A simplified fractal image compression algorithm

A simplified fractal image compression algorithm A selim*, M M Hadhoud $,, M I Dessouky # and F E Abd El-Samie # *ERTU,Egypt $ Dept of Inform Tech, Faculty of Computers and Information, Menoufia Univ, 32511, Shebin Elkom, Egypt # Dept of Electronics and Elect Communications, Fac of Electronic Eng, Menoufia Univ, 32952, Menouf, Egypt E-mails: a_seleem_87@yahoocom, mmhadhoud@yahoocom and fathi_sayed@yahoocom ABSTRACT This paper proposes a simplified fractal image compression algorithm on a block by block basis This algorithm achieves a compression ratio of up to 10 with a peak signal to noise ratio PSNR as high as 35db The idea of the proposed algorithm is based on the segmentation of the image, first, into blocks to setup reference blocks The image is then decomposed again into block ranges and a search is carried out to find the reference blocks with best match The transmitted values are the reference block values the indices of the range s match If there is no match, the average value of the block range is transmitted The advantage of the proposed algorithm is the simple computation with high PSNR achieved Keywords: Cmopression, Fractal, Decomposition, Segmentation 1 INTRODUCTION There are many techniques used to compress the image and video, one of them is the fractal which is based on the partition of the image into regions or blocks It doesn t depend on the translation of the image to the frequency domain as deferential pulse code modulation DPCM, which depends on discrete cosine transform DCT DPCM suffers from blocking artifacts which is not the case in fractal code The image compression based on fractal image compression (FIC) is one of the most popular modern image coding methods since 1988 It uses the self similarly method feature and has many characters such as, long coding time, fast decoding, high compression ratio, and decoding image has no related to resolution [1,2,3] Barnsly was the first one who gave the attractor model of the two dimension affine transforms He brought and developed methods to compress the image based on the iterated function system [8],[ 9], [10] Generally FIC is a loss compression technique In fractal image compression, the image is divided into numbers of block domains by any shape and any size (usually 16x16 to 2x2) then the image is divided again to block ranges with size less than that of block domain There are many methods to obtain attractors to decode the image Some of these methods are, block segmentation, region segmentation and cross searching; these expand self-similar fractal serial image coding Other methods are fast fractal image coding based on notably irevant check, image compression and coding method based on fractal dimension, and hierarchy fractal image compression method based on fixed vector [1], [3], [5], [6] Some of them will briefly be explained in sections (2-1), and (2-2) In our proposed algorithm the image is divided into equal squared regions, then search about reference blocks in each region The image again is divided into block ranges Then we search the reference blocks for a matched range, its indices is transmitted instead of the range itself If there is no matched reference, the average of the range is transmitted In the decoder all the pixels of the range equal the average value In this algorithm we don t use the count of the transformation which increases the complexity of the encoder; it uses the absolute difference to determine the similarity between the blocks The resulted or decoded image is acceptable with PSNR>30, at compression ratio=10 The main advantage of the proposed algorithm is that it is a simple in encoding and decoding Our proposed algorithm is explained in section 3 Some results are explained in section 4 Section 5 is for the Conclusion 2 Basics of Fractal Image Compression The word fractal was coined by Mandelbrot from the Latin word fractus, meaning broken, to describe objects that were too irregular to fit into traditional geometrical settings Several definitions have been proposed Mandelbrot defined a fractal to be a set with Hausdorff dimension strictly greater than its Euclidean dimension, ie, a set for which the only consistent description of its metric properties requires a dimension value larger than our standard, intuitive definition of the set s Ubiquitous Computing and Communication Journal 1

