Image Compression Based on Fractal with Classification by Vector Quantization

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(1)Image Compression Based on Fractal with Classi

(2) cation by Vector Quantization Lih-Ching Lin Department of Applied Mathematics. Chang-Biau Yang Kuo-Tsung Tseng Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C. Email:[email protected]. Abstract The conventional fractal encoding algorithm performs an exhaustive search to

(3) nd a close match between a range block and a large pool of domain blocks. For a large image, the domain pool increases obviously, so the encoding time will also increase. In this paper, we propose a hybrid scheme by combining the fractal image compression with the vector quantization. We use the longest distance

(4) rst algorithm to classify the domain blocks. In this way, we can reduce the range in searching the domain pool. Experiment results show that our method can e ectively speed up the encoding time about ten times. In addition, the quality of our reconstructed images is still as good as the conventional fractal algorithm.. 1 Introduction Recently, image compression becomes more and more popular with quick development of the multimedia. An image always contains a great amount of information, but some information loss is insensitive to the human eyes. Thus, we can remove such information from an image to get high compression ratio. Many image compression methods have been proposed, such as vector quantization (VQ) [5,11,15] and fractal block coding [1, 4, 8, 9, 15] and so on. Vector Quantization (VQ) [5,11,15] is a well-known method for image compression. In VQ, we

(5) rst partition the image into a set of blocks, then treat each This research work was partially supported by the National Science Council of the Republic of China under contract NSC 88-2213-E-110-012 .. block as a vector. For every vector, we

(6) nd the closest codeword from the codebook, then use the index of the codeword to represent each vector. In the decoding phase, we

(7) nd the encoded index of each vector and uses the codeword with that index to represent each vector. The fractal image compression [1,4,6,8{10,14] utilizes the existence of local self-similarity in an image to encode the image. With this way, the fractal image compression can obtain high compression ratio and good quality of the reconstructed image. Jacquin has pointed out the similarity between fractal image compression and VQ [8]. Both methods use a codebook to index each block, which is extracted from the original image. In the fractal image compression, the original image is partitioned into overlapping blocks as the codebook. Unlike the VQ, the fractal image compression needs not transmit the codebook to the decoder. The encoder

(8) nds a contractive operator whose

(9) xed point is an approximation of the original image. With this contractive operator, the decoder can use any arbitrary initial image to get an approximate image by the iterative method. Therefore, the fractal image compression has very high compression ratio. However, it takes long time to encode a fractal image. The conventional fractal encoding algorithm performs an exhaustive search to

(10) nd the best match from the codebook. In this paper, we use the longest distance

(11) rst (LDF) algorithm to classify those overlapping blocks. With our method, we can reduce the number of blocks to be searched, thus reduce the encoding time. In this paper, we focus on a fractal image compression with classi

(12) cation by vector quantization. In Section 2, we will review some related algorithms that we use in this paper. The detail of our fractal encoding.

(13) algorithm is described in Section 3. The performance of our algorithm and the experiment comparison with other fractal-based algorithms are given in Section 4. Finally, we give a conclusion in Section 5.. 2 Previous Works First, we describe a simple fractal block encoding scheme [15]. The original image is partitioned into NR nonoverlapping blocks called range blocks, denoted as Ri ; 1 i NR , with size rs rs. The original image is also partitioned into ND overlapping blocks called domain blocks, denoted as Dj ; 1 j ND , with size ds ds. Domain block size is always larger than range block size, usually ds = 2rs, so we need to contract each domain block to the size of a range block. Moreover, each range block Ri

(14) nds the minimum distortion Distfractal (tr (Dj ); Ri ), for all Dj , and output the encoded information Ii (flag; msg ). A range block is said to be smooth if the variance of all pixels in the block is small. If a range block is smooth, we don't compute the distortion and use the mean of that block to represent it. With this way, we can both reduce the encoding time and increase the compression ratio. For a given vector Qi = (b1 ; b2 ; : : : ; bN ), if P = (a1 ; a2 ; : : : ; aN ) is used to represent Qi , then the distortion between P and Qi is de

(15) ned as follows [15].. Distfractal. N X (P; Q ) = (s a i. i. k=1. + oi. k. bk ) ; 2. (1). where si and oi are de

