FINDING efficient algorithms to describe, measure, and

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1 466 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 36, NO. 3, MARCH 2014 Automatc Algnment of Genus-Zero Surfaces Patrce Koehl and Joel Hass Abstract A new algorthm s presented that provdes a constructve way to conformally warp a trangular mesh of genus zero to a destnaton surface wth mnmal metrc deformaton, as well as a means to compute automatcally a measure of the geometrc dfference between two surfaces of genus zero. The algorthm takes as nput a par of surfaces that are topologcal 2-spheres, each surface gven by a dstnct trangulaton. The algorthm then constructs a map f between the two surfaces. Frst, each of the two trangular meshes s mapped to the unt sphere usng a dscrete conformal mappng algorthm. The two mappngs are then composed wth a Möbus transformaton to generate the functon f. The Möbus transformaton s chosen by mnmzng an energy that measures the dstance of f from an sometry. We llustrate our approach usng several real lfe data sets. We show frst that the algorthm allows for accurate, automatc, and landmark-free nonrgd regstraton of bran surfaces. We then valdate our approach by comparng shapes of protens. We provde numercal experments to demonstrate that the dstances computed wth our algorthm between low-resoluton, surface-based representatons of protens are hghly correlated wth the correspondng dstances computed between hgh-resoluton, atomstc models for the same protens. Index Terms Conformal mappng, mesh warpng, Möbus transformaton, nonrgd regstraton Ç 1 INTRODUCTION FINDING effcent algorthms to descrbe, measure, and compare shapes s a central problem n mage processng. Ths problem arses n numerous dscplnes that generate extensve quanttatve and vsual nformaton. Among these, bology occupes a central place. For example, regstraton of bran anatomy s essental to many studes n neurobology [1], [2], [3]. In parallel, the belef n molecular bology that the structure (or shape) of a proten s a major determnant of ts functon has led to the development of many methods for representng, measurng, and comparng the shapes of proten structures [4], [5], [6], [7]. In general, methods that compare shapes can be classfed nto two categores: those that derve features (also called shape descrptors) for each shape separately that can then be compared usng standard dstance functons, and those that drectly attempt to map one shape onto the other, thereby provdng both local and nonlocal elements for comparson. In ths paper, we are concerned wth the latter. More specfcally, we restrct ourselves to methods that generate mappngs between two shapes that are defned by surfaces of genus zero. Surface-based shape comparson technques am at defnng drectly a map between any two surfaces that s as close to an sometry as possble. There have been many methods developed to fnd such mappngs. These methods usually rest on 1) the defnton of a dstance measure that evaluates how close the map s to an sometry, 2) choces of. P. Koehl s wth the Department of Computer Scence and Genome Center, Unversty of Calforna, Davs, CA E-mal: koehl@cs.ucdavs.edu.. J. Hass s wth the Department of Mathematcs, Unversty of Calforna, Davs, CA E-mal: hass@math.ucdavs.edu. Manuscrpt receved 22 Aug. 2012; revsed 27 Jan. 2013; accepted 20 July 2013; publshed onlne 29 July Recommended for acceptance by P. Golland. For nformaton on obtanng reprnts of ths artcle, please send e-mal to: tpam@computer.org, and reference IEEECS Log Number TPAMI Dgtal Object Identfer no /TPAMI sets of landmark ponts on the two shapes, and 3) an algorthm for fndng the mappng between these sets of ponts that mnmzes the dstance measure. Note that tem 2) s optonal, as descrbed below. An sometry, or map that precsely algns two surfaces wth no dstorton, preserves both angles and dstances. When the shapes are dfferent, then no sometry can be found, and so some metrc dstorton s necessary n any algnment. The harmonc or Drchlet energy [8], [9], the Procrustes dstance and ts contnuous varants [10], the Gromov-Hausdorff dstance and varants [11], [12], and the conformal Wasstersten dstance and varants that mmc mass transportaton [13], [14], [15] are popular metrcs used n ths context. Ideally, landmark ponts should be homologous between the two shapes, should conserve ther relatve postons, should provde adequate coverage of the shape, and should be found relably and consstently [16]. In the case of the human cortex for example, landmark ponts are usually chosen to follow ts sulc and gyr patterns (n some cases, the whole curves are consdered). The task of fndng such meanngful landmark ponts s most successfully performed manually by sklled techncans wth many years of tranng, wth the resultng danger of varablty between operators. Many methods have therefore been developed, however, to ether automate ths process or to crcumvent the need to use specfc pont correspondence n the regstraton procedure. Methods that fall n the former category rely ether on geometrc propertes of the surface such as crtcal ponts dentfed n the process of flattenng the surface [15], or on exstng knowledge, such as an atlas for the shape of nterest [17]. It s worth mentonng that these methods work on the premse that knowledge of a mappng on a small number of correspondences can be extended to gve the full map between the two surfaces of nterest [11], [18], [19]. Landmark-free methods, on the other hand, rely on a geometrc representaton of the surface n whch each vertex s assgned a sgnature, under the /14/$31.00 ß 2014 IEEE Publshed by the IEEE Computer Socety

2 KOEHL AND HASS: AUTOMATIC ALIGNMENT OF GENUS-ZERO SURFACES 467 premse that ponts wth smlar sgnatures are more lkely to correspond. Spectral technques ft n ths category. Rustamov, for example, ntroduces the global pont sgnature (GPS) of a pont on the shape, whch encodes both the egenvalues and the egenfunctons of the Laplace- Beltram operator evaluated at that pont [20]. Gubas et al. ntroduced the heat kernel sgnature, a smlar robust and multscale nvarant defned on each pont of the surface. Ths nvarant s found by solvng a partal dfferental equaton nvolvng the same Laplace-Beltram operator [21]. Fschl et al. assgned to each vertex on a mesh an average convexty, whch s then used to drve the algnment of cortcal surfaces [22], [23]. Vallant and Glaunès ntroduced a representaton of surface n the form of currents and then mposed a Hlbert space structure on t, whose norm s used to quantfy the smlarty between two surfaces [24]. In the specal case that the two surfaces to be mapped are of genus zero (.e., the surfaces are topologcally equvalent to the 2-sphere), two alternate approaches have been used to generate the closest to-sometrc mappng, namely, those that apply frst a parameterzaton (or mappng) of the surfaces onto the sphere, and those that drectly algn the surfaces n ther own coordnates (usually the 3D Cartesan coordnates). The exstence of a conformal mappng of a genus-zero surface to the round sphere s guaranteed by the Unformzaton Theorem [25], [26]. Varous methods have been proposed to generate such a conformal mappng n the dscrete case (.e., when the surface s represented by a mesh), ncludng buldng a lnear system that approxmates the Laplace-Baltram