Fractal Image Compression By Using Loss-Less Encoding On The Parameters Of Affine Transforms

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1 Fractal Image Compression By Using Loss-Less Encoding On The Parameters Of Affine Transforms Utpal Nandi Dept. of Comp. Sc. & Engg. Academy Of Technology Hooghly ,West Bengal, India Jyotsna Kumar Mandal Dept. of Comp. Sc. & Engg. University of Kalyani Nadia , West Bengal, India Abstract A way of improvement of compression ratio of fractal image compression is proposed in this paper. The improvement of compression rates is done by applying the lossless compression techniques on the parameters of the affine transformations of the fractal compressed images. The Modified Region Based Huffman and its variant are used for this purpose. The PSNR of images are remained same. The comparison of the compression ratio and time are done between fractal image compression with quardtree partitioning schemes, the same with Huffman coding and its proposed improved versions. The proposed improved fractal image compression techniques offer better compression rates most of the times keeping the PSNRs unchanged. But, the compression time of proposed techniques are significantly increased than its counterparts. Keywords- Fractal compression; Compression ratio; Quardtree partition; MRBH; MRBHM; Partitioned Iterated Function Systems (PIFS) I. INTRODUCTION The memory space needed of GUI-based application softwares increasing at a very high rate. DCT-based JPEG compression techniques [11, 12] obtain high compression rates by eliminating the high frequency components of the image. It is quite fine at low compression ratios. But, the images become blocky with the increasing compression rate and poor quality images are produced. Sometime very visible artifacts are found, particularly for sharp edges in the images. Again, it is resolution dependence. Pixels replications are done to zoom-in on a particular area of images and to magnify it. The resultant images exhibit a certain level of blockiness. To overcome this problem, the same image is stored at different resolutions wasting more memory space. Therefore, this well-established standard JPEG has its limits. An alternative technique i.e. fractal image compression [3, 4, 6, 7] is introduced to solve the problems associated with JPEG and discussed in section II. The technique offers good quality images at high compression ratios also and is resolution independent. In this paper, a way to improve the compression ratio of the same is proposed. Fractal compressed file contains a collection of affine transforms. The improvement of compression rates is done by applying the loss-less compression techniques on the parameters of the affine transformations of the compressed images. Though, any loss-less techniques can be applied, Modified Region Based Huffman with code interchange(mrbh) [1] and its variant Modified Region Based Huffman with multiple interchanging of code (MRBHM) [2] are used for this purpose here. Huffman Coding constructs Huffman tree depending on the occurrence of symbols of file. If a symbol has maximum occurrence, it is given with the shortest code. The MRBH compression procedure divides the input file into a number of regions. The number of region is chosen by a RSA program [1]. For each region, the Huffman code between the highest frequency element of that region and highest frequency element of entire file are swapped and the elements of that region are encoded. The same process is continued for each region. A MRBHM technique is based on generalized Region Based Huffman (RBH) compression by interchanging region wise multiple codes of symbols. The input stream/file is divided into a number of regions. MRSA algorithm [2] is used to select the proper values of the number of regions and the number of region wise interchanging of codes. Huffman code of more than one high frequency symbols of each region are interchanged with high frequency elements of entire file and the elements of that region are encoded. This same process is repeated for each region. The proposed improvement of this coding technique is discussed in section III in detail. Results are given in section IV and conclusions are drawn in section V. Then, the references are followed. II. FRACTAL IMAGE COMPRESSION The fractal images are used to designate images that are self-similar at different scales such as images of clouds, trees, mountains. These images have details at every scale. The fractal concept for compression of image is identified after development of the theory of Iterated Function Systems (IFS). An IFS contains a list of contractive transformations {W i : R 2 R 2 i=1,2,,n} that map the plane R 2 to itself. The list of transformations defines a map W(.)= that has list two important facts:

2 1. If the Wi in the plane are contractive, then W is a contractive in a space of (closed and bounded) subsets of the plane. 2. For a given a W on a space of images, there is a unique image called the attractor and denoted X w, that has the following properties: a. If W is used to the X w, the outcome is equal to the input and X w is termed as fixed point of W. That is, W(X w )=X w =W 1 (X w )UW 2 (X w )U..UW n (X w ). b. Given an input image S 0, S 1 =W(S 0 ), S 2 =W(S 1 )= W(W(S 0 ))=W 02 (S 0 ), and so on are obtained. The attractor does not depend on the choice of S0 and is given by X w = 0n (S 0 ). 1. Input a gray scale square image F and choose a tolerance level (quality) Q (Encoding with lower Q will have better fidelity). 2. Select minimum range R min and set the initial range R 1 =I 2 (unit square) and mark it Uncovered. 3. Select a domain pool D which must be bigger than the range to keep the contractive mapping from the domain to range. 4. WHILE(uncovered ranges R i exist), DO 4.1 From domain pool D, obtain the domain Di and the affine map Wi that well cover range R i i.e. that minimize the D rms (F (R i X I),W i (F)), where I means interval [0,1] and RMS(root mean squre) metric c. X w is unique. These properties are called the contractive Mapping Fixed- Point Theorem. These properties are used to compressed image. But, the significant compression ratios can be found only on specially constructed images using IFS and only with a human helping the compression technique. It is not possible to completely automate the compression technique. Thus, image compression using IFS is not practically possible without a person s involvement. The problem is managed with the development of the theory of Partitioned Iterated Function Systems (PIFS) where the image is partitioned into number of ranges that are non-overlapping, and range-wise IFS are found. Now, this can be automatically done. In fractal compression with PIFS, the image to be compressed itself is the dictionary. Decompression process can reconstruct the image by successive iteration of PIFS. The compression technique initially divides the image into a list of non-overlapping ranges Ri. A domain pool, D is also created by partitioning the image. Domains may overlap to each other. There are many possible partitioning schemes that can be used to select the range R i like quardtree, HV and triangular partition [4-7]. But, the quardtree partition is used here. In a quardtree partitioning scheme, a range of the image is divided into 4 same size sub-ranges when the range is not covered by any domain of the domain pool. This process continues recursively beginning from the entire image until the ranges are little enough to be covered with in a predefined RMS value. To find the match among a Dj and a Ri, the compression technique obtains s the good suitable W such that the W(Dj) is very similar to the image Ri. The affine transformations are used for this purpose. The better W from a Ri to a Dj is found by minimizing the RMS value 4.2. IF ( ( D rms (F (R i X I), W i (F)) < Q ) OR ( SIZE(R i ) <= R min )), THEN Cover range R i and output the W i to the output file. ELSE Partition range Ri into four equal sized sub-square ranges R i1, R i2, R i3, R i4 (Quardtree Partition ) that are marked as uncovered ranges. END IF; END WHILE; 5. STOP. Figure 1. Fractal compression algorithm. among Ri and W(Dj) as a function of the brightness (b) and contrast (c). To simplify the technique and to eliminate the time consuming square root operation, the Mean Square value is used rather than RMS as given in equation number 1. where n is the total pixel elements in the range and d i, r i the values of pixel elements in the domain and the range respectively. The proper values of brightness and contrast are found at the time of zero partial derivatives of E(c, b) w.r.t to b and c. The complete algorithm of the fractal image compression technique with quadtree partitioning scheme (1)

3 is shown in Fig. 1. The PIFS constructed with the technique contains a collection of affine maps. Each of which is for a particular range block. The contractive mapping is used to decompress the compressed image as contrast factor < 1 (contractive mapping). Beginning with any the decompression technique use the PIFS repeatedly. This is converged to the fixed point of the PIFS. III. THE PROPOSED SCHEME The performance in term of compression ratios obtained with the proposed fractal compression technique can easily be improved by increasing the value of quality. But, encoding image with higher quality will have poor fidelity, decreasing the PSNR of the decompressed image. Another way to enhance the compression rate is to increase the value of domain density. Therefore, the distance between two consecutive domains is reduced and the number of domain in the domain pool is significantly increased. But, the compression of image with large domain pool requires lots of time. That is, the compression time is significantly increased by a factor with the increasing of domain pool. A way of improvement of this compression technique can be done without decreasing the PSNR and increasing highly the compression time by applying any loss-less data compression technique on the parameters of affine transformations of the compressed image. The proposed technique compresses the input image using fractal technique with quardtree partitioning scheme where the image is divided into 4 same size sub-images that is given in Fig.2. The compression technique continues recursively begining from the entire image until the squares are either covered with in some specified RMS tolerance or smaller than the specified minimum range size. It produces a collection of affine transformations as shown in Fig. 3. The constants a i,j and d i,j representing scaling factor and top-left corner of domain respectively. Two constants c i and b i representing contrast and brightness factor respectively, specify the luminance part. Therefore, affine transform is basically a collection of parameters and any loss-less data compression technique can be applied on the parameters of the affine transformations that produce the ultimate compressed image as shown in Fig.4. Here, MRBH coding and MRBHM coding are applied for this purpose. The proposed fractal image compression with MRBH and MRBHM coding are termed as FIC-MBRH and FIC-MRBHM coding respectively. The process of decompression is started by decompressing the compressed affine transformations by loss-less data compression technique. Then, the fractal decompression technique generates the decompressed image based on affine transformations as shown in Fig.5. That is, begining from an any image and using the IFS repeatedly, the decompressed image is generated as shown in Fig. 6. The technique converges to a fixed image that depends only on the IFS and does not depend on the input image. Figure 2. Quardtree partitioning scheme. Figure 3. Affine transform. Figure 4. Compression process. Figure 5. Decompression process.