(16) ned in Equation 2 and Equation 3 respectively. N N N X X X N a b ( a )( b ). si =. k=1. k k. N X N a. k=1. k=1. N X b. oi = (. k=1. k. k. 2. k. N X ( a ) k=1. si. k=1. k. k=1. k. DistV Q (P; Q) =. N X (p j =1. j. qj )2. (4). The longest distance partition algorithm is as follows: Algorithm Longest Distance Partition (LDP) Input:. The splitting cluster Ci = fx1 ; x2 ; : : : ; xni g.. Two new clusters Ca and Cb and their representative codewords.. Output:. k. (2). 2. N X a )=N. Jacquin [8,9] presented a scheme for classifying the domain blocks and range blocks. It is based to the classi

(17) ed vector quantization (CVQ) [13], and classi

(18) es the blocks into three main classes i.e. shade blocks, midrange blocks, and edge blocks. Fisher [4] also used a similar scheme with more classes. Hamzaoui [6,14] proposed a hybrid scheme by combining fractal image compression with mean-removed shape-gain vector quantization (MRSG-VQ) [5, 12]. C. K. Lee and W. K. Lee also propose a simple method [10] based on the local varience to reduce the searching time on

(19) nding a close match between a range block and a large pool of domain blocks. Now, we describe the algorithm that we shall use to classify the domain blocks. The longest distance

(20) rst (LDF) algorithm [7] is a fast heuristic algorithm to generate better codebooks. It uses the longest distance

(21) rst strategy to choose which cluster should be split instead of the maximum descent criterion in the maximum descent (MD) algorithm [2,3] and it invokes the longest distance partition technique to partition one cluster into two new clusters instead of the 2-level LBG partition technique [3] or the hyperplane partition technique [2, 3]. The distance of two vectors P = (p1 ; p2 ; ; pN ) and Q = (q1 ; q2 ; ; qN ) is de

(22) ned as follows.. Calculate the centroid vi of the splitting cluster Ci .. Step 1:. (3). Ii (flag; msg) is used to represent the encoded block i, where if flag = 0 then msg = fmeang, otherwise, if flag = 1 then msg = fthe coordinate; r; si ; oi g.. The conventional fractal algorithm performs an exhaustive search to

(23) nd a close match between a range block and a large pool of domain blocks. There are some classi

(24) cation schemes [4, 6, 8{10, 14] developed to reducing the number of comparisons. Here, we will introduce some of the schemes.. Find xa such that DistV Q (xa ; vi ) max DistV Q (xj ; vi ).. =. Find xb such that DistV Q (xb ; xa ) max DistV Q (xj ; xa ).. =. Step 2: 1. jni. Step 3: 1. jni. Split cluster Ci into Ca and Cb . That is, if DistV Q (xj ; xa ) < DistV Q (xj ; xb ) then xj 2 Ca ; otherwise xj 2 Cb , 1 j ni .. Step 4:. Calculate the centroids of Ca and Cb as the two new codewords.. Step 5:.

(25) In order to reduce the partition time, the LDP algorithm uses a fast method [3] to determine which cluster x should belong to. The fast method [3] is as follows: Let xj = ( 1 ; 2 ; : : : ; N ), xa = (a1 ; a2 ; : : : ; aN ) and xb = (b1 ; b2; : : : ; bN ). Then xj is put in Ca if N X (. j. j =1. aj )2 <. N X ( j =1. So, x is placed in Ca if N X (a j =1. j. bj ) j >. j. X. 1 N 2 (a 2 j =1 j. b j )2 :. b2j ):. (5). (6). N X. 1 (a2 b2j ) in Equa2 j =1 j tion 6 are not changed during the splitting process and can be pre-calculated. The amount of computation can be reduced to N multiplications, N 1 additions and 1 comparison. So Equation 6 is used instead of Equation 5. Now, we will describe the longest distance

(26) rst algorithm. The longest distance

(27) rst algorithm chooses which cluster should be split. It

(28) nds the cluster with the maximum longest distance and applies the LDP algorithm to split this cluster. The longest distance

