COMPUTER AIDED DIAGNOSIS IN MAMMOGRAPHY BASED ON FRACTAL ANALYSIS

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1 Proc. of the 5th WSEAS Int. Conf. on Non-Lnear Analyss, Non-Lnear Systems an Chaos, Bucharest, Romana, October 16-18, COMPUTER AIDED DIAGNOSIS IN MAMMOGRAPHY BASED ON FRACTAL ANALYSIS DAN POPESCU, MAXIMILIAN NICOLAE, DANIELA ALEXANDRA CRISAN, NICOLETA ANGELESCU Department of Control an Computers POLITEHNICA Unversty of Bucharest Splaul Inepenente nr. 313, Bucharest 6, ROMANIA Abstract: An mportant tool n mecal agnoss s gtal mages analyss an nterpretaton. The obectve of the present paper s to emonstrate the utlty of fractal analyss n theoretc ecson for oubtful cases n malgnty of mammary tumours. We present the fractal an textural features able to classfy these tumours n malgn or bengn. Thus, the most mportant feature s the local connecte fractal menson calculate for eges extracte from bnary mages obtane wth fferent bnary threshols. We escrbe a tumour classfcaton algorthm base on the hstogram of the local connecte fractal menson. In orer to valate the escrbe algorthm, expermental results from 30 mages (mammography) are presente. At the work enng, some conclung remarks are presente also. Key-wors: fractal menson, local connecte fractal menson, textural features, mage segmentaton, computer ae agnoss, mammary cancer, tumour shape, mage classfcaton. 1 Introucton Image processng an mage base analyss technques have playe an mportant role n many mportant mecal applcatons base on computer ae agnoss. These applcatons nvolve the automatc mage acquston, prmary mage processng, segmentaton an extracton of features from the mage. Then the processe ata are use for a varety of classfcaton tasks, such as stngushng normal tssue from abnormal tssue, growng processes, etc. Depenng upon the partcular agnostc task, the extracte features for classfcaton process have: morphologcal propertes, colour propertes, fractal propertes or textural propertes. The textural propertes compute are closely relate to the applcaton oman to be use. Thus, Sutton an Hall [1] scusse the classfcaton of pulmonary seases by texture features. They use three types of texture features to stngush normal lungs from sease lungs: a rectonal contrast measure, an sotropc contrast measure, an a Fourer oman energy spectrum. Harms, Gunzer an Aus [2] use a combnaton between texture features (mcro-eges an sze of textons) an colour features to agnose malgnancy n bloo cells. In [3], the authors use textural features to estmate tssue scatterng parameters n ultrasoun mages. Recently, fractal geometry was use to nvestgate the relatonshp between the complexty of the epthelal/connectve texture nterface (as etermne by fractal menson) an the malgnancy of the gastrc tumour [4], [5]. In ths paper, base on gtal mage processng an classfcaton, we nten to nvestgate the possblty of breast cancer presence n oubtful cases. A mammography s classfe n a BI-RADS category (Breast Imagnng Reportng Data System) from 1 to 5. The 1-3 categores sgnfy that the probablty to be a malgnant tumour s very small; the 5 th category means that the probablty to be a malgnant tumour s very hgh. In the 4 th category the malgnancy rsk s 5-50% an, n ths case, a bopsy s necessary. The tumour aspect s opaque wth blurre eges. Thus, the nformaton

2 Proc. of the 5th WSEAS Int. Conf. on Non-Lnear Analyss, Non-Lnear Systems an Chaos, Bucharest, Romana, October 16-18, about malgnty s concentrate n the tumour contour. After mage acquston, a prmary mage processng (nose reecton, segmentaton - n orer to obtan bnary mage - an contour extracton) s necessary. The contour s reuce to a sngle pxel wth n orer to avo the effects of the thckness of the contour [10]. The algorthms for mage prmary processng are the followng (nonear type): - nose reecton (mean flter): b 2 = Me {a 2, b 1, b 2, b 3, c 2 }; - segmentaton : b 2 = 0 f b 2 <P, an b 2 =1 f b 2 P, where P s a threshol chose by the operator; - contour extracton: ' b 2 = b2 ( a 2 b1 b3 c 2 ) All these operators