Internatonal Journal of Vdeo& Image Processng and Network Securty IJVIPNS-IJENS Vol:10 No:06 1 Estmaton of Image Corrupton Inverse Functon and Image Restoraton Usng a PSObased Algorthm M. Pourmahmood, A. M. Shotorban, and R. M. Shotorban Abstract In ths paper, a new method s proposed to estmate corrupton functon nverse of a blurred mage. Ths technque can be used for restorng smlar corrupted mages. For lnear poston nvarant procedure, the corrupton process s modeled n the spatal doman by convolvng the mage wth a pont spread functon (PSF) and addton of some noses nto the mage. It s assumed that a gven artfcal mage s corrupted by a degradaton functon, represented by the PSF, and an addtve nose. Then a flter mask (as a canddate for the corrupton functon nverse) s calculated to restore the orgnal mage from the corrupted one, wth some accuracy. Calculatng a sutable flter mask s formulated as an optmzaton problem: fnd optmal coeffcents of the flter mask such that the dfference between the orgnal mage and flter mask restored mage to be mnmzed. Partcle swarm optmzaton (PSO) s used to compute the optmal coeffcents of the flter mask. Square flter masks are consdered. A comparson between dfferent exctng methods and the proposed technque s done usng smulatons. The smulaton results show that the proposed method s effectve and effcent. Snce the proposed method s a smple lnear technque, t can be easly mplemented n hardware or software. Index Term corrupton functon, flter mask, mage restoraton, partcle swarm optmzaton. I. INTRODUCTION A fundamental ssue n mage enhancement or restoraton s blur removal n the presence of observaton nose. Recorded mages almost always represent a degraded verson of the orgnal scene. There are many sources, such as the turbulence and aerosol sources n the atmosphere or spato-temporal movements and ntensty scntllatons, whch cause to mage degradaton. A systematc approach for restoraton of blurred mages models the blurred mage as a convoluton between the orgnal mage and the pont spread functon (PSF). In such a case, smple deconvoluton technques (such as a Wener flterng) can be used to restore the mage [1]. However, the man dffculty s the need for a relable knowledge of the PSF. The reason for ths dffculty s that n most practcal remote sensng stuatons ths data s not known n pror []. M. Pourmahmood s wth the Control Engneerng Department, Faculty of Electrcal and Computer Engneerng, Unversty of Tabrz, Tabrz, Iran (emal: m.pour13@gmal.com). A. M. Shotorban s wth the Department of Electrcal Engneerng, Azerbajan Unversty of Tarbat Moallem, Tabrz, Iran (e-mal: amn.mohammadpour@gmal.com). R. M. Shotorban s wth the Department of Mechancal Engneerng, Unversty of Tabrz, Tabrz, Iran (e-mal: en.ramn.sh@gmal.com). The process of restorng an mage by usng a degradaton functon, whch has been estmated n some way sometmes, s called blnd deconvoluton (due to the fact that the true degradaton functon s seldom known completely). There has been extensve works on blnd deconvoluton over the past 0 years, ncludng survey artcles [3, 4]. Exstng blnd deconvoluton methods can be categorzed nto two man classes: methods whch separate PSF dentfcaton as a dsjont process from restoraton [5, 6, 7], and methods whch combne PSF dentfcaton and restoraton n one procedure [8, 9, 10, 11]. Methods n the frst class tend to be computatonally smpler. For example, Chalmond [6] has proposed solate specfc features n the mage, such as sharp edge elements, and then estmated the PSF from them, assumng radal symmetry. However, a man drawback of ths method s the assumpton that the shapes of all the extracted edges can be modeled as deal step functons n the orgnal mage. No crteron s employed to assess the best step-edge from the set of sharp edge elements n the degraded mage. Methods n the second class use teratve methods to approxmate the blur extent [3, 4, 8-11]. They often formulate parametrc models for both the mage and the blur. The parameters are estmated n each step and used n the next teraton. A Gaussan functon s used often to approxmate the PSF [8-11]. A shortcomng of these methods s the requrement for a good ntal guess of the PSF. The resultng accuracy of the estmated PSF and the qualty of the restored mages depend on the accuracy of the ntal guess. The convergence speed of algorthm depends on ths guess. Also, n order to obtan a reasonable restored mage these methods requre one to mpose sutable constrants on the PSF and the mage. The mentoned methods have some dffcultes and restrctons. They can be just appled for some especal cases and resultng n low accuracy of the obtaned PSF and the restored mage. Also, when some unknown noses are added to the degraded mage, the performance of mentoned methods s decreased. In order to solve the mentoned problems, we propose an optmal flter mask that can produce restored mage from degraded mage and represent the nverse of the degradaton functon wth the addtve nose. For reachng ths am, frst an artfcal mage s corrupted wth some degraded functons and addtve noses. Then an optmal flter mask s calculated usng partcle swarm optmzaton (PSO) algorthm.
Internatonal Journal of Vdeo& Image Processng and Network Securty IJVIPNS-IJENS Vol:10 No:06 PSO s a populaton-based searchng technque whch has been proposed n 1995 by Kennedy and Eberhart [18]. Its development s based on the observatons of socal behavor of anmals such as brd flockng, fsh schoolng and swarm theory. Compared wth genetc algorthm (GA), PSO has some attractve characterstcs. Frst of all, PSO has memory (the knowledge of good solutons s retaned by all partcles) whereas n GA prevous knowledge of the problem s destroyed once the populaton s changed. Secondly, PSO has constructve cooperaton between partcles,.e. partcles n the swarm share ther nformaton. In PSO, each partcle makes use of the best poston encountered by tself and that of ts neghbors to poston tself toward and optmal soluton. The performance of each partcle s evaluated usng a predefned objectve functon, whch encapsulates the characterstcs of the optmzaton problem [16]. In ths paper, PSO algorthm s used to fnd the elements of a flter mask. The computed flter mask mnmzes the dfference between artfcally degraded mage and obtaned restored mage by the flter mask. In other words, the optmal flter mask s a sutable nverse of the mage corrupton caused by PSF and addtve noses, whch can be used to restore other smlar corrupted mages. Ths method s completely new and ts effectveness s shown by smulatons. The smulatons are performed usng Matlab software. The exstng blurrng functons, such as moton, average and unsharp, are used as mage degradaton functons. Salt & pepper, localvar and gaussan noses are also used to construct artfcally corrupted mages. The nverse of corrupton functon s modeled as a smple lnear square flter mask. The optmal values of the flter mask elements are computed usng PSO technque. The obtaned results are compared to the results of some wellknown exstng flters, such as geometrc mean flter, harmonc mean flter and alpha-trmmed mean flter, to demonstrate the effcency and applcablty of the proposed technque. Ths paper s organzed as follows. In Secton II, the problem of mage restoraton s brefly dscussed. In Secton III, the procedure of PSO method s explaned. Secton IV explans the proposed method for estmaton of the corrupton functon nverse. In secton V, some smulatons are used to llustrate the effcency of the proposed method. Fnally, ths paper ends wth some conclusons and future works n Secton VI. II. PROBLEM FORMULATION In ths secton, we formulate the problem of mage restoraton. As Fg. 1 shows, the mage corrupton process s modeled as a degradaton functon, whch s represented by ts PSF and an addtve nose n(x, y), operated on an nput mage f(x, y) to produce a corrupted mage g(x, y). The standard mage restoraton problem s as follows: gven g(x, y), the objectve of restoraton s to obtan an estmate fˆ (x, y) of the orgnal mage such that the estmated mage to be as close as possble to the orgnal nput mage. In general, the more knowledge about the PSF and n(x, y) wll gve closer estmaton to f(x, y). For a lnear poston-nvarant process the corrupton functon s gven n the spatal doman by orgnal mage f(x,y) g(x, y) PSF(x, y)*f (x, y) n(x, y) (1) where PSF(x, y) s the spatal representaton of the degradaton functon and the symbol "*" ndcates the convoluton operator. In many applcatons, there s no complete knowledge about the PSF and the nose. If we can frst estmate a sutable nverse of the corrupton functon (the nfluence of both PSF and nose), then we can use the obtaned corrupton model nverse to restore other degraded mages that have been corrupted n smlar ways. It can be performed as follows. Feed an artfcal (typcal) mage to the degradaton functon and add t some noses to obtan the artfcal corrupted mage. Fnd a good flter mask such that t can be represented as a sutable nverse of the corrupton functon. For lnear spatal flterng, the above process conssts smply of movng the flter mask wndow from pont to pont n the corrupted mage g(x, y).in general, lnear flterng of an mage (here the corrupted mage) of sze M N wth a flter mask of sze m n s gven by a b fˆ (x, y) w(s, t)f (x s, y t) () s a t b Degradaton functon (PSF) + Corrupton Nose n(x,y) corrupted Image g(x,y) Restoraton flter Restoraton Fg. 1. Model of the mage corrupton/restoraton process. restored Image fˆ(x,y) where a=(m-1)/ and b=(n-1)/. Fndng m n coeffcents of the flter mask by PSO s the objectve of ths paper. Therefore, n the followng secton a bref explanaton of PSO mechansm s gven. III. PARTICLE SWARM OPTIMIZATION A partcle swarm optmzer s a populaton based stochastc optmzaton algorthm modeled after the smulaton of the socal behavor of brd flocks. PSO s smlar to genetc algorthm (GA) n the sense that both approaches are populaton-based and each ndvdual has a ftness functon. Furthermore, the adjustments of the ndvduals n PSO are relatvely smlar to the arthmetc crossover operator used n GA. However, PSO s nfluenced by the smulaton of socal behavor rather than the survval of the fttest. Another major dfference s that, n PSO each ndvdual benefts from ts hstory whereas no such mechansm exsts n GA. In a PSO system, a swarm of ndvduals (called partcles) fly through
Internatonal Journal of Vdeo& Image Processng and Network Securty IJVIPNS-IJENS Vol:10 No:06 3 the search space. Each partcle represents a canddate soluton to the optmzaton problem. The poston of a partcle s nfluenced by the best poston vsted by tself (.e. ts own experence) and the poston of the best partcle n ts neghborhood. When the neghborhood of a partcle s the entre swarm, the best poston n the neghborhood s referred to as the global best partcle and the resultng algorthm s referred to as a gbest PSO. When smaller neghborhoods are used, the algorthm s generally referred to as a lbest PSO. The performance of each partcle (.e. how close the partcle s from the global optmum) s measured usng a ftness functon that vares dependng on the optmzaton problem. The global optmzng model proposed by Sh and Eberhart [16] s as follows: v 1 (G x 1 w v RAND c (P best x ) 1 1 best x ) rand c x v (4) where v s the velocty of partcle, x s the partcle poston, w s the nertal weght. c 1 and c are the postve constant parameters, Rand and rand are the random functons n the range [0,1], P best s the best poston of the th partcle and G best s the best poston among all partcles n the swarm. The nerta weght term, w, serves as a memory of prevous veloctes. The nerta weght controls the mpact of the prevous velocty: a large nerta weght favors exploraton, whle a small nerta weght favors explotaton [16, 18]. As such, global search starts wth a large weght and then decreases wth tme to favor local search over global search [16]. It s noted that the second term n equaton (1) represents cognton, or the prvate thnkng of the partcle when comparng ts current poston to ts own best. The thrd term n equaton (1), on the other hand, represents the socal collaboraton among the partcles, whch compares a partcle s current poston to that of the best partcle [17]. Also, to control the change of partcles veloctes, upper and lower bounds for velocty change s lmted to a user-specfed value of V max. Once the new poston of a partcle s calculated usng equaton (), the partcle, then, fles towards t [16]. As such, the man parameters used n the PSO technque are: the populaton sze (number of brds); number of generaton cycles; the maxmum change of a partcle velocty V max and w. Generally, the basc PSO procedure works as follows: the process s ntalzed wth a group of random partcles (solutons). The th partcle s represented by ts poston as a pont n search space. Throughout the process, each partcle moves about the cost surface wth a velocty. Then the partcles update ther veloctes and postons based on the best solutons. Ths process contnues untl stop condton(s) s satsfed (e.g. a suffcently good soluton has been found or the maxmum number of teratons has been reached). (3) IV. ESTIMATION OF THE CORRUPTION FUNCTION INVERSE BY PSO ALGORITHM The problem of fndng the corrupton functon nverse s explaned n secton. It s mentoned that for ths goal, frst an artfcal mage s passed from the degradaton functon (n practce ths degradaton functon s not mathematcally known and t s produced by moton turbulences, camera dstortons, envronmental condtons and so on) then an addtve nose s also added to get an artfcal corrupted mage. Usng these two mages and by means of PSO an optmal flter mask s found. PSO s used to fnd m n coeffcents of the flter mask, teratvely. For mplementng a PSO, a cost functon must be defned as a crteron for mnmzaton. Absolute dfference between the orgnal mage and the mage estmated by flter mask s maybe the natural cost functon. Therefore, t s defned by cost functon= ( fˆ(x,y) f (x,y)) (5) The proposed procedure for fndng the optmal mask s gven as a flowchart n Fg.. As shown n Fg., after defnng the cost functon, the populaton of partcles s ntalzed. Each partcle represents coeffcents of a canddate flter mask. The sze of the flter determnes the sze of the partcle. In ths paper, the flters are assumed to be square szed (.e. the sze of the mask s n n), so the partcle sze s equal to n. Other parameters such as ntal speed vector, populaton sze, nertal coeffcent, acceleraton constants and the value of the maxmum number of the teratons are also ntalzed. Afterwards, each partcle s used to calculate the correspondng corrupted mage fˆ (x, y), usng Eq. (). The corrupted mage s then used to compute the cost functon for each partcle, usng Eq. (5). The best partcle and the local best partcles and ther correspondng costs are saved. Then, the new postons, as the new better solutons (flter masks), are produced by Eqs. (3) and (4). Fnally the stop condton s checked; f t s satsfed the algorthm ends otherwse the process s terated. Ths process contnues untl a termnaton condton s selected (n ths paper a maxmum value for teratons as the stoppng crteron). When the algorthm s stopped, the best partcle s the best soluton and s the optmal flter mask whch can approxmate the corrupton functon nverse. Now, usng the computed flter mask, one can be restore other smlar corrupted mage. The proposed method s lnear, smple and ntellgent that ts performance s evaluated by some smulatons n the subsequent secton.
Internatonal Journal of Vdeo& Image Processng and Network Securty IJVIPNS-IJENS Vol:10 No:06 4 Start Intalzaton Generate random populaton of partcles as ntal flter masks Compute the corrupted mages usng artfcal mage and the flter masks g(x, y) f (x,y) / f (x,y) fˆ(x,y) ISNR 10log One can see that the larger ISNR, the better performance s acheved. At frst case, the lena mage s corrupted by moton functon and localvar nose. Fg. 3 shows the smulaton results. ISNR crteron results are appeared n Table 1. One can see that the results of the proposed method are better than the results of the other tradtonal flters. (6) Calculate the cost functon, for the each corrupted mage Update the global best postons and local best postons Calculate the veloctes and update partcle poston to obtan new masks No Stop condton s satsfed? Stop The best partcle s the fnal soluton (optmal flter mask) Fg.. The flowchart of PSO procedure for fndng the optmal flter mask Yes Fg.3. Results of dfferent flters and PSO based technque for restoraton of mages corrupted by moton functon and localvar nose, a) orgnal mage; b) degraded mage; c) restored mage by PSO; d) restored mage by geometrc mean flter; e) restored mage by harmonc mean flter; f) restored mage by alpha-trmmed mean flter TABLE I ISNR CRITERION FOR RESTORATION OF IMAGES CORRUPTED BY MOTION FUNCTION AND LOCALVAR NOISE Method ISNR PSO found flter mask 9.85 geometrc mean flter -1.97 harmonc mean flter -3.05 alpha-trmmed mean flter -8.07 V. SIMULATION RESULTS In ths secton, the effcency of the proposed method s evaluated usng dfferent smulatons. For ths purpose, we use Matlab software Image Processng Toolbox to produce degradaton functons and addtve noses. Moton, average and unsharp blurrng functons are selected as degradaton functons. To construct artfcally corrupted mages, salt & pepper, localvar and gaussan noses are used as the addtve noses. The obtaned results are compared to the results of some well-known exstng flters, such as geometrc mean flter, harmonc mean flter and alpha-trmmed mean flter, to demonstrate the effcency and applcablty of the proposed technque. In all smulatons populaton sze of the swarm s consdered to be 10. The maxmum number of teratons s selected equal to 100. Inertal coeffcent, w, s set as a gradually decreasng functon of teratons. Both parameters c 1 and c are chosen equal to. Snce the PSO algorthm has a random nature, t s run 100 tmes for each case study and the best result of them s selected. The mask sze s consdered to be n =5. ISNR crteron [19] s used to show the qualty of the restored mage. As Second case, the lena mage s corrupted by average functon and salt & pepper nose. Smulaton results are depcted n Fg. 4. Also, ISNR crteron results are shown n Table. It s obvous that the PSO found lnear flter mask works better than lnear and nonlnear flters. As fnal expermentaton, the effcency of the proposed method s valdated by corruptng lena mage wth unsharp functon and gaussan nose. The results of smulatons are llustrated n Fg. 5. Also, Table 3 shows ISNR crteron results. The results reveal the good performance of the PSO found optmal lnear flter mask compared to the other conventonal flters. VI. CONCLUSIONS In ths paper, a new ntellgent method for solvng mage restoraton problem s ntroduced. Usng artfcally corrupted mages, a lnear flter mask s found to act as the nverse of the mage corrupton process. For fndng the flter mask coeffcents, an optmzaton problem s formulated and solved
Internatonal Journal of Vdeo& Image Processng and Network Securty IJVIPNS-IJENS Vol:10 No:06 5 by PSO technque. Once the optmal flter mask s found, t can be used to restore other smlar corrupted mages, wthout havng any knowledge about degradaton functon or addtve noses. The performance of the proposed method s evaluated usng several smulatons. Snce the proposed method s a smple lnear procedure, t can be mplemented n hardware or Fg. 4. Results of dfferent flters and PSO based technque for restoraton of mages corrupted by average functon and salt & pepper nose, a) orgnal mage; b) degraded mage; c) restored mage by PSO; d) restored mage by geometrc mean flter; e) restored mage by harmonc mean flter; f) restored mage by alpha-trmmed mean