(29) rst algorithm is as follows: The values of (aj. bj ) and. Algorithm Longest Distance First (LDF) Input:. The codebook size and the training set.. Output:. The codebook.. Split the entire training set into two new clusters by the LDP algorithm.. Step 1:. Let the two newly formed clusters be Ca and Cb . Find the longest distances in Ca and Cb respectively.. Step 2:. Select the cluster with the maximum longest distance in all clusters. Split the selected cluster into two new clusters by using the LDP algorithm.. Step 3:. If the number of current clusters is equal to the codebook size we desire, then output the centroids of the clusters as the codebook and stop; otherwise go to Step 2.. Step 4:. The advantages of the LDF algorithm are its speed and quality. It requires much less time than other codebook generation algorithms. Moreover, the quality of the codebooks generated by LDF is very good, so we choose the LDF algorithm as a base on our fractal algorithm.. 3 The Fractal Encoding with VQ Classi

(30) cation The conventional fractal algorithm spends too much time on

(31) nding the best match between a range block and a large pool of domain blocks. In order to reduce the encoding time, we decrease the search pool by clustering all domain blocks. Our clustering algorithm is based on vector quantization (VQ). For training a local codebook in VQ, all training vectors are partitioned from an original image. In the end of classi

(32) cation, similar vectors will be put into the same cluster. Thus we utilize this concept to classify the domain blocks. Besides, an eÆcient codebook generation algorithm for VQ is very important. To generate the codebook, we choose an eÆcient method, the longest distance

(33) rst (LDF) algorithm which we have introduced in the previous section. We can classify the domain blocks eÆciently by using the LDF algorithm. First, we extract ND overlapping domain blocks from the original image and contract each domain block to the size of a range block. The elements of the training set are the domain blocks after applied the eight transformations. Thus, the size of the training set is 8ND . After applying the LDF algorithm, we get NC codewords and classify each transformed domain block into a correlative cluster. Each range block

(34) nds a nearest codeword (cluster) in the codebook and then in the cluster,

(35) nds the transformed domain block with minimum distortion to the range block. Finally, the encoded information is output. Our fractal encoding algorithm is as follows. Basic Phase. An original image.. Input:. Output:. The previous processing results.. Partition the original image into NR nonoverlapping range blocks, denoted as R=fR1 ; R2 ; : : : ; RNR g.. Step 1:. Step 2:. Extract. ND overlapping domain blocks from the original image, denoted as D=fD1 ; D2 ; : : : ; DND g. Contract each domain block to the size of a range block.. Step 3:. For each domain block Dj , calculate it's variance Dj . If Dj < T , where T is a prede

(36) ned threshold, then remove Dj from D.. Step 4:. Step 5:. 1. Use the LDF algorithm to split all tr (Dj ), 8, until the number of clusters. r .

(37) achieves NC , where NC is a prede

(38) ned codebook size. The set of all clusters is denoted as C =fC1; C2 ; : : : ; CNC g and the codeword of each cluster Ck is denoted as CWk , 1 k NC . Algorithm A Input:. An original image.. Output:. The encoding information.. Steps 1-5:. Basic Phase.. For each range block Ri , calculate its mean Ri and variance Ri . If Ri < T , then output Ii (0; Ri ); otherwise do Steps 7-8.. Step 6:. Find k such that Distfractal (CWk ; Ri ) is the minimum, where 1 k NC .. Step 7: Step 8:. Find tr (Dj ) such that Distfractal (tr (Dj ); Ri ) is the minimum, 8tr (Dj ) 2 Ck . Then output Ii (1; the coordinate ofDj ; r; si ; oi ).. In order to obtain better quality, we give a search window size W for searching more clusters in algorithm B. Experiments also show that large window size will obtain the reconstructed images with better quality, but the encoding time increases too. Thus, we again modify the algorithm by adding a threshold T . We use T to determine if a transformed domain block tr (Dj ) is good enough to represent the range block. If it is, we do not search other clusters furthermore. Algorithm Longest Distance First Fractal Encoding Input:. An original image.. Output:. The encoding information.. Steps 1-5:. Basic Phase.. De

