SEGMENTATION USING ADAPTIVE THRESHOLDING OF THE IMAGE HISTOGRAM ACCORDING TO THE INCREMENTAL RATES OF THE SEGMENT LIKELIHOOD FUNCTIONS.

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1 SEGMENTATION USING ADAPTIVE THRESHOLDING OF THE IMAGE HISTOGRAM ACCORDING TO THE INCREMENTAL RATES OF THE SEGMENT LIELIHOOD FUNCTIONS Ioannis M. Sephanais and George C. Anasassopoulos Hellenic Telecommunicaions Organizaion, GR- 8, Ahens, GREECE Medical Informaics Lab., Democrius Universiy of Thrace, GR-68, Alexandroupolis, GREECE ABSTRACT A novel algorihm for image segmenaion based on adapive hresholding of he global hisogram of an image is proposed and applied o medical images from he medical daabase of he Second Deparmen of Surgery of he Universiy Hospial of Alexandroupolis, Greece. The hreshold values are specified hrough an adapive process ha deermines he opimum number of hisogram regions and, a he same ime, aemps o minimize he opimal lielihood value obained from he specific pariion of he hisogram. The main peas of he hisogram are seleced as seeds for he iniial pariion of he hisogram. These seeds are subsequenly grown by varying heir upper and lower boundaries according o he incremenal changes of lielihood values corresponding o he differen inervals of he image hisogram. The proposed mehod provides an alernaive way of selecing he dominan peas of he image hisogram according o predefined consrains.. INTRODUCTION Image segmenaion consiss of deermining disjoin segmens of an image, denoed as I, ha are compac, feaure smooh boundaries and are homogeneous regarding he saisics of he pixel values wihin each region, { R, R, R, } P I = 3, R, where R i R j = wih i, j [, ] and i j. Image segmenaion is an essenial processing sep inheren in a variey of algorihms ha are inended for image enhancemen in he conex of acquisiion of medical imagery, oudoor and nigh vision imaging sysems ec, for auomaic paern recogniion as implemened in radar and sonar sysems, in medical diagnosic sysems or in he conex of auomaic recogniion of machine prined or handwrien exs, for shape recogniion as implemened in he conex of robo vision and low-level visio for coding of video sequences and sill images MPEG-4/H.64 and for a variey of oher image processing ass. Image segmenaion mehods may be classified ino he following caegories: Hisogram hresholding using wo or more hresholds based on he peas and he valleys of he global hisogram of an image []. Hisogram hresholding may be crisp or fuzzy [], [3]. Local filering approaches such as he Canny edge deecor [4] and similar echniques. Region-growing and merging echniques based on pixel classificaion in some feaure space [], [6]. Deformable model region growing [7]. Global opimizaion approaches based on energy funcionals [8] and/or mixure models of individual componen densiies usually Gaussians. These approaches employ such echniques as Bayesian/Maximum a-poseriori crieria [9], he Expecaion Maximizaion EM Algorihm [], propagaing frons/level se segmenaion [], [] and Minimum Descripion Lengh MDL crieria. Morphological mehods lie waersheds, morphological image analysis [3], [4] and hybrid morphological-saisical echniques []. Fuzzy/rough se mehods lie fuzzy clusering and ohers [], [3]. Mehods based on Arificial Neural Newors ANNs lie unsupervised learning and evoluionary/geneic algorihms [6]. Hybrid mehods ha aemp o unify several of he above approaches. Paricular segmenaion algorihms are, generally, no applicable o all images. Pracice shows ha a specific mehod may yield segmenaion resuls of varying qualiy when applied o images wih differen characerisics. This implies ha differen algorihms are no equally suiable for a specific applicaion.

2 . CRISP AND FUZZY HISTOGRAM THRESHOLDING Hisogram hresholding may be eiher crisp or fuzzy. A se of - hresholds, denoed as {T, T T - }, is defined in order o segmen an image ino segmens, denoed as {R, R R }, where I R : =,... I [ T, T. [ T, T is he dynamic range of he pixel values of he image. Hisogram hresholding assumes ha pixels of he image feauring comparable gray-level values are well confined wihin compac regions of he image. Since his is no always he case, furher processing of he segmened image via hresholding may be required. Deermining he appropriae hresholds ha yield an efficien segmenaion is a ey issue. The global image hisogram may be considered as a mixure of individual componen densiies, usually Gaussians. The Expecaion Maximizaion EM Algorihm [] is usually employed among oher available echniques in order o esimae he free parameers of he individual componen densiies. The dominan peas of he image hisogram dicae he number of he individual componen densiies ha comprise he image hisogram should he mixure model be adoped. The hresholds are placed a gray-level values ha correspond o deep valleys of he image hisogram. The relaive heighs of he peas of he image hisogram are usually employed in order o disinguish beween dominan and minor peas whereas fuzzy membership funcions, fuzzy mehods and various fuzzy measures are employed in order o deermine he pariion of he image hisogram ino muliple hreshold inervals [7]. Thresholds are deermined by minimizing a measure of fuzziness, lie he Shannon s Enropy or he Index of Fuzziness, over some parameerizaion of he fuzzy membership funcions ha depends upon a se of hresholds. Le I denoe he gray-level value of a pixel of an image where m= M and n= N. Shannon s enropy [8] is given as, E = S I MN ln μ, 3 all = where S is defined for each fuzzified segmen of he image a according o he following relaionship, S μ = μ ln μ. 4 Shannon s enropy assumes values from he inerval [, ]. I has a minimum value, if μ = or for all and, and a maximum value, if μ = for all and. As an alernaive o Shannon s enropy, Yager proposed in [9] a measure of fuzziness he socalled Yager s Fuzzy Index ha depends upon he relaionship beween a fuzzy se and is complemen. Several parameerizaions are possible regarding he fuzzy membership funcions provided ha μ I =. The sandard S-funcions are = usually employed in order o derive he membership funcions and, consequenly, he opimal hresholds in conjuncion wih an opimizaion algorihm ha minimizes eiher Shannon s Enropy and Yager s Fuzzy Index. Minimizaion is carried ou over he dynamic range of he pixel values. The sandard S-funcions, which are used o derive he membership funcion of image segmen as μ = S I ; c, b S ; c, b, are parameerized as,, I c b / [ I c / b + / ], c b / < I c S I ; c, b = [ c I / b + / ], c < I c + b /, c + b / < I where c is he cross-over poin of he S-funcion, i.e. he crisp hreshold beween image segmens R - and R, and b is is bandwidh seepness or fuzzy dynamical range beween he wo segmens. Obviously, he shape of he S-funcion is deermined by hese wo parameers. 3. ADAPTIVE THRESHOLDING OF THE IMAGE HISTOGRAM ACCORDING TO THE GROWING RATE OF REGIONAL LIELIHOOD FUNCTIONS 3.. Definiion of he parial lielihood funcions The proposed algorihm defines a parial lielihood funcion associaed wih each segmen of he image or, equivalenly, wih each adapively growing inerval of he image hisogram. The parial lielihood funcion for he -segmen reads, l θ R I θ dr = log p g dn = log p θ T T 6 where vecor θ holds he mean denoed as μ and he sandard deviaion denoed as peraining o R, T - and T are he hresholds defining segmen R according o Eq. and dn is he of he image wih a gray-level value of g. The proposed algorihm is iniiaed from he dominan peas of he image hisogram and increases adapively he inerval ha defines segmen R. The lower and he upper limis, which define inerval Δg corresponding o segmen R a ieraion, are denoed as