operator [27], usng crcle packng [28], solvng for degree one harmonc maps [29], [30], or mnmzng an angle-based functonal [31]. Once two genus-zero surfaces have been mapped conformally onto the sphere, the search for (near) sometres between them can be made more tractable by restrctng to a search wthn the Möbus group, the group of bjectve conformal self-mappngs of the sphere. The Möbus group acts on the sphere wth sx degrees of freedom. A sngle Möbus transformaton, and therefore a conformal mappng between the two surfaces, s determned by specfyng correspondences between exactly three ponts. Lpman and Funkhouser mplemented ths dea by samplng random trplets of ponts over each surface, computng the Möbus transformaton defned by those trplets, and by votng over the samples, usng as a rankng crtera the estmated devaton from sometry [32]. Ths dea was further extended to automatcally quantfy the overall smlarty between surfaces [15]. It was also prevously used by Tosun et al. [33] and Gu et al. [30] to mnmze sulcal dstances when algnng brans. A conformal mappng between two surfaces of genus zero can only algn exactly three ponts; exact matchng of more than three landmark ponts therefore requres relaxaton of the constrants of angle preservaton. Wang et al. [34], for example, ntroduced an approach for comparng two genus-zero surfaces C 1 ;C 2 that balanced conformablty wth landmark correspondence. They computed a conformal map f 1 : C 1! S 2 and then searched for a second map f 2 : C 2! S 2 that mnmzes an energy functon wth two terms E 1 þ E 2. The frst term E 1 s the standard Drchlet energy, whose mnmzer gves rse to a conformal map. The second term s gven by E 2 ¼ P kf 2 ðq Þ f 1 ðp Þk 2, where the norm represents the eucldean dstance of two ponts on the round sphere, and fp 2 C 1 g and fq 2 C 2 g are matchng landmark ponts. Josh et al. ntroduced a smlar extended cost functon that ncludes a stran energy (based on the Laplace-Beltram operator) and a penalty term for sulc matchng, wth the algnment beng performed on a flat surface nstead of the sphere [35]. In ths work, we are nterested n generatng a globally optmal conformal mappng between two surfaces of genus zero. In contrast to the works of Wang et al. [34] and Josh et al. [35] mentoned above, we do not relax the constrants of angle preservaton; nstead, we propose a new algorthm that fully elmnates the use of landmarks, thereby generatng a mappng based only on the knowledge of the two surface trangulatons. In ths approach, the whole mesh representng the source surface s warped onto the target surface, usng the mappng defned through the composton of dscrete conformal mappngs of the surfaces onto the sphere and the Möbus transformaton between these mappngs. The dscrete mappngs onto the sphere are generated usng the algorthm ntroduced by Sprngborn et al. [31]. The Möbus transformaton s then optmzed by gradent descent to lead to mnmal dstorton between the source mesh and ts mage, where dstorton s measured as dfference from sometry. A good choce of ntal maps s essental for success for ths knd of method, as one expects many local mnma to exst n the 6D space of conformal maps that we explore. We generate ntal algnments based on choces of trplets of ponts n each surface computed from a best ellpsod approxmaton for each. The success of such an approach s lkely to be problem dependent. We note that our optmal conformal mappng s a composton of Möbus transformatons; as each Möbus transformaton s a homeomorphsm, t s therefore a homeomorphsm between the two surfaces. Ths paper s organzed as follows: Secton 2 provdes the mathematcal background for our algorthm: dscrete conformal geometry and dstance measures between meshes. In Secton 3, we provde the detals of ts mplementaton, focusng on how to ensure ts robustness. Secton 4 presents the results obtaned by our algorthm on automatc regstratons of brans, as well as on comparng surfaces of proten structures. We conclude the paper wth a dscusson on future developments. 2 BACKGROUND 2.1 Basc Ideas Let S 1 and S 2 be two surfaces of genus zero, represented by the meshes M 1 and M 2, respectvely. Both meshes are taken to be trangular, wth M¼ðV;E;TÞ and V ¼fv g, E ¼fe j g, and T ¼ft jk g denotng the vertces, edges, and trangles, respectvely. We note that these two meshes are ndependent of each other and are lkely to have dfferent combnatorcs. Our goal s to defne a dscrete map f : M 1! fðm 1 Þ where fðm 1 Þ s a geometrc realzaton of M 1 onto S 2, that s as close as possble to an sometry,.e., that

3 468 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 36, NO. 3, MARCH 2014 The noton of dscrete conformal equvalence s defned as follows. Defnton 2. Two dscrete metrcs l and l ~ on M are dscretely conformally equvalent f, for some assgnment of numbers u to the vertces v, the metrcs are related by the formula: ~l j ¼ e ðu þu j Þ=2 l j : ð1þ Fg. 1. Globally optmal conformal mappng. mnmzes the dstorton of parwse dstances between vertces. The key observaton [32] that makes the problem tractable s that f can be understood as the composton of three dscrete conformal mappng functons, C 1, m, and C2 1 (see Fg. 1). The group of conformal self-mappngs of the sphere s well understood and s called the Möbus group. Any transformaton m n ths group s defned by specfyng the mage of three ponts, and thus has sx degrees of freedom. The mappng f s then constructed by optmzng m so that the composton C2 1 m C 1 s as near to an sometry as possble. We use ths structure to develop an automatc algorthm for comparng two genus-zero surfaces. Ths method nvolves two man steps: 1) generaton of parameterzatons C 1 and C 2 of the meshes M 1 and M 2 onto the sphere, and 2) optmzaton of the Möbus transformaton m. Mathematcal detals for each of these two steps appear n the next sectons. 2.2 Dscrete Conformal Mappng to the Sphere Whle Remann s Unformzaton Theorem guarantees that any smooth genus-zero surface S can be mapped conformally (wth angles preserved) to the unt sphere, n applcatons we are forced to work wth dscrete approxmatons of these underlyng smooth objects. The theoretcal underpnnngs of the theory of dscrete conformal maps are stll beng developed, but many methods have been developed to compute them n practce. We follow the approach proposed by Sprngborn et al., whch ntroduces a noton of dscrete conformal equvalence [31]. Whle we refer the reader to ther paper for a full descrpton of ths approach, we summarze t here to ntroduce defntons and equatons relevant to our algorthm. Dervatons of the formulas gven below can be found n ther paper. Let us consder a trangular mesh M¼ðV;E;TÞ embedded n a 2D manfold. We do not restrct ts topology or assume the presence of a boundary. The ntrnsc geometry of M s encoded n ts edge lengths. Defnton 1. A dscrete metrc on M s a functon l defned on the set of edges E that assgns to each edge e j a length l j such that the trangle nequaltes are satsfed for all trangles n T, so that no sde of a trangle has length longer than the sum of the other two sde lengths. Startng wth some dscrete metrc l, we are nterested n fndng a new dscrete metrc l ~ that s dscretely conformally equvalent and that has partcularly nce geometrc propertes. In the followng, we set l to be eucldean dstance, meanng that the relatons between lengths and angles