Mine Classification based on raw sonar data: an approach combining Fourier Descriptors, Statistical Models and Genetic Algorithms

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1 Mne Classfcaton based on raw sonar data: an approach combnng Fourer Descrptors, Statstcal Models and Genetc Algorthms I. Qudu *, J. Ph. Malasse *, G. Burel **, P. Vlbé ** (*) Thomson Marcon Sonar, Route de Sante Anne du Portzc, 960 BREST cédex, France (**) L.E.S.T. - UMR CRS 666, 6 avenue Le Gorgeu, BP 809, 985 BREST cédex, France Abstract- In the context of mne warfare, detected mnes can be classfed from ther cast shadow. A standard soluton s to perform mage segmentaton frst (we obtan bnary from graylevel mage gvng the label zero for pxels belongng to the shadow and the label one elsewhere), and then to perform a classfcaton based on features extracted from the D-shape of the segmented shadow. Consequently, f a mstae happens durng the process, t wll be propagated through the followng steps. In ths paper, to avod such drawbacs, we propose a novel approach where a dynamc segmentaton scheme s fully classfcaton-orented. Actually, classfcaton s performed drectly from the raw mage data. The approach s based on the combnaton of deformable models, genetc algorthms, and statstcal mage models. I. ITRODUCTIO Deformable contours are flexble models able to ft the data. Two technques have been wdely used: snaes and deformable templates. Snaes are actve contour models guded by nternal constrant forces and mage forces []. These free-shape models can be deformed n order to match salent mage features wthout pror nformaton on the geometry of the shape. They do not requre mage preprocessng. On the opposte, deformable templates descrbe the shape by usng a deformaton of a basc template []. The mage must be segmented frst, n order to obtan an explct contour descrpton. Our approach taes place between these two technques. Searchng the best soluton through a large space of potental solutons, a partcular optmzaton process has to be appled. Genetc algorthms are such balanced methods whch explot the best solutons whle explorng the search space. For our purpose, a set of templates (.e. ndvduals) are deformed n order to maxmze an energy functon based on the statstcal propertes of the observed mage. These ndvduals are contours ale snaes whch manage to ft the observed mage. Alteratons among ndvduals lead to a soluton characterzed by the Fourer decomposton of ts contour. The paper s organzed as follows. In Secton II, we revew brefly the prncple of the genetc algorthms. Secton III descrbes the mplementaton. In Secton IV, expermental results obtaned on both smulated and real sonar mages are provded. Fnally the concluson of our study s gven n Secton V. II. GEETIC ALGORITHMS Genetc Algorthms (GA) are stochastc search methods that mmc the natural bologcal evoluton. They operate on a populaton of potental ndvduals applyng the prncple of survval of the fttest to produce better and better approxmatons of a soluton [3] [4]. GAs are executed over a sequence of teratons on a set of coded ndvduals, the populaton, wth three basc operators : selecton/reproducton, crossover, and mutaton. In each generaton a probablstc selecton s performed based upon the ndvdual s ftness such that the best ndvduals have an ncreased chance of beng selected to reproduce n the next generaton. Genetc operators are appled on these parent chromosomes and new chromosomes (offsprng) are generated. III. DESCRIPTIO OF THE PROPOSED ALGORITHM The genetc algorthm toolbox mplemented n Matlab by Houc et al. gave us some useful functons adaptable to the gven problem [5]. Ther Matlab toolbox, named GAOT, Genetc Algorthms for Optmzaton Toolbox, provdes a group of related functons wth easy extensblty and modularty. Gven the raw mage data, the populaton converges to a contour well-ftted to the mage. Each ndvdual s characterzed by specfc Fourer descrptors related to ts shadow contour. Ths contour splts the observed mage n two homogeneous s whose statstcal propertes are used to evaluate the ndvdual s ftness. Over a sequence of teratons, GA generates better sets of ndvduals thans to selecton and reproducton (usng genetc operators) of the best ndvduals. A. Indvdual encodng Shadows are descrbed by two sets of Fourer coeffcents, correspondng to the Fourer transformatons of the coordnates of the contour pxels. Gven the set of unformly spaced pxels (x,y ) =0 - extracted from the closed contour we compute the double set of Fourer descrptors hereafter: X Y 0 0 x exp j y exp j In order to avod the effect of an arbtrary startng pont and to provde an unque descrpton for a gven contour, these coeffcents are normalzed as proposed by. Arbter [6]. Gven C x jy exp j X jy we 0 really compute the followng coeffcents : C X jy X X X C X jy C X jy Y Y Y C X jy for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths wor n other wors must be obtaned from the IEEE.

