A new Algorithm for Lossless Compression applied to two-dimensional Static Images

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1 A new Algorthm for Lossless Comresson aled to two-dmensonal Statc Images JUAN IGNACIO LARRAURI Deartment of Technology Industral Unversty of Deusto Avda. Unversdades, Blbao SPAIN Abstract: - In ths aer, a new lossless comresson algorthm method for two-dmensonal contnuous-tone mages s resented. The roosed method acheves hgher comresson ratos than unversal methods. Tradtonal methods use codng technques based on reducton of the exstng redundancy n the data or xels, manly alyng methods based on statstcal models, substtuton models, dctonary models, redctve models, wavelet transform doman, etc The roosed method s based on a model whch ncororates a new algorthm (INA Algorthm). Ths method conssts of three stes: segmentaton, comresson algorthm and encodng. In the frst, the whole mage s contnuously dvded nto blocks of a fxed sze. Each block generates two searate grous: a sequence of the dfferent xels of the block sorted n ncreasng order and the other one n a sequence formed of xels organsed n grous of ordered ars. The second one s the new develoment and man contrbuton of ths method. A new algorthm rocesses n arallel the executon of the two sequences obtaned n the segmentaton rocess by means of a data structure n the form of a bnary tree and a redctve scheme resectvely. Lastly, a fast and effcent encoder s used to generate the bnary codes. The arallel executon of ths algorthm reduces resonse tmes of the comresson and decomresson cycles. Ths method ensures a comresson rato of : to :4 wth very hgh robablty. In order to evaluate the advantages of ths method comared wth the conventonal lossless methods and the JPEG-LS standard, exermental studes have been carred out usng a set of ISO standard mages. From the exermental results we can asserted that the roosed method obtans a hgher comresson rato. Key-Words: - Lossless comresson, mage comresson, bnary tree Introducton In data comresson lterature, we can fnd the descrton of methods, technques and algorthms for lossless mage comresson. The evaluaton of these methods s resented n endless lsts that show the comresson rato acheved by each method. In order to evaluate the roosed method, we resent an alternatve classfcaton of the lossless mage comresson methods based on the crtera of the model aled. In ths secton the most relevant methods are descrbed, ncludng the JPEG-LS standard.. Lossless methods based on substtuton models The early methods used to comress mages were based more on substtuton technques than on comresson schemes. These lnear codng technques were very smle and were desgned accordng to secfcatons of comuters at the tme. These technques made use of the exstng redundancy n the data (xels) of an mage, through the relatonsh of xels wth the adjacent xels. The most sgnfcant methods used by these models are Btma Comresson [] and RLE Run Length Encodng [].. Lossless methods based on statstcal models The comresson methods based on ths model arse from the modern nformaton theory suorted by the works of C.E. Shannon, R.M-. Fano, and Davd Huffman. These methods marked a relevant ste forward regardng revous data comresson methods. The comresson method conssts of a model and an encodng technque. The statstcal model can be statc or adatve. Statstcal models exlot the redundancy n xels accordng to the robablty of occurrence n the mage. Ths means that the codec must make a revous readng of all xels before calculatng the robablty of each one. The codng s stored together wth the table of robabltes. The measure of the model s effcency ISBN:

2 s the aroxmaton to the entroy of the mage. The most reresentatve methods are Shannon-Fano Codng [3] [4], Huffman Codng [5], and Arthmetc Codng [6] [7]..3 Lossless methods based on dctonary models Models based on dctonary technques use a code to relace a strng of varable symbols. The encoder rocesses the nut xels and runs the dctonary n order to fnd ts corresondence. If the strng s found, a onter to the dctonary s used as code. Otherwse, the strng rocessed s added to the dctonary. The most relevant methods are LZ77 [8], LZ78 [9], and LZW [0]. Statstcal models and dctonary models, n ther statc model versons, requre storng the robablty table and the dctonary together wth data. Ths ncreases the comressed fle sze consderably. Adatve models avod storage of dctonares and tables of robabltes. However, the model has no avalable nformaton on the data before startng the rocess..4 Lossless methods based on satal doman models These methods use models n the sace doman to remove the satal redundancy exstng n an mage, ether locally or globally. The codng rocess s done by means of statstcal models of substtuton. The most reresentatve s the lossless JPEG standard []. The model s based on a redcton algorthm of a xel, once the revously rocessed xels are known. The result of the redcton of a xel s a value very close to zero or resdual redcton error. Ths value s encoded usng one of the two methods recommended by the standardzaton commttee: Huffman Codng and Arthmetc Codng. There are other methods n the sace doman that exlot satal redundancy locally or globally usng models based on segmentaton strateges, nterolaton and so on. In ths aer, the frst standard method develoed by the lossless JPEG standardzaton commttee based on redcton technques s evaluated. Due to the lmtatons of the comresson ratos t was reconsdered and n a few years was relaced by another standard wth a new aroach and a dfferent model. Ths standard s called JPEG-LS..5 Lossless methods based on wavelet transform These methods have ther orgns n the methods aled to loss comresson technques. Although ths currently contnues to be ts feld of ractcal alcaton, the lossless verson s ncreasng. Unlke revous methods that exlot the satal nformaton, these methods use algorthms n the transform doman to exlot the satal nformaton through frequency analyss. The most relevant methods based on wavelet transform are S- Transform [], Haar Transform [3], and Lftng Scheme [4]. Other otmzed methods lke S+P Transform [5] have acheved a sgnfcant sace n the feld of medcal mages comresson. In ths aer, the Haar Transform and Lftng Scheme are evaluated..6 JPEG-LS standard JPEG-LS s the new lossless comresson method / near-lossless roosed standard for stll mages n contnuous tone changes. Ths new standard method (ISO-4495-/ITU-T.87) [6] s based on the LOCO-I (Low Comlexty Lossless) [7]. The term "near-lossless" refers to a standard way of accetng an nsgnfcant loss of data not vsble to the human eye. A value or lmt of error s defned ror to comresson to determne the maxmum acceted error wthout losng orgnal mage qualty. Alternatvely, we roose a method wth a new aroach that acheved a better comresson rato than the tradtonal methods [8] [9]. A hgher comresson rato may be obtaned when the source mage has hgher data redundancy. Ths aer s dvded nto four sectons. In the frst one, a bref summary of the state-of art s resented accordng to a classfcaton of the methods based on the model. In the second secton, the roosed method s descrbed n detal. In the thrd secton, the exermental results are shown usng a standard test mages and the results are comared wth conventonal methods. Fnally, n the last secton, the conclusons of ths research are stated whle ontng out future work oen to researchers.. Descrton of the method roosed The method roosed n ths aer adds a dfferent aroach to the tradtonal methods. Ths method resents a new model based on the INA algorthm [0] that executes two arallel rocesses n order to dramatcally reduce comresson tme. In the frst one, the algorthm generates a seral sorted and formed by the dfferent xels of the block whch are coded usng a artcularly redctve scheme. By arallel executon, the second rocess encodes the xels of the block by means of a bnary tree structure. Ths structure s bult usng ordered ars of xels, and n turn, defnes the nodes of the bnary tree. The nodes are coded usng transversal codng. Fg. shows the comresson cycle model. ISBN:

3 Source Image Segmentaton Process,,,, 4 INA Algorthm Block Elements Predctve Codng Parallel Executon Bnary Tree Structure Bnary Codng B = {,,,,,3,,4,,, 4 } () Ste. The dfferent values of the xels are sorted n ascendng order. These values determne where each xel s laced n the seres and are used as redctors (second rocess). Block Codeword Comressed Image Fg. Comresson cycle model of the method roosed The man contrbuton of ths method s the INA Algorthm and arallel executon n order to reduce resonse tmes of the comresson and decomresson cycles. Intally, the whole mage s contnuously dvded nto blocks of 4x4 xels, from left to rght and to to bottom. The codng algorthm s aled to each block, once and agan untl the entre mage s rocessed. Fg. Segmentaton rocess The segmentaton rocess does not generate data comresson but has the followng benefts [0]: Increase the executon seed by oeratng wth small blocks of 4x4 xels. Reduce memory storage requrements. Adat the data to a bnary tree data structure for the next stes of the algorthm. Formal descrton of INA Algorthm P, P, P,3 P,4 P P P 3 P 4 P P P 3 P 4 P P P 3 P 4 Ste. The mage s dvded nto blocks of a fxed sze (4x4 xels). The xels of each block are rocessed n order to buld a bnary tree (frst rocess). < < 3 <,, < n () where, n s the number of dfferent xels. Ste 3. Parallel executon. Ste 3. Frst rocess: bnary tree Ths rocess conssts on buldng and codng a bnary tree for each block. The xels are organsed nto grous of ordered ars.,, Each ar generates a node n the bnary tree. Therefore, each block s coded usng a bnary tree that contans a maxmum of eght nodes. When the bnary tree s comlete, all the xels are reresented n the bnary tree and the branches of the tree contan at least one node [0]. (, 3) ( ) Fg. Bnary tree The codeword of each node s made u of a sequence of consecutves ones and zeros known as a transversal codng technque. The oston of the frst element n each ar defnes the number of ones and the absolute value of the dfference between the frst and the second defnes the number of zeros.,, (, 3 ) ==> 00 ( 3, 3 ) ==> 0 (3) ( 5, 6) ( 5) (4) ISBN:

4 Theorem When alyng ths roosed algorthm to each block B, the total number of bts requred to code ths block s gven by the followng exresson: [ log (n) ] 6log (n) T(B) = 8 = where n s the number of dfferent xels n a block. The comlete codng of the block s a sequence of bnary codes generated by each of the xel ars. Ste 3. Second rocess: redctve codng. A seres s made u of the dfferent values of the block, as n (). The frst xel s coded usng eght bts. Once the frst xel s coded, we know f t s even or odd. Then, the next xel can be redcted usng a resdual value (r v ). m = 8 (n ) rv Theorem The dfference of two natural numbers, x and y, n whch x-y s even, and x>y, thus the result of (x-y)/ wll be an even natural number. Theorem 3 The dfference of two natural numbers, x and y, wth ooste arty, must be odd. Thus, the result of (x-y)/ wll be a decmal number. In ths case, t s necessary to add one bt n order for t to be reversble. Table. Predctve Codng Pxel Predcton Number of Bts Pxel 8 bts P V r =( 7 - )/ N = log (( 7 - )/) (4) P 3 V r3 = ( - )/ N = log (( - )/) P 4 V r4 = ( 3 - )/ N 3= log (( 3 - )/) P 5 V r5 = ( 4-3 )/ N 4= log (( 4-3 )/) P 6 V r6 = ( 5-4 )/ N 5= log (( 5-4 )/) P 7 V r7 = ( 6-5 )/ N 6= log (( 6-5 )/) P 8 V r7 = ( 7-6 )/ N 7= log (( 7-6 )/) The comlete bnary codng of the block s the sequence of codes generated by each one of the xels and the codeword of the bnary tree. Each codeword s made u of a seres of consecutve ones and zeros, whch allows us to further reduce the codeword. = 8 m (4) (5) (6) Ste 4. Go to Ste. The rocess contnues block by block untl the segmentaton rocess of the whole mage s fnshed. The algorthm s aled block by block untl The decomresson rocess s reversble because t allows to recover the exact orgnal data from the comressed data alyng the same model of the comresson rocess. 