flter software easly. The future work can be focused on usng other ntellgent tools, such as artfcal neural networks and fuzzy logc, for mage restoraton problem. We can use an artfcal neural network to approxmate the relatonshp between a corrupted mage and orgnal mage. Then, the neural network wll be used to restore the other smlar corrupted mages. Also, we can use other PSO optmzed flters, not only the mask flters, n order to restore degraded nosy mages. REFERENCES [1] N. S. Kopeka, A System Engneerng Approach to Imagng, SPIE Optcal Engneerng Press, nd ed. Bellngham, Washngton, 1998. [] O. Shacham, O. Hak, and Y. Ytzhaky, Blnd restoraton of atmosphercally degraded mages by automatc best step-edge detecton, Pattern Recognton Letters 8, 007, pp. 094 103 [3] A. Jalobeanu, J.Zeruba, and L. Blanc-Feraud, Blnd Image Deconvoluton: Theory and Applcatons, Taylor & Francs/CRC Press. 007. [4] D. Kundur, and D.Hatznakos. Blnd mage deconvoluton, IEEE Sgnal Process. Mag. vol. 13, 1996, pp. 43 64. [5] A. S. Carasso, Drect blnd deconvoluton, SIAM J. Appl. Math., vol. 61 (6), 001, pp. 1980 007. [6] B. Chalmond, PSF estmaton for mage deblurrng, Graphcal Models Image Process, vol. 53 (4), 1991, pp. 364 37. [7] Y. Ytzhaky, R. Mlberg, S. Yohaev, and N.S. Kopeka, Comparson of drect blnd deconvoluton methods for moton-blurred mages, Appl. Opt. vol. 38 (0), 1999, pp. 435 433. [8] K. Nsh, and S. Ando, Blnd superresolvng mage recovery from blur nvarant edges n Proc. Internat. Conf. on Acoustcs, Speech, Sgnal Processng, Vol. 5, Adelade, Australa, 1994, pp. 85 88. [9] G. Pavlovc, and A. M. Tekalp, Maxmum lkelhood parametrc blur dentfcaton based on a contnuous spatal doman model, IEEE Trans. Image Process. vol.1 (4), pp. 496 504, 199. [10] A. E. Savaks, and H. J. Trussell, Blur dentfcaton by resdual spectral matchng, IEEE Trans. Image Process. vol. (), pp. 141 151, 1993. [11] D. G. Sheppard, H. Bobby, and M. Mchael, Iteratve mult-frame super-resoluton algorthms for atmospherc-turbulence-degraded magery, J. Opt. Soc. Amer. A vol. 15 (4), 1998, pp. 978 99. [1] D. L, and R. M. Merserau, Blur dentfcaton based on kurtoss mnmzaton, n: Proc. IEEE Internat. Conf. on Image Processng (ICIP), vol. 1, Genoa, Italy, 005, pp. 905 908. [13] D. P. Tuan, An mage restoraton by fuson, Pattern Recognton Vol. 34, 001. pp. 403 411. [14] F. Rooms, M. Ronsse, A. Pzurca, and W. Phlps, PSF Estmaton wth Applcatons n Autofocus and Image Restoraton, n Proc. 3th IEEE Benelux Sgnal Processng Symposum (SPS-00), Leuven, Belgum, March 1, 00. pp. 13-16. [15] B. S. Jeona, G. Choa, Y. Huhb, S. Jnb, and J. Park, Determnaton of pont spread functon for a flat-panel X-ray mager and ts applcaton n mage restoraton, Nuclear Instruments and Methods n Physcs Research A 563, 006, pp. 167 171. [16]. Sh, and R. Eberhart, A modfed partcle swarm optmzer. n Proc. of IEEE nternatonal conference on evolutonary computaton, IEEE Press, Pscataway, NJ, 1998, pp. 69-73. [17] J. Kennedy, The partcle swarm: socal adaptaton of knowledge, n Proc. of the IEEE nternatonal conference on evolutonary computaton, IEEE Press, 1997, pp. 303-308. [18] J. Kennedy, and R. Eberhart, Partcle Swarm Optmzaton, n Proc. of IEEE Internatonal Conference on Neural Networks, IEEE Press, Perth, Australa, vol. (4), 1998, pp. 194-1948. [19] M.R. Banham, and A.K. Katsaggelos, Dgtal mage restoraton, IEEE Sgnal Process. Mag. vol. 14 (), 1997, pp. 4 41. Fg. 3. Results of dfferent flters and PSO based technque for restoraton of mages corrupted by unsharp functon and gaussan nose, a) orgnal mage; b) degraded mage; c) restored mage by PSO; d) restored mage by geometrc mean flter; e) restored mage by harmonc mean flter; f) restored mage by alpha-trmmed mean flter TABLE III ISNR CRITERION FOR RESTORATION OF IMAGES CORRUPTED BY UNSHARP FUNCTION AND GAUSSIAN NOISE Method ISNR PSO found flter mask 13.08 geometrc mean flter -15.89 harmonc mean flter -17.04 alpha-trmmed mean flter -1.61