IMAGE COMPRESSION USING HIRARCHICAL LINEAR POLYNOMIAL CODING

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Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg. 11-119 Avilble Online t www.ijsm.

1 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg Avilble Online t Interntionl Journl of Computer Siene nd Mobile Computing A Monthly Journl of Computer Siene nd Informtion Tehnology IJCSMC, Vol. 4, Issue. 1, Jnury 015, pg RESEARCH ARTICLE ISSN X IMAGE COMPRESSION USING HIRARCHICAL LINEAR POLYNOMIAL CODING Rsh Al-Tmimi 1, Ghdh Al-Khfji 1, Dept. of Computer Siene, College of Siene, University of Bghdd, Bghdd, Irq 1 rshtlib1@yhoo.om ; hgkt01@yhoo.om Abstrt In this pper hierrhl modelling bsed is introdued for ompressing imges, it is bsed utilizing the lyered representtion long with the polynomil oding. The test results showed best performne of the hierrhl polynomil oding ompred to the trditionl polynomil oding. Keywords imge ompression, redundny, modeling, hierrhl sheme, polynomil oding I. INTRODUCTION In reent yers, drmti inrese in the mount of informtion vilble in the form of digitl imge dt, it beome neessry to solve the problems of storge nd time issues by utilizing imge ompression of redundny removl bsed. In generl, Imge ompression tehniques generlly fll into two tegories: lossless nd lossy depending on the redundny type exploited, where lossless lso lled informtion preserving or error free tehniques, in whih the imge ompressed without losing informtion tht rerrnge or reorder the imge ontent, nd re bsed on the utiliztion of sttistil redundny lone suh s Huffmn oding, Arithmeti oding nd Lempel-Ziv lgorithm, while lossy whih remove ontent from the imge, whih degrdes the ompressed imge qulity, nd re bsed on the utiliztion of psyho-visul redundny, either solely or ombined with sttistil redundny suh s vetor quntiztion, frtl, blok truntion oding nd JPEG [1], reviews of lossless nd lossy tehniques n be found in [],[3],[4]-[7]. Modelling or Mthemtil Model is simple desription formul utilized effiiently in imge ompression problem to remove the orreltion embedded between imge pixel neighbours (sptil/interpixel redundny. A ompression system of modelled bsed, is generlly omposed of two prts; one orresponds to mthemtil funtion (deterministi prt exploited to rete n pproximtion modelled imge tht resemble the originl imge, nd the seond prt orresponds to the error or residul (probbilisti prt s differene between originl nd the pproximted. For more detils see [8], [9], [10]. Polynomil oding is modelling bsed tehnique exploited by number of reserhers s tool to ompress imges [11], [1], [13]-[16]. The tehniques hrterized by simpliity of implementtion, effiieny in reduing imge informtion into smll effetive oeffiients. In this pper, the polynomil oding tehniques dopted hierrhlly to effiiently remove the dependeny (orreltion or redundny between neighbouring imge pixels nd between neighbouring oeffiients. The rest of the pper orgnized s follows, setion disusses the proposed tehnique in more detils; the result is given in setion , IJCSMC All Rights Reserved 11

2 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg II. THE PROPOSED COMPRESSION SYSTEM The min tken onerns in the proposed system re: 1- Polynomil oding of liner pproximtion model is exploited to ompress imge effiiently using the three oeffiients (0, 1 nd representtion tht remove the redundny between the imge itself. - The top-down lyered or hierrhl sheme is dopted to remove the redundny embedded within the oeffiients to improve the ompression rtio with preserving imge qulity. The steps below illustrted the system implnttion in more detils; Figure (1 shows the bsi steps lerly: Step 1: Lod the input unompressed imge I of size N N tht orresponds to lyer 0 or the root of the hierrhl representtion. Step : Construt the first lyer hierrhl representtion orresponds to lyer 1 using the liner polynomil oding tehniques, suh s: 1. Prtition the input imge I into non-overlpping bloks of fixed sized n n (i.e., 4 4, Find the oeffiients of the liner pproximtion model, using the equtions below [17]: n n 0 I( j...(1 n n 1 I( j ( j x ( j x...( I( j ( i ( i y y...(3 Where I(j is the originl imge blok of size (n n nd n 1 x y...(4 Here the (j-x nd (i-y orresponds to the vribles of the polynomil tht mesure the distne of pixel oordintes to the blok enter (x, y. The 0 oeffiients represent the blok men, the 1 oeffiients nd oeffiients represent the rtio of sum pixel multiplied by the distne from the enter to the squred distne in i nd j oordintes respetively. 3. Quntized/dequntized the 1 nd omputed oeffiients bove, using the uniform slr quntizer. ( 1 1Q round 1D 1Q QS1... (5 QS1 ( Q round D 1Q QS1...( 6 QS1 One quntiztion step QS 1 is dopted for the 1 nd oeffiients (the sme quntiztion step used for both of them, for the quntized 1 Q, Q /de-quntized 1 D, D oeffiients. 015, IJCSMC All Rights Reserved 113