2 Chromosomes are made of unts called genes (actually the features related to the set of Fourer descrptors) arranged n a defnte successon. Here each chromosome presents the real and magnary parts of the two sets of postve descrptors (because * X X and Y * Y ): Re ( X) Re ( X ) Im ( X ) Im( X ) gene gene gene + gene Re ( Y ) Re ( Y ) Im ( Y ) Im( Y ) gene + gene 3 gene 3+ B. Populaton Intally, a gven number p of prototypes s provded for each of the fve classes consdered : cylnders, spheres, and three stealthy mnes, the Manta and Sgeel mnes whch loo le truncated cones and the Rocan mne wth ts low and pecular profle. Ths populaton s supposed to cover a large set of possble stuatons especally under any pont of vew. Fg. shows some of the ntal ndvduals. sphere C. Evaluaton functon cylnders Manta Rocan Fg.. Some ntal ndvduals. Sgeel gene 4 The evaluaton or ftness functon s chosen n a way such that hghly ftted strngs (or chromosomes) are preferred. Durng the selecton process (see III.D..), ndvduals whose chromosomes have hgh ftness values are chosen wth hgher probablty. Gven the observed mage, ftness of a gven contour can be evaluated usng the statstcal framewor below. We am at fndng the boundary between two specfc homogeneous s of the observed mage, namely the reverberaton and the shadow of the detected object. The condtonal probablty functon of each pxel depends only on whether t belongs to the nsde or outsde of the contour:.e., all pxels nsde (resp. outsde) have a common dstrbuton characterzed by a parameter vector n (resp. out ), wth = [ n, out ] [7]. The lelhood functon to maxmze s wrtten as : PI ( ), p I(, n p I(, out, Rn Rout where I(, denotes the ntensty of pxel whose coordnates n the mage are (,, R n (resp. R out ) of pxels belongng to the shadow (resp. both the reverberaton and the echo). Components of the vector (whose length s ) are the Fourer descrptors characterzng the ndvdual (see III.A.). The correspondng contour whose coordnates are (x,y ) =... can be rebult wth: x X w where w expj / y Y w Gven, the explct contour representaton s obtaned wth a degree of smoothness defned by the order. Ths contour splts the mage n two s: the nsde (shadow) and the outsde (reverberaton and echo). The goal s to estmate and from the observed mage maxmzng the lelhood functon. For sonar mages, pxels generally have a Raylegh dstrbuton. For each, t follows: I d [ I d ] pi exp where the parameters [ d, ] have to be estmated. Appendx A gves the followng estmates: where dˆ ˆ I 4 [ I dˆ. s the number of pxels belongng to the, I s the mean of the ntenstes of these pxels, the varance of these pxels. The maxmum lelhood estmaton of and conssts n maxmzng the log-lelhood functon I, ln P, when I I(, dˆ 0. It amounts then to maxmze the followng evaluaton functon: ], for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths wor n other wors must be obtaned from the IEEE.

3 E ln I I Rn Rn ln Rn ln I I Rout Rout ln Rout The evaluaton requres the mean pont ( X 0, Y0 ) around whch the contour has to be rebult from the others descrptors ( X, Y ).... Practcally ths pont s found n the vcnty of an ntal pont (chosen by the operator for the gven observed mage) n such a way that t maxmzes the evaluaton functon E. D. Basc operators ) Selecton/reproducton procedure Durng an teraton, a fxed number of ndvduals s mantaned. Each ndvdual s evaluated thans to the above evaluaton functon to gve some measure of ts ftness. Then a new populaton s formed by selectng the best ndvduals. There are several schemes for the selecton process [8]. Based on ranng methods, normalzed geometrc ranng only requres the evaluaton functon to map the ndvduals to a partally ordered set [5]. A probablty of selecton P s assgned to ndvdual when all ndvduals are sorted: r P q q where q = the probablty of selectng the best ndvdual r = the ran of the ndvdual, where s the best p = the populaton sze q = q/(-(-q) p ) Choosng q=0.5 wth p=95, about /3 of the ndvduals have a probablty P hgher than It appears as a good compromse between a lac of dversty (for q<0.5) and large computatonal tmes (for q>0.5). ) Genetc operators Genetc operators are appled on parent chromosomes: new chromosomes,.e. offsprng, are generated. Alterng the composton of chldren, a certan dversty s preserved. Dealng wth partcular genes namely real and magnary parts of Fourer descrptors, we have mplemented some suted