Author name used in this publication: S. W. HongVersion of RecordPublishe

Chan, YH

Choy, SSO

Hong, SW

Yue, CN

Author name used in this publication: S. W. HongVersion of RecordPublishedVoR allowe

PolyU Institutional Repository

1997 IEEE International Symposium on Circuits and Systems, June 9-12,1997,1!3 Kong Regularized restaration of V compressed images with Constra:ined Least: Squares approach Steven S. 0. Choy, C. N. Yue, S. W. Hong and Yuk-Hee Chan Department of Electronic Engineering The Hong Kong Polytechnic University Hong Kong Abstract In this paper, an iterative algorithm is proposed to restore VQ-encoded images. This algorithm incorporates adaptivity into simple CLS restoration algorithm by weighting every pixel according to its expected derivation from the original. This algorithm is fully compatible with any VQ codec to improve codec 's coding perfo)w"ce. Computer simulations showed that the proposed scheme was superior to other conventional schemes in terms of SNR improvement. Besides, the image quality could also be improved subjectively by reducing the blocking effect. Introduction Most popular image compression techniques appl ied nowadays such as  t ransform coding techniques[ 11 and vector quantization techniques[2] are basically lossy approaches. These compression techniques typically allow a tole]-able distortion to appear after the compression in order to achieve a higher compression rate. When the compression rate is not too high, the distortion introduced will do little harm to the quality of encoded images since a small distortion of the image quality is generally unobservable under our human visual system. However, the story will be compb.ely different when the compression rate is sufficiently high. This is the reason why one can see the so-called blocking effect in block-based coding techniques :such as JPEG and VQ. Image restoration refers to the problem of estimating the original image from its degraded observation. Though this technique does not aim at improving the subjective look of an image as image enhancement does, it can be very helpful for one to 0-7803-3583-X/97 $10.00 01997 IEEE 1257 recover the original image from the encoded image and in turn innprove the image appearance to a certain extent as well. Amount of research has been carried out recently to eliminate blocking effect of JPEG-encoded images with various image restoration techniques[3-51. However, by far little effort has been appeared in the literature to address the restoration of VQ-encoded images. In this paper, the restoration of VQ-encoded images is formulated as a regularized image recovery problem. This work makes use of the prior knowledge of the VQ quantization distortion to recover a VQ-encoded image. Simulations shown that this proposed approach can provide a better SNR improvement compared with some other approaches adopted from lhose approaches for JPEG-decoded images. Algorithm In image restoration, an image degradation process can be generally formulated by the following matrix-vector equation g = H f + n  (1) where f and g are the lexicographically ordered original and degraded images, n is a noise vector and H represents a linear degradation operator[6]. However, this model cannot directly applied to ,a VQ process. Unlike those classical problems tackled in image restoration such as deblurring, vector quantization is not a linear degradation operation. In VQ compression, the source image ,f is partitioned into a number of subimages of equal size. Each subimage is then treated as a vector and is represented with1 a particular codeword selected from a codebook. The selection is based on the minimum Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 27, 2009 at 23:36 from IEEE Xplore.  Restrictions apply. Euclidean distance criterion, where the Euclidean distance measure between vectors and  3 is defined as d(v1,v2 ) = I1 v1 - qI . Here, without losing the generality, we lexicographically order these two-dimensional subimage vectors as one-dimensional column vectors of size D each. 44 - -3 The corresponding VQ-encoded image, say g, will certainly be distorted to a certain extent. To restore this distorted image, one possible approach is to estimate f by seeking a ,f which approximates g in a least mean squares sense under a certain constraint based on the a priori knowledge about the smoothness of typical pictures. Consider the case that the encoded image is encoded with a codebook containing N codewords, namely,  C = { 3 I i=1,2 ... N .  