are local type an the elements are n a 3x3 neghbourhoo of the central pxel b 2 (Fg.1). The result of each operator s note by b 2. a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 Fg.1.Local flterng cross type neghbourhoo base. 2 Fractal menson In orer to evaluate the malgnty rsk, one can analyse the tumour contour aspect. Thus, a metho to releve the rregularty of the contour s to calculate an combne fferent forms of fractal menson. There are many programs for countng the fractal menson n fferent forms, but the most famlar algorthm s the box countng. Many specalze papers escrbe ths algorthm. One of the man characterstcs of fractal obects s that ther measure s epenng on the scale use, fractal forms beng not easly to measure n classc geometrc context. Ther physcal propertes (length, ara, volume) are epenng on the representaton resoluton. The Hausorff menson of a set F s efne as the most effcent coverng of the set. Let conser,s R an a set of test functon f() as: N ( s) = f ( ) * s (1) where N(s) s the smallest number of spheres of s- ameter neee to cover the gven set F. When s, an unque value =D H exsts, calle the Hausorff menson of F, so that: < D > D H H N( s) N( s) 0 In equaton (1) one can conser: = lm s 0 ( N( s) ) ( f ( )) ( 1 ) s ; lm s 0 Then, the fractal menson of set F s: ( f ( )) ( 1 ) s = 0 (2) ( N( s)) DF = lm (3) s 0 (1/ s) Although smple from theoretcal vewpont, the Hausorff menson s not easly to compute. Alternatve methos are use, such as box-countng technque. The box-countng algorthm evaluates the fractal menson, functon of the evoluton of the obect sze n relaton wth the scale factor use. Ths metho s ncate n cases of homogenous structures, supplyng global, average nformaton of analyze obects, because t gnores the heterogeneous nature of mages. Such mages, wth fferent textures may have the same box-countng global menson box. The exstence, nse an mage, of fferent regons wth fferent fractal menson requres alternatve methos whch wll exten the fractal to mult-fractal noton. Tumour growth s rather complex process, ultmately epenent on the presence of tumour cells that prolferate an sprea n the host tssues. Most of the obects n bomecal mages, lke tumours, are multfractals. That means the fractal menson has a varaton n every pont of the mage accorng to the bounary of neghbourhoo. In orer to escrbe the heterogeneous nature of a tumour mage, we may compute, for every sngle pont n the mage, a local menson (box-countng menson, for nstance), lmte to a neghbourhoo of the central pxel. Thus, nstea of a sngle value meant to characterze the whole mage, we have a set of values, one for each pont n the analyze obect. The values wll be represente nto a hstogram n orer to gve emphass to the strbuton of the local rregulartes of the mage. We may conser that the global menson (of the

3 Proc. of the 5th WSEAS Int. Conf. on Non-Lnear Analyss, Non-Lnear Systems an Chaos, Bucharest, Romana, October 16-18, whole mage) s the local menson wth the hghest frequency. In some stuaton nether the local approach of the fractal menson s enough. A compact set of sconnecte ponts (0-topologcal menson) wll have a hgher fractal menson. Thus, we assocate to every sngle pont n the mage a local-connecte fractal menson, n orer to escrbe the shape structure contanng the pont, conserng only those ponts nse the neghbourhoo connecte wth the central pxel. Although the avantages of usng the local an local connecte mensons are obvous, they present an mportant savantage: the strbuton of the local an local connecte fractal mensons epens on the choce of the maxmum wnow sze. 3 Texture features use n computerze mammography If a tumour mage s analyse, we can observe fferent types of texture for malgnant, bengn or unclassfe tssues. In orer to scrmnate fferent kns of tumours, the theoretc ecson metho base on vector features extracte from texture s use. The man features whch can be utlse for tumours classfcaton are calculate from the co-ocurence matrx: correlaton, energy, entropy, homogenety an contrast. A