Jun 3, 1999 - coding performance of a vector quantisation (VQ) image compression ..... filtering edges of [251 of [I 61. CLSl. CLS2 algorithm image liitering.
Regularised restoration of vector quantisation compressed images S.S.O.Choy, Y.-H.Chan and W.-C.Siu
Abstract: The authors study the application of image restoration technology in improving the coding performance of a vector quantisation (VQ) image compression codec. Restoration of VQcompressed images is rarely addressed in the literature, and direct applications of existing restoration techniques are generally inadequate to deal with this problem. A restoration algorithm is proposed, specific to VQ-compressed images, that makes good use of the codebook to derive useful a priori information for restoration. The proposed restoration algorithm is shown to be capable of improving the quality of a VQ-compressed image to a much greater extent, compared with other existing restoration approaches. As no extra information, other than the codebook, is required to carry out the restoration with the proposed algorithm, no transmission overhead is necessary and, hence, it can be fully compatible with any VQ codec when used to improve coding performance.
1 Introduction llost popular image compression techniques applied iowadays, such as transform coding [I] and vector quanti;ation (VQ) techniques [2, 31, are basically lossy ipproaches. These compression techniques typically illow a tolerable distortion to appear after compression .o achieve a higher compression rate. When the compres;ion rate is not too high, the distortion introduced will do ittle harm to the quality of encoded images, as a small listortion of the image quality is generally unobservable mder the human visual system. However, the situation will le completely different when the compression rate is iufficiently high. This is the reason why so-called blocking :ffects can be seen in block-based coding techniques such 1s JPEG [4] and VQ. A compressed image can be considered as a degraded iersion of the original uncompressed image. Therefore mage-restoration techniques can be very helpful to recover he original image from its compressed version and, in urn, to improve the quality of the compressed image to a :ertain extent. Recently, significant research bas been :arried out to reduce coding artefacts in block transfonnlased scalar-quantised images with various image restoraion techniques [5-141. However, far less effort has been nade in the literature to address the restoration of VQ:ompressed images [15, 161. This paper is devoted to brmulating the problem of VQ-compressed image restoraion and developing a restoration algorithm specific to )ostprocessing of VQ-compressed images. ?
image blocks. Each image block is then treated as a vector. Each image vector is then compared with a set of representative image vectors called codewords, it, i = 1, 2,. . . , Ne, that are stored in a previously generated codebook. The best-matching codeword is selected, based on the minimum Euclidean distance criterion, where the Euclidean distance measure between vectors v1 and v2 is defined as d(vl, v2)= llvl - vzII. In other words, an image vector v is represented by codeword ik,if and only if d(v, Gk) 5 d(v, Gj) for all j = 1, 2,. . . , N,. Once the best-matching codewords for all image vectors of the source image have been selected from the codebook, the indices of the selected codewords are transmitted to the receiver. At the receiver, with the same codebook, the VQ-encoded image can be reconshacted based on the received indices. The decoder has a very simple structure, as it merely consists of a table look-up procedure. A block diagram of the basic VQ codec is shown in Fig. I. Compression is accomplished in VQ by using a codebook with relatively few codewords compared with the number of possible image vectors. In practice, to make codebook storage and searching feasible, the dimension of image vectors (i.e. the size of image blocks) is usually limited to about 16 = 4 x 4. For VQ at a high compression rate, the size of codebook is very small. This implies that the ratio of available codewords to the total number of
m
Image degradation in VQ encoding
n image compression using VQ, a source image x is :patially partitioned into sub-images of equal size, called
Vi, i = l ,...,Nc
: encoder
3 IEE, 1999 EE Pmceedings online no. 19990161 IOI:lO.l049/ip-~is:l9990161 'aper first received 28th July 1998 and in revised farm 8th January 1999 :he authors are with Department of Electronic and Information Engineerng, The Hong Kang Polytechnic University, IIung Hom, Kowloon, Ilong Long EE Proc.-Yis. Image Signal Procerr, Yoi. 146. Ng. 3, June 1999
codeboak j decoder
vi, 1=1....,N,
............................................................................. Fig. 1 Block diagmm
of vector
.
