A family of discrete Haar transforms

Comput. & Elect. Engng, Vol. 2, pp. 367-388. Pergamon Press, 1975. Printed in Great Britain

A FAMILY OF DISCRETE HAAR TRANSFORMS* K. R. RAO, M. A. NARASIMHAMtand K. REVULURI~ Departmentof ElectricalEngineering,Universityof Texas, Arlington,Texas 76019

(Received 17 February1975) Abstraet--A new class of discrete orthogonal transforms called generalized Haar transforms, (GHT)r is defined and developed.The base functions of (GHT)r are linear combinationsof Haar functions. Pertinent properties of (GHT)r such as, linearity,uniqueness, dyadic autocorrelation,and dyadic shift invarianceare developed.By factoringthe transform matricesinto a numberof sparse matrices,efficientalgorithmsfor fast computation of (GHT)r and its inverse are developed. By subjecting these algorithms to successive bit-reversaloperations, a single processor such as the Cooley-Tukeytype can be used for implementingall the transforms. Specificexamplesillustratingthe (GHT), its propertiesand the fast algorithmsare included. The (GHT), is appliedin digitalinformationprocessing.Its utilityand performanceis comparedwith those of other discrete transforms such as Walsh-Hadamard,Haar, slant, Fourier, Karhunen-Lo6ve etc. Digital computer programsfor fast implementationof (GHT)~ and for evaluatingsome of the performance criteria, such as variance and mean-square error are developed. INTRODUCTION In recent years, discrete orthogonal transforms have come into prominence [1-72] as a result of the developments in digital technology and in digital computers. Fast algorithms [ 1-6] resulting in reduced computational and memory requirements have further accelerated the utility and applications of these transforms. Added advantage of these algorithms is the reduced round-off error[4]. Generalized discrete transforms (GT)r and (MGT)r have been recently defined and d e v e l o p e d [ I - 3 , 5 , 6 ] . Other discrete transforms such as Haar [7-14], Walsh-Hadamard[42], slant [51, 52, 71] discrete cosine [44], Fourier [4], complex Haar [41], Karhunen-Lo6ve [31,34], Hadamard-Haar[72], and slant Haar[69] have also been developed and their effectiveness in information processing is compared in terms of the performance criteria such as variance, mean-square error, rate distortion, and classification error [34, 36]. These transforms have been applied in signal and image processing[20,24-26, 65, 68, 71, 72], Wiener filtering[36], data compression[39], feature selection in pattern recognition[32, 39, 66, 67], detection[59, 63] and analysis of dyadic invariant system[58, 61]. The objective of this paper is to develop a family of discrete Haar transforms, called, generalized Haar transforms whose base functions are linear combinations of Haar functions[7-14, 41]. This is followed by the development of fast algorithms [5, 6], dyadic autocorrelation [45], dyadic shift invariance [23, 58, 61], and applications in digital processing[39]. GENERALIZED HAAR TRANSFORM The generalized Haar transform (GHT)r of an N-periodic sampled data x(m), m = 0, 1,2 . . . . . N - 1 and its inverse are respectively defined as: 1 {(X,(n)} = ~ [H~(n)]{x(n)}

(la)

and

{x(n)}=[Hr(n)]

{Xr(n)}

r =0,1,2 ..... n-1

(lb)

where X,(m), m = 0, 1,2 . . . . . N - 1 is the rth-transform component and n = log2 N. {x(n)} and {X,(n)} are the N-dimensional data and rth-transform vectors respectively and [Hr(n)] is the *A paper based on part of this research was presentedat the 17th MidwestSymposiumon circuitsand systemsheldat the University of Kansas. September 16-17, 1974. Proc. pp. 154-168. tMember, Centred Research Labs, Texas Instruments, Inc., Dallas, Texas 75231. ~tDept. of Information Engineering,University of Illinois at Chicago Circle, Chicago, Illinois 60680. 367

368

K . R . RAO, M. A. NARASIMHAMand K. REVULURI

(2" × 2") rth-transform matrix. Other notation is described elsewhere[5, 6]. For r = O, [Hr(n )] represents the well known discrete Haar transform [28]. For other values of r, [H,(n)] becomes increasingly complex, the elements representing the linear combinations of Haar functions. The transformation in (1) is unique as [H, (n)] is unitary, i.e. [H, (n)] [H, (n)]*~- = NIt~, where IN is the identity matrix of order N. [H,(n)] can be factored into n sparse matrices as follows: n

[ H r ( n ) ] = H [H/J'(n)]

n ->r+ 1

j

where [ H,`~)(n)] = Diag{[h/°)(J)]lh,(')(J)]... Ih/~" ' "(.i)l}.

(2)

The diagonal submatrices [hfl>(j)] are unitary and they can be generated as follows:

[h/,)(l)] =

- W""]' 2

I = 0. I . . . . . ( 2 r - l )

I = 2 r + 1,(2" +2) . . . . . (2" * - I )

lh.'"(j)] = 2 .2 I2~+'®I~ i '.1=2'

(3)

= [h,">(1)] 6<)12J , ,,,o. ,~_, [hrC°)(j)] is obtained by subjecting the columns of [Ho(l)]®I2J , to bit-reversal operations. W = e -~z=m and i = ( - 1) "2. The symbol ® denotes K r o n e c k e r matrix product and .~ 1 >> is the decimal number resulting from the bit-reversal of a (n - 1)-bit binary representation of I, i.e. if 1 = 1._22 "-~ + 1._32 "-3 + - • • + 1,2' + lo2°, a (n - l)-bit binary representation, then

<~1>> =1o2 -+1,

+..-+1,

,2'+1 .

n

2~

")"

.

Using (3) the matrix factors for (GHT). and hence the transform matrices can be developed. As an example for N = 16, the matrix factors are: r=0

Haar transform ( H T )

[Ho")(4)] = IHh(1)]

r --

[ Ho<2'(4)] =

i--i,.

-

I i

®(1

-

1)[

(4)

Is-

1,),

[Ho°)(4)] =

]1

12.2

[Ho">(4)] = ~(I

where

-I)

[l _I] ,, the., am.rd ordered"adamard m.tr'x ,39. 50,

A family of discrete Haar transforms

r = 1;

369

ComplexHaar transform (CHT)

/--- ,~Q [U,("(4)] = L ~__ 2"2_/4_/~_ _

F

[H'(2)(4)] = I:®(I

_

_

1) Il

1

/, ®(1 - 1)]

[H,°)(4)] =

I I[L(I)]® L

[Hff'(4)l=[Ho")(4)]

where' [L(I)] = [I-i]

(5)

r = 2; (GHT)2 F[H~(1)]L

-1

L

~,12Is -J

I.®(1 1) I I2~@(1--1) [H:'='(4)] =[-L[L (1)] ® IL-~]~_

l ~ I[/3(1)] ® 12

]"/4®(1 1)1

[H:(3'(4)1 = IL, ® (1-l) I

L-

.