3 [low, up. The values of he parial lielihood funcions a are minimized over he parameers of he individual componen densiies, i.e. θ =[μ, ] for =,. The proposed algorihm assumes ha he slopes of he opimized, i.e. minimal, parial lielihoods, which are denoed as l, are increased monoonically. The increase rae of he parial lielihoods is he same for all segmens. A penaly parameer κ is assigned o each segmen in order o allow for segmen inerval merging. 3.. The proposed algorihm for adapive hresholding of he image hisogram The proposed algorihm is deailed in his secion as described in he following seps. A lis of he values of he parial lielihood funcions corresponding o he segmens of he image, denoed as L = { l,, l,, l3,,, l, }, is formed a. The parial lielihood funcions are opimized over heir parameers. I urns ou ha he minimum values of he parial lielihoods depend upon he opimum values of he sandard deviaions for he case of Gaussian disribuions, i.e. l, = l, θ, = N, log π, + 7 up where N = dn. This yields a corresponding lis L, low { λ, λ,, λ } = where,,,,,, Δl l + l λ = λ. The proposed,,,, = = ΔN, N, + N, algorihm assumes ha he slopes of all segmens are equal a and ha he slope values increase monoonically over ime. The corresponding lis of he gray-level inervals a is defined as g L = { Δg,, Δg,, Δg3,,, Δg, }. The seps are deailed as follows: : Iniialize he algorihm selecing he gray-level values of he hisogram peas as seeds, i.e. L g = { g Pea_, g Pea_,, g Pea_ }. : Find he minimum values of he parial lielihoods for L and deermine he lis g { λ, λ,, λ } L =.,,,,,, : Increase he hisogram inervals Δg, where =,, in such a way ha λ = λ. The, slope λ has o increase monoonically as he algorihm proceeds. 3: If some upper boundary up coincides wih lower boundary low +, se hreshold T in he se of hresholds. Adjus he gray-level values of already se hresholds in he se in order o minimize he overall lielihood. 4: Merge Δ g, wih Δ g +, Δ g = Δg, Δg +, if N log,, log + < κ N +,. Deermine +, he new s..d., denoed as m, which corresponds o Δ g m,. Reduce by one. : Sop if he inervals corresponding o he segmens of he image cover he enire dynamic range of he pixel values of he image, i.e. if U Δ g = [ T, T, = oherwise se + and go o Sep-. 4. EXPERIMENTAL RESULTS, The image shown in Fig., is an X-ray radiograph from a medical daabase ha has been developed in he Second Deparmen of Surgery of he Universiy Hospial of Alexandroupolis, Greece []. The paien is suspeced o have perforaion of a gasroduodenal ulcer. Plain radiographs are aen wih he paien in he uprigh posiion in such a case. The air escapes from he perforaion freely ino he perioneal caviy and collecs under he diaphragm. I aes ypically he form of a semilunar dar area limied beween he righ hemidiaphragm and he upper border of he liver or beween he lef hemidiaphragm and he rim of he gasric fundus. Inraperioneal perforaion of a gasroduodenal ulcer is no clearly presened in his image. Approximaely % of paiens wih perforaed gasroduodenal ulcer do no show free inraperioneal air presumably because he adjacen omenum or oher viscera seal he perforaion before a significan amoun of perioneal air escapes. Technically poor radiographs due o miscalculaed exposure condiions may also resul in false negaive images for such severely ill paiens. The absence of free air in he abdomen may lead o a misdiagnosis especially in aypical cases. Since all paiens wih perforaed gasroduodenal ulcer should undergo an emergency operaion, i is exremely imporan o mae he correc diagnosis. Therefore, false negaive radiographs in which he free inraabdominal air is no visible due o echnical reasons should be excluded. In our sudy, afer acquisiion of he plain radiographs, he films are scanned wih a Heidelberg Lynoype CPS Saphir/Opal scanner in muliple resoluions, in order o achieve he bes image according o he direcions of he surgeons. I urns ou ha furher processing of he