of trangles are those of the classcal eucldean case. Note that other metrcs are possble. Gven such a metrc l and a trangle t jk n M, the angle jk at vertex v opposte edge e jk can be recovered from the lengths of the sdes of the trangle by standard eucldean trgonometry, gvng the formula: sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff jk ¼ 2 ðl j þ l jk l k Þðl jk þ l k l j Þ tan 1 : ð2þ ðl k þ l j l jk Þðl jk þ l k þ l j Þ The curvature K at an nteror vertex v s gven by the excess angle sum: K ¼ 2 X jk : ð3þ v 2t jk The problem of mesh parameterzaton can then be stated as fndng weghts u such that the new correspondng metrc l ~ has curvature 0 at all nteror vertces n the mesh. These condtons lead to a system of nonlnear equatons, one per nteror vertex. Sprngborn et al. [31] have shown that f a soluton to ths system of equatons exsts, t can be found as the unque mnmzer of the convex energy n u: CEðuÞ ¼ X fðu ;u j ;u k Þþ X u ; ð4þ t jk 2T v 2V where fðu ;u j ;u k Þ¼ 1 ~ ~ 2 jk jk þ ~ j ~ k k þ ~ k ~ j j þ JI ~ jk þ JI ~ j k þ JI ~ k j 2 ðu þ u j þ u k Þ; wth ð5þ ~ j ¼ 2 logðl j Þþu þ u j ; ð6þ and JI s Mlnor s Lobachevsky functon: JIðxÞ ¼ Z x 0 log j2 sn tjdt: Ths can be understood by consderng the partal dervatves of the ¼ 1 X 2 ; 2 t jk 3v ~ jk so that rceðuþ ¼0 f and only f the scalng factors u solve the problem and gve a zero curvature metrc. The method descrbed above allows for the flattenng of a mesh topologcally equvalent to a dsk onto a plane. ð7þ

4 KOEHL AND HASS: AUTOMATIC ALIGNMENT OF GENUS-ZERO SURFACES 469 The parameterzaton resultng from ths algorthm s not unque. The choces made n the algorthm can lead to parameterzatons that dffer by a rotaton of the sphere. By mposng the zero mass-center (step (6)), the mappng s at least unque up to the eucldean rotaton group. Fg. 2. Dscrete conformal mappng to the sphere. The mesh representng the surface S 1 (A) s frst mapped to the plane (B) after removal of a vertex v 0 and ts open star to make t topologcally equvalent to a closed dsk. The planar layout s then projected stereographcally to the sphere (C), wth v 0 beng renstated at the North pole. The sphercal mesh s then normalzed to ensure zero masscenter (D). Note that the sulc appear clearly on the normalzed mesh. When used n combnaton wth stereographc projecton from the plane to the sphere, t can be appled to the problem of mappng meshes wth sphercal topology onto the sphere wth the followng algorthm (see also Fg. 2). Algorthm 1. Dscrete conformal mappng of a genus 0 mesh M wth dscrete metrc l to the unt sphere [31], [36]. (1) Intalzaton. Select vertex v 0 wth smallest curvature. Set u j ¼ 2log l j0 for all vertces v j n the lnk of v 0. (2) Let M 0 be M mnus the open star of v 0. M 0 s topologcally a closed dsk. (3) Flatten M 0 by solvng teratvely for u that mnmze the energy CEðuÞ defned n Secton 4. (4) Layout M 0 on the plane under the mnmzed metrc l. ~ (5) Project planar layout on the sphere stereographcally and rensate v 0 at the North Pole. (6) Apply Möbus normalzaton to ensure that the center of mass of all vertces s at the orgn. The choce of the vertex v 0 n step (1) s arbtrary; we have found that choosng the vertex wth the smallest curvature works best n practce. The mnmzaton n step (3) s performed over all nteror vertces of the mesh M 0. The vertces at ts boundary,.e., those that correspond to the lnk of v 0, have ther weghts fxed to ther values gven n step (1). The result of ths procedure, f t exsts, s a polyhedron wth vertces on the sphere that s dscretely conformally equvalent to M. It may not exst, as the set of nonlnear equatons solved n step (3) s not guaranteed to have a soluton. Indeed, whle the varatonal energy CE s convex, we do not have a convex optmzaton problem. The set of weghts u that results n new edge lengths that satsfy the trangle nequaltes s not convex. We can crcumvent to some extent ths dffculty by extendng the doman of CE: f ~ ljk > ~ l k þ ~ l j then ~ jk ¼ and ~j k ¼ ~k j ¼ 0: Ths smple fx stll does not guarantee that a soluton exsts. We wll dscuss ths further n Secton Möbus Normalzaton for Zero Mass-Center In hs elegant paper [37], Sprngborn proved that for n 3 dstnct ponts n the d-dmensonal unt sphere, there exsts a Möbus transformaton such that the barycenter of the transformed ponts s the orgn. He dd not, however, nclude a practcal method for buldng such a transformaton. We show here that searchng for ths transformaton can be formulated as a nonlnear optmzaton problem wth four varables. Let us consder n 3 dstnct ponts n the 2D unt sphere. The stereographc projecton from the North pole of ths sphere onto the xy-plane takes the pont v wth coordnates ðx ;y ;z Þ to the pont: x y P ¼ðX ;Y ; 0Þ ¼ ; ; 0 ; 1 z 1 z that can be dentfed wth the complex coordnate z ¼ X þ Y. AMöbus transformaton m that maps nfnty to nfnty n the Remann plane (.e., that maps the North pole of the sphere to tself) can be represented as a lnear form mða; bþ ¼az þ b, where a and b are complex numbers and a 6¼ 0. The pont P s mapped nto a pont P 0 wth complex coordnate z 0 ¼ az þ b. Let us defne! 0 ¼jz0 j2 ¼ðaz þ bþðaz þ bþ The nverse stereographc projecton of P 0 gves the followng formula for the coordnates of the mage v 0 ¼ ðx 0 ;y0 ;z0 Þ of v under the transformaton m: v 0 ¼ az þ az þ b þ b 1 þ! 0 ; az az þ b b ð1 þ! 0 Þ ;!0 1! 0 þ 1 : The transformaton m satsfes the zero mass center condton f and only f the functonal 0! F ða; bþ ¼ 1 X n 2! 2! 1 n 2 x 0 þ Xn y 0 þ Xn z 0 A ð9þ ¼1 satsfes F ða; bþ ¼0. The summatons extend over all ponts, and the coordnates are nonlnear functons of a and b, as gven above. We fnd the best (n the least-squares sense) transformaton mða;bþ by solvng ¼1 ða;bþ ¼ arg mn F ða; bþ: a;b ¼1 ð10þ 2.4 Warpng the Source Mesh onto the Destnaton Surface Followng the dea llustrated n Fg. 1, a mappng f between the source and destnaton surfaces S 1 and S 2 can be wrtten as the composton f ¼ C2 1 m C 1, where C 1 and C 2 are the dscrete conformal mappngs of S 1 and S 2 on the sphere, respectvely, and m s a Möbus transformaton. Any such transformaton m s characterzed by four complex numbers and s gven by the closed-form formula:

5 470 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 36, NO. 3, MARCH 2014 mðzþ ¼ az þ b wth ad bc ¼ 1: ð11þ cz þ d For convenence of notaton, we store the four complex numbers a; b; c, and d nto a vector h ~ and defne m ¼ mð hþ. ~ Note that h ~ contans eght real numbers, but corresponds to sx degrees of freedom only, as ad bc ¼ 1. Let us call M 1 and M 2 the genus-zero meshes representng S 1 and S 2, and SM 1 and SM 2 the mages on the sphere obtaned from the dscrete mappngs C 1 and C 2. We generate the warpng W m ðm 1 Þ of M 1 onto the surface S 2 by locatng the successve mages v 0, v00, and v000 of a vertex v n M 1 for the mappngs C 1,aMöbus transformaton m, and the nverse of C 2, n ths order. Both v 0 and v00 le on the sphere, wth v 00 ¼ mðv0 Þ. The key dea to fndng v000 s to locate v 00 n the sphercal trangulaton SM 2 and transfer ths locaton onto M 2. To speed up ths locaton step, we partton the unt sphere, usng unform subdvsons n the polar coordnates ð; Þ. We then fnd a representatve trangle n SM 2 for each subdvson. The algorthm for warpng M 1 onto the surface S 2 s gven n Algorthm 2. Algorthm 2. Warpng a genus 0 mesh M 1 on a destnaton surface S 2 defned by a mesh M 2. for all vertces v n M 1 do (1) Compute v 00 ¼ mðv0 Þ, where v0 s the vertex equvalent to v n SM 1. (2) Identfy the subdvson of the sphere contanng v 00 and ts representatve trangle t 0. (3) Compute barycentrc coordnates ð; ; Þ of v 00 n trangle t 0. whle <0 or <0 or <0 do (4) Identfy vertex p of t 0 correspondng to one of the negatve barycentrc coordnates. Update t 0 wth ts adjacent trangle opposte to p. (5) Update barycentrc coordnates ð; ; Þ. end whle (5) Compute poston of v 000 ¼ fðv Þ on the surface S 2 by propagatng the barycentrc coordnates ð; ; Þ onto the trangle t n M 2 that corresponds to t 0. end for The result of ths procedure s a new mesh, W m ðm 1 Þ,on S 2, wth the same combnatorcs as M 1, but dfferent geometry. Note that a dfferent warpng W m ðm 1 Þ s obtaned for each Möbus transformaton m. 2.5 Measurng the Dstance between Two Meshes The two meshes M 1 and W m ðm 1 Þ have the same combnatorcs ðv;e;tþ but dfferent geometres (encoded n the postons of the vertces), as M 1 s a representaton of S 1 whle W m ðm 1 Þ sts on the surface S 2. By constructon, W m ðm 1 Þ¼fðM 1 Þ. We defne the elastc energy L assocated wth ths mappng f as LðfÞ ¼ X kfðv j Þ fðv Þk 2 1 ; ð12þ kv e j v k j2e where k:k s the usual L 2 norm. LðfÞ s a measure of the dfferences between the two shapes represented by the surfaces S 1 and S 2. A mappng f wth LðfÞ ¼0 s called an sometry and the two meshes M 1 and W m ðm 1 Þ are then sad to be sometrc. Note that ths s a weak concept of sometry. Formally, we can only state that f f s an sometry of the underlyng surfaces, then LðfÞ ¼0. We work n the class of conformal maps, so angles are preserved. The remanng dstorton of a map from an sometry s exactly reflected n the conformal stretchng factor. The elastc energy LðfÞ defned above measures the average stretchng of the edges n a mesh and thus reflects the dstance of a conformal map from an sometry. 2.6 A General Algorthm for Mappng Two Surfaces of Genus Zero As llustrated n Fg. 1, we rely on the dea that a conformal mappng f between two surfaces S 1 and S 2 of genus zero can be wrtten as the composton of two dscrete conformal mappngs C 1 and C 2 that parameterze S 1 and S 2 onto the sphere, and a Möbus transformaton m. In optmzng the map produced from ths composton, C 1 and C 2 are fxed, whle m s varable and depends on sx degrees of freedom summarzed n a parameter vector ~ h (see (11)). The key to our approach s to choose the transformaton m to yeld the mnmum weghted dstance between the mesh M 1 representng S 1 and ts mage W m ðm 1 Þ warped onto S 2. Ths approach elmnates the need to defne correspondences between landmark ponts on the two surfaces. The weghted dstance between a mesh and ts mage under the conformal mappng f s measured by the elastc energy LðfÞ of f. For convenence of notaton, t s wrtten LðfÞ ¼ Lðmð ~ hþþ ¼ Lð ~ hþ as only m s varable. We have developed all the tools we need to perform ths optmzaton, namely: 1) an algorthm for computng the dscrete conformal mappngs C 1 and C 2 (see Algorthm 1), 2) an algorthm for generatng the warpng of a dscrete mesh onto a surface for a gven Möbus transformaton m (see Algorthm 2), and 3) a defnton of the elastc energy LðfÞ that measures ts dstance to an sometry (see (12)). Smple calculatons provde the analytcal expressons for the elastc energy functon Lð ~ hþ and ts gradent wth respect to ~ h. Ths allows us to apply a steepest descent algorthm to optmze the Möbus transformaton m. Our general algorthm for comparng the two surfaces S 1 and S 2 s then: Algorthm 3. Correspondence-free comparson of two dscrete surfaces of genus zero. Intalzaton. Apply algorthm 1 to map M 1 and M 2 onto the sphere. (1) Intalze Möbus transformaton m 0 ¼ mð ~ h 0 Þ. for n ¼ 0;...untl convergence do (2) Generate W m ðm 1 ) usng algorthm 2 where m ¼ mð ~ h n Þ. (3) Compute Lð ~ h n Þ and ts gradent rlð ~ h n Þ wth respect to ~ h n. (4) Update ~ h nþ1 ¼ ~ h n n rlð ~ h n Þ. (5) Check for convergence: f Lð ~ h nþ1 Þ < TOL, stop. end for The dampng parameter n n step (4) s obtaned by solvng the equaton Lð h ~ n þ n rlð h ~ n ÞÞ Lð h ~ n Þ usng a lne search method. The value of TOL s set to a small constant related to machne error. The result of ths procedure s a warpng of the mesh M 1 onto the surface S 2 that mnmzes dstance from an

6 KOEHL AND HASS: AUTOMATIC ALIGNMENT OF GENUS-ZERO SURFACES 471 sometry among nearby conformal maps, wth dstance measured by the elastc energy. In addton, t gves a measure of the dstance between M 1 and ts warped mage that reflects the geometrc dfferences between the two surfaces. In many cases where the surfaces S 1 and S 2 are sometrc, the procedure wll produce the sometry. 3 IMPLEMENTATION There are several ssues to address n generatng a robust mplementaton of the algorthms dscussed above that s fast enough to be usable n practce. In the followng, we descrbe wth detals the mplementatons of the man steps of the method. We do not provde a convergence or complexty analyss, and leave those for future work. 3.1 Dscrete Conformal Mappng on the Sphere Flattenng a mesh topologcally equvalent to a dsk. Followng the formalsm developed by Sprngborn et al., we have shown n the prevous secton that the dscrete mesh parameterzaton problem can be recast nto an unconstraned convex optmzaton problem wth explct formulae for the target functon CEðuÞ (see Secton 4), ts gradent, and ts Hessan (see [31]). As noted by Sprngborn et al., the Hessan s postve semdefnte wth only the constant vector n ts null-space. To fnd the vector u that mnmzes the extended energy, we use a trust regon Newton method as mplemented n the program TRON [38]. TRON s globally convergent and wll fnd a vector u that mnmzes CE. It does not guarantee, however, that the soluton s feasble: It s possble that the mnmum s reached for u that defnes a new dscrete metrc ~ l that does not satsfy the trangle nequaltes. In all the cases we have tested, we observed that the volatons of trangle nequaltes, f any, occurred n slver trangles, for whch one of the nteror angles s smaller than 5 degrees. Ths lead us to the followng strategy for elmnatng the volatons: 1. solve the convex optmzaton problem for u, 2. compute all edge lengths based on the new metrc ~ l, 3. detect all trangles whose edge lengths volate the trangle nequalty and flp the edges opposte to ther smaller angle, and 4. repeat ponts 1-3 untl all trangle nequaltes are satsfed. The same strategy of edge flppng was already consdered by Sprngborn et al. [31]. For most of the test cases descrbed n the result secton, a sngle cycle was found to be suffcent; n the remanng cases, two cycles removed all volatons. We note that ths smple strategy wll not work f the mesh contans regons wth hgh denstes of slver trangles. In such cases, remeshng s requred. Layng out the planar mesh. Once the weghts u have been found and t s verfed that the correspondng dscrete metrc ~ l s free of trangle nequalty volatons, we have new edge lengths that guarantee that the mesh M s flat,.e., t can be lad out n the plane after removal of a vertex. We assume that the mesh s orented, and that the vertex lsts for all trangles are ordered wth consstent orentaton. We generate the coordnates of all N vertces n M under the metrc l ~ usng the followng layout procedure. Each trangle n M