.

.

.

t[L(~)]®

-I _1= [H,°'(4)] I~

[H2t4'(4)] = [Hff'(4)] = [Ho")(4)] --lw/4

~

e-i3w/4

]

where,[a(1)] = [Ie-e-""J'[/3(I)] = [I-e-'~'"J

(6)

r = 3:(GHT)3 " [H~ (1)J~,

....

L~_~L__~

[H¢'(4)] = i~i

.... .....

C.A.E.E., Vol. 2, No. 4--43

~;~N

370

K. R. RAO,M. A. NARASIMHAMand K. REVULURI

[H3
IlL (1)] ~) 14

liB(i)] ® 1

_- [H2'2'(4)]

~)(1 [H3°'(4)] = tF/4 I4@ ( I - I ) I l)l [L(I)]@

I4_J=[Hz°'(4)]

(7)

[H3"'(4)] = [H2'0'(4)] = [H,'"(4)] = [Ho'4'(4)]

e '"'"] where,

e ,5.,.]

[7(1)]= [ I - e - ' ~ / ~ ] ' [6(1)]= [ I - e -'5~']

e '-~"/"] e "7"/" l [7(1)] = [1l - e.3=/sj, [r'(l)] = I l l - e ,7=/, j . The transform matrices [Hr(n )1 can be evaluated from the above matrix factors using (2). The results are: r=0 I I -

0 1

Haar transform (HT) I I I

I I I

0

I I I I I -I

0

I

-I

0

Row

0

I I I I 1 1 1 I 1 1 I I I -I -1 -1 -1 -1 -1 -I 0 0 0 0 0 0 07 I -I -I 0 0 0 0 I 1 1 I -1 -1 -1 -I_

I

I -I

-I -I 1

I -1

-1 1

1 -1

-I

I -1

[Ho(4)] =

1 -1 I -I 1 -1 1 -1

23/2

1 -1 1 -1 1 -1

0 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 (8)

The base functions of the rows of [Ho(4)] are Row

0

Base function

6° I~)1 (D21 (~22 (D31 (D32 (~33 6 34 6/)41 (b42 (~43 6 `4 645 646 647 (~)48

1

2

3

4

where 6d denote the Haar functions[55].

5

6

7

8

9

10

11

12 13 14 15

A family of discrete Haar transforms complex

Haar transform

I

1

1

1

1

1

1

1

1

1

1

1

1

1 -1

-1

-1

-1

1

1

1 -1

-1

-i

-1

-i

-i

-i

-i

1

1

1 -1

-1

-1

-1

r=

"1

!

0

0

2)'/:

371

-1

-1

0

0

0

0

1

1

0

(CHT)

1

1

i 0

-i

0

1

1 -1

1

i

i

i

i

0

0

0 -i

1

-I i

i -i

-1

row I

-1 i

-i

-i 0

-i

0

-1

1

i

-i 0

1 i

2

-i

3

0

01

i

i]

4

I

5 i

I

1 -1

0

0

I

[H,(4)] =

-1

0

0

0

0

0

1

1 -1

0

-1

i

i

0

0

-i

-1

-i

-i

0

0

0

0

i

i

0

-i

0

-i

i

8 -i

1 -1

i

9 -i

1 -1

i

10 -i

1 -1

-1

6 7

i

i

-i

11 12

1 -1

i

-i

1 -1

13 i

-i

14

1 -1

i

- i.J

15 (9)

Row

Base function

0 1

1 1 1 1 1 1 1 1 1 1 1 -1 1 1 l -l 1 -I -1 W2 1 - 1 - 1 - W2 1 -1 1 -1 'l

i I

112

2

62' - i622

3 4

62' + i62 ~ 2-'/2[63 ' -- i 6 3 3 ]

5

2-'/21632- i634]

6 7

2-'/2[63'+ i63'] 2-'/21632 + i634]

8 9

2-'/216, ' - i6:] 2-'/216+ 2 - i646]

10

2-'/2[643

il

2 - ' / 2 [ 6 , 4 - i64 s]

12 13

2 - ' / 2 [ 6 4 ' + i64 s] 2 - ' / 2 [ 6 , 2 + i 6 , 6]

14 15

2-'/2[643 + i647] 2 - ' / 2 [ 6 , 4 + i 6 , s]

-- i 6 4 7 ]

(GHT)~

r= 2

[H~(4)]

6o 6,

-1 -1

-1

1 - 1 -1

row

1 1 -1 -l W2 - W2

-1 -i

-1 -i

W6 _ W6 _ W6 We We W 2 _ W2 W 2 - W2 -W e

i i

- W2

W2

- i - i

W•

_ W6

- W2

-W" 1 - 1

1

-1 -l i i -W 2 -i -i W2 - i - i

W•

1 - I -1

-1 -l -W 2 W2

1 1

-W 6

I - 1 -1

1 I

1 -1 -i

1 -1 -i

-1 i

- W6

- We

We

-1 i

1

o

i

2 3 4 5 6 ? 8 9 lO

-1 i

1

-i W6 - W"

-i -W 6 We

-i -W e W6

i - i - i i - i - i

W2 - W2

W2 - W2

- W2 W2

- W2 W2

W6 - W"

W6 W"

-i

i

-i

i

i - i We

-1

1

-i W 's - W"

W2

W"

1

i i i

i i

i i i

1

We

- W' wW 1

- W"

"

W2

:

- W2

W -~

W 2 - W2 i - i

i - i

- W2 i - i

W2

11

12 13 14 15 (10)

K.R. RAO, M. A. NARASIMHAMand K. REVULURI

372

where W = e -n_.a6 , and the base functions are

Row

Base functions

0

f~)o

2

2-'/~[6_~'-i6~:1

3

2 -'/-]62' + iOf-]

4 5

~[ch~' + WZch,: - ich,' + W~6~"1 ~[~h,'- W2&~ z ich:, ~ W6cb, 4]

7

'zl&,' - W~&f + i6,' - W=6,"I

8

~[ch~' + W~6~ ~ - i6~ ~ + W%h~:l

~[642 +

W2644 i046 + W604¢1 ~t#'~' - W-'#'4~- i,h~'- W M J I ~[¢b4~'- W-~ch~~- i6." W~h.~l

9 10 11 12 13 14

'216~' + W%h4" + i,h~ ~ +- W2ch,'l

~lch," + w M 4 4 ~ i64" ~- w%~21 ~[6fl- W ~ & . 3 + i~h~' - W2647] ~[(~t)42- W6(])44 -L i(~4 e' W2d) 81