qsanliration image compressim
IG5
possible image vectors is also very small. For example, consider the VQ encoding of an 8-bit image at a desired bit rate of 0.5 bits per pixel where the image vectors are formed as 4 x 4 image blocks: the codebook can contain only 256 codewords. This means that on? 256 codewords are available to represent all the 256’ possible image vectors of size 4 x 4. Therefore vector quantisation introduces image degradation in the encoded images to a certain extent. In general, the higher the compression rate, the severer the distortion in the encoded image. Unlike those classical problems tackled in image restoration, VQ is not a linear-degradation operation. Thus, a VQ process cannot be described by a linear image degradation model. Alternatively, a VQ-encoded image can be simply modelled as the superposition of the original image and the noise introduced by the quantisation error. However, it is difficult to describe this quantisation noise in a precise way. Basically, the quantisation noise in VQ is codebook-dependent. As VQ is performed block by block, independently, the quantisation noise generally manifests itself as block boundary discontinuity in the encoded image. At a high compression rate, the blocking artefacts in the encoded images are very noticeable and are visually annoying to human viewers. (For illustration, close-ups of some original image segments and their VQ-compressed versions at 0.5 bits per pixel are shown in Figs 2a, b and 3 4 b). 3 Restoration of VQ-compressed images 3.1 Formulation of a priori information An image consisting of N pixels is represented by an N x 1 vector x = [ x l , x 2 , . . . , xNIr by lexicographical ordering. Consider the case that image x is encoded as y by VQ, with a codebook C containing N, codewords
C = {Gi, i = 1 , 2 , . . . , N,]
deviation from the corresponding pixel bounded. In formulation. we have bb.i
- z6,i)
2
5
xb,i
in
xb
is also (5)
eb,i
where B ~ is, the ~ bound for the uncertainty of pixel yb,i in encoded image y. As we have Y h = 8,, Yb, = Ck,; holds and 1 By the bound E,,; can be estimated to be a function of bk,f assuming2 that eh,; is proportional to & we have E ~=, Kuk,;, ~ where IC is a constant. Note that this assumption can easily be satisfied as long as IC is sufficiently large. In particular, by assuming a normal distribution, IC % 10 is a reasonable estimate, as the range of [Bk,i - 3ak,;, Bk,; h k , i ] covers more than 99% of the possible values of x ~ , ~ . As vector quantisation is a many-to-one operation, many images can be encoded into the same compressed image. However, we can exploit the information carried by u&’s to define a set of images whose VQ-compressed version is y. In restoring a VQ-compressed image, this set becomes very useful, as it provides us with an important constraint for seeking the original image x. To formulate this set of images, we first rewrite eqn. 5 as
+
wa.ibb,, - x 6 , i )
2
5 IC*
(6)
where a&= l/u&. The overall distortion in y can then be described as
where N, is the total number of sub-images partitioned. In matrix-vector formulation, we have
IIAW-dIII’ 5 Ed
(8)
where Y=M!Y;
(1)
i
and each codeword is of size D.According to VQ theory, these N, codewords partition the whole vector space concerned say K into N, non-overlapped Voronoi regions. For a particular Voronoi region
=
I . . . .
