Subsampling-Adaptive Directional Wavelet Transform for ImageCoding

Jizheng Xu and Feng WuMicrosfot Research Asia

No. 49 Zhichun road, Haidian district, Beijing 100190, P. R. ChinaEmail:{jzxu,fengwu}@microsoft.com

Abstract

In lifting-based directional wavelet transforms, different subsampling patterns mayshow significant difference for directional signals in image coding. This paper in-vestigates the influence of subsampling in directional wavelet transform. We showthat the best subsampling depends on the direction and the directionality strength ofthe signal. To improve the coding performance, we further propose a subsampling-adaptive directional wavelet transform, which can use different subsampling patternsadaptively and according to the local characteristics of the image. To handle theboundary transition when subsampling changes, a phase completion process is appliedto ensure that wavelet transform with various subsampling can be performed withoutintroducing boundary effects and performance loss. Experimental results show that theproposed transform can achieve significant coding gain in image coding compared toother existing directional wavelet transforms.

1. Introduction

Natural objects and scenes are often inhomogeneous, which leads to directionality of images.For a certain part of an image, usually the correlation is obviously stronger along one directionthan others. However, conventional wavelet transforms may not able to exploit such directionality,despite that they are successfully used in image coding. In conventional wavelet transforms, a2D transform is consisted of two 1D wavelet transforms. One is horizontal and the other isvertical. For other directions that are neither horizontal nor vertical, the representation may notbe efficient.

Recently, lifting-based directional wavelet transforms [2]–[4] have been proposed to bringdirectionality into the transforms. In [1], it has been shown that a wavelet transform can befactorized into several lifting steps. The basic idea to construct a directional wavelet transformis to perform all the lifting steps of that transform along a certain direction. Such a designpreserves the advantages of wavelet transforms, e.g., multiresolution analysis, critical sampling.The coefficient structure of a directional transform is also the same as that of a conventional oneso that it can be easily integrated into the existing wavelet-based image coding systems. Exper-imental results have been shown that directional wavelet transforms can perform significantlybetter than conventional ones in image coding, both in PSNR and in visual quality, especiallyfor those images which contain much anisotropic content, e.g., sharp edges and strong textures.

2010 Data Compression Conference

1068-0314/10 $26.00 © 2010 IEEE

DOI 10.1109/DCC.2010.15

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Both in conventional and directional wavelet transforms, a subsampling process is appliedto identify the positions of low-pass coefficients after the transform. Basically the subsamplingpattern decides how the low resolution image, generated by the wavelet transform, is placedrelated to the original image. It also influences the energy distribution between low-pass andhigh-pass coefficients. Thus, it also decides the coding efficiency of the wavelet transform. Aswhat we will present in the following sections, we have observed that in directional wavelettransforms, different subsampling patterns may lead to significant differences in term of thecoding performance.

In this paper, besides identifying the subsampling problem in directional wavelets, we alsoanalyze the influence of different subsampling patterns on the performance of directional wavelettransforms. We show that different subsampling patterns favor different directional content. It alsoturns out that the relation between the direction of the content and the most suitable subsamplingpattern is complicated. The most suitable subsampling depends on the direction and the strengthof the directionality, of the signal. Since an image may contain different directional information,to more efficiently represent the signals according to local characteristics, we argue that fordifferent parts of the image, the corresponding optimal subsampling patterns should be used.However, such a simple idea may not be easy to implement, because the boundary between eachtwo subsampling patterns may break the transform structure. Another contribution of this paperis that we present a subsampling-adaptive directional wavelet transform that can easily supportdifferent subsampling patterns for different parts of an image.

The rest of this paper is organized as follows. Section 2 briefly introduces directional wavelettransform with different subsampling patterns. In Section 3, the coding gain of those directionalwavelet transforms are analyzed and the influence of subsampling is discussed. In Section4, we present the design of the proposed subsampling-adaptive directional wavelet transform.Experimental results are shown in Section 5 to verify the effectiveness of the proposed transform.We conclude the paper in Section 7.