co-occurrence matrx s a two-mensonal array C n whch both the row an the column runnng numbers represent a set of possble mage values V. For grey-tone mages V can be the set of possble grey tones an for colour mages V can be the set of possble colours. The value of C(, ncates how many tmes value co-occurs wth value n some esgnate spatal relatonshp. For example, the spatal relatonshp may be that value occurs mmeately to the rght of value. To be more precse, we wll look specfcally at the case where the set V s a set of grey tones an the spatal relatonshp s gven by a vector that specfes the splacement between the pxel havng value an the pxel havng value. Let be a splacement vector (r, e) where r s a splacement n rows (ownwar) an e s a splacement n columns (to the rght). Let V be a set of grey tones. The grey-tone co-occurrence matrx C for mage I s efne by C (, = {((r,s), (t,v)): I(r,s) =, I(t,v) = } where (r,s), (t,v) N x N, (t,v) = (r+x, s+y),, V,an * s the carnalty of a set. Fgure 2 llustrates ths concept wth a 6 x 8 mage 1 an two fferent co-occurrence matrces from 1: C (1,0),an C (-1,0). There are two mportant varatons of the stanar grey-tone co-occurrence matrx. The frst s the normalze grey-tone co-occurrence matrx N efne by (4) [I] = [C (1,0) ]= [C (-1,0) ]= Fg. 2. Two fferent co-occurrence matrces for a grey-tone mage [I]. C (, N (, = (4) C (, Ths relaton normalzes the co-occurrence values to le between zero an one an allows them to be thought of as probabltes n a large matrx. The secon s the symmetrc grey-tone co-occurrence matrx S (, efne by S (, = C (, + C (, (5) whch groups pars of symmetrc aacences. Co-occurrence matrces capture propertes of a texture, but they are not rectly useful for further analyss, such as comparng two textures. Instea, numerc features are compute from the cooccurrence matrx that can be use to represent the texture more compactly. The followng are stanar features ervable from a normalze co-occurrence matrx: Energy N 2 (, (6) = Entropy = N (, log N (, (7)

4 Proc. of the 5th WSEAS Int. Conf. on Non-Lnear Analyss, Non-Lnear Systems an Chaos, Bucharest, Romana, October 16-18, ( 2 Contrast N (, (8) = Homogenety = Correlaton = N (, 1 + ( μ )( μ ) N (, σ σ (9) (10) where µ, µ are the means an σ, σ are the stanar evatons of the row an column sums N () an N (, whch are efne by relatons (11),(12). N ( ) = N (, (11) N ( = N (, (12) For the textural mages the colour an the texture are more mportant of perceptual vewpont because there are not group of obects. The regons of textural mages ten to spear n whole mage, n tme that the non-textural mages are usual partton n group regons. 4 Algorthm for mammary cancer agnoss. Expermental results The oute of each mage was analyze by estmatng the global fractal menson, the local fractal menson an local connecte fractal menson. For ths purpose we use an orgnal software package escrbe n etal elsewhere [6]. In bref, the fractal menson of each oute was measure by the box-countng algorthm an the local fractal menson an local connecte fractal menson were estmate accorng to the algorthms publshe n [8] an [6]. The mages were analyze by estmatng ther local mass scag propertes. The computer program measure the total number of pxels locally connecte n a wnow of ncreasng sze, centre at a pont [4]. Locally connecte relates to all pxels wthn the largest box use whch belong to the cluster connecte to the pxel where the box s centre. The algorthm for ths proceure s: 1. Conser the current pont P; 2. Mark all the ponts connecte wth P wthn a growng s-sze wnow centre at P (s s uner a fxe s max value whch may not be mofe urng the analyss of the whole mage: 32, for example). Notce that every pont has eght neghbour pxels (N, NW, V, SW, S, SE, E, NE). 3. Count every tme how many ponts N(s) of the analyze obect are wthn the wnow; 4. Usng the least square metho, compute the slope of the log-log curve compose by the ((N(s),(s)) ponts. a) b) c) ) e) f) Fg.3. Mammary tumour BI-RADS 4. a) Intal mage, 256 grey levels; b)130 threshol segmentaton; c)140 threshol segmentaton; )150 threshol segmentaton; e)160 threshol segmentaton; f)170 threshol segmentaton. Most of the nformaton about the malgnty of a tumour s contane n the contour of the tumour shape. Doubtful tumours are characterze by blurre contours whch are changng by fferent threshol use to separate the tumour from backgroun segmentaton. The mage was 1024x1024 pxels, 256 grey levels, bmp format, wth