(4.4,. . . ,
r“’
Y$‘
(9) (10)
A,
R, = {vlv E V and d(v, Q k ) 5 d(v, Cj) f o r i = 1 , 2 , . . . , N c ] (2)
and
the associated variance vector is given as +
uk
2
= (‘k,l
3
2 ak.23
2
’
, , I uk.D)
t
r
(3)
Ob.1
where 1
= -C”.R,(Vrn No.k
- %,m)z
(4)
Here, No,kis the total number of vectors in Rh, v, is the mth element of v, and QCm is the mth element of 8,. In practice, to reduce the realisation complexity, some typical images can be selected to form a training set Cl,, and then i?k can be approximated with vectors in Q , f l R k instead of R k . Note that all ak are solely determined by the codebook and, hence, can be precomputed once the codebook is given, and stored as an extension of the codehook for future application. Let x6 and yb be the D-dimensional vectors representing the bth sub-image of the original image x and the encoded image y, respectively. Suppose Gk is the codeword used to represent x6. In that case, we have yb = e h , and the distance between y b and xb is bounded by the boundary of the Voronoi region Rk. For any particular pixel yb,; in yb its 166
Here, b E { 1, 2, . . . , N3}. Note that both A and Ab are diagonal matrices. Vectors x’ and y’ can also he formulated as x’=Px and y’ respectively, where P is the permutation matrix that permutes an image vector from lexicographical order to the order in eqns. 9 and 10. Then, eqn. 8 can be written as IIAPO, - x)llz 5 cd. According to the preceding formulations, the constraint set for the restoration solution .t of a VQ-compressed image is described by
=e,
J&) = llAP(j~--S)lI2 5 (13) Typical images would generally have weak high-kequency components, as the intensity of neighbouring pixels is highly correlated. This feature can he exploited as additional a priori information in the restoration of VQcompressed images. To incorporate this information into /€€Proc.-Yir. Image Signal Process, Vol. 146, No. 3, June IYY9
restoration, we apply the following smoothness measure of image P:J*(P)= IIHP112, where H is a linear highpass operator. The smoothness constraint for the restored image can then be described by
where E~ is the upper energy bound of the high-frequency components.
3.2 Formulation of restoration algorithm In the preceding section based on the a priori information concerning the VQ encoding process and the original image, two constraint sets have been defined for the restoration of VQ-compressed images. According to the regularisation principle for image restoration [17-20], among images that satisfy both of these two constraint sets, a physically reasonable choice for the restored image is the image minimising
+
J ( i ) = IlAPb -.?)I12 a\lHiIl2 (15) where a = E&,. The minimisation of J(2) with respect to P leads to the normal equation [IY, 201 (P'A'AP
+ aH'H).? = P'A'APv
(16) In practice, direct computation of 2 based on this equation is not feasible, as the inverse of matrix (P'A'AP+ EH"), which is of huge size N x N, has to be computed. It is a prohibitive task even for an image of moderate size. An iterative technique using a steepest-descent algorithm can be applied successively to approximate the solution [21]. This leads to the following iteration for the restoration solution: ;k+,
=i
k
+ B(P'A'APO, - ik) - U"Hik]
(17)
where k is the iteration index (k= 0, 1,2, . . .), and fi is the relaxation parameter. A sufficient condition for the convergence of this iteration is 0 < fi e 2/,l,pIax, where A,, is the largest eigenvalue of matrix (P'A'APlaH'H) [21]. It is common to set the initial estimate Po to be y, although theoretically no restrictions are imposed on the initial estimate. We now address various issues in the practical implementation of the iterative restoration algorithm. First, matrix P'A'AP can be precomputed prior to the restoration. This can be done easily by using coding indices A of the encoded image y and variance vectors ;k associated with the codebook. Recall that the variance vectors are merely codebook-dependent and, hence, can be precomputed, once the codebook is given, and stored for restoration purposes. At formulation level, diagonal matrix A'A is constructed by placing
in the bth segment of diagonal place of A'A. Then, matrix P'A'AP is simply a diagonal matrix that consists of the same set of elements as in A'A in a different arrangement. Specifically, suppose that the jth pixel of the image, in lexicographical order, corresponds to the ith pixel of the bth block of the image, and this block is encoded with the kth codeword, then the jth diagonal element of P'A'AP is equal to I/&. In practical implementation, we can use a 2-D array of the same dimensions as the processed image to store these diagonal elements. Parameter CI also needs to be determined prior to performing the iteration. Recall that parameter a is defined IEE Pmc.-yis.Image Signal Process, Vol. 146, No. 3, June 1999
