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Frequency-refined multiresolution decomposition using wavelet splitting
1 January 2000
Optics Communications 173 Ž2000. 81–94 www.elsevier.comrlocateroptcom
Frequency-refined multiresolution decomposition using wavelet splitting Zikuan Chen, Mohammad A. Karim
)
The UniÕersity of Tennessee, Department of Electrical and Computer Engineering, 414 Ferris Hall, KnoxÕille, TN 37996-2100, USA Received 16 August 1999; accepted 26 August 1999
Abstract Frequency resolution of wavelet transform is further refined by splitting wavelets along selected directions into wavelet packets. Since the spectral distributions of wavelet packets lie within that of the wavelets, the narrowness of their spectral support contributes to improved frequency resolution. Experimental results show that such wavelet splitting offers better compression and recovery scores than the standard wavelet transform. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Multiresolution analysis; Wavelet transform; Wavelet packets; Subband coding
1. Introduction The wavelet transform involves joint time– frequency representation with locality and, thus, it is useful in non-stationary signal analysis w1–3x. The wavelet transform manifests as a hyperbolic time– frequency tiling, for example, in a spectrogram representation w2x. In particular, high frequency analysis is adapted to narrow time-width while low frequency analysis uses wider time-width. The adaptive property of a standard wavelet transform is fixed, i.e.,
) Corresponding author. Fax: q1-423-974-5483; e-mail: [email protected]
given a scaling function it accommodates all applications in identical manner. In other words, a standard wavelet transform, except for its inherent time– frequency adaptability, is unable to adapt itself to specific application. An irregular tree decomposition is expected to yield better performance than a standard wavelet tree decomposition w4–7x. Accordingly, an embedded zero-tree based on dyadic wavelet decomposition has already been proposed to prune a wavelet tree w4x. In this paper, we explore identifying an arbitrary subband decomposition tree by splitting the wavelets into wavelet packets, offering improved adaptability to specific applications. A wavelet transform can be described by its wavelet orthonormal basis generated from a mother wavelet by dilations and translations w1,3,8x. The
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 4 7 9 - 4
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Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
orthonormal wavelet basis typically provides a framework for studying images simultaneously at different levels. Using a multiresolution analysis, the successive ‘fine-to-coarse’ approximations can be represented by a nested sequence of subspaces while the information differences between two resolution levels are described by a sequence of complement subspaces w9x. Multiresolution analysis can be considered as a recursive filtering using a pair of wavelet filters and, eventually, by the scaling function as determined by two-level equations w1,3,8x. While the sequence of nested subspaces associated with a multiresolution analysis is described by dilations and translations of a scaling function, the sequence of complement subspaces is described by dilations and translations of a mother wavelet w8,9x. A wavelet function can be considered as a highpass signal. It has bandpass characteristic since frequency localization requires that a high-pass signal decays rapidly with increasing frequency. The narrower the bandpass bandwidth, the better the frequency resolution. The spectral support of a dyadic wavelet is a dyadic interval whose size changes as 2 j where j is the resolution level. However, the exponential behavior of dyadic support is fixed. For a higher-resolution wavelet, one makes use of the next higher resolution level. Alternatively, in this paper, we refine the wavelet frequency resolution at a level without going to the next level. We demonstrate the advantages of wavelet splitting over a standard wavelet transform in terms of compression and recovery scores. Since wavelet splitting uses the same wavelet filters as in the multiresolution decomposition process, its implementation involves applying recursively wavelet filtering to the coefficients of wavelet transform. Wavelet splitting is rendered on selected direction at specific levels and, therefore, can be used for adaptive multiresolution image decomposition. This paper is organized as follows. We derive wavelet filters from the set of two-level equations associated with a multiresolution analysis and apply them to splitting the wavelets into wavelet packets in Section 2. In Section 3, we consider wavelet splitting along selected directions at specific levels. The experimental results are presented then in Section 4 and, finally, conclusions are summarized in Section 5.
2. Review A multiresolution analysis is a process of recursively applying wavelet filtering on the approximation Žtrend. at each level. On the other hand, wavelet splitting is realized by applying wavelet filtering on the detail Žfluctuation. of wavelet transform. The wavelet filters are obtained from two-level equations associated with a multiresolution analysis.
2.1. Two-leÕel equations A wavelet basis function cm nŽ t . is given by
cm n Ž t . s 2 m r2c Ž 2 m t y n . , m,n g Z
Ž 1.
where c is a function in a square integrable space L2 Ž R ., the so-called ‘mother wavelet’. It is constructed by dilations and translations of the mother wavelet where the dilation factor is 2. The wavelet basis can be constructed also using a scaling function. A multiresolution analysis of L2 Ž R . associated with the scaling function f Ž t . is defined as a nested sequence of closed subspaces Vj 4j g Z such that w9x
Ž i . PPP Vy2 ; Vy1 ; V0 ; V1 ; V2 PPP
Ž 2a .
Ž ii . f Ž t . g Vj
Ž 2b .
Ž iii .
mf Ž2 t . gV
jy1
F Vj s 04 , D Vj s L2 Ž R . jgZ
Ž iv . f Ž t . g V0
Ž 2c .
jgZ
m f Ž t y n. g V , n g Z 0
Ž 2d .
Ž Õ . V0 sspan f Ž t y n . , n g Z 4
Ž 2e .
where the overbar denotes a closed vector space and the sequence of nested subspaces is indexed by level j. The resolution at level j is 2 j, which is considered as the sampling rate of a sampling grid. The ‘fineto-coarse’ multiresolution analysis suggests a sequence of nested grids that go from the ‘fine grid’ Žhigh resolution. to the ‘coarse grid’ Žlow resolution. with decreasing j. The subspace V0 is referred to as a reference subspace whose basis functions are constructed by translating the scaling function. The sub-
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
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space Vj at resolution 2 j is spanned by an orthogonal basis given as
where the overhead cap denotes the corresponding Fourier transform, and
Vj sspan f jn Ž t . , n g Z 4
m0 Ž j . s
sspan 2 j r2f Ž 2 j t y n . , n g Z 4
Ž 3.