Accord ing  to V Q  theory, these N codewords partition the whole vector space concerned, say V ,  into N non-overlapped Voronoi regions. For a particular Voronoi region R, = ( T? &V and d ( 2  $J 5 d ( 2  $ for j=1,2..N}, the associated variance vector is given as o~=(o,,~, i + 2  Here, No,k is the total number of vectors in R,, + th x, is the mth element of x and Y , , , ~  is the m element of y,. In practice, in order to reduce the realization complexity, one may select some typical images to form a training set  R, and then approximate 3 with vectors in R,n R, instead of R,. Note that all 8 ’ s  are solely determined by the codebook and hence can be precomputed once the codebook is given and stored as an extension of the codebook for future application. 4 Let and be the bth subimages of g and f respectively. Suppose 2 is the codeword used to represent In that case, we have gb=yk and the distance between $ and  is bounded by the boundary of the Voronoi region R,. For  any particular pixel gb,i, its deviation from fb,i is also bounded. In formulation, we have 4 4  where gb,i and fb,i are respectively the ith elements of g, and i$ respectively, and, is the bound for the uncertainty of pixel gb,i in the observed image. Since we have gb=yk, gb,i=yki holds and the bound can be estimated to be a function of o&. In this paper, that is proportional to is assumed. In that case, we have = K (T~ ,~ ,  where K is a constant. Note that this assumption can easily be satisfied as long as K is sufficiently large. In practice, by assuming a normal distribution, K=10 is  a reasonable estimate as the range of [ Y,,~- 3 o ~ , ~ ,  yk,i+ 3 G ~ , ~  ] covers more than 99% of the possible values of fb.i. -+ 3 + +  2 Adaptivity is then introduced to weight the contribution of a particular pixel to the total expected image distortion according to i ts  amount  of uncertainty. To achieve this, we first rewrite eqn (2) as 2 -2 where obtained by = o ~ , ~ .  The overall distortion can then be Ns D Jd = a& ( gb,i - fb,i )” 5 N,D K = E d b=l I=I (4) where N, is  the total number of subimages partitioned. In matrix form, we have Jd = ll A( g ’ - f  ) 112 ( 5 )  0 Both A and Ab are diagonal matrixes. Typical images would generally have weak high frequency components as the intensity of neighboring pixels is highly correlated. This feature can be used as an additional constraint to regularize the restoration solution. In practice, a smoothness Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 27, 2009 at 23:36 from IEEE Xplore.  Restrictions apply. 2 measure J, is defined as J, = II S f II , where S is generally a highpass  fi l tering operator.  The smoothness constraint can then be defined as J, 5 E,, where E, is the upper energy bound of the high frequency components. The computation of the regularized solution then reduces to the minimization of (4) subject to J, 5 E,. This is equivalent to the minimization of J = II AP( g - f ) 112 + a II Sf 112 (6) ,  where a = E ~  /E, and P is the permutation matrix that permutes a lexicographically ordercd image f to f'. The minimization of J with respect to f leads to the normal equation [ PTATAP + ~ S T S  If = PTATAP~ (7) In general, f cannot be evaluated directly from this equation as it requires the inversion of an huge matrix. An alternative approach is to use a steepest descent algorithm to obtain f iteratively. This approach leads to the following iterative equation: where f k  is the estimate o f f  afte.r the kth iteration and p is the contraction parameter for the iteration. Convergence can be guaranteed if p satisfies the following condition: 2 o < p <  - %" (9) where hmm is the largest eigenvalue of the matrix (PTATAP+aSTS As matrix A is  codebook- dependent and S can be predetermined, both aSTS and PTATAP can be precomputed and hence p can also be well-defined. This saves quite a bit of effort in  real iz ing (8).  The  init ial  estimate f, = PTATAPg is generally used to obtain the solution. 1 Simulations Simulation has been performed to evaluate the performance of the proposed restoration scheme on a set of 256 level gray-scale digital images of size 256x256 each. In our simulation, four standard images "House", "Girl", "Couple" and "Germany" were used as the training set to generate a codebook with LBG algorithm. The codebook is of size 256 and the codeword size is 16 (a block of 4x4 pixels). Images encoded with this codebook were then restored with the proposed scheme. In our simulation, variance vectors were approximated with the training vectors as we mentioned in previous section, and, a and fi were set to be (1011 Sg ]I2) and 1 respectively. 