dimension A fractal has a fractional dimension; thus some people say we get the word fractal from fractional dimension According to Bamsley, a fractal is a geometric form whose irregular details recur at different scales and angles which can be described by affine or fractal transforms Various other definitions have been proposed, but they are not complete in that they exclude a number of sets that clearly ought to be regarded as fractals Falconer proposed that it is best to regard a fractal as a set that has properties such as those listed below, rather than to look for a precise definition which will almost certainly exclude some interesting cases Typical properties of a fractal are 1 It has a fine structure, ie, details on arbitrarily small scales 2 It is too irregular to be described in traditional geometrical language, both locally and globally 3 It usually has some form of self-similarity, perhaps approximate or statistical 4 Its fractal dimension (Hausdorff dimension) is usually higher than its Euclidean dimension 5 In most cases of interest, a fractal is defined in a very simple way, perhaps recursively In block segmentation (BS) fractal compression methods, the image of size (N x M) is decomposed into partitions 2 Types of segmentations 21 Block based FIC In block segmentation (BS) fractal compression methods the image of size (N x M) is decomposed into partitions (blocks in this case) called block domain pool D, each block domain D i of size (a x b) is; D i ; where 0 <i N 1 ; and N 1 is the number of the domain blocks N 1 =(NxM)/(axb) (1) To code the image we decompose it again into block ranges R i with size (aa x bb) less than the size of D i Each block range R i has the image transform τ i, which is also called the compressive code of the range block, and contains two parts: gray (T i ), and geometry (S i ) transform or; τ i =T i +S i (2) and the total transformation of the total ranges of the image τ; these transformations saved in or transmitted to decoder Instead of saving or transmitting the whole block range[1],[4], where τ is τ = τ i (3) Where 0< i (N/aa x M/aa) is the number of block ranges The processing of FIC is for each range block R i, search the domain pool for a block domain which has minimum distortion (d i ) after the transformation and the transform τ i which has the minimum d i is the compression code 22 Region segmentation In this method the image is decomposed into regions -instead of blocks in the BS method-these regions are called R i with any size and any shape, τ i is the affine transform from D i (block domain) to R i and τ; is the total transformation defined by equ3 The FIC processing is to find affine transform τ i (D i ) such that the distortion between D i and R i is defined by d(r i,τ i (D i ))<τ i (4) Then the transformations τ are transmitted or saved, which can be used to decode the image at the decoder [1,5] 3 The proposed algorithm The proposed algorithm is based on block and region segmentation The steps of the algorithm are as follows; 1- Load an image of size N x M, if (N/a), and (M/b) are not integer numbers, we extend the image to become of reasonable size for each block 2-Decompose the image into block domains D (i, j) with size (a x b), (a=b=16 or 8) as in fig1 Where, i=1:n/a, and j=1:m/b 3-Compare all block domains in each region segmented (which are colored with red, green ) to detect which of the domain blocks may be used as a reference The test is based on the distortion which is given by d(d(i,j),d((i+t),(j+w)); (5) Where t, and w=-1, or 1 This compared to a threshold (thsh) Fig2 4- Make a domain pool to be transmitted or saved [D 1, D 2 D i1 D NRB ]=[RB(1,1),RB(1,2), RB( i, j ), RB(N/a,M/b)] (6) Where NRB is the Number of Reference Blocks 5-Decompose each reference block domain into blocks called reference ranges R ref of size (aa, bb), Ubiquitous Computing and Communication Journal 2