s assgned a flag, vt, ntally set to zero; a smlar flag vv s assgned to each vertex v. We create an empty master lst L. We pck a trangle t 0 at random n the mesh structure. One of ts vertces s set at the orgn of the plane, and a second one j s placed on the x-axs, wth ts dstance to the orgn set to l ~ j. The trangle t 0 s added to the lst L and ts flag s set to one; the flags vv of the two vertces and j are set to one. The layout algorthm proceeds as follows. Algorthm 4. Planar layout of the parameterzed mesh. Set nv ¼ 0 whle jlj 6¼ 0 do t ¼ popðlþ). (1) Let t ¼ð; j; kþ and SðtÞ ¼vvðÞþvvðjÞþvvðkÞ. f SðtÞ < 3 then (2) Let be the vertex wth vvðþ ¼0. Buld usng standard geometry; vvðþ ¼1; nv :¼ nv þ 1. end f for all t n adjacent to t wth vtðt n Þ¼0 do vtðt n Þ¼1; L ¼ L S ft n g. end for f nv%10;000 ¼ 0 or nv ¼ N then (3) Regularze layout end f end whle In step (1), the trangle t s ether t 0 or a neghbor of a trangle that was already lad out; n both cases, at least two of ts vertces have already been consdered and SðtÞ 2. The constructon process n step (2) s then very smple: The postons of two vertces of t n the plane are known, and the dstances from these vertces to the thrd vertex are gven by the new edge lengths of the mesh. Ths s a smple geometrc problem that has two solutons, one for each orentaton. When buldng the frst trangle, one soluton s pcked at random. For all other trangles, only one soluton s feasble as the orentaton of t must match the orentatons of ts neghborng trangles. In theory, ths breadth-frst approach wll lay out meshes of any sze. We have observed n practce, however, that accumulaton of numercal errors can lead to sgnfcant dstortons, especally for meshes wth large length ratos. To crcumvent ths problem, we have ntroduced a regularzaton process as step (3). At a stage n the process wth nt trangles already bult, we can compute a dstorton error as follows: DE ¼ X ðx x j Þ 2 þðy y j Þ 2 l ~ 2 2; j ð13þ e j where the summaton extends over all edges e j n the nt trangles, and ðx ;y Þ are the planar coordnates of vertex v.ifde s small (we use a cutoff of 10 8 ), the layout s consdered correct. Otherwse, we perform a nonlnear optmzaton of the coordnates of the vertces to reach a mnmum of DE. Smple calculatons provde explct formulae for the gradent and Hessan of DE, and we can use the Newton trust regon method as mplemented n TRON to perform ths optmzaton. To save computng tme, regularzaton s only performed at multples of 10,000 vertces and when the layout s complete. In all

7 472 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 36, NO. 3, MARCH 2014 examples descrbed n the results secton, ths procedure yelds worst relatve length errors bounded by Möbus normalzaton for zero mass-center. As descrbed n Secton 2, the normalzaton s cast nto a nonlnear optmzaton problem (see (10)). An explct formula for the dependence of F ða; bþ on the parameters a and b of the Möbus normalzaton s avalable (see (9)). It s also easy to compute ts gradent and Hessan. We can therefore agan use the Newton trust regon method as mplemented n TRON to perform ths optmzaton. Typcal examples ndcate that we only need up to 10 teratons wth TRON to reach krf ða; bþk Warpng the Source Mesh onto the Target Surface The method we propose for warpng a mesh M 1 onto a surface S 2 s qute smple. Startng wth a vertex v n M 1, we dentfy ts correspondng pont v 0 n the sphercal mesh SM 1. We then locate ts mage v 00 ¼ mðv0 Þ on the sphercal mesh SM 2 and transfer ths locaton onto the mesh M 2. The mplementatons of these steps s straghtforward, wth the excepton of the pont locaton problem. As ths procedure needs to be repeated for all vertces n M 1 (whch can be on the order of hundreds of thousands), and subsequently for many trals for the Möbus transformaton m, we need t to be fast and relable. Our approach s akn to the jump-and-walk algorthm used for pont locaton n 2D or 3D trangulatons. Gven a pont v 00 on the sphere, we frst jump to a tral trangle t n the sphercal mesh SM 2. If the (sphercal) barycentrc coordnates of v 00 wth respect to t are all postve, v 00 s nsde t and the algorthm stops. If one of the coordnates s negatve, we dentfy the correspondng vertex n t and walk toward the trangle adjacent to t that s opposte ths vertex. The procedure s then terated untl the correct trangle s dentfed. It s easy to see that ths method s guaranteed to converge. It may not be fast, however, as ts speed depends on the qualty of the jump,.e., on how close the ntal tral trangle s from the actual trangle contanng. Standard jump-and-walk technques pck the ntal trangle from a random subset. We propose an alternatve approach that requres preprocessng. A smlar method was proposed by Wu et al. [39]. We frst put a grd G on the sphere based on sphercal coordnates. We chose 400 dvsons for the azmuthal angle and 200 dvsons for the polar angle, correspondng to a total of 80,000 rectangular subdvsons on the sphere. We numercally defne the coverage of a trangle t n v 00 SM 2 by samplng unformly 20 ponts nsde t and computng ther polar coordnates. The subdvsons of the grd G that contan these ponts are assgned t as a representatve. Ths procedure s repeated for all trangles n SM 2. Any rectangular subdvson s that does not have a representatve at ths stage s processed further. Frst, we fnd the closest subdvson to s wth a representatve, t 0, usng a breadth-frst algorthm. The trangle contanng the center of s s then located wth the jump-and-walk procedure descrbed above, usng t 0 as the ntal tral. Ths trangle s set as the representatve of s. As a result of the ths procedure, all subdvsons n the Gaussan grd are assgned a representatve trangle n SM 2. The ntal tral trangle for a vertex v 00 s then taken to be the representatve trangle of the subdvson that contans v Optmal Möbus Transformaton for Near-Isometry The Möbus transformaton that leads to a closest to sometrc mappng among conformal maps between the two surfaces of nterest s obtaned as the soluton of a nonlnear optmzaton problem (see Algorthm 3). We have adapted a steepest descent approach to solve ths problem. Steepest descent methods are usually fast and easy to set up. They are, however, very senstve to local mnma and hghly dependent on the qualty of the ntal guess for the soluton. A random or a trval ntal guess (such as the dentty transformaton) s lkely to lead to a local mnmum. We have therefore developed a smple procedure to automatcally generate better startng ponts. The dea s to use the best ellpsod approxmaton to each surface to gve the ntal algnment. Whle ths algnment s specfed by settng the mages of three ponts, we do not rely on user selected landmarks or on local geometrc features to select these ponts. Instead, we smply compute the prncpal components of the sets of ponts representng the two shapes. For the mesh M 1, we compute the covarance matrx over ts set of vertces: C 1 ¼ X v ð~v ~Þð~v ~Þ T ; ð14þ where ~v s the vector of coordnates for vertex v, s the center of mass of the vertces, and the summaton runs over all vertces. The ellpsod whose axes are defned by the unt egenvectors of C 1, scaled by the assocated egenvalues, s the best ft ellpsod to the mesh M 1. These three axes cut the surface represented by M 1 n three pars of ponts, ða 1 ;A 0 1 Þ, ðb 1;B 0 1 Þ, and ðc 1;C1 0 Þ. Usng the same procedure on M 2, we get three correspondng pars of ponts, ða 2 ;A 0 2 Þ, ðb 2 ;B 0 2 Þ, and ðc 2;C2 0 Þ The three ponts ða 2;B 2 ;C 2 Þ defnes a drect orentaton for M 2. There are four choces of trplets of ponts on M 1 whose correspondences to these three ponts lead to an algnment of the axes of the two ellpsods wth proper orentaton; these are ða 1 ;B 1 ;C 1 Þ, ða 1 ;B 0 1 ;C0 1 Þ, ða 0 1 ;B0 1 ;C 1Þ, and ða 0 1 ;B 1;C1 0 Þ. Each of these correspondences defnes a unque Möbus transformaton. One of the advantages of choosng these ponts s that they are well separated, whch leads to stablty n the correspondng transformaton. Each of these transformatons s then used as an ntal guess for the steepest descent approach. Ths leads to four dfferent optmzatons. The soluton wth the lowest resultng elastc energy L s chosen to defne the optmal Möbus transformaton. We have mplemented the whole procedure nto a Fortran program, MatchSurface. 