2. Directional wavelet transforms with different subsampling patterns

Usually to perform a lifting-based wavelet transform on an image I , a subsampling processis used to divide the pixels into two separate subsets, IL and IH , where IL

⋃IH = I and

IL

⋂IH = ∅. In a typical lifting-based wavelet transform on I , the samples of IL are used to

predict those of IH to generate the high-pass coefficients H . Then H will be updated to IL toget the low-pass coefficients L. As shown in figure 1, there are

H = IH − P (IL), (1)

L = IL − U(H), (2)

where P () and U() are prediction and update operators. In more complicated wavelet transforms,more lifting steps may be involved between those two subsets [1]. Similarly, the other wavelettransform is applied on L and H respectively to complete a whole 2-D wavelet transform. Thelow-pass and high-pass coefficients of L are LL and LH . For H , the corresponding outputs areHL and HH .

In a conventional 2D wavelet transform, the first transform is performed on each column ofpixels and then the second one is on each row. For the first transform, IL contains all even-number rows and IH contains all odd-number rows. Thus, in figure 2(a), all black and deep gray

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circles denote IL, others are of IH . Since the transform is performed among rows, we call itrow transform. After the row transform, the even-number rows become low-pass coefficients andthe odd-number rows are the high-pass coefficients. Then the second transform, called columntransform, is perform on each row, i.e., among different columns. The column transform furtherdivides the signals into four different subbands, LL, LH , HL and HH , as shown in figure 2(a).For a corresponding directional wavelet transform, it has the same subband arrangement. Thedifference is that in a conventional transform, the lifting steps are vertical for the row transformand horizontal for the column transform; while in the corresponding directional transform, thelifting steps can be along other predefined directions. For example, for the row transform, in [3],nine directions shown in figure 3(a) are used. To predict a signal in IH , we follow one directionand find correspondences in the previous and next rows and use their average as the result of P().The update step is done in the similar way. The conventional row transform equals to alwaysperforming lifting steps along d4. When following a given direction, e.g., d1, the correspondencesare not in integer positions, an interpolation process will be applied to estimate the value of thepixels at fractional positions. It should be noted that the interpolation can only involve the pixelsin IH , because according to (1), all the information used in P () should be from IH . The casefor the update step is similar.

In the transform mentioned above, the subsampling process divides rows into different subsetsfor the first transform and then divides columns for the second transform. We call such asubsample pattern ’row-column’. An alternative, called ’column-row’, is to divide columns forthe first transform and then rows for the second transform. In such a case, the column transformwill be performed before the row transform. The subbands will be arranged in a different way, asshown in figure 2(b). For the conventional wavelet transform, however, both subsampling patternsgenerate the same results, since the row and column transforms are commutative. However, fora directional wavelet transform, it is not the case. The transform results can be much different.So are the image coding results. Actually to perform wavelet transform in the column-rowsubsampling is identical to rotate the image by 90◦ and then perform the transform in the row-transform subsampling. It is reported in [2] that the coding performance of directional transformmay change much when the image is rotated by 90◦, which indicates that the row-column andcolumn-row subsampling can yield different coding performances.

Another subsampling described in the literature of directional wavelet transforms is quincunx[2]. In the quincunx subsampling, IL contains all signals whose sum of the row and columnnumbers is even. The others are of IH . The output subbands are shown in figure 2(c).

Figure 3 shows different direction setting for different subsampling patterns. Among them, thedirection setting of the first transform in the row-column sampling is from [4]. For the column-row subsampling, the directions are simply got by rotating those in figure 3(b). The setting offigure 3(c) is from [2]. For the direction settings of the second transform and more other details,readers are referred to [4] and [2]. Although some results show that the quincunx subsamplingmay not be suitable for natural image coding [7], a superior performance using the quincunxsubsampling than those of the row-column and column-row subsampling patterns is shown indirectional wavelet coding [2], which indicates that in some cases the quincunx subsamplingmay be effective. In the following sections, we will analyze the factors that may influence theperformances of different subsampling patterns and show the conditions when some subsamplingis better than others.

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Subsample P()

+

-

U()

+

+

IL

IH

I

Low-pass

High-pass

Figure 1. The diagram of a lifting-based wavelet transform.

(a) Row-column (b) Column-row (c) Quincunx

LL LH HHHL

Figure 2. Four subbands of the 2D directional wavelet transform with different subsam-pling patterns.

d0d1d2d3d4d5d6d7d8

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d1

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d1

d2

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L H

Figure 3. Direction settings for different subsampling patterns.

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3. Coding gain analysis for different subsampling patterns

We assume that the signals are static, which means that the signal distribution is not related tothe position of that signal. In such a case, for a 2D conventional or directional wavelet transform,the distribution within an output subband is also static. Let σ2

LL, σ2LH , σ2

HL and σ2HH denote the

signal variances in the LL, LH , HL and HH subbands respectively. We further assume thatthe input signals have a unit variance without loss of generality. Then the effectiveness of anormalized transform can be measured by the coding gain [8] as

G =1

(σ2LL · σ2

LH · σ2HL · σ2

HH)14

. (3)

However, even for a normalized wavelet transform, its corresponding directional transform maynot be normalized. The reason is because the directional wavelet transform may involve aninterpolation process, where the energy or l2 norm may be changed. Thus, to make the calculationof the coding gain normalized, we calculate the basis vectors bLL, bLH , bHL and bHH for thefour subbands. The basis vector can be got by performing the inverse transform on a discretedelta function with the non-zero signal placed at the corresponding subband. Then the codinggain is

G =

(‖bLL‖22 · ‖bLH‖2

2 · ‖bHL‖22 · ‖bHH‖2

2

σ2LL · σ2

LH · σ2HL · σ2

HH

) 14

. (4)

Since a directional wavelet transform is still a linear system, it can be denoted by convolutionfilters. For a 2D wavelet transform, we calculate the convolution filters vectors fLL(p), fLH(p),fHL(p) and fHH(p), where p ∈ Z2, corresponding to the four subbands. The convolution filtercan be calculated by performing the 2D transform on a discrete delta function with the non-zerosignal placed at different places and getting the output at a fixed position of the correspondingsubband. In practical calculation, the non-zero signal can be varied within a window since thefilter of the directional wavelet transform has a limited support. As long as the window coversthe support, the calculation result is the same.

For a directional signal s(q), where q ∈ Z2, we use the anisotropic correlation model in[4], where it assumes that the correlation between two signals is determined by their relativepositions. The correlation is static, defined as

Rωa,ωb,θ(τ) = σ2e−√

τT Γ(ωa,ωb,θ)τ , (5)

where τ = [δx, δy]T ∈ Z2 is the displacement between two correlated signals. In the aboveequation,

Γ(ωa, ωb, θ) =

(cos θ sin θ− sin θ cos θ

)(ω2

a 00 ω2

b

)(cos θ sin θ− sin θ cos θ

)T

, (6)

where ω2a and ω2

b decide the correlations along and perpendicular to direction θ.Then given a correlation function Rωa,ωb,θ(τ), the output variance of the LL subband can be

calculated as

σ2LL = ‖E(fLL · s)‖2

2

= E2[Σp∈Z2fLL(p) · s(p)]

= Σp,q∈Z2fLL(p) · fLL(q) · E[s(p) · s(q)]

= Σp,q∈Z2fLL(p) · fLL(q) · Rωa,ωb,θ(p − q). (7)

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Figure 4. Coding gain comparison with different correlation models.

In practical calculation, a window can be used to limit the computation due to the limit supportof the convolution filter. σ2

LH , σ2HL and σ2

HH can also be calculated in the same way. Thus, withthe basis vectors of four different subbands, the coding gain can be got by (4).

To simplify the description, in the following, we will use DWRC transform, DWCR trans-form and DWQC transform to denote the directional wavelet transforms using the row-column,column-row and quincunx subsamplings, respectively. As shown in figure 3, a directional wavelettransform can select many directions. For the first transform, there are up to nine directions. Thatis also the case for the second transform. Thus, for a 2D directional wavelet transform, there areup to 81 direction settings when combing all the directions of the first and second transforms.Given a correlation model Rωa,ωb,θ(τ), to find the best coding gain a 2D directional wavelettransform can achieve, we get the coding gains for each direction setting and determine themaximum coding gain among all direction settings.

