Reconstruction of Deformed Bite Marks using Affine Transformations.

Transcription

1 econstruction of Deformed Bite Mrks using Affine Trnsformtions. Gerrit Stols, h.d 1.; Hermn Bernitz, h.d 2. Corresponding uthor: rof Hermn Bernitz, Deprtment of Orl tholog nd Orl Biolog, School of Dentistr, Universit of retori,.o. Bo 66, retori 0001, Guteng, South Afric. UNNING HEADING: Deformtion of Bite mrks 1. Deprtment of Science, Mthemtics nd Technolog Eduction, Universit of retori, South Afric 2. Deprtment of Orl tholog nd Orl Biolog, Universit of retori, South Afric

2 3 ABSTACT Bite mrks inflicted on nimte nd innimte objects will undergo certin degree of deformtion. This fct remins one of the biggest stumbling blocks when nlsing evidence for court presenttion. It hs been demonstrted tht the relibilit of pttern ssocition nlsis will not be ffected b miniml degrees of wrping, shrinkge nd distortion. In this stud ffine trnsformtions were pplied to bite mrks to estblish if miniml distortions would ffect the mthemticll determined reltionships of the defined fetures. In rel cse stud, it ws then tested whether the distorted bite mrks mtch the dentition of the perpetrtor b ppling n ffine trnsformtion. This ws confirmed to be the cse. Affine trnsformtions will thus not ffect the reltionships of the individul fetures found in bite mrks. The numericl clcultions vlidte the relibilit of pttern ssocition nlsis in the presence of miniml mounts of wrping, shrinkge nd distortion. KEYWODS: Forensic Odontolog, bite mrk deformtions, ffine trnsformtions. Mn different techniques hve been used to nlse bite mrks found on nimte nd innimte objects(1-5). Irrespective of the techniques used, the degree of wrping, shrinkge nd distortion present in bite mrk ptterns remins one of the biggest stumbling blocks when nlsing evidence for court presenttion(1, 6). For the purposes of this pper, deformtion will include wrping, shrinkge nd distortion. The epert will never know the ect position of the victim during the biting episode, the ect degree of moisture loss s result of dehdrtion of bitten pple ehibit or the precise percentge of butterft which oozed from smple of cheese(4). Although the deformtion m be microscopic in nture, it cretes degree of uncertint when epert evidence is given in bite mrk-ssocited court cses(7). Bernitz(1) hs shown tht smll degree of wrping, shrinkge nd distortion will not ffect the pttern ssocited nlsis of bite mrks. He demonstrted the concept b compring set of fcil portrits tken of Gr ler t 45 nd 75 ers of ge, nd ltered with digitl distortion. The golfer ws esil identified despite the induced digitl chnges nd the smll mount of deformtion tht tht hd tken plce during the 30 er period. The ese of recognition ws however due to the constnt nd unchnged reltionships of his fcil fetures. The purpose of this stud is to nlse mthemticll the 3

3 4 deformtion of bite mrks, so s to demonstrte the constnt numericl reltionships tht eists during these deformtion processes. The mthemticl proof of the cognitive method s described b Bernitz(1) will be of invluble help to epert witnesses when giving evidence in bite mrk cses. Mterils nd Methods Affine trnsformtions cn be used to reconstruct bite mrk tht ws distorted s result of stretching, shrinking, enlrgement, reduction, rottion, shering or n combintion of the bove. In order to reconstruct trnsformed bite mrk, three points on the suspect s dentition tht correspond with three points on the inflicted bite mrk re needed. If three points on bite mrk imge re in stright line, the will remin in stright line fter the trnsformtion. This tpe of trnsformtion preserves collinerit, midpoints of segments, nd segment division rtios. An ffinit is the product of sher, strin nd similrit. A rnge of possible distortions tht could occur in bite mrk re illustrted in Fig 1. These imge distortions preserve both imge reltionships nd ke chrcteristics. The ffine trnsformtions do not, however, preserve distnces. Appling n ffine trnsformtion to uniforml deformed bite mrk cn correct for the enlrgement, reduction, stretching nd shering of the mrk(8). To demonstrte this concept the following mthemticl eplntion is offered. AffineTrnsformtion (Affinit) An ffinit mps ech point (, ) of the Eucliden plne to point ccording to the eqution = A where A nd where the mtri A is invertible. Cederberg(9) eplins tht for n three non-colliner points, nd, there is n ffinit tht mps the points to,, nd ( Fig 2). Using the three points, nd on the inflicted bite mrk nd three corresponding points on the suspect s dentition, nd it is possible to determine mtri A with vlues 11,, 13, 21, 22, nd 23. To determine if the bite mrk is cused b the 4