(39) ne a search window size, W , and a threshold, T .. Step 6:. For each range block Ri , calculate its mean Ri and variance Ri . If Ri < T , then output Ii (0; Ri ); otherwise do Steps 8-9.. In Algorithm A, if a codebook of large size is built, it needs more time in Step 5 and Step 7. However, the search time can be reduced in Step 8. In addition, the quality of the reconstructed image increases very little with a large codebook, so we generate a codebook with a median size. The time required for algorithm A is little, but we

(40) nd that the quality of the reconstructed image is not good enough. Thus, we modify algorithm A to algorithm B by increasing the search window on the codebook. That is, when the close match tr (Dj ) is found, more than one cluster is searched.. Step 7:. Algorithm B. The longest distance

(41) rst (LDF) fractal encoding algorithm e ectively reduces the encoding time with the threshold T . Small threshold T will obtain the reconstructed images with better quality, but the reduced time is not as much as that with large threshold. We use the LDF fractal encoding algorithm to compare with other fractal encoding algorithms in this paper. Our experiment results are listed in the next section.. Input:. An original image.. Output:. The encoding information.. Steps 1-5: Step 6:. Basic Phase.. De

(42) ne a search window size, W .. For each range block Ri , calculate its mean Ri and variance Ri . If Ri < T , then output Ii (0; Ri ); otherwise do Steps 8-9.. Step 7:. Find a set S = fs1 ; ; sW g such that Distfractal(CWk ; Ri ) Distfractal(CWsm , Ri ), 8k 62 S and 8sm 2 S .. Step 8:. Find tr (Dj ) such that Distfractal(tr (Dj ); Ri ) is the minimum, 8tr (Dj ) 2 Csm , m = 1; ; W . Then output Ii (1; the coordinate ofDj ; r; si ; oi ).. Step 9:. Find a set S = fs1 ; ; sW g such that Distfractal(CWk ; Ri ) Distfractal(CWsm , Ri ), 8k 62 S and 8sm 2 S .. Step 8:. Step 9:. Find tr (Dj ) such that Distfractal (tr (Dj ); Ri ) < T or Distfractal(tr (Dj ); Ri ) is the minimum, 8tr (Dj ) 2 Csm , m = 1; ; W . Then output Ii (1, the coordinate of Dj , r, si , oi ).. 4 Experiment Results and Performance Analysis In this section, we show our experiment results and analyze the performance of our algorithms. Our algorithm is implemented by Borland C++ Builder on PC with Intel CeleronTM processor 300A MHz and 64 MB RAM. Our testing images include "Lena", "F16",.

(43) "Pepper" and "Baboon". All of these images are of 256 gray levels with resolution 256 256. To measure the quality of the reconstructed image, we use the peak signal-to-noise ratio (PSNR), which is de

(44) ned as: PSNR = 10 log10 [. PLi P255Lj (xij 2. 1. LL. =1. =1. x^ij )2. ];. where L L = size of image, xij = pixel value of the original image at coordinate (i; j ), and x^ij = pixel value of the reconstructed image at coordinate (i; j ) [3, 7]. All decoding process in this paper uses an image with the initial value of each pixel is 128. And si and oi in Equation 2 and Equation 3 are quantized to 3 bits and 7 bits respectively. Now, we would like to show some of our experiment results. Table 1 shows the PSNR and the encoding time of our algorithm with various parameters. We

(45) nd that large search window size will get better quality. However, a small search window size will reduce the encoding time. According to the expriment results, we can get near PSNR if the threshold T = 9 N . Thus, the best parameters in our algorithm are that codebook size = 500, window size = 15 and T = 9 N . Table 2 shows the comparison of the PSNRs in iterations 1 through 9 for the conventional fractal algorithm, the local variance fractal algorithm and our algorithm. The PSNR of our algorithm converges after the sixth iteration but the local variance algorithm converges after the 8th or 9th iteration. Finally, the performance analysis is summarized in Table 3. The relative speedup of our algorithm to the conventional exhaustive search algorithm is about ten times. Not only our algorithm is faster than conventional fractal algorithm, but also the quality of our reconstructed images is as good as that of the conventional fractal algorithm. Our algorithm is also faster than the local variance algorithm. We also test for the 512 512 Lena image and the experiment results are shown in Table 4. In Table 4, we