genetc operators mang good use of Fourer descrptors propertes. In order to eep constant the number of ndvduals durng each teraton, crossover taes two ndvduals and produces two new chldren whle mutaton alters one ndvdual to produce a sngle new chld. ote that c (resp. p ) wll refer to a chld (resp. a parent): - crossover A crossover operaton recombnes genetc materal of two parent chromosomes to produce offsprng for the next generaton. We mplement four dfferent operatons (best(p,p ) means the best of (p,p ) n terms of ftness ):. Random exchanges occur between the descrptors of the parents wth a probablty equal to 0. c p p c p 0.5 p c best( p, p) c 0.5 p p same operaton as. but only on the coeffcents X and Y (related to the prncpal axes) The three last crossovers occur wth a probablty equal to 0./3. On the whole, the crossover probablty s fxed as mutaton Based on the general dea of mutaton, the frst operaton happens one tme per teraton and conssts n addtonnng a whte gaussan nose on one of the descrptors. The second operaton occurs more frequently wth a probablty equal to 0. and operates on the whole chromosome. It conssts n dong an affne transformaton of the Fourer descrptors (see appendx B). Chld s contour appears le the affne transformaton of the parent s as f t was seen through a dfferent pont of vew. Sx affne transformatons are possble usng a matrx A: 0 scalng A, wth the random 0 parameter 0.4; 0.4\ 0 elongaton along the rows of the mage 0 A, wth the random parameter 0 0.4; elongaton along the columns A, wth 0 0.4; 0.4 the random parameter rotaton cos 40 A sn 40 ; the random parameter sn 40, wth cos 40 sew transformaton on the rows A, wth 0 0.; 0. the random parameter 0 sew transformaton on the columns A, 0.; 0.. wth the random parameter After a genetc operaton, each chld s characterzed by ts normalzed descrptors (normalzaton has to be ensured after any genetc operaton) and ts ftness value. E. Termnaton crteron The genetc algorthm s termnated when one of the followng condtons s met: - no mprovement n the best ndvdual happens durng three consecutve generatons, - the specfed maxmum number of generatons taen as 0 s reached. for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths wor n other wors must be obtaned from the IEEE.

4 F. Classfcaton step Classfcaton can be performed by comparng the Fourer descrptors of the wnner wth the Fourer descrptors of the ntal prototypes. Thans to the unqueness of the normalzed Fourer descrptors, we compute a classfcaton crteron assgnng the fttest ndvdual to the same class as ts nearest neghbour accordng to a mnmal dstance. A dstance d j can be computed from genes of the fttest ndvdual and those of the j th ndvdual of the ntal 4 populaton such that j d g g 4 j p where p s the weght related to the maxmal value allowed for the gene g j g... 4 are the estmated genes and g... 4 are the genes of the j th ndvdual s chromosome of the ntal populaton. Each ndex j refers to a specfc prototype mne whose class s nown. The ndex j Arg mn d j (p s the sze of the j... p populaton) s then related to the nearest prototype. otce that ndvduals consttutng the ntal populaton s arranged as follows: 9 cylndrcal mnes, 9 sphercal mnes, 9 Manta mnes, 9 Sgeel mnes and 9 Rocan mnes. Remember that genes are related to the real and the magnary parts of the normalzed Fourer descrptors accordng to the ndvdual representaton (see III.A.). IV. EXPERIMETAL RESULTS To mprove the robustness of the optmzaton, an mage normalzaton s performed to provde a new mage as t would be seen through a grazng angle of 45 degrees preservng shape ratos. For each real and smulated example, we gve the contour rebult from the estmated descrptors and the best dstance d j as defned above. Smulated sonar mages : d 38 = 0.0 sphere d 8 = 0.60 cylnder d 84 = 0.09 rocan Synthetc aperture sonar mages [9]: d 37 = sphere d 58 = sgeel d 4 = 0.33 cylnder d 55 = 0.8 manta d 56 = 0.08 manta d 5 = cylnder V. COCLUSIO In the context of mne warfare, shapes of possble mnes are well-nown [0]. In terms of shadow recognton, one need only to consder a lmted set of cases dependng on the geometry of mnes. Genetc algorthms are such technques for searchng through a space (populaton) of potental ndvduals. In ths paper, we characterze each ndvdual by for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths wor n other wors must be obtaned from the IEEE.