where f jnŽ t . has a definition analogous to Eq. Ž1.. An array detector can detect a signal with only a finite number of pixels, which correspond to a representation on a subspace at certain level. The original signal is considered as a full-resolution representation in the reference subspace V0 for the purpose of multiresolution analysis. Further, we assume that the resolution of V0 is equal to 1 Žs 2 0 .. Beginning with a full resolution signal, one can find information in reduced data using ‘fine-to-coarse’ processing. A practical ‘fine-to-coarse’ multiresolution analysis is described by a sequence of finite nested subspace Vj 4 , j s yJ, yJ q 1, . . . , 0 where J ) 0 represents the number of levels. The coarsest approximation is obtained at level yJ which corresponds to the multiresolution analysis depth. With successive approximations from level j to level j y 1, the difference information between the successive levels is collected into a complement subspace Wjy1 of Vj . In general, Vj at level j can be split into two orthogonal complement subspaces at the next level j y 1. For a sequence of approximations from level 0 to level yJ, we have
1
h w n x exp Ž yi n j . '2 Ý n
Ž 7.
is a trigonometric polynomial. On the other hand, when W0 is spanned by an orthonormal basis c Ž x y n.,n g Z 4 , we have a different two-level relation since c Ž x . g W0 ; V1:
c Ž x . s Ý g w n x f Ž 2 x y n. n
s Ý ² c Ž x . , f Ž 2 x y n . :f Ž 2 x y n .
Ž 8.
n
or
cˆ Ž j . s m1 Ž jr2 . fˆ Ž jr2 .
Ž 9.
where m1 Ž j . s
1
g w n x exp Ž yi n j . . '2 Ý n
Ž 10 .
Since f g V0 ; V1 , and V1 is a subspace spanned by f Ž2 x y n., n g Z 4 as given in Eq. Ž3., one obtains a two-level relation given by
A multiresolution analysis, thus, leads to two two-level equations which can be interpreted as bank filtering w8x, i.e., m 0 and m1 can be considered as a pair of wavelet filters. The coefficients hw n x4 and g w n x4 of the wavelet filters are obtained from the two two-level equations. A multiresolution analysis, thus, consists of two sequences of subspaces Vj 4 and Wj 4 generated from a single scaling function f Ž t .. It can be interpreted in terms of orthogonal projections as well. Let P Vj and P Wj denote the orthogonal projections on subspace Vj and Wj , respectively w9x:
f Ž x . s Ý hw n x f Ž 2 x y n .
P Vj f Ž t . s Ý ² f Ž u . , f jn Ž u . :f jn Ž t .
0 V0 s VyJ [ Ž [jsyJ Wj . .
Ž 4.
n
n
s Ý ² f Ž x . , f Ž 2 x y n . :f Ž 2 x y n .
Ž 5.
n
where ² P , P: denotes an inner product. In frequency domain, we have
fˆ Ž j . s m 0 Ž jr2 . fˆ Ž jr2 .
Ž 6.
s Ý a j w a x f jn Ž t .
Ž 11a.
n
P W j f Ž t . s Ý ² f Ž u . , c jn Ž u . :c jn Ž t . n
s Ý d j w n x c jn Ž t . . n
Ž 11b.
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Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
The coefficients a j w n x,n g Z 4 and d j w n x,n g Z 4 represent respectively the approximations and details of f Ž t . at resolution 2 j. The algorithm for finding coefficient d j is referred to as the wavelet transform Žor standard wavelet transform herein. and d j is called a wavelet coefficient. A ‘fine-to-coarse’ multiresolution analysis with depth J Ž J ) 0. results in the following orthogonal wavelet representation w9x: f Ž t . : Ž ayJ w n x ,d j w n x ,y J F j F y1, n g Z . .
Ž 12 .
Due to orthogonality of the wavelet functions, it can be interpreted as a decomposition of the original signal into a set of independent constituent signals.
With wavelet filters hw n x4 and g w n x4 , the coefficients a jy14 and d jy1 4 at level j y 1 can be obtained from that at level j using wavelet filters. This leads to the following decomposition algorithm: a jy1 w n x s Ý h w k y 2 n x a j w k x
Ž 13a.
k
bjy1 w n x s Ý g w k y 2 n x a j w k x k k
s Ý Ž y1 . h w 1 y k q 2 n x a j w k x .
Ž 13b.
k
In order to generate the wavelet coefficients, we need to first calculate a 0 w n x4 for a signal representa-
Fig. 1. Standard wavelet transform: Ža. an octave-band tree; and Žb. its spectral partition.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
tion in V0 , then work down to successively coarser levels using Eqs. Ž13a. and Ž13b., with error at each successive level corresponding to the wavelet coefficients. Fig. 1Ža. shows a scheme for multiresolution decomposition obtained by iterating Eqs. Ž13a. and Ž13b. where H and G stand for filtering using hw n x4 and g w n x4 , respectively. It represents a standard wavelet tree and its subband decomposition scheme is given in Fig. 1Žb.. Both H and G filtering operations involve convolution and decimation. 2.2. WaÕelet splitting Consider the frequency bands of a wavelet c Ž t . to approximately equal to wy2 D,y D x jw D,2 D x where D denotes its passband-width. Its dilated version at level j will have the band support of wy2 jq1D,y 2 jD x j w2 jD,2 jq1D x. A multiresolution analysis realizes a binary partition given by
Ž 0,` . s
then f 1 Ž t y 2 k ., f 2 Ž t y 2 k .;k g Z 4 is an orthonormal basis for space E. According to this lemma, wavelet filters can be used to split any space spanned by the orthonormal functions f Ž t y n.;n g Z 4 into two parts: f 1 Ž t y 2 k .;k g Z 4 and f 2 Ž t y 2 k .;k g Z 4 . Since m 0 and m1 , hw n x4 and g w n x4 , exist at different frequency ranges, wavelet-splitting corresponds to dissecting the spectral support of f Ž t . into slices and allotting alternate slices to f 1 and f 2 , which have better frequency localization. If f Ž t . in Eqs. Ž15a. and Ž15b. is substituted by f Ž t ., then Eqs. Ž15a. and Ž15b. reduces to the two-level equations, Eqs. Ž5. and Ž8. respectively, when f 1 s fy1 and f 2 s cy1. The splitting of the wavelet into two wavelet packets can be rendered on a dilated wavelet at specific level:
c j1 Ž t . s Ý h w n x c j Ž t y n .
j
D Ž 2 D ,2
jq1
Dx .
Ž 14 .
jsy`
Ž 16a.
n
c j2 Ž t . s Ý g w n x c j Ž t y n .