5 Proc. of the 5th WSEAS Int. Conf. on Non-Lnear Analyss, Non-Lnear Systems an Chaos, Bucharest, Romana, October 16-18, a wnow grey level for tumour representaton. Dfferent contours, for fferent threshols are represente n the fgures 3b, 3c, 3, 3e, an 3f. Fg 3a represents the ntal mage corresponng to the mammography an the area of nterest selecte by the specalst. We can observe that the contours are fferent for fferent bnary threshols: 130, 140, 150, 160, an 170. The box countng fractal menson spectrum s nearly constant n the oman (Fg.4). Also the maxmum frequency for local connecte fractal menson Df: 1.42, 1.38, 1.33, 1.36 (Fg.4, Fg.5, Fg.6, Fg.7), fluctuates very lttle. Fg.5. Local connecte fractal menson for Fg.2, Df=1.33. Fg.8. Local connecte fractal menson for Fg.3e, Df=1.36. Fg.4.Box countng fractal menson Fg.9. Local connecte fractal menson for Fg.3f, Df=1.12. Fg.5. Local connecte fractal menson for Fg.3b, Df=1.42. Fg.6. Local connecte fractal menson for Fg.3c, Df=1.38. A set of 30 mages (mammography BI-RADS 4), from Funen Ccal Insttute, Bucharest, was processe an, n each case, the meum local connecte fractal menson was calculate, for fferent segmentaton threshols. The expermental values are presente n Table1 an n Fg.10 an a agnoss threshol equal to 1.4 n meum fractal menson were establshe. A meum fractal menson less than 1.4 represents a bengn case an a meum fractal menson excee 1.4 represents a malgnant case. After tumour evoluton nvestgaton 18 cases was classfe lke bengn an other 12 cases was classfe lke malgnant. The percentage of correct agnoss n the malgnant case was approxmately 92%. Table 1. Expermental values Meum fractal menson Cases 16 (89%) Bengn (18 cases) Malgn (12 cases) <1.4 >1.4 <1.4 >1.4 2 (11%) 1 (8%) 11 (92%)

6 Proc. of the 5th WSEAS Int. Conf. on Non-Lnear Analyss, Non-Lnear Systems an Chaos, Bucharest, Romana, October 16-18, Fg.10. Mean of the fractal menson: left- bengn cases, rght- malgnant cases. References [1] Sutton, R.; E. L. Hall, Texture Measures for Automatc Classfcaton of Pulmonary Dsease, IEEE Transactons on Computers, vol.c-21, 1992, pp ; [2] Harms, H., U. Gunzer, an H. M. Aus (1996), Combne Local Color an Texture Analyss of Stane Cells, Computer Vson, Graphcs, an Image Processng, vol.33, 1996, pp ; [3] Insana, M. F., R. F. Wagner, B. S. Garra, D. G. Brown, an T. H. Shawker, Analyss of Ultrasoun Image Texture va Generalze Rcan Statstcs, Optcal Engneerng, vol.25, 1996, pp ; [4] Dobrescu, R., M. Dobrescu an F. Talos, Multfractal mecal mage analyss usng fractal menson, n: R. Dobrescu, C. Vaslescu (Es) Interscpary Applcatons of Fractal an Chaos Theory, Etura Acaeme Romane, Bucurest, 2004; [5] Vaslescu, C., A. Herlea, B. Ivanov, R. Dobrescu, F. Talos (2004), A survey on fferences between ntestnal an ffuse type of gastrc carcnoma, n : R. Dobrescu, C. Vaslescu (Es) Interscpary Applcatons of Fractal an Chaos Theory, Etura Acaeme Romane, Bucurest, 2004; [6] Crsan, A.D., Image processng usng fractal technques, Ph.D Thess, Poltehnca Unversty Bucharest, 2005; [7] F.Talos, Usng local fractal menson n multfractals analyss, Proceengs of the "Interscpary approaches n fractal analyss", IAFA 2003, Bucharest, Romana, 2003; [8] G. Lann an J.W. Rppn, How mportant s tumor shape? Quantfcaton of the epthelalconnectve tssue nterface n oral lesons usng local connecte fractal menson analyss, J. Pathol., 179, 1996, pp ; [9] Barnsley, M., Fractals Everywhere, Acaemc Press, 1988; [10] Popescu, D., Dobrescu, R, Mocanu, S., Decate Prmary Image Processors For Moble Robots, WSEAS Trans. on Systems, Issue 8, Vol.5, August 2006, p ;