15

Factoring of the transform matrix [H,(4)], r = (1, 1.2.3 into the sparse matrices (4-7) leads directly to the signal flow graphs (Figs. 1-4) for efficient implementation of these transforms. The corresponding signal flow graphs for the inverse transformation based on (lb) and (8-11) are shown in Figs. 5-8. It may be observed that significant reductions in arithmetic operations and consequent savings in computational times are achieved through these fast algorithms. These flow graphs do not, however, have the 'in-place' structure. This property can be realized by subjecting the columns of [H,(n )] to successive bit reversal operations as described for HT and CHT. As an example, the bit reversal operations are illustrated for N = 8 and r = I Column~

!] I

2

3

1

1

1 -1

1 -1

4

1

1 -I

5 1

-i

--1

6

base f u n c t i o n oh,,

1

-i i

row 0

7

i i

i

3

2 ' n [ 6 f + i63]

O] "

4

2 '/216~'- i6,31 2 '"~[6, ~-- i6J]

6

2 '21¢t),' + i6?1

[H,(3)I = 3112 0

0

0 I

1

0

0

0 -i

-

1

I

(I

0

i

-- i

0

0

0

0

1 - I

0

0

i

- i

1

I

-I

O i l

(12)

Rearrange the columns of [H,(3)] in (12) in bit reversal order for N' = 8. That is, {0, 1.2.3.4, 5. 6, 7}~{0, 4, 2, 6. 1, 5.3, 7} and hence [H,(3)] becomes I 1 I 1

[H,t3)I=

--i i

1 1 1 1 I -1 1 1 1 -i I i I i --I -i

I 1 I --1

/rlt -i 0 ~,/~

10

"

L column

1 1

1" 0

0

1 - i

i

0

0

0

1

~

1

1

I

2

3

I'' i]I

2,, 2

0

l/ - 1

1 - i

0

0

1

o

(13)

A family of discrete Haar transforms

373

Transform data ' ~ _ xo(O) Xo(I) • ~2 Xo(2) xo(3)

Input

data x(O) x(I) x(2) x(3) -I x(4)

/ /

x( 5 ) -I! x(6) x(7)_

x,(5) XI( 6 )

x2(2) x2(3) x2(4) x2(5) X2(6)

x,(7)/-,- ~ - -

x2(7)

x(8) <

x~(8) - 2V-E xl(9 ) 2J-2 x~(lO) 2~-2 x=(ll) "2 vr2

Xo(8 ) Xo(9 ) xo(lO] xo(ll)

xl(12) xi(13) xl(14) x,(15)

Xo(12) Xo(13) Xo(14) Xo(15)

x( 9 ) _1( x(lO) x(ll) x(12) x(13) x(14)

I

I

-I

x~(2) xi( 3 )

"

Xl(4)

.

=,2,J'2 =2v~ m,2 ~ =2VE

Xo(4) Xo(5) Xo(6) xo(7)

Fig. I. Signal flow graph for (GHT)., N = 16.

Transform data ~ x,(O) -I xj(i)

Input

data x(O)

x,(O) x,(I) / x,(2) / x r(3)

x ( l ) -I x(2) x(3) -I x(4)

x2(O) ~ x2(I)~_, x2(2) ~ - ~ " x2(3)~_1

~

~'(2) "

x(5) -I x(6) x(7) x(8) x(9) x(lO) x(ll) x(12) x(I3) x(14)

~ \ \ \ \

_

x(IS)

~ ~ ~ ~

x I (7) x t (8) Xl(9) Xl(lO) Xl(ll) X l(12) xl (13) X~(14) Xl(15)

xl

(4)

x I (7) x~_/ x2(8) ~:~-'~'~7, X2(9) ~E7~_7-/ x2(lO) k~/~x--7 -/'/x2(ll ) Z ~ X2(12) X2(15) ~ X2(14) / '/: \ x2(15)

" 2 =~ ,- 2 m, 2 . 2 " -- 2 =2

xl(8) Xl(9) x,(IO)

XI(II) X1(12) Xl (13) XI(14) Xl(15)

Fig. 2. Signalflow graphfor (GHT),, N = 16. Input dora x(o) \ x( I )-I'

x(2) x( 3)_i x(4) x(5) x(6)

x( 7)_1,

x(8)

Transform data x3(0)~;7 x2(O) x3(I)~ x2(I) XS(2)-~=~;:: ~ X2(2) x 3(3) -~-C~1~ x2(3) ~X3(4)"~::~W2 X2(4) x3( 5 ) ~ x2( 5 ) x3(6) -<~:~7w6xz(6) x3(7)~x2(7 )

x~(O) xp(I) Xl(2) X~( 3 ) Xr(4) Xl ( 5 ) x,(6) xl(7)

x/O) x2(f) X2(2) X2( 3 ) X2(4) x2( 5 ) x2(6) x2(7)

~ _i<~:~ < - ~ "~ / -- -# ~ ~"~, ~'~Z-i ~ ~=/~"~

x i ( l l ) ~ ?v~2 ~ xd(12 ) Z~. ~ x,(13) Z ~ _ ~ . x, (14-) Z 7 q ' ~ x~tlS)Z i ~

x2(ll) x2(12) x2(13) x2(14) x2(ts)

/~,.2 ~ x3(ll)~ x3(12) ,. ~ x3(,3) . ~W?x3(14) ~ ~x_we 305) --

<

x(9) x(lO) x( 11)

x(12) x(13) x(14) x(lS) f -: -I

Fig. 3. Signal flow graph for (GHT)2, N = 16.

v/2 V~" ~ J'2 "rE

x2(ll) x2(12) x2(13) x2(14) x2(15)

374

K. R. RAO, M. A. NARASlMHAM and K. REVULURI

Input data

Transform data

.3,,,

*,,,

x(4) _

X(7) 1

~

i

X2(7) ~"

X(8) < ~ ~ ' X(9)

Xl(8) N *1(9)

x(121

~ . ~ . . . . . ~ . ~,.~.

X ( I5 )

~: -

I

-

2

-'- ~

X2(14)~"~WX2(15) / " = " ~ _W-6

=/

-

r

~

/

X2(8) - - . ~ ~ _ 2 X 3 ( 8 )

x2(121

X ' ( ~5 )

-W

X3(7)

""

\i

*1(14) / --

-,'

3

-~

~..~.~_~-w

X3(8

-~'~

x3( i l )

.:-"--

. _6x3(12) ,,,, 3(14) ~..,," ~.~r 7 X3(14) ,3(15) ~ X~(i5) _W7

Fig. 4. Signal flow graph for (GHT),, N = 16.