as (N,DIC)/E,~, where E," is the bound of the smoothness constraint set defined in eqn. 14, N,D is the image size, and IC is a constant to make eqn. 5 hold. In practice, as x is unavailable, we set to be IC'IlHyl12, where IC' is a tuning parameter for image smoothness. As for IC, as we have mentioned previously, IC= 10 is a reasonable estimate. 4
Simulation and comparative study
Simulation has been carried out to evaluate the performance of the proposed restoration algorithm on a set of VQ-compressed images. In our simulation, a number of test images were first encoded with a codebook of 256 codewords of size 4 x 4 each. This results in a compression rate at 0.5 bits per pixel. The test images applied are a set of de facto standard 256 grey-level images of size 256 x 256 each. The codebook used was generated with the LBG algorithm [22]. The training set consists of subimages of four standard images, namely, 'house', 'girl', 'couple' and 'Germany'. The proposed restoration algorithm was then applied to restore the compressed images. Some other restoration schemes were also evaluated for comparison. Most of them were originally proposed for restoring JPEG-encoded images. Note that restoring a VQ-compressed image is more or less the same as restoring a JPEG-encoded image, in view of their common interest in deblocking. Hence, they were also adopted here and simulated for comparative study, as few schemes had been proposed for restoring VQcompressed images. After applying various algorithms to restore the VQcompressed images, we applied the SNR improvement (SNRI) as a performance index to evaluate objectively the restoration results obtained. Mathematically, the SNRI is defined as
where xi, yi and iiare the ith pixels in the original, compressed and restored images, respectively. This measure provides us with an idea of how much has been improved in the image's fidelity to its original, in the sense of mean square error (MSE), after restoring the degraded image. To provide a more comprehensive comparative evaluation of the proposed restoration algorithm, we also used the restoration performance measure proposed in [23] in our comparative study. This measure is termed restoration score (RS) and was found to be better than the SNRI as a performance index for image restoration. The measurement is based on the weighted sum of fidelity improvement of each image pixel, and the main properties of the human visual system are incorporated through the weighting factors. The results of the SNRI obtained with the implemented algorithms are furnished in Table 1, and Table 2 gives the results of the RS. In these two Tables, the first and second columns are the restoration results of filtering approaches to reduce blocking artefacts [24]. In spatial filtering, a 3 x 3 spatial filter was applied to the entire image, whereas, in block-edges filtering, this 3 x 3 filter was applied merely to block boundaries. The filter we applied is the same as in [24]. Specifically, the kernel of this filter is defined as
h(m, n) = h,(m)h,(n), m,n = -1,O, 1
(20)
where ha( - 1) = h,(l) = 0.274 and h,(O) = 0.452 167
Table 1: SNR improvements of various algorithms in restoring VQ-compressed images
Filtering 1241 Spatial filtering
Girl
Germany couple Sailboat Tiffany Face Hat
Average
Constrained least square
Block edges liitering
Algorithm of [251
- 0.106 0.455 - 0.550
- 0.069 0.573 - 0.137
- 0.114 0.034 - 0.509
0.066 0.478 0.234
0.214 0.755 0.138 - 0.233 0.307 0.875 1.273 1.654 0.396
- 0,019
0.109 0.327
0,280
image Baboon Peppers House Lenna
POGS
~
0.776 0.239 - 0.042 0.342 0.905 1.279 1.622
0.497
0.058 0.232 0.235 - 0.228 0.502 0.680 0.121
Algorithm of [I 61
0.682 0.379 0.290 0.363 0.640 0.900 1.071 0.489
Algorithm CLSl
Algorithm
[I21
[el
0.090 0.334 0.259 0.235 0.398 0.289 0.279 0.308 0.368 0.475 0.522 0.323
0.083 0.472 0.377 0.329 0.577 0.405 0.407 0.415 0.521 0.71 1 0.804 0.464
CLS2
Proposed algorithm 0.076 0.802
0.608 0.443 0.991 0.549 0.603 0.542 0.890 1.468 1.818 0.800
All values for SNR improvement are given in dB
Table 2: Restoration scores of various algorithms in restoring VQ-compressed images
Filtering [24]
POCS
Spatial filtering
Block edges filtering
Algorithm of [25]
0.097
0.051
0.185
0.102 0.054 0.075 0.109 0.067 0.055
0.129 0.198 0.190
0.180 0.215
0.081
0.188
0.098 0.158
0.169 0.232 0.259 0.199
Image Baboon Peppers House Lenna Girl
Germany Couple Sailboat Tiffany Face Hat Average
0.107 0.142 0.199 0.122 0.105 0.149 0.177 0.277 0.314 0.170
0.181
0.094
= 0 if U
+ v > 2561
0.186
0.143 0.156 0.204 0.142 0.141 0.155 0.170 0.249 0.278 0.175
0,202
0.223
(21)
and
C4
= (610 5 e, 5 255 for any e, E b]
(22)
respectively, where e, is a pixel, b is any image of size 256 x 256, and B[u,v] is a 2-D transform coefficient of b, as defined in [25]. The algorithm of Park, et al. [I61 is based on lowpass filtering followed by a projection onto the so-called p-scaled hypercube, which is a much simplified vector-quantisation constraint region. In its simulation, : concerned is determined with the same the value of 1 training set we applied in implementing our algorithm. The 168
Algorithm of 1161 0.096
The third and fourth columns show the restoration results of projection-based restoration approaches. Narayan and Doherty’s algorithm 1251 is a heuristic POCS algoand i4 in 1251 rithm. In our simulation, the convex sets i3 were defined as
C3 = {blB[u,VI
Constrained least square Algorithm
Algorithm
GLSl
CLSP
[I21
[el
0.098 0.121 0.093 0.105 0.124 0.105 0.104
0.116 0.159 0.126 0.144 0.170 0.141 0.140
0.116
0.150
0.114 0.142 0.132 0.114
0.147 0.193 0.187 0.152
Proposed algorithm 0.116 0.252
0.192 0.214 0.266 0.198 0.193 0.214 0.215 0.319 0.346 0.230
value of pi for each codeword is then set to he 10% of pr, as suggested in [16]. The last three columns of Tables 1 and 2 show the results obtained with three restoration algorithms based on the constrained least square approach, including the proposed algorithm. The CLSl and CLS2 algorithms were devised based on the two postprocessing deblocking methods proposed in 1121 and 181, respectively. Specifically, in the CLSl algorithm, the solution was obtained as
and in the CLS2 algorithm, the solution was obtained as
where C I ~ is a parameter that controls the degree of smoothness of the solution, and R is a diagonal matrix IEE Pmc.-yis. Image Signal Piocess, Yo/. 146, No. 3, June I999
b
a
I
0
I
h
:ig. 2 Comparison of restoration performance of various apprnaches in reslon’ng VQ-compressed ‘Lema’ , Original d Black edges filtering g Algorithm CLSl
, VQ compmssed at 0.5 bits per pixel Spatial filtering
e Narayan’s algorithm fPar!& algorithm
h Algorithm CLSZ i Proposed algorithm
lefined in [XI. As usual, the iterative approach was applied o obtain these two restoration solutions. In implementing he CLSl and CLS2 algorithms, parameter CI, was set to be :qual to Ilx -yIIz/IIHxl12. In the implementation of our roposed algorithm, parameter IC’ was set to he 10. In all hree iterative algorithms, the highpass operator H was set o be the 2D discrete Laplacian operator [26], relaxation iarameter /3 was set to he 0.5, and the termination rule was Iik+- ik112/112k112 < 1 x 10 All of the algorithms :onverged in the experiments performed. Among all the implemented algorithms for restoring JQ-compressed images, the simple space-invariant filterng and the block-edges filtering techniques are non-iteraive and require the least computational effort. The other ive algorithms are all iterative and hence, demand much iigher computational cost than the simple filtering technilues. The computational costs of these five iterative algoithms are comparable with one another and are generally 0-30 times higher than the non-iterative simple filtering nethods. From all the restoration results furnished in Tables 1 and !, it can be seen that the proposed restoration algorithm iutperforms the other algorithms, in terms of both the iNRI and RS,. On average, by applying the proposed lgorithm to restore the VQ-compressed images, an iNRI of 0.800 dB in image quality was achieved, and an CS of 0.230 was obtaineQ whereas the average SNRI ihtained with the six other approaches was just 0.497 dB
’,
EE Pmc.-V&.Image Signal Process. yOL’146,No. 3, June 1999
at best and 0.121 dB at worst. In terms of average RS, the best is 0.199, and the worst is just 0.094. In the light of these results, it can he concluded that the proposed approach is most effective in restoring VQcompressed images among the various restoration techniques. For visual evaluation of the restoration results, closeups of original images, their VQ-compressed versions and corresponding restored images obtained with the various approaches are provided in Figs. 2 and 3. Fig. 2 shows the restoration results of ‘Lema’, and Fig. 3 shows those for ‘peppers’.