Figure 4 compares the coding gains of different 2D wavelet transforms with different Rωa,ωb,θ(τ)models. We list the curves of the coding gains for five sets of (ωa, ωb). For each set of (ωa, ωb), wevary θ from 0◦ to 90◦ and get the coding gains for the DWRC, DWCR, DWQC and conventionalwavelet transforms. Due to the symmetry of these wavelet transforms, all the curves are periodicwith the cycle of 90◦. More different ωa and ωb are, more anisotropic the signal is. Whenωa = ωb, the signal is isotropic, which is the case of figure 4(a). In figure 4(a), all the curves arehorizontal lines because varying θ does not change the correlation function. The coding gainsof the DWRC, DWCR and conventional wavelet transforms are actually the same. However, thecoding gain of the DWQC transform is lower than others. It means that for isotropic signals, thequincunx subsampling is not so effective as other subsampling patterns. In figure 4(b), the signalbecomes a little anisotropic. Then it can be seen that the coding gain changes along with the

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angles. In most cases, all directional wavelet transforms are better than the conventional, whilenear 0◦ or 90◦, the DWQC transform is still worse than the conventional. The conventionaltransform performs equally to the DWRC and DWCR transforms when θ is 0◦ or 90◦. For otherangles, the conventional becomes worse, especially when θ = 45◦. The other observation is thatfrom 0◦ to 45◦, the DWRC transform is better than the DWCR transform. While from 45◦ to90◦, the DWCR transform is better. From figure 4(a) to figure 4(e), the signal becomes more andmore anisotropic. The coding gain difference between the DWRC and DWCR transforms alsobecomes larger. But it still follows the observation in figure 4(b), i.e., the DWRC is better from0◦ to 45◦ and the DWCR is better from 45◦ to 90◦. Both the DWRC and DWCR transforms havepeaks at 0◦, 45◦ and 90◦. For the DWQC transform, the situation is more complicated. When thesignal becomes isotropic, the DWQC starts to perform better than the conventional. Althoughat 45◦, its performance is still much worse than those of the DWRC and DWCR transforms,it can outperform than the others at some other angles, especially when the signal is stronglyisotropic. It makes sense because such a phenomenon can be explained when considering thedirection settings of different transforms. For example, in figure 3(c), when the signal has strongercorrelation along d1, i.e., d6 in figure 3(b), we compare the correlation between the center pixeland the pixels which d1 points at in figure 3(c) and the correlation between the center pixel andthe pixels which d6 points at in figure 3(b), which is interpolated by the upper and lower pixels.When the correlation is much stronger along that direction than along others, the content is muchsharp. However, since the interpolation is basically a low-pass filtering, it may smooth the signaland lead to low correlation in figure 3(b); while in figure 3(c) the transform does not suffer fromthe interpolation. On the other hand, when the correlation along that direction is just a littlestronger than along others, the correlation in 3(b) may be higher than that in 3(c), since in 3(b)the prediction distance is much shorter. Thus, the performance of the DWQC transform relatedto those of other transforms depends not only on the angle of the signal’s direction, but it alsodepends on the strength of the anisotropy of the signal. That explains with some pictures, theDWQC transform may get superior performance than others [2], while in some other literatures,it is reported that the quincunx subsampling may not suitable for natural image coding.

4. Subsampling-adaptive directional wavelet transform

In the previous section, we mentioned that for a directional signal, some subsampling patterncan be better than others. It depends on both the direction of the signal and the strength ofthe anisotropy. In an image, all kinds of content may exist. For some content, the edges canbe very sharp, which are much directional and anisotropic. For some other content, it mayshow directionality but the contrast is not so much. There is still some content, which is muchhomogenous. Thus, using only one subsampling is not enough and apparently not effectivefor coding different parts of an image. A simple idea is to apply different subsamplings ondifferent regions of the image. We can divide the image into blocks and for each block, weselect one subsampling. Such a block-based partition is widely used in image/video coding dueto its simplicity. However, such a simple idea may not be easy to implement. An example isshown in figure 5, where the left 4x4 block uses the row-column subsampling and the rightone uses the quincunx subsampling. Different subsamplings lead to different subset partition inthe wavelet transform. As also shown in the figure, when we apply the first transform along d0

defined in figure 3(c) on pixel v, which is of subset IH , we will use pixel w, which belongs toIH too. However, such a situation is not allowed in lifting-based wavelet transforms. When the

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LL

LH

HH

HL

vw

IL

IH

Figure 5. An illustration of different subsamplings for different blocks.

interpolation is involved in the transform, more pixels of IH are to be used, which makes theproblem more complicated. We can also make the transform for each block independent, i.e.,the lifting steps should not cross the block boundary. However, this will keep the transform fromexploiting inter-block correlation, which may cause significant performance loss and blockingartifacts, especially when the block size is small.