4 EXPERIMENTAL RESULTS 4.1 Bran Surface Mappng We demonstrate the feasblty of our algorthm by applyng t to the bran surface matchng problem. We consder a set of cortcal surface models extracted from n-vvo MRI on 38 anonymous subjects [40]. These are the same models that were used by Yeo et al. to evaluate the performance of ther

8 KOEHL AND HASS: AUTOMATIC ALIGNMENT OF GENUS-ZERO SURFACES 473 Fg. 3. Automatc bran surface mappng. The best regstratons of the left and rght hemspheres of the bran of Subject1 (Source) from the Anon database [40] onto the correspondng hemspheres of the template bran of Subject39 (Target) are shown on the nflated cortcal surfaces of the target. The boundares between the 35 regons obtaned by manual parcellaton (see text for detals) are shown n red and green for the target and the mages of the Source, respectvely. Note that these are shown for llustratons only, as they were not used n the regstraton process. Sphercal Demons algorthm [41], wth the excepton that we removed Subject6, as ts database of manual parcellaton s ncomplete. Ths data set covers sgnfcant anatomc varablty as t contans young, mddle aged, and elderly subjects and Alzhemer s patents. For each subject, the left and rght cortcal surface models are provded as dscrete genus-zero surface meshes, as well as nformaton on ther manual parcellaton nto 35 regons [42], as descrbed n [41, Table II]. The left and rght hemspheres of Subject39 are consdered as templates (or Targets) on whch all the other correspondng surfaces (Sources) are regstered followed the procedure outlned n Algorthm 1; the optmzaton of the Möbus transformaton from the unt sphere to tself was performed four tmes, each wth a dfferent ntal guess derved from the prncpal components of the mesh (see mplementaton above). Usually, three of these optmzatons led to local mnma whle the fourth one provded a good algnment between the template and source meshes, as llustrated n Fg. 3 for Subject1 n the database. To assess the performance of our new algorthm, we repeated the Source to Target regstratons descrbed above wth two wdely used landmark-free regstraton methods as well as wth a modfed verson of MatchSurface that uses landmarks. We frst appled the FreeSurfer regstraton algorthm [23] usng the default FreeSurfer settngs. We also used the Sphercal Demon algorthm [41]. As the latter only performs a regstraton of the sphere on tself, we used as nput the sphercal representatons of the hemspheres generated by FreeSurfer. Fnally, we used a modfcaton of MatchSurface n whch the landmark-free energy functon LðfÞ defned n (12) s replaced wth a functon CðfÞ that computes the dscrepances between the postons p T of landmark ponts on the Target and the mages fðp S Þ of the correspondng ponts on the Source meshes: CðfÞ ¼ X f p S p T 2: ð15þ Landmark ponts were set to sample unformly the boundares of all 35 regons. Each boundary was represented wth 70 to 500 ponts, for a total of 9,136 ponts. We note that all four methods nclude a mappng to the sphere. To measure the qualty of the cortcal regstratons provded by all four methods (MatchSurface, Freesurfer, Sphercal Demon, and the modfed MatchSurface whch we refer to as Landmark), we used a modfed Hausdorff dstances [43]. We projected each of the 35 regons from each of the Source subjects onto the Target surfaces and computed the mean dstance between ther boundares. Fg. 4 dsplays the average dstance per structure for the four algorthms, for the left and rght hemspheres. Standard errors are shown as bars. The numberng of the structures correspond to [41, Table II]. Of the three landmark-free methods, the Sphercal Demon algorthm performs best, followed by FreeSurfer, then by MatchSurface. The mproved performance of Sphercal Demon compared to Freesurfer was already notced [41]. Here we report that these two methods perform better than MatchSurface. We observe, however, that the dfferences between the three methods are small. It should be noted that FreeSurfer and Sphercal Demons are desgned to provde hgh-qualty local algnments, whle MatchSurface generates a global algnment of the two surfaces. It s therefore sgnfcant that a fully conformal correspondence gves landmark matchng that s of comparable qualty to nonconformal correspondences. The local versus nonlocal behavor s also sgnfcant to understand the results of the landmark-based method. Landmark performs sgnfcantly worse than the two local landmark-free methods, and slghtly worse than MatchSurface. Whle Landmark s desgned to obtan a good algnment of the landmarks, the optmzaton s performed globally n the space of conformal maps of the sphere to tself,.e., n the space of Möbus transforms. Any such map can only algn exactly three ponts between two representatons of the sphere. When more landmarks are present, t provdes a best-ft soluton to the algnment problem wth the constrant of mposng conformalty, leadng to some structures beng poorly algned. Ths lmtaton has already been descrbed [34]. In Table 1, we compare the runnng tmes of the three landmark-free regstraton methods. MatchSurface s sgnfcantly faster for generatng the mappng of a genus-zero surface onto the sphere as t solves ths problem by mnmzng a functonal energy. MatchSurface s slower n generatng the regstraton. We note, however, that our current mplementaton ncludes four mnmzatons of the elastc energy wth four dfferent ntal condtons; we are workng on elmnatng ths requrement, whch would reduce the runnng tme for regstraton by 4. The total runnng tme of each method s obtaned by summng the regstraton runnng tme wth twce the tme needed for sphercal mappng (as t s performed on the target and source meshes). MatchSurface s then found to be sgnfcantly faster than the two other methods. 4.2 Morphodynamcs of Proten Structure Surfaces Dfferent expermental technques provde dfferent representatons of proten structures. For example, hgh-resoluton X-ray and NMR technques generate atomstc models of protens that are accurate at the Angstrom level, whle technques based on electron mcroscopy (EM) provde much