3. Exermental results In ths secton we resent comresson results obtaned wth the basc confguraton of the roosed method descrbed above. These results are comared wth those obtaned wth other conventonal methods reorted n the lterature []. The average comresson ratos for each model are shown n the Table. The rato obtaned s hgher than the one obtaned usng conventonal methods. Table. Average Comresson Rato and b. Methods Comresson BPP ratos,03 7,79 Substtuton models Arthmetc models, 7, Dctonary models,3 6,53 Predctve models,57 5,09 Transformed models,76 54 JPEG-LS Standard,75 57 Method roosed,77 5 The exermental studes have been carred out usng a collecton of ISO standard mages. Lena Baboon Peers Boats Entroy: Entroy: Entroy: Entroy: -7.9 Mean Value: 05 Mean Value :9,4 Mean Value:00 Mean Value: 9,70 Cameraman Barbara Goldhll Zelda Entroy: Entroy: -.7,63 Entroy:-7.48 Entroy:-7,6 Mean Value:8,7 Mean Value: 7,39 Mean Value:0 Mean Value: 9,6 Fg. 4 A set of ISO standard mages ISBN:

5 4. Conclusons A new method for the lossless comresson of statc mages based on INA Algorthm has been develoed and resented n ths aer. Its major contrbutons are: Imrovement of rato comresson as comared to conventonal methods. Parallel executon Faster codng-decodng rocess. The method roosed s sutable for comresson of colour mages and grayscales mages. The dentfyng features of ths new method allow the blocks to be rocessed n arallel. In future research the method roosed should be mroved. Imrovement wll be orented towards comactng of the codeword. References: [] H. Glbert and M. Thomas. Data comresson: technques and alcatons, hardware and software consderatons. John Wley and Sons Ltd; New York, 987,. 06. ISBN: [] J. A. Storer, Data Comresson: Methods and Theory Comuter Scence Press, Rockvlle, MD, 988. [3] C.E. Shannon, A Mathematcal Theory of Communcaton", Bell System Techncal Journal, vol. 7, , July, October, 948. [4] R.M. Fano, The transmsson of Informaton Techncal Reort 65, Reseca Laboratory of Electroncs, MIT, Cambrdge, M.A [5] D.A. Huffman, A method for the constructon of Mnmum Redundancy Codes Proceedngs of the I.R.E., vol. 40 No. 9, , ISSN: [6] J.J. Rssanen and G.G. Langdon, Arthmetc Codng IBM Journal of Research and Develoment, Vol. 3 No March 979. [7] I. Wtten, H. Neal, M. Rradford, J.G. Cleary, Arthmetc Codng for Data Comresson Communcatons of the ACM, vol. 30, No 6, June 987. [8] J. Zv and A. Lemel, A Unversal Algorthm for Sequental Data Comresson IEEE Trans. Inform. Theory 3 (May), May 977. [9] Zv, J., and A. Lemel, Comresson of Indvdual Sequences va Varable-Rate Codng IEEE Trans. Inform. Theory Setember 978. [0] T. A. Welch, A Technque for Hgh- Performance Data Comresson IEEE Comuter. vol. 7, No. 6, June 984. [] G.K. Wallace, The JPEG Stll Pcture Comresson Standard Communcatons of the ACM, Vol. 3 No [] A. Sad and W. Pearlman, "A new fast and effcent mage codec based on set arttonng n herarchcal trees" IEEE Trans. on Crcuts and Systems Vdeo Technology, 6, 996, [3] A. Haar, Zur Theore der orthogonalen Funktonensysteme, Mathematsche Annalen, 69, , 90. [4] I. Daubeches and W. Sweldens, "Factorng wavelet transforms nto lftng stes," rernt, Bell Laboratores, Lucent Technologes, 996. [5] A. Sad and W. A. Pearlman, "Reversble Image Comresson va Multresoluton reresentaton and redctve codng" IEEE Trans. Image Processng, vol Nov [6] Informaton Technology, Lossless and Near- Lossless Comresson of Contnuous-Tone Stll Images, ISO/IEC 4495-, ITU Recommend. T [7] M. J. Wenberger, G. Serouss, and G. Saro, LOCO-I: A low comlexty, context-based, lossless mage comresson algorthm, Data Comresson Conference, Snowbrd, UT, Mar. 996, [8] J.I. Larraur, "A New Lossless Comresson Algorthm for Statc Color Images", Proc. IS&T, [9] J.I. Larraur and E. Kahoraho, "A Fast Bnary Tree Encoder for Lossless Image Comresson", Proc. Sgnal Proc., Pattern Recog. & Alcatons", 00 [0] J.I. Larraur and E. Kahoraho, A new Method for Real Tme Lossless Image Comresson aled to Artfcal Vson PCS003 Proceedng of Pcture Codng Symosum, INRIA, , Arl 003. ISBN [] A.E. Savaks, Evaluaton of algorthms for lossless comresson of contnuous-tone mages Journal of Electronc Imagng (), January 00. ISBN: