Proceedings of SPIE, Vol.3164, San Diego, CA, USA, 30 July { 1 August 1997, pp.279{290
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Predictive Image Coding With Adaptive Wavelet Transform Tilo Strutz, Heiko Schwarz and Erika Muller University of Rostock Institute of Communications and Information Electronics Richard-Wagner-Str.31, 18119 Rostock, FRG
ABSTRACT
This paper presents a new method for an ecient coding of image data. Based on the wavelet transform, the algorithm utilizes the remaining correlation between subbands. A successive approximation of wavelet coecients yields a hierarchical symbol stream, which is highly compressed with a prediction of signicant descendants (PSD). The coding algorithm distinguishes itself by a high adaptivity to the image content. The resulting bitstream contains all image information in order of its signicance. Therefore it is possible to truncate the bitstream at any point, without endangering the decoding process. The advantages of such an embedded bitstream are spatial and rate-distortion scalability. Further improvement is obtained using a new adaptive wavelet transform known as wavelet packets. Contrary to earlier techniques, relevant statistical properties (mainly the correlation) of the current subband are rst analyzed. Dependent on that, the decomposition decision is made, whether the subband should be decomposed or not. This procedure yields not to a best-basis selection but to a near-optimal decomposition structure. The main advantage is the reduction of computational cost. Keywords: image coding, data compression, hierarchical prediction, wavelet transform, wavelet packets
1 INTRODUCTION
Meanwhile, wavelet-based compression techniques are well-established in the area of still image coding. In many publications in recent years, it was shown that wavelet codecs achieve higher compression rates than the standard technique JPEG. Since the last qualitative step in wavelet-based image coding was taken through the idea of zerotree coding by Shapiro,7 there are only quantitative changes to note. The important new step was the combination of a successive approximation of wavelet coecients with a coding algorithm, that allows a bit transfer in an embedded fashion. This enables a progressive transmission of the image content. Since then, many contributions have concerned with modications of the zerotree coding,2,4 and some techniques have been presented, which are similar to the zerotrees, and also have a high compression performance.6,10,11 Furthermore, there are some other approaches with very good results, which, however, do not have the possibility to produce an embedded bitstream.3,9,12 Comparing the results of all these techniques, it seemed that a saturation point was reached in the increasing of the coding performance. Only, if the restriction to dyadic wavelet transform is lifted and arbitrary decomposition structures are used, a further improvement of wavelet codecs can be noticed. For some images, a signicant rise in objective image quality can be achieved, if wavelet packets are applied. Using wavelet packets, no bounds to dyadic structures exist, but the decompositions are dependent on the image content. In adaptive wavelet packets schemes, rst a full decomposition is applied. Beginning at the deepest level of the decomposition tree, the branches are pruned, where the parent branch gives a lower cost representation than the children.4,8,13 This yields to optimal solutions in the rate-distortion sense, but the procedure is very expensive due to the high computation complexity. To lower the computational costs, other adaptive approaches have been developed, which reduce the full wavelet packet library, for instance, using a complexity constraint energy criterion.5 In this contribution a new wavelet-based coding scheme is introduced, which uses the advantage of progressive transmission of image information via a successive renement of wavelet coecients. The coecients are quantized in a layered manner. In every layer, data symbols are assigned to the coecients and these symbols are grouped to quarternary trees (Fig. 2a). The compaction is performed using a prediction of signicant descendants (PSD) instead of zerotrees. To increase the compression performance, some modications are presented. On the one hand they Email:
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Proceedings of SPIE, Vol.3164, San Diego, CA, USA, 30 July { 1 August 1997, pp.279{290
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improve the adaptation of the arithmetic coder used, and on the other hand they perform a sorting of PSD-trees and of data symbols corresponding to their signicance. The resulting bitstream enables a free scalability of the codec in image quality, image resolution, and bitrate. Therefore, the new algorithm is also suitable for video coding. Based on this coding strategy, a new adaptive wavelet transform is presented. In contrast with the known techniques, rst relevant statistical properties (mainly the correlation) of the current subband are analyzed. Dependent on that, the decomposition decision is made, whether the subband should be decomposed or not. This procedure yields not to a best-basis selection but to a near-optimal decomposition structure. The main advantage is the reduction of computational cost. The following Section enter into the idea of successive approximation. Section 3 is concerned with the prediction of signicant descendants (PSD). After that, the coding of the PSD-trees is described in Section 4. Several optimization strategies are discussed in Section 5. In Section 6, the new approach to adaptive wavelet transform is explained. The performances of the PSD- and the zerotree coding are compared in Section 7. The last Section includes the presentation of compression results and a conclusion.