4 gradually decrease with the increasing quality values for all techniques. TABLE I. File name COMPARISON OF COMPRESSION RATIOS OF IMAGES FIC- FIC- FICQP FIC-H MRBH MRBHM CHEETAH.GS LISAW.GS ROSE.GS MOUSE.GS CLOWN.GS Figure 6. Decoding of image. IV. RESULTS For experimental purpose, five grey scale image files having size Bytes (200 X 320) have been taken. The GS, suffix is for non-formatted grey-scale image files. The comparison of compression ratios, compression times among fractal image compression technique with quardtree partitioning scheme (FICQP), the same with Huffman coding (FIC-H) and the proposed fractal techniques FIC- MRBH and FIC-MRBHM have been made in table I with a domain density value of 1 and image quality 1 for five gray scale images. The compression times are given in table II for the same. The pictorial representations of compression ratios, compression times are illustrated in Fig. 7 and Fig. 8 respectively. The proposed all two techniques offer comparatively better rate of compression than its counterpart for all five images. The performance in terms of compression ratio of FIC-MRBHM is much better than FIC- H and FIC-MRBH. But, the compression times taken by the proposed techniques are much more than existing techniques for all five images. FIC-MRBHM takes much more compression time than others. However, the PSNR of proposed techniques are same with existing techniques as given in table III. The comparison of compression ratios and PSNR with increasing quality values (Here encoding with lower quality offers better fidelity) for a particular image (MOUSE.GS) for all four techniques has been done in table IV. For all four techniques, the compression ratios increase with the increasing quality values. The pictorial representation of that is illustrated in Fig. 9. But, the PSNRs of the image gradually decrease with the increasing quality values as illustrated in Fig. 10. The comparison of compression times with increasing quality values for the same MOUSE.GS image for all four techniques have been made in table V and the pictorial representation of that is illustrated in Fig. 11. The compression times of the image Figure 7. TABLE II. Figure 8. The graphical representation of comparison of compression ratios of images. COMPARISON OF COMPRESSION TIMES OF IMAGES FIC- FIC- FICQP FIC-H File name MRBH MRBHM CHEETAH.GS LISAW.GS ROSE.GS MOUSE.GS CLOWN.GS The graphical representation of comparison of compression times of images.

5 Quality Factor(Q) TABLE III. PSNR OF IMAGES File name PSNR (in db) CHEETAH.GS LISAW.GS ROSE.GS MOUSE.GS CLOWN.GS TABLE IV. COMPARISON OF COMPRESSION RATIOS AND PSNR WITH VARYING QUALITY FACTORS FOR MOUSE.GS IMAGE FIC- FIC- FICQP FIC-H MRBH MRBHM PSNR (in db) TABLE V. Quality Factor(Q) COMPARISON OF COMPRESSION TIMES WITH VARYING QUALITY FACTORS FOR MOUSE.GS IMAGE FICQP Time FIC-H FIC- MRBH FIC- MRBHM Figure 11. The graphical representation of comparison of compression times with varying quality factorss for MOUSE.GS image. Figure 9. The graphical representation of comparison of compression ratios with varying quality factors for MOUSE.GS image. V. CONCLUSION The proposed Fractal Image Compression by using loss-less encoding techniques MRBH and MRBHM coding on the parameters of affine transform FIC-MRBH and FIC- MRBHM improve the compression rates of images significantly. Though, the PSNR of existing and proposed techniques are same, the encoding and decoding time of images are increased in the proposed improved versions. Among two proposed compression techniques, FIC- MRBHM technique offers better compression rates. But, it takes more compression time than its counter parts. Farther investigation is needed to reduce the compression time of the proposed techniques without loss of PSNR. The same concept can also be applied to Adaptive Quardtree and HV partitioning schemes. ACKNOWLEDGMENT Figure 10. The graphical representation of PSNR with varying quality factors for MOUSE.GS image. Both the authors extend thanks to the CSE Dept., University of kalyani and PURSE Scheme of DST, Government of INDIA and CSE Dept., Academy Of Technology (AOT), WEST BENGAL, INDIA for providing the infrastructures.

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