5 Conclusions Very little research has been carried out to address the restoration of VQ-compressed images. Furthermore, direct applications of existing restoration techniques are generally inadequate to deal with the problem. This paper has proposed a restoration algorithm specific to VQcompressed images. This algorithm makes good use of the codebook to derive useful a priori information for restoration. It has been demonstrated by detailed simulation results and comparative study that the proposed restoration algorithm is capable of improving the quality of a VQ-compressed image to a much greater extent, compared with other existing restoration approaches [X, 12, 16, 24, 251. 169
d
, Fig. 3 Comparison of restoration performance of various appmches in restoring VQ.compressed >eppers' a
Original
b VQ compressed at 0.5 bits per pixel c
Spatial filtering
d Block odgcs filtering e Narayan's algorithm f Parrs algorithm
g Algorithm CLSl h Algoritlim CLSZ
i Proposed algorithm
As no information other than the codebook is required to carry out the restoration with the proposed approach, no transmission overhead is necessary and, hence, it can be fully compatible with any VQ codec when used to improve coding performance. Also note that the proposed algorithm is not tailor-made for a particular VQ scheme, and, hence, no a priori knowledge about the construction of the codebook is required during the restoration. This means it will always be able to provide a reasonable restoration performance, with whatever VQ scheme it works.
6 CHOY, S. S. O., CHAN, Y. H., and SIU, W C.: 'Reduction of block-transform image coding artifacts by using local statistics of transform coefficients'. IEEE Signal Pmcess. Lett., 1997, 4, (I), pp.
6 Acknowledgment
This work was supported by a Hong Kong Polytechnic University Research Grant, project account no. 340.8 13.A3.420. 7 References 1 PENNEBAKER, WB., and MITCHELL, J.L.: 'JPEG still image data compression standard' (Van Nostrand Reinhold New York, 1993) 2 CHAN, YH., and SKI, W.C.: 'In search of the optimal searching sequence for VQ encoding', IEEE Trans. Commun., 1995, 43, (121, pp. 2891-2893 3 GERSHO, A., and GRAY, R. M.: 'Vector quantization and signal compression' (Kluwer Academic Publishers, 1992) 4 WALLACE, G. K.: 'The JPBG still-pichlre compression standard', Commun. ACM, 1991,34, pp. 3 0 4 4 5 CHOY, S. S . O., and CHAN, Y. H.: 'Reduction of coding artifacts in transform image coding by using local statistics of tlnnsform coefficients'. IBEE Int. Symposium on Circuits and systems, ISCAS'97, June 1997, Hong Kong, pp. 1089-1092 170
908 14 ZAKHQR, A.: 'Iterative procedures for reduction ofblocking effects in transform image coding', IEEE Trans. Cimuirs @sr, video Technol,, 1992.2, (I), pp. 91-95 15 CHOY, S. S. O., YUE, C. N., HONG, S. ,W, and CHAN, Y. H.: Regularized restoration of VQ compressed images with constrained least squares approach'. IEEE Int. Symposium on Circuits and systems, ISCAS'97, J u e 1997, Hong Kong, pp. 1257-1260 IEE Pme.-Yis. lmmage Signal Process, Yo/. 146, No. 3, June 1999
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