9 474 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 36, NO. 3, MARCH 2014 Fg. 4. Comparsons of MatchSurface, FreeSurfer, and Sphercal Demons n regsterng cortcal surfaces. The mean modfed Hausdorff dstances computed over the boundares of the 35 structures are shown for (A) the left hemsphere, and (B) the rght hemsphere. lower resoluton (typcally Angstroms) 3D electron densty maps for the protens of nterest. Whle hgh resoluton s preferred, the complextes of applyng the correspondng technques on large molecular systems and the comparatve ease wth whch low-resoluton EM and MS technques can be used on the same system mean that low-resoluton models are often avalable long before ther hgh-resoluton counterparts. Many technques have been developed to acheve more detals from EM and MS data by fttng atomc-resoluton models nto the low-resoluton TABLE 1 Runnng Tmes for Landmark-Free Regstraton densty maps. These technques work well f such models are avalable. However, such nformaton s not avalable for many proten complexes, especally those that are dynamc. There s, therefore, a need to develop technques that can analyze and measure the densty maps drectly. As these maps defne surfaces, we can test the method descrbed above and compare proten structures at a coarse level. We llustrate these tests on one proten, calmoduln. Results are shown n Fg. 5. Calmoduln s a calcum bndng proten expressed n all eukaryotc cells. It s a small proten that conssts of two small domans separated by a lnker regon. The flexblty of ths lnker s key to the ablty of calmoduln to bnd to a wde range of substrates [44]. We consder two statc structures for calmoduln correspondng to two dfferent conformatons, the apo (lgandfree) and holo (lgand-bound) forms, wth PDB codes 1CLL and 1A29, respectvely. We generated two trajectores between these two conformatons that mmc the correspondng structural transton wth MnActonPath. Ths program calculates the most probable trajectory between two known structural states, n the sense of maxmum lkelhood or mnmum acton [45]. The trajectory s descrbed wth 50 all-atom conformatons. We then measured the evoluton

10 KOEHL AND HASS: AUTOMATIC ALIGNMENT OF GENUS-ZERO SURFACES 475 the program smesh, descrbed n detal n [47] and [48]. The correspondng trangular meshes have smlar szes for all protens, wth, on average, approxmately 40,000 vertces and 70,000 trangles. We compared these meshes usng MatchSurface. Results of these calculatons are shown n Fg. 5. We note that t was not possble to perform the comparsons for all conformatons, as for some ther skn meshes have a genus larger than zero, and our method s currently lmted to surfaces wth genus zero. Hgher genus s ntroduced when two nonadjacent patches of a proten surface come nto contact and create a new handle. Geometrc changes n the surface also occur when two adjacent patches fold together and no longer form part of the proten s boundary surface. Whle not changng the genus, ths foldng can cause abrupt changes n the elastc energy requred to deform one surface onto another. We do observe that for all genus-zero conformatons for whch the comparsons were possble, the dstances measured based on ther surfaces (elastc energy, low resoluton) correlate well wth the dstances measured based on the atomc coordnates (crms, hgh resoluton), wth correlaton coeffcents above Fg. 5. Coarse and hgh-resoluton analyses of the dynamcs of calmoduln. We have generated a trajectory wth 50 conformatons between the lgand-free structure and a lgand-bound structure of calmoduln. Ths proten undergoes a consderable change of conformaton as t transforms nto the bound state, as s evdent n the rghtmost fgure. The crms dstances between the hgh-resoluton structures of these conformers and the apo structure (conformaton 0) are plotted versus the conformaton number as a red sold lne. We also plot the elastc energy between the surface meshes representng the same conformers and the surface mesh of the lgand-free proten as red dots; note that data are mssng for some conformatons as ther skn meshes have genuses greater than zero. We note a hgh level of correlaton (0.96) between these two measures. The same behavor s observed when comparng the 50 conformatons wth the lgand-bound structure for crms (dashed blue lne) and wth the surface of conformer 47 for the surface-based comparson (blue dots); we could not use the surface of the lgand-bound structure as ts genus s greater than 0. The surfaces reconstructed from the warped meshes and cartoon representatons of the hgh-resoluton structures are shown for a few conformatons along the trajectory. The red dots on the proten surfaces llustrate the postons of landmark ponts manually pcked on the lgand-free structure on the successve warped meshes. of the structures of calmoduln along these trajectores by computng ther coordnate root mean square devatons (crms) computed over C to the ntal and to the fnal conformatons. Results of these calculatons are shown n Fg. 5 as sold and dashed lnes, respectvely. Note that these measures are based on knowledge at hgh resoluton, as they are computed from the postons of the atoms. In parallel, we compared the structures along the trajectores based on ther skn surfaces [46]. We use the standard model n chemstry of representng a structure as a unon of balls, wth each ball correspondng to an atom. The skn surface of a proten s computed from the boundary of the unon of these balls, where the center of a ball s gven by the coordnates of the atom, and ts radus s set to ts van der Waals radus plus a probe radus of Ra ¼ 1:4 A. We generated qualty meshes for the skn surfaces of all proten conformatons along the trajectores for calmoduln usng 4.3 MatchSurface Is Robust wth Respect to the Qualty of the Dscrete Representatons of the Surfaces The mplementatons of the algorthms presented n Secton 2 nto the program MatchSurface were desgned to be fast and robust. However, we do not control the qualtes of the nput meshes (both source and destnaton). To measure the senstvty of MatchSurface to the mesh qualty, we consdered agan the calmoduln proten and generated four dfferent meshes for the same skn surface, from a fne mesh wth 32,721 vertces and 65,438 trangles, to a much coarser mesh wth 3,523 vertces and 7,042 trangles. The three coarse meshes were generated from the fne mesh wth a procedure that mantans topology [49]. We then compared the skn surface wth tself usng MatchSurface, for all four possble source meshes and all four possble destnaton meshes, for a total of 16 comparsons. Results are shown n Fg. 6. We note frst that all elastc energes for the 16 comparsons are small, much smaller than the values recorded for dfferent confguratons of the surface (see Fg. 5). Whle ths s expected as we are bascally comparng a surface to tself, t remans a reassurng result as t llustrates the robustness of the method under change of mesh, as long as the mesh remans fne enough to accurately represent the geometry of the surface. The elastc energy can only be compared drectly for the same source mesh, as t depends on the number of edges n the mesh, and ncreased elastc energy for fner meshes does not reflect ncreased geometrc dfferences. For coarse source meshes, the qualty of the destnaton mesh has lttle mpact on the result. For a fne source mesh however, there s a loss of qualty f the destnaton mesh s coarse. Ths s expected, as a coarse destnaton mesh provdes a poor representaton of ts surface and the fne mesh wll need to be dstorted to ft on ths coarse surface. Ths s seen n Fg. 6 for the case where the source mesh has 32,721 vertces, but not for the other cases.