2 QUANTIZATION OF WAVELET COEFFICIENTS USING SUCCESSIVE APPROXIMATION 2.1. Wavelet transform
The wavelet transform is performed in such a way that for an optimal coding a low-pass band with a minimal size of 9x9 pixel should remain. The number of decomposition levels is calculated by z
z
= log2
size
9
(1)
whereby the lower value of row and column size is. The operator produces the highest integer lower than or equal to . Since the new coding strategy utilizes the correlation of wavelet coecients between the subbands, it is favourable to have a phase shift, which is as small as possible. Therefore the use of biorthogonal lter banks with linear phase is advantageous. For the investigations in this paper, theP9/7-lters rst introduced by Antonini et al. are used.1 To avoid any gain in the analysis stage of the lter bank, n ] = 1 is used for the analysis low-pass lter ]. size
bxc
x
g n
g n
2.2. Fundamentals
The principle of successive approximation of wavelet coecients was proposed for the rst time by Shapiro.7 It means a layered assembling of amplitudes, through which, starting with the coarsest resolution, the coecient values are rened in every layer. This is comparable with the resolution of binary numbers. Consider a binary number, which is represented by eight bits, the domain of denition contains 256 dierent values. Starting with the most signicant bit, a doubling of the resolution is performed by bisecting the intervals of quantization. The rst bit splits the domain of denition in the two intervals 0128) and 128256), the following bit subdivides, dependent on the rst bit, either the lower interval into the domains 064) and 64128) or the upper interval into the domains 128192) and 192256) and so on. In the new coding scheme, the quantization of the wavelet coecients i is made in a similar manner. First, the domain of denition of the coecients is computed with d
d
jd je max = dmax i i
(2)
:
The operator denotes the lowest integer greater or equal to . For every layer , the length of the quantizer intervals l is calculated by l = max with l = 2l =1 2 (3) dxe
x
d
l
K
K
l
l
::::
l is the number of intervals, has a starting value of 2, and is doubled at every layer due to the binary subdivision.
K
Proceedings of SPIE, Vol.3164, San Diego, CA, USA, 30 July { 1 August 1997, pp.279{290 I
P/N .. ..
1
r0
I
-
.....
r1
2
r0
dmax
P/N .. ..
.I...... ..
.....
r1
3 P .. .. .....
r2
-
r3 dmax
Figure 1: First and second step of the successive approximation
2.3. Assignment of data symbols
Regarding the interval length l as threshold, it is possible to assign attributes to the coecients corresponding to their alation to the intervals. If a coecient is greater than or equal to l , it is called signicant, if it is lower, then it is insignicant. Whereas in the example of Subsection 2.2. the dierent bits decided the signicance, now data symbols are used for the successive approximation. These data symbols are necessary for the further coding process. If a coecient is insignicant an I-symbol is assigned. Two symbols have to be used to describe the signicance, because the coecients also can be negative. P marks positive signicant coecients and N marks negative signicant coecients. To determine the coecient values from the data symbols, for every interval k ) with = l and = ( + 1) l ( = 0 1 2 ) a reconstruction value k is to be dened. Supposing the distribution of the wavelet coecients is uniform, then it is suitable to choose the central value l (4) k = + 2 = l + 2 for the minimizing of the reconstruction error. However the analysis of wavelet coecients of natural pictures shows that their distribution is highly concentrated close to zero. That is why all coecients falling in the quantizer interval 0 0 l ) are assigned to null. ( 0 for = 0 (5) k= l l + 2 otherwise Figure 1 illustrates the successive renement distinctly. The threshold 1 is calculated according to equation (3). To every coecient with an absolute amplitude of i 1 , the reconstruction value 0 = 0 and the data symbol I are assigned. The other coecients receive a reconstruction value of 1 = 1 51 , as well as a P- or an N-symbol dependent upon their sign. With that, the sign information is given and has not to be regarded in the following layers. Simultaneously, the reconstruction values serve as reference for the renement process of the coecient values. In the next layers 1, all coecients with a reconstruction value equal to zero are treated in the same fashion as in the rst layer with the distinction that the threshold is reduced according to equation (3). All coecients, which already were assigned to a signicance interval in a former layer, are processed in a dierent way. In this case, only a comparison of the original value and the reconstruction value is made. If the original value is lower, then the coecient is insignicant (I) in this layer and the reconstruction value is decreased by 0 5l . In the other case, the signicance symbol P is assigned and the reconstruction value is increased by 0 5l. In Figure 1 (right), both these new reconstruction values are named