3 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg Step 3: Construt the seond lyer hierrhl representtion orresponds to lyer from the previous lyer oeffiients (lyer 1 oeffiients, the 0 orresponds to men (verge of the imge, the liner polynomil oding tehniques utilized, s follows: 1. Prtition the omputed 0 from lyer 1 into non-overlpping bloks of fixed sized n n (i.e., 4 4, 8 8, the size of 0 is equl to N/n N/n.. Find the oeffiients of 0 of the liner pproximtion model, using the equtions below: n n 00 0( j......(7 n n 0( j ( j x 01...( 8 ( j x 0( j ( i y 0 ( i y...( 9 Where 0 (j is the men of originl imge of blok of size (n n nd n 1 x y......( 10 The 00, 01 nd 0 oeffiients orrespond to lyer onstruted using the 0 oeffiients from lyer 1 tht regrded s n imge. 3. Quntized/dequntized the 00, 01 nd 01 omputed oeffiients bove, using the uniform slr quntizer. ( 00 00Q round 00D 00Q QS00...( 11 QS00 ( 01 01Q round 01D 01Q QS01... (1 QS01 ( 0 0Q round 0D 0Q QS01...( 13 QS01 Two quntiztion steps QS 0, QS 1 dopted one for the 00 oeffiients, nd one for 01 nd 0 oeffiients, for the quntized 00 Q, 01Q, 0Q /dequntized 00 D, 01D, 0D oeffiients. 4. Determine the deterministi prt (funtion formul 0 of mthemtil liner model bse using the dequntized oeffiient nd the vribles. D D( j x D( i y...(14 5. Find the probbilisti prt or error (residul s differene between the modelled pproximted imge 0 nd the originl one 0. 0E ( , IJCSMC All Rights Reserved 114

4 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg Quntized/dequntized the error, using the uniform slr quntizer. ( 0 0EQ round E 0ED 0EQ QS0E...( 16 QS0E QS Where 0 E is the error quntiztion step for the quntized 0 EQ /dequntized 0E D oeffiients. Step 4: Build the pproximted up lyers from the subsequent lyers, nmely onstrut lyer 1 from lyer nd lyer 0 from lyer 1, suh s: 1. Build the modeled pproximted â 0 orresponds to lyer 1, using the two modeling prts, pproximted 0 nd the error 0 E D. ˆ 0 0 0E D......( 17. Determine the deterministi prt I of mthemtil liner model bse using the dequntized oeffiient of lyer 1 &lyer, nd the vribles I ˆ0 D( j x D( i y...( Find the probbilisti prt or error (residul s differene between the modelled pproximted imge I nd the originl one I. IE I I......( Quntized/dequntized the error IE, using the uniform slr quntizer. IE IEQ round( IED IEQ QSIE...( 0 QSIE Where QS IE is the error quntiztion step for the quntized IEQ /de-quntized IED oeffiients. 5. Build the modeled pproximted Î orresponds to lyer 0, using the two modeling prts, pproximted I nd the error IED. ˆ I I IED......( 1 6. Enode the lyer informtion of quntized oeffiients ( 00 D, 01D, 0D nd the quntized error ( 0 E D long with the lyer 0 informtion of quntized oeffiients ( 1 D, D nd the quntized error (IE using LZW oding tehniques. The tehniques, worked reversely from subsequent lyers, to onstrut up lyers, mens using the oeffiients ( 00, 01, 0 of lyer to onstrut pproximted lyer 1 ( â 0 nd then using the lyer 1 oeffiients ( â 0, 1, to onstrut the pproximted imge Î. 015, IJCSMC All Rights Reserved 115

reversely from subsequent lyers, to onstrut up lyers, mens use lyer informtion to onstrut pproximted lyer 1 nd then using the lyer 1 informtion to onstrut the pproximted imge Î Reonstrut ed Imge Fig.