5 a normalzed double Fourer transformaton of the coordnates of the contour pxels. Gven a statstcal crteron, namely the evaluaton functon, ndvduals converge towards an optmal one. From raw mage data, a Maxmum Lelhood approach allows to rate ndvduals n terms of ther ftness. Furthermore specfc genetc operators tae advantage of the double Fourer descrptors: whle crossovers gve offsprng smlar to ther parents, new offsprng appears thans to mutatons preservng dversty. Fnally, the fttest ndvdual s classfed comparng ts descrptors and those of the ntal populaton. Our approach stands out aganst classcal classfcaton processng. Dealng wth raw data, we dscard some undesrable steps. Indeed, wthout sequental processng, a punctual perturbaton wll have less repercusson on the fnal result. Instead of dong mage segmentaton frstly, feature extracton secondly and performng classfcaton from these features lastly, each ndvdual acts as a potental soluton dentfable thans to the Fourer decomposton of ts contour. In ths way, the whole of the contours move n order to match the shadow resultng n a dynamc classfcaton process. APPEDIX A Probablty densty functon ( x, ) of Raylegh dstrbuton s defned as x d [ x d] ( x, ) px exp whose parameters are ( x, ). d stands for the shft from the orgn and s the scale parameter. If x=(x,x,,x ) s a realzaton of the varate X=(X,X,,X ) consderng random varables (pxels of the observed mage), ndependent of each other and obeyng a Raylegh law ( x, ), formally the maxmum lelhood estmate ˆ s ˆ arg max ln Thans to ndependence, ln P ( x X ) = ln P X ( x ) x d exp x d = lnx d ln x d from whch we obtan the estmate x x d 0,. x ˆ wth To fnd the shft d, one can use the statstcal propertes of the centered Raylegh law (d=0) that s to say the mean and the varance defned as : and For the general Raylegh law (d0) the mean d depends on d such as d d. Consequently for pxels X obeyng to the same law, d corresponds to the dfference between the mean of these pxels ntenstes and a term dependng on ther varance : d x d d d wth 4 d ( x d ) APPEDIX B An affne transformaton of the contour s acheved dong ths operaton on the double set of descrptors. Indeed, gven the followng affne transformaton of the descrptors (except translaton): U X a b X A, =,, V Y c d Y Computng the coordnate values of the correspondng U ax by contour pxels from we have, wth V cx dy w exp j /, u v U w 0 0 V w 0 0 so u v a c ax cx by dy w w ax cx b x x A d y y by dy what proves that the correspondng contour undergoes the same affne transformaton. REFERECES [] M. ass, A. Wtn and D. Terzopoulos, Snaes: actve contour models, Internatonal Journal of Computer Vson, pp. 3-33, 988. [] A.. Jan, Y. Zhong and S. Lashmanan, Object matchng usng deformable templates, IEEE Trans. on Pattern Analyss and Machne Intellgence, Vol. 8, o. 3, March 996. [3] D.E. Goldberg, Genetc Algorthms n Search, Optmzaton and Machne Learnng, Addson-Wesley, Readng, MA, 989. for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths wor n other wors must be obtaned from the IEEE.

6 [4] Z. Mchalewcz, Genetc Algorthms + Data Structures = Evoluton Programs, Sprnger-Verlag, Berln, 99. [5] C.R. Houc, J.A. Jones, M.G. ay, A genetc algorthm for functon optmzaton : a Matlab mplementaton, CSU-IE Techncal Report 95-09, 995. [6]. Arbter, Affne-nvarant Fourer descrptors, n From Pxels to Features, J.C. Smon (ed.). Amsterdam, The etherlands: Elsever Scence, 989. [7] M.A.T. Fgueredo, J.M.. Letão, and A.. Jan, Adaptatve parametrcaly deformable contours, Proceedngs of Computer Vson and Pattern Recognton, 997. [8] D.E. Goldberg and. Deb, A comparatve analyss of selecton schemes used n genetc algorthms, Foundatons of Genetc Algorthms, Ed. G.J.E. Rawlns, 99. [9] D. Bllon and F. Fohanno, Theoretcal performance and expermental results for synthetc aperture sonar selfcalbraton, n OCEAS 98 MTS/IEEE, pp , September 998. [0] Jane s Underwater Warfare Systems, Ed. By Anthony J. Watts, Tenth Edton for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths wor n other wors must be obtaned from the IEEE.