`
85
Ž 16b.
n
or, equivalently, j
jq1
Each interval Ž2 D,2 D x represents a dyadic band and all the disjoint bands cover the whole frequency range as shown in Fig. 1Žb. for an ideal bandpass wavelet with D s pr2. As the resolution level decreases Žincreases. from one level to the next, the dyadic interval becomes halved Ždoubled.. Such octave-band decomposition tree is not always desirable since its exponential behavior is too rapid to be optimal in some cases. An irregular tree decomposition, on the other hand, results in improved performance w7x. Based on the octave-band tree, we hereby construct an arbitrary tree by splitting wavelets into wavelet packets. With wavelet filters hw n x4 and g w n x4 given by Eqs. Ž5. and Ž8., Daubechies identifies a lemma for ‘wavelet-splitting’ w4x. Lemma: For any function f Ž t . so that f Ž t y n.,n g Z 4 is an orthonormal basis for space E, define
cˆj1 Ž j . s m 0 Ž j . cˆj Ž j .
Ž 17a.
cˆj2 Ž j . s m1 Ž j . cˆj Ž j . .
Ž 17b.
f 1 Ž t . s Ý hw n x f Ž t y n .
With wavelet splitting rendered at all levels, a signal can be expressed as a double sequence of wavelet packets:
Ž 15a.
n
f 2 Ž t . s Ý g w n x f Ž t y n. n
Ž 15b.
At level j, the construction of wavelet packet appears as a change of orthonormal basis inside Wj providing Wj s Wj1 [ Wj2 . The orthogonal projections on wavelet packets are given by P W j1 f Ž t . s Ý ² f Ž u . , c jn1 Ž u . :c jn1 Ž t . n
s Ý d 1j, n c jn1 Ž t .
Ž 18a.
n
P W j2 f Ž t . s Ý ² f Ž u . , c jn1 Ž u . :c jn2 Ž t . n
s Ý d j,2 n c jn2 Ž t . .
Ž 18b.
n
f Ž t . : Ž ayJ , n ,d 1j, n ,d j,2 n y J F j F y1,n g Z . .
Ž 19 .
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Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
Eq. Ž19. represents a tree structure, however, it is not necessary to split wavelet at every level. Fig. 2Ža. shows an example of wavelet packet decomposition at level y2 where the wavelet coeffi1 w x4 2 w x4 cient dy2 w n x4 is split into dy2 n and dy2 n . Fig. 2Žb. shows the corresponding spectral decomposition. The wavelet packet c 1 Ž t . is the low frequency slice of c Ž t . while c 2 Ž t . is the high frequency slice. The highlighted area shown in Fig. 2Žb. denotes the spectral support of wavelet packets. For clarity, the bandpass characteristics of c Ž t ., cy1Ž t . and cy2 Ž t . are illustrated in Fig. 3Ža.. The spectral support for wavelet splitting associated with
cy1Ž t . is shown in Fig. 3Žb.. It is seen that wavelet packets are more compact than the wavelet. Also, the spectral support for each wavelet packet takes only a sliced part of its wavelet implying that the wavelet packet has a better frequency resolution.
3. Theory One popular way to constructing 2-D multiresolution analysis involves tensor product of two 1-D multiresolution analyses, where the dilations of the
Fig. 2. Ža. A tree of wavelet splitting; and Žb. its spectral partition.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
Fig. 3. The spectral supports for: Ža. wavelet c Ž t . and its dilations; and Žb. wavelet splitting for dilated wavelet cy1 Ž t .. 87
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Fig. 4. Ža. A 3-level 2-D standard wavelet decomposition. The wavelet splitting representations at level y3 for: Žb. horizontal; Žc. diagonal; and Žd. vertical directions.
resulting orthogonal wavelet basis control both variables simultaneously since Vj2 s Vj m Vj s Ž Vjy1 [ Wjy1 . m Ž Vjy1 [ Wjy1 . s Vjy1 m Vjy1 [ Ž Vjy1 m Wjy1 . [ Ž Wjy1 m Vjy1 . [ Ž Wjy1 m Wjy1 . 2 2 s Vjy1 [ Wjy1
Ž 20 .
where the superscript ‘2’ denotes two dimensions w9x. A 2-D subspace at resolution 2 j is the sum of subspace at resolution 2 jy1 and its complement. The 2 2-D complement subspace Wjy1 consists of three operands: Vjy1 m Wjy1 , Wjy1 m Vjy1 , and Wjy1 m Wjy1. The 2-D scaling function and the three 2-D wavelets are defined as
F Ž x, y. sf Ž x. f Ž y.
Ž 21a.
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Fig. 5.Ža. An original image; Žb. its 1-level wavelet transform; Žc. its 2-level wavelet transform; Žd. its 3-level wavelet transform; Že. the wavelet splitting results along horizontal direction at level 2; and Žf. the wavelet splitting along diagonal direction at level 2.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
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Fig. 5. Žcontinued..
C h Ž x, y. sf Ž x. c Ž y. C v Ž x, y. sc Ž x. f Ž y. d
Ž 21b. Ž 21c. Ž 21d.
F j; n1, n2 Ž x , y . s 2 jf Ž 2 j x y n1 . f Ž 2 j y y n2 . .
Ž 23b.
C Ž x, y. sc Ž x. c Ž y. where superscripts Žh,v,d. denote horizontal, vertical and diagonal directions, and
Cj;ln1, n2 Ž x , y . s 2 jC l Ž 2 j x y n1,2 j y y n2 . ,
Vj2 sspan F j; n1, n2 Ž x , y . ;n1,n2 g Z 4
Just as in Eq. Ž5., the two-level equation associated with the 2-D scaling function is given by
Wj2 s
[ lsh ,Õ , d
Ž 22a.
Wj2 l
l s Ž h,v,d . .
F Ž x, y. s
sspan Cj;ln1, n2 Ž x , y . ;n1,n2 g Z, l s Ž h,v,d . 4
Ž 22b.
Ž 23c.
Ý
h n1, n2F Ž 2 x y n1,2 y y n2 . ,
Ž 24 .
n1, n2
where the coefficients determine a trigonometric polynomial:
where m 0 Ž j ,h . s
Wj2 l sspan Cj;ln1, n2 Ž x , y . ;n1,n2 g Z 4 ,
l s Ž h,v,d . ,
Ý
h n1, n2 exp Ž yi Ž j n1 q h n2 . . .
n1, n2
Ž 23a.
Ž 25 .
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
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Fig. 5. Žcontinued..
There are three two-level equations associated with the three 2-D wavelets, i.e., l
C Ž x, y. s
Ý
l l g n1, n2C
n1, n2
or
Ž 26 .
or
j h j h , Fˆ , , l s Ž h,v,d . . 2 2 2 2
ž /ž /
Ž 27 . Wavelet is split next into wavelet packets at level j by
Cˆ j l1 Ž j ,h . s m 0 Ž j ,h . Cˆ j1 Ž j ,h .
Ž 29a.
Cˆ j l2 Ž j ,h . s m1l Ž j ,h . Cˆ j1 Ž j ,h . .
Ž 29b.
Since Wj2 l s Wj2 l [ Wj2 l, the orthogonal projections onto wavelet packets are given by P W j2 l1 f Ž x , y . s
l1
Cj Ž x , y . s
l l g n1, n2Cj Ž x y n1, y y n2 .
Ž 28b.
n1, n2
Cˆ Ž j ,h . s ml
Ý
Ž 2 x y n1,2 y y n2 . ,
l s Ž h,v,d .
l
Cj l2 Ž x , y . s
Ý
l
h n1, n2Cj Ž x y n1, y y n2 .
Ý
1 ² f Ž u,Õ . ,Cj;ln1, : l1 n2 Ž u,Õ . Cj; n1, n2 Ž x , y .
n1, n2
n1, n2
Ž 28a.
1 s Ý d j;l1n1, n2Cj;ln1, n2 Ž x , y .
n
Ž 30a.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
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Fig. 5. Žcontinued..
P W j2 l 2 f Ž x , y . s
Ý
2 ² f Ž u,Õ . ,Cj;ln1, : l2 n2 Ž u,Õ . Cj; n1, n2 Ž x , y .
n1, n2 2 s Ý d j;l2n1, n2Cj;ln1, n2 Ž x , y .
Ž 30b.
n
where the coefficients are referred to as the wavelet packet coefficients. With wavelet splitting rendered at all levels, a signal can be expressed as a double sequence of wavelet packets: f Ž x , y . : Ž ayJ ; n1, n2 ,d j;l1n1, n2 ,d j;l2n1, n2 , j g yJ ,y J q 1, . . . ,y 1 4 ,n g Z,
l s Ž h,v,d . . .
Ž 31 .
Here again, instead of wavelet splitting at all levels for all the orientations, one can split the wavelet at selected directions Ži.e., horizontal, vertical, or diagonal. at specific levels. For an image, the horizontal, vertical and diagonal edges show up in wavelet packet coefficients d jh , d jv and d dj , respectively. If the original image consists of an N = N array, then every array d jh consists of Nr2 j = Nr2 j elements. Accordingly, d jh can be represented by an image one quarter the size of h d jq1 . The pixel values represent the magnitudes of the coefficients. Fig. 4Ža. shows the corresponding scheme for the case of three multiresolution levels. Also shown are the cases for horizontal, vertical and diagonal wavelet splitting at level y2 respectively in Fig. 4Žb. –Fig. 4Žd.. The lowest frequency compo-
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nent is a lower resolution version of the original scene while the high frequency bands typically pertains to information about the contours, edges, and other finer details. Since most coefficients in high frequency bands are insignificant with respect to a threshold, such subband coding results in dominating zeros for image compression.
4. Results The experimental results obtained for a standard wavelet transform and cases of wavelet splitting is
Fig. 7. Recovered image for a threshold value 20 using: Ža. standard wavelet transform; and Žb. wavelet splitting.
Fig. 6. Standard wavelet transform and wavelet splitting in terms of: Ža. compression score; and Žb. recovery score.
shown in Fig. 5. Fig. 5Ža. shows the original image ‘facets’ while Fig. 5Žb. –Fig. 5Žd. show the respective results of level 1 through 3 standard wavelet decompositions. Fig. 5Že. and Fig. 5Žf., on the other hand, show the respective results obtained using the wavelet splitting schemes along horizontal and diagonal directions, respectively at level 2. Let C wv and C wp denote the coefficient vectors for a standard wavelet decomposition and a wavelet packet decomposition, respectively, and CXwv and CXwp denote the corresponding versions obtained after quantization. Standard wavelet and wavelet packet
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decompositions can be compared using compression and recovery scores as given by Scom s
a C w n x / 04 a C w n x 4
= 100%, C s Ž C wv ,C wp .
Ž 32 . and Srec s
5 CX w n x 5 5Cw nx5
= 100%, C,CX s Ž C wv ,C wp .
Ž 33 .
where a C w n x s 04 represents the number of nonzero coefficients in C w n x, a C w n x4 represents the total number of coefficients in C w n x, and < < P < < represents the vector norm. A high Scom indicates a high compression rate while that for Srec indicates low distortion. Fig. 6 shows the results of compression and recovery scores, for example, for the wavelet transform and wavelet packet transform shown in Fig. 4Žc.. Since their values depend on the thresholds used in quantization, we obtain compression scores and recovery scores for a wide range of thresholds. The results of Fig. 6 show the advantage of using wavelet splitting over standard wavelet decomposition for both image compression and recovery. Wavelet splitting when compared to standard wavelet transform results in higher compression scores as well as recovery scores. At low threshold, in particular, wavelet splitting contributes to higher compression score and almost the same recovery score as the standard wavelet transform. At high threshold wavelet splitting is found to be superior to standard wavelet transform in terms of both compression and recovery scores. For comparison, we consider recovered images for both standard wavelet transform and wavelet splitting. Fig. 7Ža. shows the recovered image corresponding to the standard wavelet transform of Fig. 4Ža. for a threshold of 20. Its compression and recovery scores are 83.50% and 99.80%, respectively. Fig. 7Žb. shows the recovered image corresponding to the wavelet splitting scheme of Fig. 4Žc. for the same threshold. The corresponding compression and recovery scores are 90.16% and 99.73%, respectively.
5. Conclusion Wavelet splitting is realized by applying wavelet filtering to dilated wavelet function at selected frequency resolution. The process amounts to dividing the subspace associated with a wavelet into that for the wavelet packets. Since the spectral support of wavelet packet takes only a slice of the wavelet, it offers better frequency resolution. A better frequency resolution can be achieved by splitting wavelets at selected orientations. In general, wavelet splitting offers improved compression and recovery performances. An adaptive decomposition can be obtained by merging or pruning a tree structure. An arbitrary tree decomposition is achieved by splitting the corresponding wavelets into wavelet packets. The resulting irregular subband decomposition tree is an acceptable compromise between the dyadic wavelet tree and binary full tree. The resulting enhanced time-frequency adaptability leads to optimal solutions in terms of resolution levels and orientation selectivity.
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Optics Communications 173 Ž2000. 81–94 www.elsevier.comrlocateroptcom
Frequency-refined multiresolution decomposition using wavelet splitting Zikuan Chen, Mohammad A. Karim
)
The UniÕersity of Tennessee, Department of Electrical and Computer Engineering, 414 Ferris Hall, KnoxÕille, TN 37996-2100, USA Received 16 August 1999; accepted 26 August 1999
Abstract Frequency resolution of wavelet transform is further refined by splitting wavelets along selected directions into wavelet packets. Since the spectral distributions of wavelet packets lie within that of the wavelets, the narrowness of their spectral support contributes to improved frequency resolution. Experimental results show that such wavelet splitting offers better compression and recovery scores than the standard wavelet transform. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Multiresolution analysis; Wavelet transform; Wavelet packets; Subband coding
1. Introduction The wavelet transform involves joint time– frequency representation with locality and, thus, it is useful in non-stationary signal analysis w1–3x. The wavelet transform manifests as a hyperbolic time– frequency tiling, for example, in a spectrogram representation w2x. In particular, high frequency analysis is adapted to narrow time-width while low frequency analysis uses wider time-width. The adaptive property of a standard wavelet transform is fixed, i.e.,
) Corresponding author. Fax: q1-423-974-5483; e-mail: [email protected]
given a scaling function it accommodates all applications in identical manner. In other words, a standard wavelet transform, except for its inherent time– frequency adaptability, is unable to adapt itself to specific application. An irregular tree decomposition is expected to yield better performance than a standard wavelet tree decomposition w4–7x. Accordingly, an embedded zero-tree based on dyadic wavelet decomposition has already been proposed to prune a wavelet tree w4x. In this paper, we explore identifying an arbitrary subband decomposition tree by splitting the wavelets into wavelet packets, offering improved adaptability to specific applications. A wavelet transform can be described by its wavelet orthonormal basis generated from a mother wavelet by dilations and translations w1,3,8x. The
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 4 7 9 - 4
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Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
orthonormal wavelet basis typically provides a framework for studying images simultaneously at different levels. Using a multiresolution analysis, the successive ‘fine-to-coarse’ approximations can be represented by a nested sequence of subspaces while the information differences between two resolution levels are described by a sequence of complement subspaces w9x. Multiresolution analysis can be considered as a recursive filtering using a pair of wavelet filters and, eventually, by the scaling function as determined by two-level equations w1,3,8x. While the sequence of nested subspaces associated with a multiresolution analysis is described by dilations and translations of a scaling function, the sequence of complement subspaces is described by dilations and translations of a mother wavelet w8,9x. A wavelet function can be considered as a highpass signal. It has bandpass characteristic since frequency localization requires that a high-pass signal decays rapidly with increasing frequency. The narrower the bandpass bandwidth, the better the frequency resolution. The spectral support of a dyadic wavelet is a dyadic interval whose size changes as 2 j where j is the resolution level. However, the exponential behavior of dyadic support is fixed. For a higher-resolution wavelet, one makes use of the next higher resolution level. Alternatively, in this paper, we refine the wavelet frequency resolution at a level without going to the next level. We demonstrate the advantages of wavelet splitting over a standard wavelet transform in terms of compression and recovery scores. Since wavelet splitting uses the same wavelet filters as in the multiresolution decomposition process, its implementation involves applying recursively wavelet filtering to the coefficients of wavelet transform. Wavelet splitting is rendered on selected direction at specific levels and, therefore, can be used for adaptive multiresolution image decomposition. This paper is organized as follows. We derive wavelet filters from the set of two-level equations associated with a multiresolution analysis and apply them to splitting the wavelets into wavelet packets in Section 2. In Section 3, we consider wavelet splitting along selected directions at specific levels. The experimental results are presented then in Section 4 and, finally, conclusions are summarized in Section 5.
2. Review A multiresolution analysis is a process of recursively applying wavelet filtering on the approximation Žtrend. at each level. On the other hand, wavelet splitting is realized by applying wavelet filtering on the detail Žfluctuation. of wavelet transform. The wavelet filters are obtained from two-level equations associated with a multiresolution analysis.
2.1. Two-leÕel equations A wavelet basis function cm nŽ t . is given by
cm n Ž t . s 2 m r2c Ž 2 m t y n . , m,n g Z
Ž 1.
where c is a function in a square integrable space L2 Ž R ., the so-called ‘mother wavelet’. It is constructed by dilations and translations of the mother wavelet where the dilation factor is 2. The wavelet basis can be constructed also using a scaling function. A multiresolution analysis of L2 Ž R . associated with the scaling function f Ž t . is defined as a nested sequence of closed subspaces Vj 4j g Z such that w9x
Ž i . PPP Vy2 ; Vy1 ; V0 ; V1 ; V2 PPP
Ž 2a .
Ž ii . f Ž t . g Vj
Ž 2b .
Ž iii .
mf Ž2 t . gV
jy1
F Vj s 04 , D Vj s L2 Ž R . jgZ
Ž iv . f Ž t . g V0
Ž 2c .
jgZ
m f Ž t y n. g V , n g Z 0
Ž 2d .
Ž Õ . V0 sspan f Ž t y n . , n g Z 4
Ž 2e .
where the overbar denotes a closed vector space and the sequence of nested subspaces is indexed by level j. The resolution at level j is 2 j, which is considered as the sampling rate of a sampling grid. The ‘fineto-coarse’ multiresolution analysis suggests a sequence of nested grids that go from the ‘fine grid’ Žhigh resolution. to the ‘coarse grid’ Žlow resolution. with decreasing j. The subspace V0 is referred to as a reference subspace whose basis functions are constructed by translating the scaling function. The sub-
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space Vj at resolution 2 j is spanned by an orthogonal basis given as
where the overhead cap denotes the corresponding Fourier transform, and
Vj sspan f jn Ž t . , n g Z 4
m0 Ž j . s
sspan 2 j r2f Ž 2 j t y n . , n g Z 4
Ž 3.
where f jnŽ t . has a definition analogous to Eq. Ž1.. An array detector can detect a signal with only a finite number of pixels, which correspond to a representation on a subspace at certain level. The original signal is considered as a full-resolution representation in the reference subspace V0 for the purpose of multiresolution analysis. Further, we assume that the resolution of V0 is equal to 1 Žs 2 0 .. Beginning with a full resolution signal, one can find information in reduced data using ‘fine-to-coarse’ processing. A practical ‘fine-to-coarse’ multiresolution analysis is described by a sequence of finite nested subspace Vj 4 , j s yJ, yJ q 1, . . . , 0 where J ) 0 represents the number of levels. The coarsest approximation is obtained at level yJ which corresponds to the multiresolution analysis depth. With successive approximations from level j to level j y 1, the difference information between the successive levels is collected into a complement subspace Wjy1 of Vj . In general, Vj at level j can be split into two orthogonal complement subspaces at the next level j y 1. For a sequence of approximations from level 0 to level yJ, we have
1
h w n x exp Ž yi n j . '2 Ý n
Ž 7.
is a trigonometric polynomial. On the other hand, when W0 is spanned by an orthonormal basis c Ž x y n.,n g Z 4 , we have a different two-level relation since c Ž x . g W0 ; V1:
c Ž x . s Ý g w n x f Ž 2 x y n. n
s Ý ² c Ž x . , f Ž 2 x y n . :f Ž 2 x y n .
Ž 8.
n
or
cˆ Ž j . s m1 Ž jr2 . fˆ Ž jr2 .
Ž 9.
where m1 Ž j . s
1
g w n x exp Ž yi n j . . '2 Ý n
Ž 10 .
Since f g V0 ; V1 , and V1 is a subspace spanned by f Ž2 x y n., n g Z 4 as given in Eq. Ž3., one obtains a two-level relation given by
A multiresolution analysis, thus, leads to two two-level equations which can be interpreted as bank filtering w8x, i.e., m 0 and m1 can be considered as a pair of wavelet filters. The coefficients hw n x4 and g w n x4 of the wavelet filters are obtained from the two two-level equations. A multiresolution analysis, thus, consists of two sequences of subspaces Vj 4 and Wj 4 generated from a single scaling function f Ž t .. It can be interpreted in terms of orthogonal projections as well. Let P Vj and P Wj denote the orthogonal projections on subspace Vj and Wj , respectively w9x:
f Ž x . s Ý hw n x f Ž 2 x y n .
P Vj f Ž t . s Ý ² f Ž u . , f jn Ž u . :f jn Ž t .
0 V0 s VyJ [ Ž [jsyJ Wj . .
Ž 4.
n
n
s Ý ² f Ž x . , f Ž 2 x y n . :f Ž 2 x y n .
Ž 5.
n
where ² P , P: denotes an inner product. In frequency domain, we have
fˆ Ž j . s m 0 Ž jr2 . fˆ Ž jr2 .
Ž 6.
s Ý a j w a x f jn Ž t .
Ž 11a.
n
P W j f Ž t . s Ý ² f Ž u . , c jn Ž u . :c jn Ž t . n
s Ý d j w n x c jn Ž t . . n
Ž 11b.
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The coefficients a j w n x,n g Z 4 and d j w n x,n g Z 4 represent respectively the approximations and details of f Ž t . at resolution 2 j. The algorithm for finding coefficient d j is referred to as the wavelet transform Žor standard wavelet transform herein. and d j is called a wavelet coefficient. A ‘fine-to-coarse’ multiresolution analysis with depth J Ž J ) 0. results in the following orthogonal wavelet representation w9x: f Ž t . : Ž ayJ w n x ,d j w n x ,y J F j F y1, n g Z . .
Ž 12 .
Due to orthogonality of the wavelet functions, it can be interpreted as a decomposition of the original signal into a set of independent constituent signals.
With wavelet filters hw n x4 and g w n x4 , the coefficients a jy14 and d jy1 4 at level j y 1 can be obtained from that at level j using wavelet filters. This leads to the following decomposition algorithm: a jy1 w n x s Ý h w k y 2 n x a j w k x
Ž 13a.
k
bjy1 w n x s Ý g w k y 2 n x a j w k x k k
s Ý Ž y1 . h w 1 y k q 2 n x a j w k x .
Ž 13b.
k
In order to generate the wavelet coefficients, we need to first calculate a 0 w n x4 for a signal representa-
Fig. 1. Standard wavelet transform: Ža. an octave-band tree; and Žb. its spectral partition.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
tion in V0 , then work down to successively coarser levels using Eqs. Ž13a. and Ž13b., with error at each successive level corresponding to the wavelet coefficients. Fig. 1Ža. shows a scheme for multiresolution decomposition obtained by iterating Eqs. Ž13a. and Ž13b. where H and G stand for filtering using hw n x4 and g w n x4 , respectively. It represents a standard wavelet tree and its subband decomposition scheme is given in Fig. 1Žb.. Both H and G filtering operations involve convolution and decimation. 2.2. WaÕelet splitting Consider the frequency bands of a wavelet c Ž t . to approximately equal to wy2 D,y D x jw D,2 D x where D denotes its passband-width. Its dilated version at level j will have the band support of wy2 jq1D,y 2 jD x j w2 jD,2 jq1D x. A multiresolution analysis realizes a binary partition given by
Ž 0,` . s
then f 1 Ž t y 2 k ., f 2 Ž t y 2 k .;k g Z 4 is an orthonormal basis for space E. According to this lemma, wavelet filters can be used to split any space spanned by the orthonormal functions f Ž t y n.;n g Z 4 into two parts: f 1 Ž t y 2 k .;k g Z 4 and f 2 Ž t y 2 k .;k g Z 4 . Since m 0 and m1 , hw n x4 and g w n x4 , exist at different frequency ranges, wavelet-splitting corresponds to dissecting the spectral support of f Ž t . into slices and allotting alternate slices to f 1 and f 2 , which have better frequency localization. If f Ž t . in Eqs. Ž15a. and Ž15b. is substituted by f Ž t ., then Eqs. Ž15a. and Ž15b. reduces to the two-level equations, Eqs. Ž5. and Ž8. respectively, when f 1 s fy1 and f 2 s cy1. The splitting of the wavelet into two wavelet packets can be rendered on a dilated wavelet at specific level:
c j1 Ž t . s Ý h w n x c j Ž t y n .
j
D Ž 2 D ,2
jq1
Dx .
Ž 14 .
jsy`
Ž 16a.
n
c j2 Ž t . s Ý g w n x c j Ž t y n .