5 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg Input Imge I orrespon ds to Construt lyer 1 using input imge by utilizing the polynomil oding: 0 1 Construt lyer using 0 imge by utilizing the polynomil oding: The tehniques, worked reversely from subsequent lyers, to onstrut up lyers, mens use lyer informtion to onstrut pproximted lyer 1 nd then using the lyer 1 informtion to onstrut the pproximted imge Î Reonstrut ed Imge Fig. (1: The proposed hierrhl polynomil ompression system in prtil exmple. Experimentl Results Experiments were done to ompre the performne of the suggested the hierrhil polynomil oding with the trditionl polynomil using fixed blok of size 4 4, with vrious quntiztion steps for errors (residul imges in lyer 1 nd lyer, wheres the quntiztion steps for the oeffiients dopted the s the identil for the both lyers (i.e., use the sme quntiztion step for the oeffiients in lyer 1 nd lyer 0, 1,, 00, 01 nd 0. All the imges used re stndrds (see Figure for n overview of 56 gry levels (8bits/pixel of size The Compression rtio (rtio of originl size to the ompressed size in byte nd the Pek Signl to Noise Rtio (PSNR dopted s n objetive fidelity mesure between the originl imge I nd the deoded imge Î s in eqution (. 55 PSNR 10 log 10...( 1 N 1 N 1 [ Iˆ( x, y I( x, y] N N x0 y01 015, IJCSMC All Rights Reserved 116

tht showed tht the performne improved using hierrhil polynomil oding tehniques in terms of ompression rtio bout

mening overburden problem tht onsuming extr lrge number of bits with the preserving the imge qulity.

third lyer where s no orreltion embedded between lyer oeffiients.

where for high qulity imge, low ompression rtio hieved, tht impliitly mens smll blok size utilized with low

(: Tested imges ( Len, (b Cmermn (Rose nd (d Pper, gry sle imges of size 56 56. Cse1 CR=10.1339 PSNR=30.

6469 PSNR= 9.7171 CR= 13.85 PSNR= 30.1770 CR= 11.474 PSNR= 9.3641 Fig.

6 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg The experimentl results re listed in tbles (1 nd ( for trditionl nd hierrhl polynomil oding respetively, tht showed tht the performne improved using hierrhil polynomil oding tehniques in terms of ompression rtio bout one nd hlf on verge long due to the redution of 0 resolution (i.e., 0 orresponds to men of the imge impliitly mening overburden problem tht onsuming extr lrge number of bits with the preserving the imge qulity. Generlly, two lyers onstrution is suffiient to remove the redundny, tully there s no need to extend the work to third lyer where s no orreltion embedded between lyer oeffiients. Lstly, there s trde off between ompression rtio nd the qulity ffeted by the quntiztion step nd the blok size, where for high qulity imge, low ompression rtio hieved, tht impliitly mens smll blok size utilized with low quntiztion step, nd vie vers, figure(3 shows n exmple of deoded imges. b d Fig. (: Tested imges ( Len, (b Cmermn (Rose nd (d Pper, gry sle imges of size Cse1 CR= PSNR= CR= PSNR= CR= PSNR= CR= PSNR= Cse CR= PSNR= CR= PSNR= CR= PSNR= CR= PSNR= Fig. (3: Deoded imge using the hierrhl polynomil oding, using quntiztion oeffiients equls to 1 for both lyers, quntiztion step of error lyer1 error is equl to 50, with (Cse1 quntiztion step of error lyer is equl to nd (Cse quntiztion step of error lyer is equl to 0 015, IJCSMC All Rights Reserved 117