4 5 5 suspect s dentition we ppl the inverse of the mtri A, clled A -1 on ll the coordintes of the bite mrks nd determine if the corresponds with the suspect s dentition. The question, however, is how to find the mtri A of the ffinit from the vlues of (, ) (, ), (, ) (, ), nd (, ) (, ). To Find the Mtri epresenttion of n Affinit To be ble to define n ffinit A we need to identif three pirs of corresponding points in both imges. In Fig 2 we use the following pirs: (, ) (, ), (, ) (, ), nd (, ) (, ). Substituting these pirs of corresponding points into the eqution = A will result in the following equtions: Eqution Eqution Eqution Eqution Eqution Eqution 6 Using equtions 1, 3 nd 5 nd ppling Crmer s rule for sstem in three vribles we cn find the vlues of 11,, nd 13 : Using equtions 2, 4 nd 6 nd ppling Crmer s rule for sstem in three vribles we cn find the vlues of 21, 22, nd 23 :

5 6 The mtri representtion of the ffinit tht mps point to, to, nd to is T = Use the inverse of mtri T nd multipl it with the coordintes of the end points of ech segment of the bite mrk. It will correspond with the endpoints of the coordintes of ech segment of the suspect s dentition in the cse of uniforml distorted bite mrk. Emple of n Actul Cse A rel cse stud ws used to demonstrte the degree of correltion tht eists fter the ffine trnsformtion of bite mrk. The lower dentl stud models of the suspect were used nd represented the originl imge. The bite mrk present on the brest of the victim represented the trnsformed imge. The following nlticl clcultions were crried out to demonstrte tht the trnsformtions present in the bite mrk on the victim did not significntl ffect the reltionships of the individul fetures. We need three pirs of points to define n ffinit, Fig 3. We cn use the following pirs: A(4.05, 4.39) A(2.62, 3.04), E(10.5; 2.41) E(9.5, 6.93), nd H16.22, 4.63) H(17.59, 3.94). Using the method s illustrted in the previous prgrph the mtri tht represent this ffinit is T = This mtri tht defines the ffine trnsformtion ws determined b mpping the point A, E nd H on the suspects dentition to the points A, E nd H on the bite mrk (Fig 3 & 4). If the bite mrk on the victim ws inflicted b the suspect s dentition, the inverse of the ffinit T 1 must mp points A, B, C, D, F, G, nd H on the suspects dentition which is indicted b A, B, C, D, F, G, nd H. When ppling the inverse of the ffinit T 1 to the points the clcultions show tht the coordintes of the trnsformed 6

6 7 bite mrk re A(4.05, 4.39), B(5.91, 2.97), C(6.94, 2.99), D(9.7, 2.46), E(10.5, 2.41), F(13.43, 2.78), G(13.85, 3.18), nd H(16.22, 4.63) (Fig 4). To determine if the bite mrk ws cused b the suspect s dentition we compre the coordintes of the points A, B, C, D, F, G, nd H of the suspect s dentition with A, B, C, D, E, F, G nd H respectivel. esults Affine trnsformtions will not ffect the reltionships of the individul fetures found in bite mrks. Mthemticll defined points will mintin constnt reltionship during minor deformtions. The rel cse stud clerl demonstrted tht mthemticll clculble ffine trnsformtion hd tken plce during the biting process nd tht the bite mrk ws in fct distorted imge of the originl suspect s dentition. Discussion This stud shows tht the cognitive method described b Bernitz in 2005(1) cn be mthemticll vlidted. Minor distortions in the mrks left b the suspect s dentition will be present in ll bite mrks inflicted on skin nd innimte objects during the biting process nd during the nlsis of the bite mrks. Vritions in tissue structure, dehdrtion nd photogrphic technique will induce these distortions. The trnsformtions observed in bite mrks will thus never be perfectl uniform. The results of the mthemticl model pplied to the rel cse stud showed tht minor distortions did not ffect the bilit the show positive concordnce between the suspect s dentition nd the bite mrk. The effect on the pttern ssocition nd ffine trnsformtion nlsis is negligible s long s the deformities re miniml. Distortions encountered in routine bite mrk nlsis cn generll be considered to be miniml. When nlsing minimll distorted bite mrk the presence of distem between two centrl teeth might vr in width between the dentition of the perpetrtor nd the bite mrk present on the victim, but the spce between the centrl teeth will be present in both cses nd hve similr reltionship to the djcent teeth. 7