`
85
Ž 16b.
n
or, equivalently, j
jq1
Each interval Ž2 D,2 D x represents a dyadic band and all the disjoint bands cover the whole frequency range as shown in Fig. 1Žb. for an ideal bandpass wavelet with D s pr2. As the resolution level decreases Žincreases. from one level to the next, the dyadic interval becomes halved Ždoubled.. Such octave-band decomposition tree is not always desirable since its exponential behavior is too rapid to be optimal in some cases. An irregular tree decomposition, on the other hand, results in improved performance w7x. Based on the octave-band tree, we hereby construct an arbitrary tree by splitting wavelets into wavelet packets. With wavelet filters hw n x4 and g w n x4 given by Eqs. Ž5. and Ž8., Daubechies identifies a lemma for ‘wavelet-splitting’ w4x. Lemma: For any function f Ž t . so that f Ž t y n.,n g Z 4 is an orthonormal basis for space E, define
cˆj1 Ž j . s m 0 Ž j . cˆj Ž j .
Ž 17a.
cˆj2 Ž j . s m1 Ž j . cˆj Ž j . .
Ž 17b.
f 1 Ž t . s Ý hw n x f Ž t y n .
With wavelet splitting rendered at all levels, a signal can be expressed as a double sequence of wavelet packets:
Ž 15a.
n
f 2 Ž t . s Ý g w n x f Ž t y n. n
Ž 15b.
At level j, the construction of wavelet packet appears as a change of orthonormal basis inside Wj providing Wj s Wj1 [ Wj2 . The orthogonal projections on wavelet packets are given by P W j1 f Ž t . s Ý ² f Ž u . , c jn1 Ž u . :c jn1 Ž t . n
s Ý d 1j, n c jn1 Ž t .
Ž 18a.
n
P W j2 f Ž t . s Ý ² f Ž u . , c jn1 Ž u . :c jn2 Ž t . n
s Ý d j,2 n c jn2 Ž t . .
Ž 18b.
n
f Ž t . : Ž ayJ , n ,d 1j, n ,d j,2 n y J F j F y1,n g Z . .
Ž 19 .
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Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
Eq. Ž19. represents a tree structure, however, it is not necessary to split wavelet at every level. Fig. 2Ža. shows an example of wavelet packet decomposition at level y2 where the wavelet coeffi1 w x4 2 w x4 cient dy2 w n x4 is split into dy2 n and dy2 n . Fig. 2Žb. shows the corresponding spectral decomposition. The wavelet packet c 1 Ž t . is the low frequency slice of c Ž t . while c 2 Ž t . is the high frequency slice. The highlighted area shown in Fig. 2Žb. denotes the spectral support of wavelet packets. For clarity, the bandpass characteristics of c Ž t ., cy1Ž t . and cy2 Ž t . are illustrated in Fig. 3Ža.. The spectral support for wavelet splitting associated with
cy1Ž t . is shown in Fig. 3Žb.. It is seen that wavelet packets are more compact than the wavelet. Also, the spectral support for each wavelet packet takes only a sliced part of its wavelet implying that the wavelet packet has a better frequency resolution.
3. Theory One popular way to constructing 2-D multiresolution analysis involves tensor product of two 1-D multiresolution analyses, where the dilations of the
Fig. 2. Ža. A tree of wavelet splitting; and Žb. its spectral partition.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
Fig. 3. The spectral supports for: Ža. wavelet c Ž t . and its dilations; and Žb. wavelet splitting for dilated wavelet cy1 Ž t .. 87
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Fig. 4. Ža. A 3-level 2-D standard wavelet decomposition. The wavelet splitting representations at level y3 for: Žb. horizontal; Žc. diagonal; and Žd. vertical directions.
resulting orthogonal wavelet basis control both variables simultaneously since Vj2 s Vj m Vj s Ž Vjy1 [ Wjy1 . m Ž Vjy1 [ Wjy1 . s Vjy1 m Vjy1 [ Ž Vjy1 m Wjy1 . [ Ž Wjy1 m Vjy1 . [ Ž Wjy1 m Wjy1 . 2 2 s Vjy1 [ Wjy1
Ž 20 .
where the superscript ‘2’ denotes two dimensions w9x. A 2-D subspace at resolution 2 j is the sum of subspace at resolution 2 jy1 and its complement. The 2 2-D complement subspace Wjy1 consists of three operands: Vjy1 m Wjy1 , Wjy1 m Vjy1 , and Wjy1 m Wjy1. The 2-D scaling function and the three 2-D wavelets are defined as
F Ž x, y. sf Ž x. f Ž y.
Ž 21a.
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Fig. 5.Ža. An original image; Žb. its 1-level wavelet transform; Žc. its 2-level wavelet transform; Žd. its 3-level wavelet transform; Že. the wavelet splitting results along horizontal direction at level 2; and Žf. the wavelet splitting along diagonal direction at level 2.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
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Fig. 5. Žcontinued..
C h Ž x, y. sf Ž x. c Ž y. C v Ž x, y. sc Ž x. f Ž y. d
Ž 21b. Ž 21c. Ž 21d.
F j; n1, n2 Ž x , y . s 2 jf Ž 2 j x y n1 . f Ž 2 j y y n2 . .
Ž 23b.
C Ž x, y. sc Ž x. c Ž y. where superscripts Žh,v,d. denote horizontal, vertical and diagonal directions, and
Cj;ln1, n2 Ž x , y . s 2 jC l Ž 2 j x y n1,2 j y y n2 . ,
Vj2 sspan F j; n1, n2 Ž x , y . ;n1,n2 g Z 4
Just as in Eq. Ž5., the two-level equation associated with the 2-D scaling function is given by
Wj2 s
[ lsh ,Õ , d
Ž 22a.
Wj2 l
l s Ž h,v,d . .
F Ž x, y. s
sspan Cj;ln1, n2 Ž x , y . ;n1,n2 g Z, l s Ž h,v,d . 4
Ž 22b.
Ž 23c.
Ý
h n1, n2F Ž 2 x y n1,2 y y n2 . ,
Ž 24 .
n1, n2
where the coefficients determine a trigonometric polynomial:
where m 0 Ž j ,h . s
Wj2 l sspan Cj;ln1, n2 Ž x , y . ;n1,n2 g Z 4 ,
l s Ž h,v,d . ,
Ý
h n1, n2 exp Ž yi Ž j n1 q h n2 . . .
n1, n2
Ž 23a.
Ž 25 .
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Fig. 5. Žcontinued..
There are three two-level equations associated with the three 2-D wavelets, i.e., l
C Ž x, y. s
Ý
l l g n1, n2C
n1, n2
or
Ž 26 .
or
j h j h , Fˆ , , l s Ž h,v,d . . 2 2 2 2
ž /ž /
Ž 27 . Wavelet is split next into wavelet packets at level j by
Cˆ j l1 Ž j ,h . s m 0 Ž j ,h . Cˆ j1 Ž j ,h .
