A Fast Fractal Image Compression Algorithm Using Predefined Values for Contrast Scaling

Transcription

1 Proceedngs of the World Congress on Engneerng and Computer Scence 007 WCECS 007, October 4-6, 007, San Francsco, USA A Fast Fractal Image Compresson Algorthm Usng Predefned Values for Contrast Scalng H. Mar Nam, M. Salaran Electrcal Engneerng Faculty Unversty Of Mazandran Tel-Fax: Emal: h_mare@nt.ac.r, M_Salaran@nt.ac.r Abstract In ths paper a new fractal mage compresson algorthm s proposed n whch the tme of encodng process s consderably reduced. The algorthm explots a doman pool reducton approach, along wth usng nnovatve predefned values for contrast scalng factor, S, nstead of scannng the parameter space [0,]. Wthn ths approach only doman blocks wth entropes greater than a threshold are consdered. As a novel pont, t s assumed that n each step of the encodng process, the doman block wth small enough dstance shall be found only for the range blocks wth low actvty (equvalently low entropy). Ths novel pont s used to fnd reasonable estmatons of S, and use them n the encodng process as predefned values, mentoned above. The algorthm has been examned for some well-known mages. Ths result shows that our proposed algorthm consderably reduces the encodng tme producng mages that are approxmately the same n qualty. eywords: Image compresson, fractal codng and mult resoluton.. Introducton Fractal mage compresson s wdely used n mage processng applcatons such as mage sgnature [], texture segmentaton [], feature extracton [3], mage retrevals [4,5] and MR, ECG mage processng [6]. However, ths method suffers from a long encodng tme as ts man drawback. Ths long encodng tme arse from very large number of doman blocks that must be examned to match each range block. The number of range blocks wth sze of n n, n an N N mage, s ( N / n), whle the number of doman blocks s ( N n + ). Consequently t can easly be shown that the computaton for matchng range blocks and doman blocks has complexty of 4 O ( N ) [7]. Thus reducng ths encodng tme s a focus of research wth practcal ramfcatons. Several methods have been proposed to overcome ths problem. One common way s the classfcaton of blocks n a number of dstnct sets where range and doman blocks of the same set are selected for matchng. Here, the encodng tme s saved at cost of mage qualty. Reducng the sze of doman pool s another method that has been employed n several manners. In some approach doman blocks wth small varance [3] and n some others doman blocks wth small entropes were deleted from the doman pool [7] In addton to the sze of the doman pool, the computatonal cost of matchng a range block and a doman block has an mportant role n encodng tme. We reduced ths cost by estmatng the approxmate optmum values for contrast scalng factor, S, nstead of searchng for t. Combnng these two novel ponts, we propose a new fractal mage codng that has a consderable shorter encodng tme than the next fastest algorthm [7]. In secton we present a bref descrpton of the fractal mage codng. The proposed algorthm s presented n secton 3. In secton 4 the methodology and the results are presented and compared wth the next fastest algorthm. Fnally, n secton 5 conclusons are presented and some future works are addressed.. Fractal Image Codng: A Bref Revew At the frst step n fractal codng an mage s parttoned nto none overlappng range blocks of sze B B, where B s a predefned parameter [4,5,8]. Then a set of doman blocks are created from the orgnal mage, takng all square blocks of sze B B wth nteger step L, n horzontal and vertcal drectons. Wthn each member n the doman pool, three new doman blocks are created by clockwse rotatng t 90º, 80º and 70º, also these three and the orgnal doman block all are mrrored. Here, n addton to the orgnal doman block, we have seven new doman blocks. These blocks are added to the doman pool. After constructng doman pools (related to each range block) we must select the best doman block from doman pool and fnd an affne transformaton that maps the selected doman block wth mnmum dstance. The mentoned dstance between a range block, R, and a decmated doman block, D, both wth n pxels s defned as follows: n E( R, D) = ( sd + o ) () = r The best coeffcent S and O are [9]: ISBN: WCECS 007

2 Proceedngs of the World Congress on Engneerng and Computer Scence 007 WCECS 007, October 4-6, 007, San Francsco, USA < R R., D D. > s = () D D. o = R sd (3) <, >, R, D, R and D are nner product, range block, doman block, mean of R and mean of D respectvely. Because of hgh computatonal cost of (), t s convenent to search S across a pre-sampled set of [0,], nstead of calculatng (). Along the matchng process, the best found transformatons are only saved for range blocks whch have been mapped wth an acceptable error. The remanng range blocks are splt nto 4 new smaller range blocks, and the matchng process s restarted for the new set. For example, f range blocks ntally have a sze of 6 6 pxels, the range blocks of the succeedng steps wll have a sze of8 8, 4 4 and respectvely, that leaves a four step algorthm. Two strateges were used to reduce the encodng tme n fractal codng algorthms. In hs research, Saupe found that the doman pool s not necessary to nclude all of possble doman blocks and only the hgh varance blocks are suffcent [3,0]. In another work, the entropy measure was used nstead of varance [7]. Ths entropy based method s superor to the Saupe method so we compare our algorthm to the entropy based one. 3. The proposed algorthm In ths paper we use two novel ponts to reduce the encodng tme. The frst pont s restrctng the doman pool to hgh entropy doman blocks. Ths causes the total evaluaton tme for fndng related doman block of a range block to become shorter. The entropy of a block s defned below. Suppose N be a block of an Image as shown n fgure Fgure, a doman block of sze In the above fgure g s the grey level of the pxel at locaton (, ). Suppose g for, =,,..., n vares n{ L, L,..., L }. Also suppose the number of observatons of L over the pxels s q. So the probablty of L s defned as equaton 5, q q (4) p = = n q = The entropy s defned as below: entropy = pln( p) = (5) Here s some example block wth sze of Fgure four doman blocks and ther related entropy As evdent from fgure, low entropy blocks are smoother and consequently have lower nformaton contents. Lackng hgh frequency nformaton, low entropy blocks cannot cover hgh entropy range blocks. On the other hand, hgh entropy blocks may cover all range blocks. To cover low entropy rang blocks we can smply reduce nformaton of the doman blocks. 3. The effect of contrast scalng factor, s Another mportant parameter that was nvestgated s the contrast scalng factor s. To do ths, a large number of experments wth exhaustve search for s were performed. Hstograms of the best selected values of s are shown n fgure 3 for all four steps respectvely. To analyss the effect of s, t wll be helpful to recall the operaton of s. As mentoned n secton, doman block pxels are multpled by s and then the nteger part s consdered. Indeed, s maps nteger values of doman pxels to nteger values of range pxels (0 < s < ). Here a smple proposton s presented that helps us nterpret the presented hstograms n fgure 3. Proposton Suppose 0< s < s < and X, X and Y are three sets of postve nteger values wth the same sze. If X = [ sy ] and X = [ sy ] then Entropy( X) Entropy( X ) *[ x] s the bggest nteger less than x. Proof: There s a smple proof as follows 0 < s < s < 0 < X = [ sy ], 0 < X = [ sy ] Max( X) Mn( X) < Max( X ) Mn( X ) < Max( Y ) Mn( Y ) (Here 0 < s < has a contractve role). On the other hand we have N( X) = N( X) = N( Y) Here N(X) s the sze of the set X. Thus X and X have the same sze, but wth dfferent doman of varaton and also are processed from the same set through a smple multplcaton. It s obvous that the redundances of X wll be larger than the ones of X. Ths smply concludes the results. At step range blocks are 6 6 or of sze 56 pxels. Consder now a block wth a determned entropy or nformaton. It s obvous that all permutatons ISBN: WCECS 007

3 Proceedngs of the World Congress on Engneerng and Computer Scence 007 WCECS 007, October 4-6, 007, San Francsco, USA constructed by rearrangng pxels of the block have the same entropy as the orgnal. A smple and qualtatve measure or as a lower bound for the number of these permutatons s as follows: N P 56! n! n! Ln! = (6) k where n s the number of pxels wth grey level of, bg n means the block has more redundances and equvalently low entropy. Large values for n are ndcatve of small dstnct permutatons. As a result, at step only range block wth small entropy wll have the chance to be coded and consequently s has small values (recall the proposton above). If the entropy of a block s hgh at step one, then the number of blocks wth that entropy wll be hgh ( n s are small) so the probablty that t could not be coded at ths step would be hgh. Therefore, we expect that s would have a small value. At lower steps, 3 and 4 block szes are 8 8, 4 4. Followng the same logc we see that N P drastcally decreases. As a qualtatve comparson we wrte: NP 64! 0 NP 56! Agan, wth a smlar dscusson t may be shown that, blocks of hgher entropy at level are encoded so s s left at greater values. Ths wll also happen n lower steps. The hstogram of the best s n lower steps, accordng above dscusson, wll be shfted to the rght, as shown n fgure 3. In each step of exstng algorthms, all members of a 0-member set of s, sampled from [0, ], are evaluated. It can be seen from fgure 3 that all values of s need not be evaluated and we can restrct s to one or two dstnct values. Obvously, restrctng the sze of s to a -member set wll decrease the search and hence the encodng tme consderably. To fnd a true estmaton of s, a large number of experments wth an exhaustve search for s were performed. One can easly see that at step the optmal s value s often less than 0., ndependent of the mage, so for ths step we may let s = 0.. At step the optmum value of S s less than 0.5 so here we choose s to be {0., 0.4}. For step 3, s has approxmately a unform dstrbuton across [0,], so to determne some dstnct values here we choose S from {0.3, 0.8}. For step 4 as shown n fgure 3d, s the hgher value n [0 ]. Here s s chosen from{0.5, 0.9}. In ths step (3a) (3b) (3c) (3d) Fgure 3 Hstogram of s at a) step b) step c)step 3 d)step 4 blocks sze are that cause to be encoded very well. We reduced the set of values of s to a twomember set that leaves three cases for range blocks. The frst case s where the selected value s the same value as obtaned from the exhaustve search of s here sn t any problem. The second case s where the ISBN: WCECS 007

4 Proceedngs of the World Congress on Engneerng and Computer Scence 007 WCECS 007, October 4-6, 007, San Francsco, USA selected s s not the best value, but the error s less than the threshold and the range blocks are coded approxmately optmal here the encodng tme s saved but the s somewhat damaged. In the thrd case, the selected value causes the encodng error to become so large that range blocks can not be encoded. Hence, the range blocks are splt and the encodng s done n followng steps. Ths means a better at the cost of small degradaton of tme and compresson rato. 4. Experments and results Several experments were performed to evaluate the proposed algorthm and compare t wth the exstng entropy based methods. Computer programs employed n these experments were wrtten n C++ runnng on a Pentum (450MHz) wth 56 MB RAM. Comparson results are shown n fgure 4a, b, for dfferent pool szes and the Lena mage. To have a reasonable comparson, the two algorthms are compared n fxed. Fgure 4a,b show the compresson rato and encodng tme for =35.07db. In these fgures compresson rato and encodng tme are plotted versus pool sze wth the as the parameter. evdent that the proposed algorthm s superor especally n encodng tme economy. The results of proposed algorthm for Lena mage are presented n fgure 6. Table the comparson results for Baboon Prop Entropy Pool sze (S) Table the comparson results for F6 Proposed Entropy based Pool sze (S) CR proposed entropy_based pool_sze (4a) Tme(s) proposed entropy_based pool_sze (4b) Fgure 4, Compresson rato and encodng tme of proposed algorthm and entropy based versus pool sze a, b) at fxed =35.07db Fgure 5 Orgnal Image To gan a greater percepton of proposed algorthm the results for two other famlar mages are presented n tables and. Comparng the two algorthms, t s Fgure 6 Com.Rat=.7 Tme(8. s) =33.57db ISBN: WCECS 007

5 Proceedngs of the World Congress on Engneerng and Computer Scence 007 WCECS 007, October 4-6, 007, San Francsco, USA 5. Conclusons and future works In ths paper we presented a new method for fractal mage compresson to reduce encodng tme. Centrally, our algorthm employed predefned values for contrast scalng factor rather than sweepng the entre parameter space durng search. Expermental results ndcate a superor performance level n comparson to the exstng entropy based methods. In the future we ntend to further develop ths approach n frequency doman applcatons and produce quanttatve comparsons wth other hybrd methods. References [] N.T.Thao, A hybrd fractal DCT codng scheme for mage compresson, n Proceedng ICIP-96 (IEEE Internatonal Conference on mage processng), Lausanne,Swtzerland,Sept. 996, vol., pp [] M. aplan and C.-C. J. uo, "Texture segmentaton va haar fractal feature estmaton,"j. Vs. Commum. Image Represent, vol.6, no. 4, pp , 995. [3] D. Saupe. Lean Doman Pools for Fractal Image Compresson. Proceedngs IS&T/SPIE 996 Symposum on Electronc Imagng: Scence & Technology Stll Image Compresson II, Vol. 669, Jane 996 [4] G. E. Qen, S. Lepsqy, and T. A. Ramstad, An nner product space approach to mage codng by contractve transformaton s, n Proceedng ICASSp-9 (IEEE - Internatonal Conference on Acoustcs Speech and Sgnal Processng) Toronto, Canada, May 99, vol.4, pp [5] T. Tan and H. Yan, "Obect recognton usng fractal neghbor dstance :Eventual convergence and recognton rates, " n Proc. ICPR000, Vol., 000, pp [6] M. Barnsley and L. Hurd. Fractal Image Compresson. On Image Processng: Mathematcal Methods and Applcatons. pp. 83-0, Clarendon Press, Oxford, 997. [7] M.Hassaballah,M.M.Makky and Y.B. Mahdy, A Fast Fractal Image Compresson Method Based entropy, Electronc Letters on computer Vson And Image Analyss 5():30-40,005 [8] D.M.Monro, Class of fractal transforms, Electroncs Letters,vol.9,no.4,pp ,Feb. 993 [9] Y. Fsher. Fractal Image Compresson: Theory and Applcatons. Sprnger-Verlag, New York, 994. [0] D. Saupe, Fractal mage compresson va nearest neghbor search, n: Conference on Proceedngs of NATO ASI Fractal Image Encodng and Analyss, Trondhem, Norway, 995. ISBN: WCECS 007