Transform

Input data * O) r)

data

.o,lo> - - - . . ¢ ~

.o2( O ~ - - . _ ¢ _ /

.o3CO~ - - " - - : - - ~ t

Xo( Xo(

2)

Xo(3)

Xo2(3)

~ x / < ~ Xo2(3)

Xo(4)

2

Xo2(4)'~.~

Xo3(4),~~2

Xo(5 )

=2

x02(5),,. ~

x03(5) X ' ~ l ~ " 2 " I x

Xo(7 )

,2

2-Ix

3)

x

4) 5) 6)

Xo3(7 )

(7) I~ (8)

Xo(8)

,,

Xo3(8)

Xo(9 ) Xo (10)

,~2vt'~" 2vr~

x03(9 ) Xo3(lO)

Xo (11)

,_2ur~

x03(ll) ~

x 0 (12)

'~2urn"

Xo3(12)

Xo (13) x o (14} x o (15)

2./'2 ,, 2~" 2vt~

(9) -I x

(~0)

i (;i) (121

Xo3(13) Xo3(14) _ Xo3(15)

(13)

=

, -I

(J4)

(15)

Fig. 5. Signal flow graph for (IGHTL, N = 16.

Rearrange the columns of the (4 x 4) submatrices enclosed by [] in bit-reversal order for N = 4, i.e. {0, 1, 2, 3}--, {0, 2, 1, 3}

1 1

1 -1

1

[/~/,(3)1 : -~,t2

-

El

1 1 1 -I

1 1 1 -1

1 1 1 -1

'11 '1 °' i] [:i °' 1 -i

1

0

0

i

1

0

-

1

2,/2

i

- I

1 -i

0

O-i

-1

0 -i

"

(14)

375

A family of discrete Haar transforms Input data

Transform data

•,( o ) ~

x,,(.o)

x,2( o )

x (I) --'I x (2)Z~ ~

x,~( I ) xu(2)

~,2( ' ) ~ ~ _ i ~ , ~ (

x (3)

_/

xli(3)

.4"2

x,,(4

x (5)

~-

x.( 5

x (6)

" /'2"=

xll(6 x (7

Xi2(6)~

x(8 x(9

~,2( e ) \ /

x

(4)

x (7)

x(O)

-'---i.../ x'3( o

7-ix ( t )

'

x(2)

x,2( 3 ) , , ~ , , , ~

_

x,3( 3

x12( 4 ) / z ~

II x ( 3 ) x(4)

>X'z'(4

~1 x ( 5 )

x,2(7 ) / ,

x (10)

x(6)

~Xi3(6

\

\x,3(7

~-I x ( 7 ) > x(8) x(9)

/ x,3(e

.

> x(Io)

x=3(lO)

I

x (11)

~-I x ( l l )

x (12)

>

X x(I3)

x13(15)

\ x(15)

(,,) Z , / \ " < x,2 / \ ; x,~(,~) j ~ . _ ~

x (14) x (15)

X12(15) - - ;-i

--

)

(13)

X12(13)

x (13)

x(12

- -I

x(14)

Fig. 6. Signal flow graph for (IGHT),, N = 16. Transform data

Input data ~_ x(O) 22( I )

x2(2) / x2(4) X2(5)

X21(2)

~

~

x2,(4)

" ~

X21(5)<"~'~

~_w_%J

x2,(7)

w-

7-1 x23( I ) ;7 X23(2)

22(2)

/

~ " /

x22(4)

>

,. X22(5 )

I x(I)

/ x(2) 2-I x(3) 7 x(4)

x23(4)

~ X23(5)

>-I x(5) x(6)

w

x2(7) X2(8)

='

X21 (8)

/

=-i -:~

~ x22(7)

" /-

X22(8)

,~(9)

.4

x2(~o) x2(ll)

- . x2,(Io) _~5/v-...."x22(Io) ~ /E¢~ x2,(IIi'w~/=-W-':-'~ ~ x22(II)

x2(12) x2(13)

=, ¢d

, (9)'-'-'~"-.-~..,

x23(7)

~-I x ( 7 )

=~

X23(8 )

>

\\v// /

,9,

,,,-,×

_ - \

\

x(8)

>-I x(9)

-

X2~(12) ,"~,~= ¢,./ X22(12) x2, (13)w~o'.,.p~_j x22(13)

x23(~o)

> x(lO)

x23(11)

~-I x( If )

X23(12) x23(13)

x(12) x(13) x(14) x(15)

-I

Fig. 7. Signal flow graph for (IGHT)2, N = 16.

Rearrange the columns of the (2 x 2) submatrices enclosed by [] in bit-reversal order for N = 2, i.e. {0, 1}~ {0, 1}. This, of course, results in no change. It can be observed that [H,(3)] is the transform matrix of MCBT (modified complex BIFORE transform)[40]. The matrix factors for [H,(3)] and [H,(3)] respectively are [H,(3)I=[H,

(I)

(2)

(3)

(3)1[H, (3)][H, (3)1

where

[H(2,(3)]=Diag IF1

L, -

[U,(')(3)]=Oiag

l] FI-~ u,

FF 1 1

LL,_ 1

L1 1

]

,j, v ( = , , )

~ ,FI

1, 11

1-1J, L

1

-i i

1

-I

.I] 13 I

.

(15)

376

K. R. RAO, M. A. NARASIMHAMand K. REVULURI Tronsform dat'o

x3(O)

x3( I ) 3

Input

~

dato

x3,(O) -I

xsl( I )

-/-

St(3) xsr(4)

X3(4)

- I "~

x32(0) xs2(I)

xs3(O) x3s( I )

x32(2) x32(3)

x33(2) ] *33(3) x33(4) x3s(5) x33(6) xs3(7)

X32(4) X32(5) Xs2(6) x32(7)

i" /

X3(9)

~,,,_1 ~ ~

x~r(9 )

xs('O)

~-¢v

xs, iO)w-~- / ~ . C ' 2 ~ "

x32(8) x32(9) X52(10) xz2(ll) x32(12)

x(I) x(2) tl x(3) x(4) x(5)

i, x(6)

x(7) x(8) x(9) x()O) x(II)

x 3s(8 ) xsz,(9) X33(10) xss( t l ) x33(12)

xs()J) : ~xs,(r) ) w_2-. x3(12) ~ x3,(12) ~ ) xs(13) W ~ x 3 ( 1 3 ) W~6~"~>~. _ x32(13) x3(14) ~ X ~ l ( 1 4 ) -- / ~ ~ / " "~' X32(14) x 305) W ~ xs,(t5 )-W~'/'-X32(15) _W-7 _W-6 -

x(O)

xss( 13) xs3(I4) x33(15)

w

-L

x(12) x(13) x(14) x(tS)

Fig. 8. Signal flow graph for (IGHT),, N : 16. I

1 I

1 I

1

[H~(3>(3)] =

1

(15)

I

I-I I-I |-I l "-- I

and

1

] I

I

I2 I

~]

[H.(3)] =

(16)

LLJj____

1 , 7 ( 2 ) i~

Based on (16) and (15) the signal flow graphs for (GHT), are shown in Figs. 9 and 10 respectively. The 'in-place' structure in Fig. 10 can be observed. The 'Cooley-Tukey' type flow graph is also developed for (GHT), and (IGHT),, r = 0, 1.2 for N = 16 in Figs. (11-13) and in Figs. (14-16) respectively.