which will be saved or transmitted directly without processing 6-Count the number of pixels of the transmitted reference blocks (Ref Pix); Ref Pix= NRB x (a x b) (7) 7-Decompose the image again into block ranges R of size (aa x bb) (usually aa=bb=4) 8-For each block range R i search the reference block ranges in the region of this range, if there is no matched range, search other reference block ranges R ref in the other regions 9-Find the indices of this matched reference (d x, d y ), transmit or save it 10-Count the number of pixels which is used to transmit or save the range block (Range pix); Range pix=2x(number of matched ranges) (8) Transmit the average of the block range 12-Count the number of pixels which is used to transmit averaged blocks (av Pix); avpix = number of averaged blocks (9) 13-The total number of transmitted or saved pixels, (eqs6, 7, 8); dn = avpix + Range pix + ref pix (10) 14-The compression ratio is; CR=(NxM)/dN (11) 15-After decoding the image, compute the peak signal to noise ratio PSNR PSNR = 10 log ( N,M (original image s pixels decoded image s pixels)) / (number of the original image s pixels) (12) 4 Experimental Results It is obvious that the algorithm doesn t search for the affine transforms of each range block; it is directly search for similarity between the range block and a reference block in a reference domain based on the distortion ( eq5) -We used two of Mat Lab images Saturn and pout of sizes ( 328 x 438)and (291 x 240 ) respectively These sizes are extended to (336x448) and (304x240) respectively The sizes of the block domain used are a = b = 16 and 8, the size of the range blocks is, aa x bb=4x4, and the threshold takes values from 001 to 10 The decoded pout images are shown in figs4 to 13, and the original image in Fig3 The original Saturn image is shown in fig 14, and its decoded images are shown in Fig15 to 25 -The resulted compression ratios are change from 3 to 10, where PSNR s are ranges from 12 to 35 for Saturn - For pout image the CR changes from35 up to 7 and PSNR from 13 up to 31 -The decoded images are accepted especially at higher PSNR (> 20) - Fig26 and Fig27 show the relation between PSNR versus CR for the two images - As expected the PSNR decreased by increasing the threshold which means that we are increase the distortion in the decoded image and this is a logical relation The curves which shows the relation between the PSNR and the threshold are shown in Fig28 and Fig29 The resulted decoded images are acceptable with a CR =10 or 7 in the tow cases This algorithm mainly gives the simplification in encoding the image which is complexity and consumes a lot of times Usually the relation between the PSNR and CR is, decreasing in PSNR with increasing in CR, but in our results the PSNR increases with increasing in CR The explanation of this case is; For each range we are searching the domain pool (reference domain) for a matching reference range using a certain threshold (thsh), if the distortion between the range and it s matched range less than the threshold, we replace the range by its matched reference range and transmit it It is not the same but the distortion between them is less than the threshold Which decrease the PSNR If there is no matched range with distortion less than the threshold we transmit the average of the range The average is one value (also the number of the range which has a matched range is decreased) so CR increase with increasing PSNR We can consider that it is a processing of searching for an optimum values to compress the image, then we use only this point or the value of threshold We can consider that it is a processing of searching for an optimum values to compress the image, and then we use only this point or the value of threshold This algorithm gives a new method to encode the image in which instead of use the old iterated methods we transmit the average of the range for simplification The resulted CR is more than that resulted by some other methods as spiral architecture [3] The time consumed to encode and decode the image by the proposed algorithm is 16 sec in case of block size of 16x16 and 46 sec in case of 8x8 It is very fast if we compare this time with times consumed by Ubiquitous Computing and Communication Journal 3

many other fractal methods which may be thousands of sec [2] 5 Conclusion The paper proposed a simplified fractal image compression algorithm The computational complexity of this algorithm is small The achieved compression ratio using this algorithm is moderate Experimental results show that algorithm can be used as a fast fractal based compression algorithm with high efficiency 6 References 1- Erjun Zhao Dan Liu: `Fractal image compression methods: A review` (ICITA`05) 0-7695-2316-1/05 $2000 2005 IEEE 2- Kin-wah ching Eugene and Ghim-Hwee Ong: `A two-pass improved encoding scheme for fractal image compression` (CGIV`06) 0-7695-2606-3/06 $2000 2006 IEEE 3- Xiangjian He, Huaqing Wang: `Fractal image compression on Spiral Architecture` (CGIV`06) 0-7695-2606-3/06 $2000 2006 IEEE 4- Barnsly M F A D: A Better way to compress image byte, 1988 5- A Is M, Clarkson T: Survey of block based fractal image compression and its application 6- AEJacquin,: image coding based on fractal theory of Iterated Contractive image transformations, IEEE Trans On image processing, 18-30,1992 7- Y Fisher (editor),: fractal image compressiontheory and applications Springer-verlag, new york, USA, 1995 8- Ning lu, :Fractal imaging, academic press, new york,1997 9- Brent Wohlberg and Gerhard de Jager: A review of the Fractal Image Coding Literature, IEEE trans On image processing, vol8,no12, December 1999 10-Barnsley M F, Sloan A DA: better way to compress images Byte,1988 Ubiquitous Computing and Communication Journal 4