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-encoded image is more or less the same as restoring a JPEG-encoded image in view of their  common interest  in deblocking. Hence, as few schemes had been proposed for restoring VQ-encoded images, they 'were also adopted here and simulated for comparison study. Tab le  1 shows the S N R  improvenient (APSNR) obtained with various approaches, where APSNRE10 loglo ( II g - f 1 1 2  / I lf- f 1 1 2  ). One can see that the proposedl scheme is much superior to others in terms of this objective criterion. The first and second columns are the restoration results of filtering approaches[3]. A 3x3 spatial filter was applied to the entire image in spatial filtering but to the bmlock boundaries only in block edges filtering. Narayan's algorithm[7] is a heuristic POCS algorithm. In our simulation, the convex sets c3 and c4 concerned in [7] were defined as C3={bl B[u,v]=O if u+v>256} and C4={b I 0 5  e n < 2 5 6  f o r  any en€ b } respectively, where en is a pixel, b is any image of size 256x256 and B[u,v] is a 2D transform coefficient of b as defined in [7]. The CLSl approach were basically the same as that proposed in [4] and the CLS2 approach were adopted from [5] .  In particular, solutions were obta.ined by minimizing objective functions II f-g1I2+al II Sfl12 and II R ( f - g  ) II + a2 II S f II respectively. Iterative approach was applied. Since direct implementation of [SI may cause convergence problem in restoring VQ-encoded images, the determination of R was modified in a way that all its (diagonal elements were mapped to a range of [0.5,1 .IO] linearly with mapping f(r)=(r+l)/2 after following the procedures defined in [5] to get R. As usual, a 3x3 Laplican operator was used as Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 27, 2009 at 23:36 from IEEE Xplore.  Restrictions apply. ~~~ Germany 0.2732 0.0923 0.2539 0.3230 0.5497 -0.0203 Girl 0.4884 0.1120 0.4033 0.571 1 0.901 1 0.2409 House 0.1528 0.1183 0.2657 0.2407 0.6616 0.2217 S in all CLS approaches including the proposed scheme. The terminating criterion for all iterative processes was I I ~ ~  -fk-l II / I I ~ ,  II * 5 I X ~ O - ~ .  Conclusions In this paper, a CLS scheme is proposed to restore VQ-encoded images. This scheme incorporates adaptivity into simple CLS restoration scheme[4] by weighting every pixel according to its expected derivation from the original. Computer simulations showed that the proposed scheme could achieve a much better restoration performance in terms of SNR improvement  compared  with other  exis t ing schemes[3-5,7]. Besides, the image quality could also be improved subjectively by reducing most of the blocking effect. Since no extra information other than the codebook is required to cany out the restoration with the proposed scheme, no transmission overhead IS necessary and hence it can be fully compatible with any VQ codec to improve its 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 makes it be always able to provide a reasonable restoration performance whatever VQ scheme with which it works, which is quite different from the schemes that are dedicated to restore images encoded with particular type of VQ schemes. References [ I ]  W.B. Pennebaker and J.L. Mitchell, JPEG still image data compression standard, Van Nostrand Reinhold, New York, 1993. [2] A. Gersho and R. M. Gray, Vector quantization and signal compression, Kluwer Academic Publishers, 1992. [3] H. C. Reeve and J. S. Lim, "Reduction of blocking effect in image coding," Optical Engineering, Vo1.23, No.1, pp.34-37, Jan 1984. [4] Y. Yang, N. P. Galatsanos and A. K. Katsaggelos, "Regularized reconstruction to reduce blocking artifacts of block discrete cosine transform compressed images," IEEE Trans. CAS for Video Technology, Vo1.3, No.6, pp.421-432, Dec 1993. [SI S. W. Hong, Y. H. Chan and W. C. Siu, "An adaptive constrained least square approach for removing b locking  ef fec t , "  IEEE I S C A S ' 9 5 ,  Vol.11, pp.773-776, 30 Apr - 3 May 1995, Seattle, USA. [6] J. Biemond, R. L. Lagendijk and R. M. Mersereau, "Iterative methods for image deblurring," IEEE Proceedings of the IEEE, Vo1.78, No.5, pp.856-883, May 1990. [7] A. Narayan and J. F. Doherty, "Convex projections based edge recovery in low bit rate VQ", IEEE Signal Processing Letters, Vo1.3, No.4, pp.97-99, Apr 1996. Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 27, 2009 at 23:36 from IEEE Xplore.  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Regularized restoration of VQ compressed images with constrained least squares approach

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