DYADIC AUTOCORRELATION

The dyadic autocorrelation of {x(n)} can be expressed as ]

d(h)=~

N-I

~] x(rn)x*(m(~h),

h =0,1,2 ..... N-1

(17)

m =o

where (m (~h) denotes modulo 2 addition of the binary representation of m and h. The (GHT)r of {d(n)} is {Dr(n)} = 1 [Hr(n)]{d(n)}.

(18)

377

A family of discrete Haar transforms

Trandai sfo-ram ~ . i xl(O)

Input data x ( O ) . ~ x ( t ) ~

xl(I)

_

~

x(2)-i~~ x ( 3 ) _ ~

xl(2) -

-I

xi(3) ,/"Z

x(4) ~ , ~

"

Xl(4)

x(5) ~ / ~

=

Xl(5)

-

xl(6)

•-

x I (7)

/

-I

Fig. 9. Signal flow graph for (GHT),, N = 8.

Inputdata '(0) ~ / ~ / /

x,(o)- i ~ x2(O~ )

"(t)

~(2)

x~(I)

x2(t)

x,(2)

~

xi(3)

_~

_

x~( I

~

)

x,(2) xt(3)

x2(4) -i

r(5',

x,(o)

-I

~ ~

r(4;

,,,"( 6: r(7

Transform data

-I

,= x~(4)

xz~5)

~

x,(5)

x2(6)

-

xJ6)

xz(7 )

,P2" -

x~[7)

Fig. 10. Cooley-Tukey type signal flow graph for (GHT),, N = 8.

Input

Transform data

data

x(O) x(I)

xa(O) Xl(1)

x(2)

xi(2) x~(3)

x(3) X(4) x(5) x(6) x(7)

x2(O)

',,X//

xl(5) x=(6) xi (7)

x(8) x(9) x (I0) x(rt) :x(12)

x1(8) xj(9) xl (10)

x(13) x(14) xCIS)

x I (14] x I (141 xi (15

Xl(ll)

i=

2 2~B

2~

D

2~ 2~2

BR

L

xt(12] 2~2 D

IlL

2~2 2~2

j

rz~=) r2(2) r2(3) -I 2 ,, r2(4) 2 rz~5)~ 2. rz(6) 2. r2(7)

x, o) x3(1)

I-

xo(i) xoC3)

xd4) xo(5) xo(6) Xo(7)

Xo(8) xo(9) XoOO) xdH) XoO2) Xo03)

Xo(J4) Xo(JS)

Fig. 11. Cooley-Tukey type signal flow graph for (GHT)., N = 16 ( - indicates bit reversal).

378

K. R. RAO,M. A.

and K. REVULUR]

NARASIMHAM

Transform data

I n p u t data

x(I) x(2) x(3)

x(4) X (5

x(5)i

n.

x(6)

- -

~

-I

8R

X3(4) -

~

~

x3(6)

--

x~(7)

--~-~"---

,.

- x,(8)

x(71] x(8)

~'l ( 8

x(9)

~1( 9

x (10)

~'/10:

x(ll) x(12) z(/3) ~,(m)

-,7///kkV

/// ?

;e(15)

x(8)

\\Z

' / x2(9 )

2.

xl(9 )

/ - / x (10)

x (10)

rl( II

. 2 Z/. X2( II )

~

r102:

x2(12 )

2

x (12)

x2(13)

2 ,- ._

xl(i3)

x2(14)

2

x,(/4)

x2(i 5)

2 ,,

x~(t5)

rl(13:

\

rI (14:

\

r~ 05:

-I

-__

'/;~ " - XI(4)

-/ x3(5)

2 "

x,(5) f2

"

x (6)

x(7)

Xl(If)

k'ig. 12. Cooley-Tukey type signal flow graph for (GHTh N :: It, Input data x(O)

xl(O )

/

=

(3)

x,(2)

x (4)

( (

xi(3) x,(4)

x(5)

I

x,(5)

x(6)

/

x,(6)

x

x(7)

x(8) \

x('°,

\,

x(14)

-: -I

&XX/

~ " =

x2(2)

- ~ ~

x2(3)

f

X3( I )

-

.i

x~(8)

~

\,:.A \ WA.

x2(9 )

w \ ~/'~"~

") X2[" -IO

~ 2 -
W%/

x1(12)

X2(12)

..

X (14)

~/~..W

xl(13)

\

xl(15)

-/

.=

xl(II)

Xl(14)

~3(3~--2~=.~-"--._

X3(4)


2(2) ,

~

W2

x2(3) X2(4)

~.

""- xgT)

~

x2(7)

2 x3(8) I,,I/ ~. x3(9)

-W v~l,

X2(8)

---

"/2~,

x2(9)

---

W

xdlo)

72 72 ,,, 72

x2(11)

X3(14) __

~p

X2(14)

x3(15) .

7"2.

x2(15)

W2 x.(Io) o

~. `

-W2 ,, ~ W ~ we

2 _W 6 - ~ X2(15) ~ =_W6~

/i\

:2 ( I ) -

-W 2

xz(H)-~

\

~

N=4 ~

I-~-'-I X2(8)

\

~'- _[= ~

J~ ~ ;e~,-~ - / ~"

D

lxo(7)l

xr(9 )

,.~ x 3 ( O ) ~ ; 7

X2( I )

xl(7)

x I (IO)

x(12)

x(15) /

x2(O)

x4(I)

x(I)

x(2)

\-/

Transform data x2(O)

xs(fl) x 3(12)

__ --

X2(12)

Fig. 13. Cooley-Tukey type signal flow graph for ((]HT)> N = 16. From (1, 17 and 18) the relationship between D, (m) and X, (m) can be established as follows: D,(0) = Ix,
D,(j)-

IN," . (' ,)u,, ,,

K I

, E

j= 2 4 ( K

I)2

(J)

l~l (19)

~()'1 ~ r where U ( 1 - r ) = [ l ; l > r .