(46) nd that the performance of our algorithm with the 512 512 Lena image is also very good.. 5 Conclusion In this paper, we propose a fast encoding algorithm for fractal image compression based on vector quantization. In the conventional fractal encoding algorithm, each range block performs an exhaustive search to

(47) nd a best match from the domain pool, so a large domain pool will signi

(48) cantly increase the search time. Thus, we propose a scheme to classify. Table 1: The PSNR and time of our algorithm with various parameters. NC : codebook size, W : search window size, T : threshold, N : range block size range block size. Test image is Lena 256 256. (a) Domain block size: 16 16; range block size: 8 8; bpp: 0.379. (b) Domain block size: 8 8; range block size: 4 4; bpp: 1.232.. Time (sec) PSNR LDF Fractal clustering encoding 27.4351 197.735 52.212 27.9438 197.870 179.116 28.0895 197.824 337.957 28.1305 197.850 473.223 28.1149 198.784 302.135 28.0427 198.815 253.136 (a) Time (sec) Parameters PSNR LDF Fractal clustering encoding C = 500 =1 32.8442 93.449 115.315 =5 33.5797 93.735 376.951 = 10 33.7773 93.760 723.731 = 15 33.8678 93.751 1004.328 = 15 33.6956 86.066 407.394 =9 (b) Parameters C = 500 =1 =5 = 10 = 15 = 15 =9 = 15 = 16 N. W W W W W T. N. W T. 249.947 376.986 535.781 671.073 500.919 451.951. N. N. W W W W W T. Total. ;. Total 208.764 470.686 817.491 1098.079 493.460. N. the domain blocks by using the longest distance

(49) rst (LDF) algorithm. In average, our method can reduce window size = W at least, the search space to search codebook size NC 15 the percentage is 500 = 3% in this paper. Theoretically, we can also reduce the same percent of the encoding time. That is, we should reduce the encoding time about 33 times when the percentage is 3%, but actually it is only about ten times. The reason is, the overhead in training the codebook and

(50) nding the closest codeword set. Thus, how to decrease the overhead is one of our future works. Experiment results show that our method is faster than the conventional fractal encoding method and the local variance method, and we still have good quality of the reconstructed images under the same compression ratio.. References [1] M. F. Barnsley, Fractals everywhere. San Diego, USA: Academic Press, Inc., 1988..