Ž 29a.
Cˆ j l2 Ž j ,h . s m1l Ž j ,h . Cˆ j1 Ž j ,h . .
Ž 29b.
Since Wj2 l s Wj2 l [ Wj2 l, the orthogonal projections onto wavelet packets are given by P W j2 l1 f Ž x , y . s
l1
Cj Ž x , y . s
l l g n1, n2Cj Ž x y n1, y y n2 .
Ž 28b.
n1, n2
Cˆ Ž j ,h . s ml
Ý
Ž 2 x y n1,2 y y n2 . ,
l s Ž h,v,d .
l
Cj l2 Ž x , y . s
Ý
l
h n1, n2Cj Ž x y n1, y y n2 .
Ý
1 ² f Ž u,Õ . ,Cj;ln1, : l1 n2 Ž u,Õ . Cj; n1, n2 Ž x , y .
n1, n2
n1, n2
Ž 28a.
1 s Ý d j;l1n1, n2Cj;ln1, n2 Ž x , y .
n
Ž 30a.
Z. Chen, M.A. Karimr Optics Communications 173 (2000) 81–94
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Fig. 5. Žcontinued..
P W j2 l 2 f Ž x , y . s
Ý
2 ² f Ž u,Õ . ,Cj;ln1, : l2 n2 Ž u,Õ . Cj; n1, n2 Ž x , y .
n1, n2 2 s Ý d j;l2n1, n2Cj;ln1, n2 Ž x , y .
Ž 30b.
n
where the coefficients are referred to as the wavelet packet coefficients. With wavelet splitting rendered at all levels, a signal can be expressed as a double sequence of wavelet packets: f Ž x , y . : Ž ayJ ; n1, n2 ,d j;l1n1, n2 ,d j;l2n1, n2 , j g yJ ,y J q 1, . . . ,y 1 4 ,n g Z,
l s Ž h,v,d . . .
Ž 31 .
Here again, instead of wavelet splitting at all levels for all the orientations, one can split the wavelet at selected directions Ži.e., horizontal, vertical, or diagonal. at specific levels. For an image, the horizontal, vertical and diagonal edges show up in wavelet packet coefficients d jh , d jv and d dj , respectively. If the original image consists of an N = N array, then every array d jh consists of Nr2 j = Nr2 j elements. Accordingly, d jh can be represented by an image one quarter the size of h d jq1 . The pixel values represent the magnitudes of the coefficients. Fig. 4Ža. shows the corresponding scheme for the case of three multiresolution levels. Also shown are the cases for horizontal, vertical and diagonal wavelet splitting at level y2 respectively in Fig. 4Žb. –Fig. 4Žd.. The lowest frequency compo-
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nent is a lower resolution version of the original scene while the high frequency bands typically pertains to information about the contours, edges, and other finer details. Since most coefficients in high frequency bands are insignificant with respect to a threshold, such subband coding results in dominating zeros for image compression.
4. Results The experimental results obtained for a standard wavelet transform and cases of wavelet splitting is
Fig. 7. Recovered image for a threshold value 20 using: Ža. standard wavelet transform; and Žb. wavelet splitting.
Fig. 6. Standard wavelet transform and wavelet splitting in terms of: Ža. compression score; and Žb. recovery score.
shown in Fig. 5. Fig. 5Ža. shows the original image ‘facets’ while Fig. 5Žb. –Fig. 5Žd. show the respective results of level 1 through 3 standard wavelet decompositions. Fig. 5Že. and Fig. 5Žf., on the other hand, show the respective results obtained using the wavelet splitting schemes along horizontal and diagonal directions, respectively at level 2. Let C wv and C wp denote the coefficient vectors for a standard wavelet decomposition and a wavelet packet decomposition, respectively, and CXwv and CXwp denote the corresponding versions obtained after quantization. Standard wavelet and wavelet packet
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decompositions can be compared using compression and recovery scores as given by Scom s
a C w n x / 04 a C w n x 4
= 100%, C s Ž C wv ,C wp .
Ž 32 . and Srec s
5 CX w n x 5 5Cw nx5
= 100%, C,CX s Ž C wv ,C wp .
Ž 33 .
where a C w n x s 04 represents the number of nonzero coefficients in C w n x, a C w n x4 represents the total number of coefficients in C w n x, and < < P < < represents the vector norm. A high Scom indicates a high compression rate while that for Srec indicates low distortion. Fig. 6 shows the results of compression and recovery scores, for example, for the wavelet transform and wavelet packet transform shown in Fig. 4Žc.. Since their values depend on the thresholds used in quantization, we obtain compression scores and recovery scores for a wide range of thresholds. The results of Fig. 6 show the advantage of using wavelet splitting over standard wavelet decomposition for both image compression and recovery. Wavelet splitting when compared to standard wavelet transform results in higher compression scores as well as recovery scores. At low threshold, in particular, wavelet splitting contributes to higher compression score and almost the same recovery score as the standard wavelet transform. At high threshold wavelet splitting is found to be superior to standard wavelet transform in terms of both compression and recovery scores. For comparison, we consider recovered images for both standard wavelet transform and wavelet splitting. Fig. 7Ža. shows the recovered image corresponding to the standard wavelet transform of Fig. 4Ža. for a threshold of 20. Its compression and recovery scores are 83.50% and 99.80%, respectively. Fig. 7Žb. shows the recovered image corresponding to the wavelet splitting scheme of Fig. 4Žc. for the same threshold. The corresponding compression and recovery scores are 90.16% and 99.73%, respectively.
5. Conclusion Wavelet splitting is realized by applying wavelet filtering to dilated wavelet function at selected frequency resolution. The process amounts to dividing the subspace associated with a wavelet into that for the wavelet packets. Since the spectral support of wavelet packet takes only a slice of the wavelet, it offers better frequency resolution. A better frequency resolution can be achieved by splitting wavelets at selected orientations. In general, wavelet splitting offers improved compression and recovery performances. An adaptive decomposition can be obtained by merging or pruning a tree structure. An arbitrary tree decomposition is achieved by splitting the corresponding wavelets into wavelet packets. The resulting irregular subband decomposition tree is an acceptable compromise between the dyadic wavelet tree and binary full tree. The resulting enhanced time-frequency adaptability leads to optimal solutions in terms of resolution levels and orientation selectivity.
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