&4 - 3

A i +j, i +j =6

0

^

A k, i +k =

,

6 -2

0 &2 - 2 &2 - 2

^ (

0

0 &6 - 2

0

0

^

0

&6 - 2 6 - 2

&6 - 2

4 -3

A i +j, 0=3

0

_ & _ & _ & _ & _ & _ & _ & _ & _ & _ & &- 2 0 - 2 &- 2

0

0

0

0

0

&- 3

0

0

0

0

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2

0

&- 2 - 2

0

0

0

^

^

A i +j, k =6

,

0 &4 - 3

0

0

&- 2 0 - 2

0

0

0

0

0

6 -2

&6 - 2

0

6 -2 0

0

0

0

0

0

0

0

&- 3

-3

0

&- 3

0

-3

&- 3

0

-3

0

&4 - 3

4 -3

0 &4 - 3

,

^

^ ^

A i +j, i +k =

105

OSIRIS WAVELETS IN THREE DIMENSIONS 0

0

^

^ ^

A i +j, j +k =

0

0

&6 - 2 &6 - 2 0

0

0

0

0

0

0

0

0

4 -3

(

^

^

^ ^

^

A i +j, i +j +k =3

,

&4 - 3 &4 - 3 0

^

^

^

^

^

0

0

0

0

0

0

0

0

-3

-3

&- 3

0

0

-3

0

0

0

0 &- 3 &- 3

0

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2 -2

&6 - 2

6 -2

6 -2

0

0

0

0

0

-3

-3

0

0

0

0

0

^

^

A i +k, i =3

,

0

0

0

0

&- 3

0

-3

-3

&- 3

4 -3

0

-3

-3

&- 3

&- 3

&4 - 3 &4 - 3 0 -3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6 -2

&6 - 2

&6 - 2 0

^

A i +k, k =

,

0

0

0

0

0

0

0

0

0

0

0

-3

0

&- 3

-3

0

0

0

0

&- 3

0

-3

&- 3

0

0

0

0

4 -3

0 &4 - 3

4 -3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 &6 - 2

6 -2

- 2 &- 2 0

^

^

A i +k, i +k =6

0 0

-2

0

0

0

0

-3

0

&- 3

-3

-3

&- 3

0

0

0

0

4 -3

0

0

0

0

0 &4 - 3

,

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6 -2

^

0

&- 3

4 -3

^

&2 - 2

-3

&- 3 0

A i +k, j +k =

0

2 -2

&- 3

6 -2

A i +k, i +j =

0

2 -2

-3

&- 2 - 2 0 &- 2

A i +k, j =6

0

0

&- 3

0 &2 - 2 &2 - 2

A i+k, 0=3

0

_ & _ & _ & _ & _ & _ & _ & _ & _ & _ & 6 -2

&6 - 2 &6 - 2 0

0

0

0

0

&- 3

-3

-3

0

4 -3

&- 3

&4 - 3 &4 - 3 0 -3

-3

0

0 2 - 2 2 - 2 &2 - 2

,

^

^

^

A i +k, i +j +k =3

0

0

0

0

0

0

0

0

0

-3

-3

&- 3

0

-3

-3

&- 3

106

GUY BATTLE 0

(

^

A j +k, 0=3

0

^

^

^

^

^

A j+k, i +j =

^

^

^

(

0

0

_ & _ & _ & _ & _ & _ & _ & _ & _ & _ & 0

0

0

0

0

0

0

0

0

0

0

0

0

2 -2

2 -2

&2 - 2

^

^

A j +k, i =6

,

-2

&- 2 &- 2

0

0 &- 3 &- 3

-3

0 &- 3 &- 3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6 -2

&6 - 2 0

6 -2

^

A j +k, k =

,

-3

&- 3

0

&4 - 3

4 -3

0 &4 - 3

-3

&- 3

0

6 -2

-3

0

0 &6 - 2

6 -2

&4 - 3

0

4 -3

-3

0

&- 3

-3

&- 3

0

-3

&- 3

-3

&4 - 3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&6 - 2 0

6 -2

&6 - 2

&6 - 2

6 -2

&4 - 3

4 -3

&4 - 3

-3

&- 3

0

-3

&4 - 3

4 -3

0 &4 - 3

&- 3

-3

&- 3

0

0

0 &- 3

^

^

A j +k, i +k =

,

0 &6 - 2 -3

0

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 2 - 2 - 2 0

^

^

^

A j +k, i +j +k =3

,

0 &2 - 2

-3

&2 - 2 2 - 2

0

0

0

0

0

&- 3

&- 3

-3

0

0

0

0

0

&- 3

&- 3

-3

0

0

0

0

0

0

0

0

0 &- 2 &- 2 - 2

A i +j+k, 0=6

0

0

&- 3

^

0

0

-3

A j +k, j +k =6

0

0

0

A j +k, j =

0

0

&2 - 2 2 - 2 2 - 2 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&- 3

-3

-3

0

0

0

0

0

&- 3

-3

-3

0

0

0

0

0

0

0

0

0

,

^

^

^

A i +j +k, i =3

0

107

OSIRIS WAVELETS IN THREE DIMENSIONS &2 - 2

^

^

^

A i +j +k, j =3

^

^

^

^

_ _ _

A i +j +k, i +j =3

2 -2 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

&- 3 0

-3

-3

&- 3 0

-3

0 &2 - 2

2 -2

2 -2 0

0

0

0

0

0

0

0

0

0

0

0

-3

0

&- 3

-3

0

0

&- 3 0

^

^

^

A i +j +k, j +k =3

0 &2 - 2

0

2 -2

0

0

-3

&- 3

&2 - 2

0

0

0

0

0

0

0

0

&- 3

-3

-3

&- 3

-3

-3

0

0

0

&2 - 2

& _ & & _ & & _ & ^

^

A i +j +k, k =3

,

0

0

0

0

0

0

0

0

0

0

-3 0

0

0

0

0

-3

&- 3

2 -2

^

^

^

A i +j +k, i +k =3

,

0

-3

0 &- 3

&- 3

&2 - 2 &2 - 2 0

0

2 -2

0

0

&2 - 2 0

2 -2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-3

&- 3

0

-3

-3

&- 3

0

-3

0 -2

- 2 &- 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

,

^

^

^

^

A i +j +k, i +j+k =6

0 0

0

0

0

0

0

0

0

0

0

REFERENCES 1. G. Battle, in ``Wavelet Transforms and TimeFrequency Signal Analysis'' (L. Debnath, Ed.), Birkhauser, Basel, in press. 2. G. Baker, Phys. Rev. B 5 (1972), 2622. 3. P. Collet and J. Eckmann, ``A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics,'' Springer-Verlag, New York, 1978. 4. H. Koch and P. Wittwer, Comm. Math. Phys. 164 (1994), 627. 5. K. Wilson, Phys. Rev. B 4 (1971), 3184. 6. G. Golner, Phys. Rev. B 8 (1973), 339. 7. J. Dixmier, ``Les Algebres d'Ope rateurs dans l'Espace Hilbertien,'' GauthiersVillars, Paris, 1957.