Actually, such a problem is because the subset partition is not uniform between neighboringblocks. In the first transform, the image is divided into IL and IH , or two different phases.When we use different subsamplings in different blocks, the positions of those two phases arenot regular any more. Thus, a lifting step which is on one phase may need to use other pixelsof the same phase. To resolve this problem, we propose to use a phase-completion process toestimate one phase from the other so that we can guarantee that during a lifting step, only onephase’s information is used. For example, in the prediction step in figure 5, besides IL, pixelw ∈ IH is also used. However, if we can derive the value of w using pixels of IL, we can ensurethat the prediction can be got from information only from IL. Thus, such a subsampling-adaptivedirectional wavelet transform can be represented by

H = IH − P (C(IL)), (8)

L = IL − U(C(H)), (9)

where C() is the phase-completion process. In all the three subsampling patterns shown in theprevious sections, IL and IH are interleaved. Since usually the local correlation within an imageis strong, estimation of IH from IL is feasible. In this paper, we use a simple method, whichuses the average of all the pixels of IL within a window to get the estimation. For example, infigure 5, a 3x3 window centering at w is used to get the estimation of w.

When finishing the first transform, the second transform is performed on different blocks. Inthe second transform, information of the pixels at positions of LL is used to predict those at thepositions of LH . As we can see from figure 2 and figure 5, the positions of LL are uniformlyplaced in all the three subsmapling patterns. Thus, the prediction step in the second transformdoes not suffer from the problem in the first transform and can be done as usual. As for theupdate step, we can use the energy distributed update [9] according to the prediction step, whichcan also be done without considering the influence of different subsamplings.

By introducing a phase-completion process in the directional wavelet transform, we can mix updifferent subsamplings and fully exploit each subsampling’s strongpoint to improve the efficiencyof the transform.

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5. Experimental results

We have conducted extensive experiments to verify the effectiveness of the proposed subsampling-adaptive wavelet transform in image coding. We integrate the proposed into JPEG2000 vm9.0software. Similar to [3], we use blocks with variable sizes from 128x128 to 4x4 to partition theimage. For each block, one of the three subsampling patterns is selected and then one directionis chosen. The criterion to select subsamplings and directions is also similar to that in [3]. Wecompare the proposed transform and directional transforms with only one subsampling enabledin terms of coding performance in figure 6. The performance gain of the proposed subsampling-adaptive transform and the DWRC, DWCR, DWQC and conventional wavelet transforms areshown by the curves labeled by ’SA-RC’, ’SA-CR’, ’SA-QC’ and ’SA-J2K’ in each sub figures atvariant bit-rates. The corresponding gains of the proposed transform over the DWRC, DWCR,DWQC and conventional wavelet transforms can be up to 4.6dB, 1.2dB, 1.2dB and 1.5dB,respectively. In most cases, the proposed transform outperforms others. Only at some points,there is little performance loss, i.e., the y-value is smaller than 0 in the curves. That may bebecause of the overhead bits to represent the subsampling patterns for each block.

6. Conclusions

This paper analyzes the influence of subsampling on directional wavelet transforms. It isshown that different subsampling may be suitable for signal with different directional properties.However, the theoretical analysis indicates that the relation between subsampling and the codinggain of the corresponding directional transform is complicated. It depends on the direction and thestrength of the signal anisotropy. Based on the above observations, we propose to use differentsubsamplings for different regions in image coding. To handle the non-uniform subsamplingamong neighboring blocks, we propose the subsampling-adaptive directional wavelet transformusing a phase completion process. Actually the idea not only enables us to mix up differentsubsamplings, but it may also allow us to use different transforms for different parts, whichis left for further work. Extensive experimental results verify the effectiveness of the proposedtransform.

References

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Figure 6. Coding performance comparison between the proposed transform and thetransforms using different subsampling patterns.

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