D(1,1) D(1,2) D(1,3) D(1,4) D(1,5) D(1,6) D(1,7) D(2,1) D(2,3) D(2,3) D(2,4) D(2,5) D(2,6) D(2,7) D(3,1) D(3,2) D(3,3) D(3,4) D(3,5) D(36) D(3,7) D(4,1) D(4,2) D(4,3) D(4,4) D(4,5) D(4,6) D(4,7) D(5,1) D(5,2) D(5,3) D(5,4) D(5,5) D(5,6) D(5,7) D(6,1) D(6,2) D(6,3) D(6,4) D(6,5) D(6,6) D(6,7) D(7,1) D(7,2) D(7,3) D(7,4) D(7,5) D(7,6) D(7,7) Figure 1: Block segmentation and block reference searching RB(1,1) RB(1,2) RB(2,1) RB(2,2) Figure 2: Reference blocks in each region Ubiquitous Computing and Communication Journal 2

Figure 3: The original image Figure :4 Thsh=001, CR=7 PSNR=31, a=b=16, aa=bb=4 Figure 5: Thsh=01, PSNR=3095, a=b=16, CR=618 aa=bb=4 Figure 6: Thsh=05, CR=422 PSNR=296, a=b=16, aa=bb=4 Figure 7: Thsh=1, CR=413 PSNR=266, a=b=16, aa=bb=4 Figure 8: Thsh=2, CR=38 PSNR=215, a=b=16, aa=bb=4 Figure 9: Thsh=1, CR=26 PSNR=267 a=b=8 aa=bb=4 Figure 10: Thsh=2, CR=32989 PSNR =206, a=b=8, aa=bb=4 Figure 11: Thsh=05, CR =21 PSNR=292, a=b=8, aa=bb=4 Ubiquitous Computing and Communication Journal 3

Figure 12: Thsh=01, CR =52 PSNR =305, a=b=8, aa=bb=4 Figure 13: Thsh=001,CR =72 PSNR =346, a=b=8, aa=bb=4 Figure 14: The original image Figure 15: Thsh=001, CR=986, PSNR=357,a=b=16,aa=bb=4 Figure 16: Thsh=5,CR=92, PSNR=275,a=b=16,aa=bb=4 Figure 17: thsh=4, CR=934, PSNR=309, a==b=16, aa=bb=4 Figure 18: thsh=1, PSNR=357, a=b=16, CR=971, aa=bb=4 Figure 19: Thsh=8,CR=87, PSNR=177, a=b=16, aa=bb=4 Figure 20: thsh=01, CR=98, PSNR=357, a=b=16,aa=bb=4 Figure 21: thsh=05, CR=975, PSNR=358, a=b=16,aa=bb=4 Figure 22: Thsh=10, CR=84, PSNR=131, a=b=16, aa=bb=4 Ubiquitous Computing and Communication Journal 4

Figure 23: Thsh=05, CR=999, PSNR=346, a=b=8, aa=bb=4 Figure 24: Thsh=01, CR=10, PSNR=346, a=b=8, aa=bb=4 Figure 25: Thsh=00,1 CR=101, PSNR=346 a=b=8 aa=bb=4 40 cr vs PSNR for "saturn" image 31 cr vs PSNR for "pout" image 30 35 29 30 28 27 PSNR 25 PSNR 26 25 20 24 15 23 22 10 84 86 88 9 92 94 96 98 10 CR 21 35 4 45 5 55 6 65 7 CR Figure 26: CR versus PSNR for saturn image Figure 27: CR versus PSNR for pout image Ubiquitous Computing and Communication Journal 5

31 thsh vs PSNR for "pout" image 40 thsh vs PSNR for "saturn" image 30 35 29 28 30 PSNR 27 26 PSNR 25 25 24 20 23 15 22 21 0 02 04 06 08 1 12 14 16 18 2 thsh Figure 28: PSNR versus threshold for saturn image 10 0 1 2 3 4 5 6 7 8 9 10 thsh Figure 29: PSNR versus threshold for saturn image Ubiquitous Computing and Communication Journal 6