(51) Table 2: Comparison of the PSNRs after executing 1 through 9 iterations. (a) Domain block size: 16 16; range block size: 8 8; bpp: 0.379. (b) Domain block size: 8 8; range block size: 4 4; bpp: 1.232.. # of Conventional iterations fractal 1 2 3 4 5 6 7 8 9. 19.135 22.594 25.470 27.319 27.982 28.145 28.153 28.142 28.136. (a). # of Conventional iterations fractal 1 2 3 4 5 6 7 8 9. 21.031 25.769 29.989 32.680 33.687 33.950 33.982 33.967 33.959. (b). PSNR LDF fractal 19.416 22.914 25.737 27.444 28.044 28.167 28.144 28.125 28.115 PSNR LDF fractal 21.710 26.457 30.482 32.757 33.514 33.711 33.710 33.702 33.696. Local variance fractal 18.077 21.260 24.118 26.155 27.140 27.396 27.464 27.478 27.479 Local variance fractal 20.010 24.254 28.316 31.397 32.903 33.421 33.541 33.559 33.551. [2] C. K. Chan and C. K. Ma, \Maximum descent method for image vector quantization," Electronics Letters, Vol. 27, No. 19, pp. 1772{1773, Sep. 1991. [3] C. K. Chan and C. K. Ma, \A fast method of design better codebooks for image vector quantization," IEEE Transactions on Communications, Vol. 42, No. 2/3/4, pp. 237{243, Feb./Mar./Apr. 1994. [4] Y. Fisher, and. Fractal image compression:. .. application. Verlag, 1994.. theory. New York, USA: Springer-. [5] A. Gersho and R. M. Gray, Vector quantization and signal compression. Boston, USA: Kluwer Academic Publishers, second ed., 1992. [6] R. Hamzaoui, \Codebook clustering by selforganizing maps for fractal image compression," Fractals, Vol. 2, No. 0, 1994. [7] M. C. Huang and C. B. Yang, \Fast algorithm for designing better codebooks in image vec-. Table 3: Performance of various fractal algorithms. LDF fractal is our algorithm; the window size of local variance fractal I is 3000. The window size of local variance fractal II in (a) is 10000 and in (b)is 24000. (a) Domain block size: 16 16; range block size: 8 8; (b) Domain block size: 8 8; range block size: 4 4. Conventional Algorithm Lena bpp= 0.379 F16 bpp= 0.339 Pepper bpp= 0.461 Baboon bpp= 0.400. PSNR Time (sec) PSNR Time (sec) PSNR Time (sec) PSNR Time (sec). Algorithm Lena bpp= 0.379 F16 bpp= 1.176 Pepper bpp= 1.759 Baboon bpp= 1.316. PSNR Time (sec) PSNR Time (sec) PSNR Time (sec) PSNR Time (sec). fractal. fractal. LDF. Local variance fractal I. Local variance fractal II. 28.136 5262.174. 28.115 500.919. 27.159 541.105. 27.479 895.134. 25.792 4369.355. 25.630 425.703. 25.113 451.424. 25.293 746.070. 26.957 5722.120. 26.884 610.994. 26.211 586.935. 26.484 972.130. 23.149 7073.116. 23.075 637.094. 22.801 717.960. 22.918 1173.390. (a) Conventional LDF fractal. fractal. Local variance fractal I. Local variance fractal II. 33.959 5345.645. 33.696 493.460. 32.331 536.320. 33.551 2081.676. 32.195 4895.065. 32.025 552.324. 30.290 500.595. 32.000 1927.050. 33.341 6032.285. 33.147 616.644. 31.964 589.800. 32.912 2277.370. 27.106 9580.355. 26.952 1200.984. 25.388 980.895. 26.514 3636.090. (b). tor quantization," Optical pp. 3265{3271, Dec. 1997.. Engineering. , Vol. 36,. [8] A. E. Jacquin, \Image coding based on a fractal theory of iterated contractive image transformations," IEEE Transactions on Image Processing, Vol. 1, No. 1, pp. 18{30, Jan. 1992. [9] A. E. Jacquin, \Fractal image coding: a review," Proceedings of the IEEE, Vol. 81, No. 10, pp. 1451{1465, Oct. 1993. [10] C. K. Lee and W. K. Lee, \Fast fractal image block coding based on local variances," IEEE Transactions on Image Processing, Vol. 7, No. 6, pp. 888{891, June 1998. [11] Y. Linde, A. Buzo, and R. M. Gray, \An algorithm for vector quantizer design," IEEE Transactions on Communications, Vol. C-28, No. 1, pp. 84{95, Jan. 1980. [12] T. Murakami, K. Asai, and E. Yamazaki, \Vector quantization of video signals," Electronics Letters, Vol. 7, pp. 1005{1006, 1982. [13] B. Ramamurthi and A. Gersho, \Classi

(52) ed vector quantization of images," IEEE Transactions on Communications, Vol. 34, No. 11, pp. 1105{ 1115, Nov. 1986. [14] D. S. Raouf Hamzaoui, Martin Muller, \VQEnhanced fractal image compression," IEEE International Conference on Image Processing. , pp. 1{4.. (ICIP'96), Lausanne, Sept. 1996.

(53) Table 4: Performance of various fractal algorithms with the 512 512 Lena image. the parameters in our algorithm are that codebook size = 2000, window size = 15 and T = 9 N . (a) Domain block size: 16 16; range block size: 8 8; bpp: 0.380; the window size of local variance fractal is 40000. (b) Domain block size: 8 8; range block size: 4 4; bpp: 1.289; the window size of local variance fractal is 96000. Algorithm. PSNR. Conventional fractal LDF fractal Local variance fractal. (a). (b). Time (sec). PSNR. Time (sec). 29.990. 131968.480. 35.058. 88763.863. 29.860. 20665.195. 34.817. 7297.361. 29.379. 34826.097. 34.587. 31865.435. [15] S. C. Tai, Data compression. Taipei, Taiwan: Unalis, second ed., 1998..

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