4 images by he applicaion of he proposed algorihm faciliaes he diagnosis. The hisogram of he image is given in Fig..a. The gray-level values corresponding o he hree salien peas of he hisogram a., 69.3 and., which are mared wih he dashed lines, are used o iniiae hree differen segmens of he image hisogram. The parial lielihoods of he hree segmens sar from zero, l, =l, =l 3, = and grow wih he same slope lambda which increases monoonically. Each segmen of he image hisogram is defined by a lower and an upper hreshold, which are moving owards lower and higher pixel values respecively as he segmen grows driven by he increasing slope of is parial lielihood. The inner hresholds mee a 9. for λ=4.8 and a 69. for λ=6.9 as depiced in Fig..b, where he lielihood slopes vs. he levels of he moving hresholds are illusraed. The parial lielihoods corresponding o he hree segmens of he hisogram are esimaed from he sandard deviaions s..d. of he pixel values wihin each segmen. Fig. 3 presens he evoluion of he corresponding sandard deviaions,,3, he parial lielihood values and heir slopes for he hree segmens. Segmenaion of he original image using he obained hresholds is given in Fig. 4. The Lagrange mulipliers, denoed as λ,, have o increase monoonically during growh. I urns ou ha a careful selecion of he seeds ha iniiae he segmenaion algorihm yields region-growing according o such a crierion in mos pracical applicaions of he proposed mehod. This is illusraed in Figs. 3. Should Shannon s Enropy be used in order o deermine he hresholds of he image in Fig. according o he heory in Secion, he esimaed hresholds are, in order of imporance: 7, 3, 3 and oher gray-level values ha lead o less significan reducions of he Shannon s energy as deermined by he local minima in Fig.. Sandard S-funcions wih b =4 are used o derive he membership funcions. These hreshold values compare well agains our resuls using he proposed segmenaion algorihm.. DISCUSSION The proposed algorihm may be applied slighly modified in he conex of convenional region-growing. The parial lielihood funcions are defined upon growing regions of he image per se insead of adapive inervals of is hisogram. Mainaining monooniciy of he lielihood slopes in such a case urns ou o be a ricy as. Thus applicaion of proper spliing and merging rules during he execuion of a generalizaion of he proposed algorihm becomes imperaive for a proper segmenaion. ACNOWLEDGMENT This wor was parially suppored by Gree Minisry of Educaion and he European Union, under program PYTHAGORAS REFERENCES [] N. Osu, A Threshold Selecion Mehod From Gray-Level Hisograms, IEEE Trans. Sys. Man Cyberne., Vol. 9, 979. [] E. E. erre and M. Nachegael Eds.. Fuzzy Techniques in Image Processing. Series Sudies in Fuzziness and Sof Compuing, vol.. Springer-Verlag,. [3] H. R. Tizhoosh. Fuzzy Image Processing: Inroducion in Theory and Applicaions. Springer-Verlag, 997. [4] J. R. Canny, A Compuaional Approach o Edge Deecion, IEEE Transacions on Paern Analysis and Machine Inelligence, vol. 8, no. 6, pp , 986. [] R. Adams and L. Bischof, Seeded Region Growing, IEEE Trans. Paern Analysis and Machine Inelligence, vol. 6 6, pp , 994. [6] J. R. Beveridge e al., Segmening Images Using Localizing Hisograms and Region Merging, In l J. Comp. Vision, vol., 989. [7] T. McInerney and D. Terzopoulos, Deformable Models in Medical Image Analysis: A Survey, Medical Image Analysis, vol., pp. 9 8, 996. [8] D. Mumford and J. Shah, Opimal Approximaions by Piecewise Smooh Funcions and Associaed Variaional Problems, Comm. Pure Appl. Mah., vol. 4, pp , 989. [9] S. Geman and D. Geman, Sochasic Relaxaion, Gibbs Disribuions and he Bayesian Resoraion of Images, IEEE Transacions on Paern Analysis and Machine Inelligence, vol. 6, pp. 7-74, 984. [] A. Dempser, N. Laird and D. Rubin, Maximum Lielihood From Incomplee Daa Via The EM Algorihm, Journal of Royal Saisical Sociey, Series B 39, pp. 38, 977. [] S. Osher and J. A. Sehian, Frons Propagaing wih Curvaure Dependen Speed: Algorihms Based on Hamilon- Jacobi Formulaions, Journal of Compuaional Physics, 79, pp. -49, 988. [] T. Brox and J. Weicer, Level Se Based Image Segmenaion wih Muliple Regions, Paern Recogniion, pp. 4-43, Augus 4.

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