379

A family of discrete Haar transforms

Input data x ° ( O ) ~

,o(,) __, .,'o(,)

£ ¢2

" /

x°(O)"=~

x02(O) X

xo,(,)~

xo:,

~ . ~ / _

2

Xo(4) ._.

:q

"~

~ 2 ~

Xo~ _, xo(,,, Xo(7)

x°3(O)

/

,o~(,, x _ ' ~ / / , o , , , ,

~<~_ Jxo,(3)~!

"

Transform data x(O) --~~ x(I)

2

--2 ,,,

,:,.~./.~ &/,,/'v"

x(2) x(3)

,o~,~)

.,'o~(~) vv':v

.,'o~(~)

F ~ I ,.,I

xo.~(4) x ...

-I/'..z'~/~ /__/~/~ \

( 4 1 x(5)

IXo~q~/~\ xo~ I . . . . ,1~=4/ / - \ \ x (6) i,,o~,~ / _ , \ o~ [Xod7~ : -I"- ' Xo:~(7) . xo (8) 2~ _-.= x o (9)

2,/2

x o (10) Xo(ll)

_- --

x(6) x(7) x(8)

ro3(9;

x(9)

2v~ .. 2f2.

ro3(lO

x(lO)

~'o3(H

7//~\\\

'~ '')

x 0 (12)

2v~.,

~'o3(12

//

x(12)

Xo(13)

2,/'2 ,

Zo3(1:5

~

Xo (14)

22~=, 2 f"2

ro3(14 ro3(I 5

xo (15)

:I

x(13) x(14) x(15)

-

-I

Fig. 14. Cooley-Tukey type signal flow graph for (IGHT)o, N = 16 ( - indicates bit reversal).

Transform data x,(O)

x,(') - ' ~ _ 1 xl

x,(O) ~ " x,,(') ~ .

I.~: xB2(O) "~k " / / ~ Xl2(2) .~. v .

(3)

X1(4) -/ ~m-

XH(4)

/.~

m,/

x t (5)

~ ~

x (5)

~ -

~ :-

x I (6)

~ -

x (6)

_/'~=~/'~

xl (12)

2 =

xl(13)

2 =,

x=(14) xl(15)

2 = 2

x~3(2) xt3(3)

/ x(I)

\\ ~// ~, x(2) \\\\//1

f.'~2',~/h?x\xj3(6)

I',:i;l / !-,\ xw2(8)

D 2 =

x~3(O) x~3( I )

x j3(4) x,3(5)

x, o i iii XI (lO) Xl(ii)

Input / x dora (O)

\

z"

q¢ 8 :

,rr3(II rr3(12;

Xl2(15) /

/

\ x(lO) lx(r2)

xl3( 141 x

xt3 (15

x(6) x(7)

~2?//X~\~ x ( l l )

r=3(131

,, ,~) Z / \ X

~' x(4) x(5)

~ ~

x~3(7) q:3(9) q3(lO:

i x(3)

-I

\ x03) \ x(r4) ~ x(15)

Fig. 15. Cooley-Tukey type signal flow graph for (IGHT),. N = 16 ( - indicates bit reversal).

This can be illustrated for n ---4 and r = 0, 1, 2, 3. Haar transform (HT)

r=0

Do(0) = [Xo(0)[ 2 D o ( l ) = [Xo(1)[ 2 Do(2) + Do(3) =

j=4

Do(j) = ~

E" Do(j)=

j =8

2 -''2

2-'/2]Xo(2) + Xo(3)1 z Xo(j)

E'~

j =a

Xo(j) 2

(20)

380

K. R. RAO, M. A, NARASIMHAMand K. REVULURI Transform

Input data

data

x(O)

x(I) x(2)

I?Z'>P~"."~ ...(~,~...XX/. "(" \\\; I/t I ~' I -I ~,2 V V V x23(3) ~\\\W/// ,, , ~ ~ ..I~, ~ \ ~ V / t

""> , 7 ~ "'<" -"-

x(3) x(4) x(5)

x(6) 2

x(7)

_W_ 6 -

X2(8)

v£'~--'-

X2'(8)

~,,~"

A¢/" X22(8)iW,,

x2(9)

~'~-'-- x2,(9) W - ~ . ~ ¢ ~

_'\ " \ I / / ' *

-

x tl x i W - 6 ~ /

......

Y- l i ~

' " ~' W " ' - - ~ -'~ " . . . . - , 7 / \ , x2(14)

~

"

x (14)

/_

v

~.

x

(14)

/

/

=- \

~ x(8)

x2~(9)

x(9)

xo3(lO) - I Z ~ ~

x(lO) ~(II)

~/,f,

- w-..

vr2

x23(8) - I Z ~ / ~ / ~ "

x22(9) ~

.... W~'~/'~

x2(13)

"--7

P////Z~,\~'

....

x(12)

~ "v

,:~(,3~

\

x'2~,(14)

x(13) x(14) x(lS)

Fig. 16. Cooley-Tukey type signal flow graph for (IG HTg. N = 16 ( r = 1

indicates bit re,,ersal).

Complex Haar transform (CHT)

D , ( 0 ) = [X,(0)l 2

D,(i)

=

Ix,(~)l'

D d 2 ) + D , ( 3 ) = IX,(2)i: + IX,(3)] z

~ f

D , ( j ) = 2 '/:IIX,(4) + X,(5)i-" + iX,(6) + Xff7)l:] 4

Y~

-(.i)=7

+

(21)

•

r=2

o.(o) = Ix:(o)l: D:(I) = Ix.(W D:(2) + D:(3) = IX:(2)I 2 +

Ix2(3)1:

L D:(j)-L- IX:(J)I2 j.:4

i 4

15

~., D : ( j ) =

'~ - 'r-[iX,(8) . +X:(9)]2+!X:(IO)+X:(I1)I + LX:(12)+ X.( 13)!-" + !X.( 14)+

2

x.(~5)l'1

(22)

r=3 D3(0) = [X3(0)12 D3(1) = IX3( 1)[2 D3(2) + D~(3) =

Ix~(2)!' + ix~(3)b:

L D~(i). = L IXM)I: i ,~4

j =4

2i ::~ ~,(J):j2 Ix~(j,:. =8

(23)

Afamilyof discreteHaartransforms

381

DYADIC SHIFT INVARIANT SPECTRUM The (GHT)r spectrum is not invariant to cyclic shift of {x(n)}. However, if the data sequence {x(n)} is shifted dyadically, the power spectrum invariant to dyadic shifts can be developed. If {x(d'~(n)} is {x(n)} shifted dyadically by 1 places, then {x~d')(n)} the (GHT)r of {x(dt~(n)} is related to {X,(n)} the (GHT)r of {x(n)} as follows: {x(al)(n) } = 1 [Hr(n)] [L'd')I{x (n)} 1 = -~ [H,(n )1 [x ~"(n)]

= 1 [H, (n)] [i ed,)][Hr (n)]* r {Xr (n)} = [s/d')( n )1{Xr (n)}

(24) !=0,1 ..... N-1

where [IN(dl)] is IN whose rows are shifted dyadically by 1 places and l

[s,(d'~(n)] = ~- [Hr(n)] [i,,/d,,] [Hr(n)],r

(25)

is the l-th dyadic shift matrix relating {x, Cd'~(n)}and {Xr(n)}. As an example for N = 8, r = 0, 1, 2, and 1= 0, 1. . . . . the dyadic shift matrices are r=0 [So"'(3)] =

Haar transform (HT) -I2.__J

[Soa~(3)l =

1

0

0

0 0 0

0 0 1

0 1 O_

-1 o o

0 o o

0 1

0 0 "12 I [So~3)(3)] =

-- ,,-?--fi-1 1

0 II-1 'o I I

J 0

[So")(3)] =

0

[So")(3)] = Diag f , -

-1

0

Li . o

0 1 0 0 0 0 1 1 0 0 0

0

1,[~

I

1] 0J'

0 o -1

0

=

f -°111 _

Diag

0

,

I-o ~-1,~°o o o ,11 l, Ll L0 01 00

ro

o-1_1,o-i]

_0

0

0

I

1 0 L o - 1

0 o

O[ oj,

382

K. R. RAO, M. A. NARASIMHAMand K. REVULURI

[So~6)(3)1 = Diag

1.-1,

[

FO

0

0

0

0

1

0-1]

L-1

ol'

0

roo r o-,1

1,-1,~_,

/

o I ool/ LI

[S#7~(3)] = Diag

117 O[

0

0 AI]

o-,]]

o

oJ, l O _ l

o-~

O l/

(26)

o o, j

L-1 0 0 0 ] Complex Haar transform (CHT)

r=l F1~ I

]

ts,'"(3)]: U-T--rJ

I

[S(2)(3)] = Diag

[S,'S'(3)] = Diag

~o,o oq 0 LO0 0 0

I2,-12,

L12,-I2, rI -1o-i o o7] 0 0 0l. /I

r [S,'4)(3)] =

Diag

[S,'~(3)] = Diag

[S,(6'(3)] = Diag

[$1':'(3)1 Diag

0 01 0 1[ 1 O]

[

o-ii

o

o

0

0 - 1

0-,]

r00

oJ 0-i00 - ~ 7 7

',-',i,

o~, L; ,° o° o~°I/

.

7 , ol,,

1,-1,[

r 0-iq [ 0 0 0 i 0J"-i 0 0 L 0 -i 0 [

0

1.-1.1_i

i1

o o

O-i

0]']0 Li

[,, ,,,., r

rI o0

o

L-i

i 0

o

o

OJ'l o - i L-i

iq7 0 I

0 0 (1 0 J J

I- o o o

i]

0

0/_1

i

o o 0

0

]]'

i o[ 0

(GHT)2

r=2

~4 5_____] I- I,_J

[S2'"(3)1= L

[$2 (3)]=Diag

I2.--I2.2 112

t

i

[$2(3'(3)] = Diag [Iz. - 12.2 .2 [L-i-1

1

"

i ],

2_,,,2 [

I

L i

L-i

lJ

, /]] -I

(27)

383

A familyof discrete Haar transforms

[0 1 ' 0 '0 J '2 1+i L-l+/

0 0

1-i -1-i

-1+i

0-i] [$2(')(3)] = Diag f , - 1, [

i]

I

0i

[Sz('~(3)] = Diag f , - l [

0

0

0 _1..I

0

-1+i

1- i -1-i

-1-i 1-i 0

[$2(')(3)] = Diag f , - 1, [ _ ~

i ] 9_1/2 0J'-

o0

Diag

f

-

r0

- 1 + i]']

o

o,j

0

0 ,I

0

0

o

+0

l-i 0

L-l+/

[52(7)(3)]

H

0

1+i

0

0

- l - I]'] 1-i| /

- (1+ i)7] // o

Oo,, j

0

0 0 ,+ill

i

-1t2 I 0

0

-1-i

0

]

0

- i+ i

0

0 0

L 1-i

0

0

0

The structure of the dyadic transformations described in (26-28) indicates that each dyadic shift matrix is block diagonal. Also each block diagonal submatrix of [S/d')(n)] is unitary. This leads directly to the following dyadic shift invariant power spectrum: ix/~,(o)l 2 = Ix,(o)l =, Lx/',(l)12 = Ix,(1)] 2 3

Ix/~l'(m)l 2= ]~ rn = 2

[X,(m)] 2

rn = 2

7

~

Ix/""(m)l 2-m =4

IX,(m)ll 2 m =4

1=1,2 . . . . . 7. The dyadic spectrum can be generalized for any N = 2 " , 1,2 . . . . . N - 1, as follows: ix/'.,(o)]

r =0, 1. . . . . n - 1 ,

and 1=

= = ix~(o)l =

2K--I

2K--I

Ix/"(m)l == m =2 K -I

Y~

IX,(m)l =

m =2 K -I

r=0,1 ..... n-l,

1=1,2 . . . . . N - l ,

K = l , 2 . . . . . n.

APPLICATIONS

Discrete orthogonal transforms have been utilized in a number of diverse disciplines including spectral analysis filter simulation, convolution and correlation processes, bandwidth compression, spectroscopy, optics, acoustic wave propagation, speech and image processing, data compression, sequency multiplexing, pattern recognition, and Wiener filtering. The utility of the generalized Haar transforms developed here is investigated in terms of their performances in some of these application areas. (i)

F e a t u r e selection in p a t t e r n recognition

The data domain covariance matrix of a random vector {x(n)} is defined as [~O(n)]

= E[({x(n)} - {2(n)})({x(n)} -

{2(n)}) T]

where E represents the expected value operator and the bar above the vector indicates the mean

384

K.R. RAO,M. A. NARASIMHAMand K. REVULURI

or expected value of the vector i.e. {£(n)} = E[{x(n)}] also

[O(n)] =

I cr2~,.,cr~,. "~ 2 2 O'xmO'xu

cr~.~,. ,,

-J

0"~,,,. ~,

where the diagonal elements are the variances of the individual random variables and the off-diagonal elements are the covariances of the random variables x(I) and xlm ). The transform domain covariance matrix of {x(n)} is given by [~(n)l = [A(n )110(n )1 [A(n )l *T where [A(n)] is the transform matrix, and z 2

2

o~.,o'L,

or{,,. ~,

[ q ' ( n ) ] =

.

. . . . . . ~_ (]'X(N

.

.

I

.

I

....

I)00"X(N 111

0"~1/% " i}(~ ~_i *

Andrews[20] has developed a criterion for eliminating the features i.e. components of a transform vector, which are least useful for classification purposes. For a first order Markov process signal (data) in the presence of a white noise, the data and noise covariance matrices are respectively given by [34-36].

[~,.(n)]

=

p2 p 1

pN N p N p

I

P

p 2 p

1 p

p N-I

pN 2 pN 3

J 2 3

I

where p is the adjacent element correlation and [6w(n)] = Diag[koK,, •. - Ko] where 1/Ko is the signal-to-noise ratio. The effectiveness and performance of the (GHT)r is checked in terms of the variances of the transform domain covariance matrix for p = 0.9. This is shown in Tables 1-3 for r = 0, 1,2, and n =3,4,5. (ii) Wiener filtering [36, 73] The application of discrete transforms in Wiener filtering is described in Fig. 17.

t [G(n)] Fig. 17. GeneralizedWienerfiltering. {z(n)} is an N-dimensional input vector which is the sum of the data vector (random process) {x (n)} and an uncorrelated white noise vector {w (n)} with zero mean. The Wiener filter [G(n)] is in the form of a (N × N) matrix, and {~(n)} is the estimate of {x(n)}. The objective of Wiener

385

A familyof discreteHaartransforms Table 1. Variancedistributionfor a I orderMarkovprocess, p = 0.9 and Ko = 1 (N = 8) i

(GHT)0

(GHT)1

(GHT)2

l

6.1855

6.1855

6.1855

2

0.8635

0.8635

0.8635

3

0.2755

0.2755

0.2755

4

0.2755

0.2755

0.2735

5

0. I000

O. lO00

0.0963

6

0,I000

O. lO00

0.I037

7

O.lO00

O. lO00

0. I037

8

O. lO00

O. lO00

0.0963

Table 2. Variance distributionfor a I order Markov process, P = 0.9 and Ko = 1 ( N = 16) i

(GHT)0

(GHT)l

(GHT)2

l

9.8375

9.8346

9.8346

2

2.5364

2.5364

2.5364

3

0.8638

0,8635

0.8635

4

0.8638

0.8635

0.8635

5

0.2755

0.2755

0.2488

6

0.2755

0.2755

0.3021

7

0.2755

0.2755

0.3021

8

0.2755

0.2755

0.2488

9

O. lO00

O. lO00

0.0966

I0

0.1000

O. lO00

0.0966

II

O. lO00

O. lO00

0. I033

12

O. lO00

O. lO00

0.1033

13

0.1000

O. lO00

0,I033

14

O. lO00

O. lO00

0.1033

15

O. lO00

O. lO00

0.0966

16

0. I000

0.1000

9.0966

filtering is to design the filter matrix [G(n)] such that the expected value of the mean square error between {x (n)} and {£ (n)} is minimized. Pearl [35] has shown that eo the mean square estimation error due to scaler filtering ([G(n)] is constrained to be diagonal matrix, eo can be expressed as N--I

eo = 1 - 1

~o (--°'---~'-'

\Orx. 4- O'wnl/

where trw,, are the diagonal elements (variances) of the transform domain covariance matrix of

{w(n)}. The values of the error estimate for p = 0.9 and ko = I for (GHT), are compared with other discrete transforms as a function of n in Table 4. CONCLUSIONS

A new class of discrete orthogonal transforms called generalized Haar transforms (GHT)r is defined and developed. The base functions of these transforms are linear combinations of Haar functions with increasing complexity in their relationships. Thus, these transforms represent C.A.E.E., Vol. 2, No. ¢ - H

386

K. R. RAO, M. A. NARASIMHAMand K. REVULURI Table 3. Variancedistribution for a I order Markov process,#= 0.9and i

K . = t ( N = 32)

(GHT)o

(GHT)o

(GHT)o

13.5703

13.5688

13.5681

6.1015

6.1011

6.1011

2.5364

2.5364

2.5364

2.5364

2.5364

2.5364 O. 7068

5

0.8635

2.8635

6

0.8635

2.8635

1.0199

7

0.8635

0.8635

1.0199

8

0.8635

0.8635

0,7068

9

0.2755

0.2755

0.2562

I0

0.2755

0.2755

0.2562

II

0.2735

0.2755

0.2945

12

0.2755

0.2755

0.2945

13

0.2755

0.2755

0.2945

14

0,2755

0.2755

0.2945

15

0,2755

0.2755

0.2562

16

0,2755

0.2755

0.2562

17

0. I000

0.1000

0.0976

18

0.I000

0.1000

0.0976

19

0.I000

0.1000

0.0976

20

0.I000

0.1000

0,0976

21

0.I000

0.1000

0.I024

22

0.I000

0.1000

0.I024

23

0.I000

0.1000

O. 1024

24

0.I000

0.I000

O. 1024

25

0.I000

0. I000

O. 1024

26

O. lO00

0.1000

0.I024

27

O.lO00

0.1000

0.1024

28

O. lO00

0.1000

0.I024

29

O. lO00

0.1000

0,0976

3O

0.1000

0. I000

0.0976

31

0. I000

O. lO00

0.0976

32

0. I000

0.1000

0.0976

Table 4. Mean square error for first order Markov process when p

= 0 . 9 a n d K = 1 for

16

scalar Wiener filtering 32

(GHT)0

0.29426

0.26498

0.25893

0.25816

(GHT) l

0.29426

0.26498

0.25893

0.25816

0.26497

0,25886

0.25766

(GHT)2 DCT

0.29200

0.25460

0.23740

0.22820

DFT

0.29640

0.27060

0.25920

0.24410

WHT

0.29420

0.26490

0.25820

0.25820

KLT

0.29150

0.25330

0.23560

0.22680

decomposition of the input data in terms Haar functions and their linear combinations. Fast algorithms, based on matrix factoring, for eificient computation of (GHT)r are developed. Signal flow graphs illustrating these algorithms are presented. The 'in-place' property of the fast algorithms can be restored by introducing successive bit-reversal operations in these flow graphs. The dyadic autocorrelation theorem for the (GHTL is defined and developed. It is shown that the (GHT)r power spectrum is invariant to dyadic shifts of the data sequence {x(n)}. Digital computer programs for implementing fast (GHT)r and its inverse and for computing variance and mean-square error as applicable in feature selection and Wiener filtering are developed. The effectiveness of the (GHT)r is compared with other standard transforms in terms of the

A family of discrete Haar transforms performance

c r i t e r i a s u c h as v a r i a n c e , a n d m e a n - s q u a r e

387

e r r o r . T h e ( G H T ) r d e v e l o p e d in this

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