7 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg Tble 1: Trditionl polynomil oding with quntiztion step equl to one for ll the oeffiients Imge Qnt. CR PSNR Error Len Rose Ppper Cmer mn Tble : Hierrhl polynomil oding with quntiztion step equl to one for ll the oeffiients in lyer 1&lyer uses seleted se from the trditionl polynomil oding when quntiztion of error is equl to 50. Imge Qnt. CR PSNR Error Len Rose Ppper Cmer mn , IJCSMC All Rights Reserved 118

8 Rsh Al-Tmimi et l, Interntionl Journl of Computer Siene nd Mobile Computing, Vol.4 Issue.1, Jnury- 015, pg Referenes [1] Ghdh, Al-K, Imge Compression bsed on Qudtree nd Polynomil. Interntionl Journl of Computer Applition s,vol. 76,No. 3,pp.31-37,013 []Khobrgede, P. nd Thkre, S. Imge Comprssion Tehniques-A Review Interntionl Informtion Tehnologies, Vol.5,No.1,pp. 7-75,014. Journl of Computer Siene nd [3] Mrimuthu, M. nd Swminthn, P. Review Artile: An Overview of Imge Compression Tehniques. Reserh Journl of Applied Siene, Engineering nd Tehnology, Vol.4,No.4,pp ,014 [4]Gonzlez, R. C., Digitl Imge Proessing", Interntionl Soiety for Optil Engineering (SPIE, Edition, 00. [5] Shin, D. A Review of Imge Compression nd Comprison of its Algorithms. Interntionl Journl of Eletronis & Communition Tehnology, Vol.,No.1,pp-6, 011. [6] Anith, S. D Imge Compression Tehnique-A Survey. Interntionl Journl of Sientifi & Engineering Reserh, Vol.,No.7, pp.1-6,011 [7] Amrut, S.G. nd Snjy L.N, A Review on Lossy to Lossless Imge Coding. Interntionl Journl of Computer Applitions (IJCA,Vol. 67,No.17,pp ,013 [8]Ghdh,Al-K, Intr nd inter frme ompression for video streming.phdthesis,extrunion,uk.01 [9] Hud, M. Lossless Imge Compression Using Predition Coding nd LZW Sheme. High Diplom Disserttion, Bghdd University.011 [10] Sinn,D, Medil Imge Compression. High Diplom Disserttion, Bghdd University.014 [11]George, L. E. nd Sultn, B.. Imge Compression Bsed on Wvelet, Polynomil nd Qudtree. Journl of Applied Computer Siene & Mthemtis, Vol.11,No.5,pp. 15-0,011 [1]George, L. E. nd Ghdh, Al-K. Fst Lossless Compression of Medil Imges bsed on Polynomil. Interntionl Journl of Computer Applitions Vo. 70, No.15,pp ,013 [13] Ghdh, Al-K.. Wvelet Trnsform nd Polynomil Approximtion Model for Lossless Medil Imge Compression. Interntionl Journl of Advned Reserh Computer Siene nd Softwre Engineering, Vol.4,No.1 pp ,014 [14] Hider, Al-M., Seletive Bit Plne Coding nd Polynomil Model for Imge Compression, Interntionl Journl of Advned Reserh in Computer Siene nd Softwre Engineering,Vo.4,No.4 pp ,014 [15] Hider, Al-M., nd Zinb, Al-R,.Lossless Imge Compression bsed on Preditive Coding nd Bit Plne Sliing,Interntionl Journl of Computer Applitions, Vo. 93, p1,014 [16] Ghd,Al-K, Hierrhil Autoregressive for Imge Compression, Journl of College of Edution for Pure Sienes, Vol. 4 No.1,pp 36-41,014 [17]George, L. E. nd Dhnnon.B.N. Imge Compression Using Polynomil nd Qudtree Coding Tehniques.Interntionl Journl of Sientifi & Engineering Reserh, Vol. 4, No 11,pp , , IJCSMC All Rights Reserved 119

Image Compression based on Quadtree and Polynomial

Image Compression based on Quadtree and Polynomial Interntionl Journl of Computer Applitions (0975 8887 Imge Compression sed on Qudtree nd Polynomil Ghdh Al-Khfj Ph.D Dept. of Computer Siene, Bghdd University, College of Siene. ABSTRACT In this pper, n