11 476 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 36, NO. 3, MARCH 2014 Fg. 6. Senstvty to mesh qualty. We performed all parwse comparsons of four dfferent dscrete meshes representng the same skn surface of the proten calmoduln. Three of these meshes (two coarse and one fne) wth 3,523, 7,810, and 3,2721 vertces are shown on top. The correspondng elastc energes are plotted as a functon of the number of vertces n the destnaton meshes, for the dfferent source meshes. We note that all energy values are small (see Fg. 5 for comparson). Ths s expected as the underlyng surface s the same; t s stll encouragng as t llustrates robustness. 4.4 MatchSurface Is Senstve to Small Varatons n the Surfaces To assess the senstvty of MatchSurface to varatons n the detals of the surfaces, we generated dfferent skn surfaces for the proten calmoduln by varyng the probe radus Ra. A low probe radus defnes a surface that follows closely the envelope of the unon of atoms, also referred to as the vdw surface of the proten, whle a large probe radus generates a much smoother surface, as all atoms have been sgnfcantly nflated (see Fg. 7). The skn surface obtaned wth Ra ¼ 1:5 s the closest match to the accessble surface of the proten [50]; we use t as a reference. All the skn surfaces are then compared to ths reference usng MatchSurface. The resultng elastc energes are plotted aganst the probe radus Ra n Fg. 7. In these calculatons, the reference mesh s used as a source, and warped around the dfferent skn surfaces. Clearly, MatchSurface s very senstve to the coarseness of the surface. The senstvty s more pronounced as the source mesh s warped onto a surface wth more detals (see panel (B) of Fg. 7). We note that the dfferences measured between these dfferent skn surfaces are much larger than the dfferences observed between dfferent representatons of the same surface (see Fg. 6). 5 SUMMARY AND CONCLUSIONS We have developed a new method for automatcally generatng a conformal map between two surfaces of genus zero. Ths new approach leads to flexble regstraton of the two surfaces and accurate measurements of ther geometrc dssmlartes, wthout the need for the selecton of landmark ponts. Its mplementaton wthn the program MatchSurface s based on fast and robust numercal methods, makng surface comparsons feasble for a wde range of data sets. We have llustrated ts use for bran surface mappng and proten surface comparsons. We have Fg. 7. Senstvty to surface fluctuatons. (A) The elastc energy values obtaned when comparng skn surfaces for calmoduln wth the reference skn surface are plotted aganst the probe radus Ra used to generate these surfaces. We show three examples of these skn surfaces, for Ra ¼ 0.5, 1.5, and 3.0 A. (B), (C), and (D) (bottom row) show the surfaces generated by the reference mesh after t has been warped on these three skn surfaces. compared MatchSurface wth FreeSurfer and Sphercal Demon, two landmark-free methods for bran surface mappngs. We have shown that a fully conformal correspondence generates surface regstraton that s of comparable qualty to a non-conformal correspondence. We have demonstrated that the dstances computed wth our algorthm between low-resoluton, surface-based representatons of protens are hghly correlated wth the correspondng dstances computed between the hgh-resoluton, atomstc models for the same protens. Ths paper, however, s just a frst step toward achevng automatc, landmark-free regstraton of general surfaces. The current method has lmtatons that suggest drecton for future work. Frst, the method apples only to surfaces of genus zero. The dscrete conformal mappngs from the surfaces to the sphere rely on ths property. In addton, we use the fact that a conformal self-mappng of the sphere belongs to the group of Möbus transformatons, whch provdes sgnfcant smplfcaton as a closed analytcal form s avalable for members of that group. The concept of dscrete conformal equvalence can be extended to surfaces wth arbtrary topology, ether through the ntroducton of cone sngulartes [31], or through the defnton of a dscrete conformal equvalence between a eucldean trangulaton on the surface and a hyperbolc trangulaton [36]. These alternate defntons lead to (dscrete) mappngs of the surface onto a doman n hyperbolc space (whch can be represented by the Poncare dsk model.) In general, there are no conformal maps between two surfaces of hgher genus, though there are varous constructons of

12 KOEHL AND HASS: AUTOMATIC ALIGNMENT OF GENUS-ZERO SURFACES 477 Fg. 8. The problem of sngulartes. We have used MatchSurface to compare two human models, one wth one arm bent at nnety degrees (source), and the second wth both arms extended and parallel to the body (target). The surface (S on T) reconstructed from the mesh obtaned by warpng the source mesh S on the target surface T s dstorted, as t s mssng the hands and feet. Vertces n these regons were assgned large, negatve scalng factors u; ths s llustrated on the rght panel that shows the surface S wth all vertces whose factors u are lower than 10 colored n red. closest-to-conformal mappngs. Fndng closest-to-sometrc mappngs for such surfaces s a topc for future studes. Second, our algorthm works well for surfaces wth unform geometry wth no extreme protrusons or spkes. Ths s usually the case for bran surfaces as well as for proten surfaces, but t s not true n general. In partcular, MatchSurface does not perform well on models of humans, as llustrated on Fg. 8. The problem can be ascrbed to the sngulartes ntroduced by the arms and legs of the models. Ther dscrete conformal mappngs to the sphere ntroduce very large negatve scalng factors on the vertces located at the hands and feet (see the left panel n Fg. 8, whch n turn lead to nfntesmally small edge lengths n the projected metrc ~ l and consequently large numercal errors. Ths problem s not specfc to our method, as t appears n many conformal mappng procedures. In some cases approxmatng by a conformal map appears to be too restrctve for accurate algnment. One soluton s to ntroduce cone sngulartes n the regons wth the worst dstortons (see, for example, [31]). Ths brngs us back to the frst lmtaton dscussed above and ts possble resoluton through the use of hyperbolc geometry. Fnally, our algorthm s lmted to fndng global matchngs between a par of surfaces of genus zero. 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Molecular Bology, vol. 55, pp , Patrce Koehl was traned as an engneer at the Ecole Centrale de Pars, France, where he graduated wth a master s degree n boengneerng n He receved the PhD degree n molecular bology from the Unversty Lous Pasteur, Strasbourg, France, n He joned the French Natonal Center for Research (CNRS) the same year as a senor scentst. In 1997, he vsted the Department of Structural Bology at Stanford Unversty; he extended hs stay more than 7 years, becomng a senor research assocate n that department. In 2004, he joned the Unversty of Calforna, Davs, where he s currently a professor of computer scence and assocate drector of bonformatcs at ts Genome Center. He s a recpent of the Bronze medal of the CNRS and s an Alfred P. Sloan fellow. Joel Hass s currently a professor and char of the Department of Mathematcs, Unversty of Calforna, Davs, where he has been snce Pror to that he held postons at the Unversty of Mchgan, Hebrew Unversty of Jerusalem, and the Unversty of Calforna at Berkeley. He has held vstng postons at the Unversty of Melbourne, Insttute for Advanced Study n Prnceton, and the Mathematcal Scences Research Insttute n Berkeley. He s an Alfred P. Sloan fellow.. For more nformaton on ths or any other computng topc, please vst our Dgtal Lbrary at