7 8 This concept cn be pplied to rotted teeth, missing teeth, teeth out of the rch, intercnine widths, incisl grooves or n other recognisble tooth fetures(2). The rel cse stud clerl demonstrted high degree of concordnce fter the common points within the bite mrk nd suspect s dentition were evluted mthemticll nd confirmed to be ffine trnsformtions. The epert witness is required to demonstrte tht the tooth mrks present on the victim s bod or innimte object nd the suspect s dentition show similr dentl fetures present in the sme position, in reltion to the sme teeth, in the sme shped rches nd hve similr size rtios(1-5, 10). The difficulties eperienced b epert witnesses regrding miniml mounts of wrping, shrinkge nd distortion present in bite mrks cn now be scientificll nullified b the bove numericl eplntion. Conclusion Smll mounts of wrping, shrinkge nd distortion will not ffect the reltionships of fetures within bite mrk. The pttern ssocition nlsis of bite mrks cn be pplied to skin nd innimte objects s long s the deformtions re miniml. Affine trnsformtions cn be pplied to skin nd innimte objects s long s the deformtions re miniml. The mthemticl limits of these deformtions which will significntl ffect the reltionship of dentl fetures hve not et been determined. Acknowledgements The uthors would like to thnk rof WF vn Heerden nd Dr Z Bernitz for their ssistnce with the finl proof reding of the mnuscript 8

8 9 FIG. 1- Emples of ffine trnsformtions Similrit Strin Sher FIG.2- Three pirs of corresponding points define n ffinit (, ) (, ) Suspect s dentition (, ) (, ) (, ) (, ) Distorted bite mrk 9

9 10 FIG.3- Coordintes of tooth structure A, (4.05, 4.39) B, (5.90, 2.80) C, (7., 3.33) D: (9.42, 2.65) E, (10.50, 2.41) F, (13.28, 2.46) G: (13.63, 2.99) U, (16.22, 4.63) ~,,,,,,: )... ) CI (5.45,.$.61) D: (1.55, -4.77) E: (9.50. ~. 93) F: (13.26, -4.64) G: (13.97, -4.03) H: (IB ) 10

10 11 FIG.4 - Trnsformed bite mrk A&A 32 B B Bite mrk on victim (distorted) Bite mrk fter trnsformtion Suspect s dentition C C 31 D E&E D 41 F 42 G&G F H&H A H B C 31 D E 41 F G 42 11

11 eferences 1. Bernitz H. Concepts to elucidte the pttern-ssocited nlsis of tooth mrks in court. J Dent Assoc S Afr 2005;60: Bernitz H, vn Heerden W F, Solheim T,Owen J H. A technique to cpture, nlze, nd quntif nterior teeth rottions for ppliction in court cses involving tooth mrks. J Forensic Sci 2006;51: Bernitz H, Owen J H, vn Heerden W F,Solheim T. An integrted technique for the nlsis of skin bite mrks. J Forensic Sci 2008;53: McNmee A H, Sweet D,rett I. A comprtive relibilit nlsis of computer-generted bitemrk overls. J Forensic Sci 2005;50: Sweet D,Bowers C M. Accurc of bite mrk overls: comprison of five common methods to produce eemplrs from suspects dentition. J Forensic Sci 1998;43: rett I A. The brriers to chieving n evidence bse for bitemrk nlsis. Forensic Sci Int 2006;159: Shesb D,McDonld D G. A forensic clssifiction of distortion in humn bite mrks. Forensic Sci Int 2001;2: Jin A. Fundmentls of Digitl Imge rocessing rentice-hll Cederberg J N. A course in modern geometrics. Springer Blckwell S A, Tlor V, Gordon I, Ogleb C L, Tnijiri T, Yoshino M et l. 3-D imging nd quntittive comprison of humn dentitions nd simulted bite mrks. Int J Legl Med 2007;1:9-17.