with 2 and 3 for the second layer. The successive approximation of the wavelet coecients is performed for the dierent subbands in a graduated manner. Whereby in this section, the three detail components (horizontal, vertical, diagonal) at every resolution level are treated together as one subband. Considering subband = 0 as the component with the highest frequencies, the length of the quantizer intervals are obtained with lj = 2 lj+1 (6) This leads to following processing steps. was the number of decomposition levels. In the rst layer, all coecients of the lowest subband = 1 are divided up, as described above. The other subbands have a threshold of 1j max , no subdivision is made, and the coding of data symbols is not necessary. In the second layer, the resolution of the coecients of the subband = 1 is doubled, whereas the approximation starts at this point for the subband = 2. The remaining subbands = 3 to =0 are not taken into consideration again. At every layer, a further subband is integrated, until all subbands join the approximation. The connections are shown in Table 1 with the example of four subbands. I
u
k
o
k
k
r
:::
o;u
u
u o
r
k
I
k
r
k
jd j
:
:
r
r
j
:
z
j
z;
d
j
j
z;
z;
j
j
z;
Proceedings of SPIE, Vol.3164, San Diego, CA, USA, 30 July { 1 August 1997, pp.279{290
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layer subband max lj coding l
j
d
=
1 2 3 4 5 0123 0123 0123 012 3 01 2 3 0 0 1 2 0 1 2 4 1 2 4 8 2 4 8 16 4 8 16 32 {{{x {{xx {xxx xxx x xx x x
Table 1: Graduation of the successive approximation
a)
b)
Figure 2: a) Quarternary tree structures in the wavelet domain b) PSD-trees corresponding to the same region The coecients of the low-pass band are processed similar to the wavelet coecients. The domain of denition is xed to 0 i 255. The distribution is assumed to be uniform, and the reconstruction values are initialized with 127 5. Therefore, the decision on rst signicance is not necessary, and the values are rened at every layer using a 2-symbol-alphabet. a
:
3 PREDICTION OF SIGNIFICANT DESCENDANTS
Using the successive approximation, it is simple to integrate the correlation between coecients of adjacent subbands into the coding algorithm. Regions with high contrasts as edges, are composed of dierent frequencies. For that reason, they can be localized in two or more subbands. In opposition with it, smooth regions have small coecients in all subbands. Connecting the coecients, which contain information of the same region, one obtains quarternary tree structures (Fig. 2.a). With the exception of the highest frequency subband, every coecient at a given subband can be related to four coecients at the next ner subband. These structures oer good conditions for an ecient information compaction. The symbol alphabet I, P, N indroduced in the preceeding section is now expanded using additional data symbols, with which a prediction is possible, whether a wavelet coecient in the subband = has signicant descedants in the subbands or not. The three new symbols are P SD. . . coecient is signicant, positive, and has at least one signicant descendant N SD. . . coecient is signicant, negative, and has at least one signicant descendant SD. . . coecient is insignicant, but has at least one signicant descendant The SD-extensions stand for \signicant descendant". Applying these data symbols, it is possible to construct PSD-trees, whereby three PSD-trees represent an own image region in each case (Fig. 2.b). The total number of PSD-trees is given by the number of coecients in the lowest subband = 1. First of all, the three symbols I, P, and N are assigned as described in Subsection 2.3.. Then, beginning with the subband of highest frequency, the symbol of the parent coecient receives an SD-extension, if the current coecient does not have the I-symbol. For the xation of the symbol positions, a second pass is required. If a coecient has an I-symbol and its parent has an SD-marker, then the I-symbol is changed to an F(ill)-symbol. The dierence is that the F-symbol has to be coded, but not the I-symbol. f
g
j
j
:
12
Proceedings of SPIE, Vol.3164, San Diego, CA, USA, 30 July { 1 August 1997, pp.279{290 Barbara (512x512)
Entropy 7.632
Clown (512x512)
7.739
Salesman (352x288)
6.836
Lena (512x512)
7.446
Goldhill (512x512)
7.478
bpp 0.0625 0.1250 0.2500 0.5000 1.0000 dyadic 23.03 24.70 27.25 30.81 35.71 adaptive 23.43 25.76 28.63 32.16 36.54 EZW7 23.10 24.03 26.77 30.53 35.14 S+P6 23.35 24.86 27.58 31.39 36.41 dyadic 25.16 27.84 31.51 35.44 39.56 adaptive 25.88 28.66 32.16 36.21 40.18 S+P 25.57 28.23 31.95 35.93 40.05 dyadic 25.08 26.94 29.08 32.38 36.80 adaptive 25.44 27.06 29.54 32.94 37.42 dyadic 28.04 30.67 33.61 36.78 40.05 adaptive 28.21 30.76 33.78 36.90 40.07 EZW 27.54 30.23 33.17 36.28 39.55 S+P 28.39 31.10 34.11 37.21 40.40 dyadic 26.34 28.21 30.28 32.66 36.04 adaptive 26.49 28.51 30.30 32.75 36.07 S+P 26.73 28.48 30.56 33.13 36.55
Table 2: Performance of the new codec for several test images (in terms of PSNRdB]) in comparison with other codecs
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