Computing geodesic paths on manifolds

Transcription

1 Proc. Natl. Acad. Sci. USA Vol. 95, pp , July 1998 Applied Matematics Computing geodesic pats on manifolds R. Kimmel* and J. A. Setian Department of Matematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA Communicated by Alexandre J. Corin, University of California, Berkeley, CA, May 4, 1998 (received for review Marc 20, 1998) ABSTRACT Te Fast Marcing Metod is a numerical algoritm for solving te Eikonal equation on a rectangular ortogonal mes in O(M log M) steps, were M is te total number of grid points. In tis paper we extend te Fast Marcing Metod to triangulated domains wit te same computational complexity. As an application, we provide an optimal time algoritm for computing te geodesic distances and tereby extracting sortest pats on triangulated manifolds. 1. Introduction Setian s Fast Marcing Metod (1), is a numerical algoritm for solving te Eikonal equation on a rectangular ortogonal mes in O M log M steps, were M is te total number of grid points in te domain. Te tecnique inges on producing numerically consistent approximations to te operators in te Eikonal equation tat select te correct viscosity solution; tis is done troug te use of upwind finite difference operators. Te structure of tis upwinding is ten used to systematically construct te solution to te Eikonal equation troug an optimal ordering of points during te update. Te optimal ordering is executed using a eap operator to extract te next point in te update sweep. As suc, te tecnique is a reminiscent of Dijkstra s metod (2); owever, te resulting approximation is consistent in tat it produces te correct sortest pat on an ortogonal grid. For details about Fast Marcing Metods, see refs. 1 and 3. Bart and Setian (4) ave recently constructed operators for viscosity solutions to bot te Eikonal equation and Hamilton Jacobi equations on arbitrary triangulated domains. Tese operators exploit te upwind nature to construct te correct entropy-satisfying weak solutions. In tis paper, we extend tese ideas and build a Fast Marcing Metod for triangulated domains. As an application, we provide an optimal algoritm for computing te geodesic distances and tereby extracting sortest pats on triangulated manifolds. Te outline of tis paper is as follows. First, we review te Fast Marcing Metod for ortogonal grids. Ten, for motivational reasons, we analyze te structure of tis metod on a triangulated planar grid constructed directly from an ortogonal grid. We ten follow wit a general procedure for computing te solution of te Eikonal equation on arbitrary acute triangulated domains, followed by an extension to general (nonacute) triangulations. As an application, we compute geodesic distances and minimal geodesic pats on manifolds. 2. Te Fast Marcing Metod on Ortogonal Grids Here, we briefly review te Fast Marcing Metod for computing te solution to te Eikonal equation; for details, see Te publication costs of tis article were defrayed in part by page carge payment. Tis article must terefore be ereby marked advertisement in accordance wit 18 U.S.C solely to indicate tis fact by Te National Academy of Sciences /98/ $2.00/0 PNAS is available online at ttp:// ref. 1. Te goal is to solve te equation T =F x y [1] Te key idea is to build an approximate to te gradient term tat correctly deals wit te development of corners and cusps in te solution. It is well known tat te above Eikonal equation becomes nondifferentiable, and an appropriate weak solution must be built. Te appropriate weak solution comes from satisfying te entropy condition; see, for example, ref. 5. Te crucial point in tis (or any suc appropriate) numerical sceme is te correct direction of te upwinding and treatment of sonic points. For details and an extensive review, see ref. 3. Te Fast Marcing Metod exploits tis idea by carefully considering te nature of upwind, entropy satisfying approximations to te Eikonal equation. In more detail, consider one particular upwind approximation to te gradient, given by (see ref. 6) [ ] max D x ij T D +x ij T 0 2 1/2 + max D y ij T D +y = F ij [2] ij T 0 2 Te central idea beind Fast Marcing Metods is to systematically advance te front in an upwind fasion to produce te solution T. Te key is te observation tat te upwind difference structure of Eq. 2 means tat information propagates one way, tat is, from smaller values of T to larger values. Hence, te algoritm rests on solving Eq. 2 by building te solution outward from te smallest T value. Te algoritm is made fast by confining te building zone to a narrow band around te front, motivated by te narrow band introduced by Copp (7), used in recovering sapes from images in Malladi, Setian, and Vemuri (8), and analyzed extensively by Adalsteinsson and Setian (9). Te idea is to sweep te front aead in an upwind fasion by considering a set of points in a narrow band around te existing front, and to marc tis narrow band forward, freezing te values of existing points and bringing new ones into te narrow band structure. Te key is in te selection of wic grid point in te narrow band to update. Te Fast Marcing Metod algoritm is as follows: First, we tag points in te initial conditions as Alive. We ten tag as Close all points one grid point away. Finally, we tag as Far all oter grid points. Ten te loop is as follows: 1. Begin loop: Let Trial be te point in Close wit te smallest T value. 2. Add te point Trial to Alive; remove it from Close. 3. Tag as Close all neigbors of Trial tat are not Alive; if te neigbor is in Far, remove it from tat list and add it to te set Close. 4. Recompute te values of T at all neigbors according to Eq. 2 by solving te quadratic equation, using only values of points tat are Alive. 5. Return to top of loop. Tis algoritm works because te process of recomputing te T values at upwind neigboring points cannot yield a value * ron@mat.lbl.gov. To wom reprint requests sould be addressed. setian@mat. berkeley.edu. Also, see ttp://mat.berkeley.edu/~setian/level_set.tml. 8431

2 8432 Applied Matematics: Kimmel and Setian Proc. Natl. Acad. Sci. USA 95 (1998) Tere is a real solution T, wit T, T A and T T C, to te one-dimensional update (degenerate quadratic) T T A 2 = 2 F 2 i j For eac possible up down/left rigt pair, we construct all possible real solutions; we ten accept as te updated point te one tat produces te smallest value of T. Judicious programming and careful attention to Far values makes te above searc extremely fast. Tis is te Fast Marcing Metod described in refs. 1 and 3. Fig. 1. Upwind construction of accepted values. smaller tan any of te accepted (Alive) points. Tis is known as a monotone property and means tat we can marc te solution outward, always selecting te narrow band grid point wit minimum trial value for T, and readjusting neigbors, witout aving to go back and correct an accepted value (see Fig. 1). Anoter way to look at tis is tat eac minimum trial value begins an application of Huygen s principle, and te expanding wave front touces and updates all oters. Te speed of te algoritm comes from a eapsort tecnique to efficiently locate te smallest element in te set Close. If tere are M total points in te computational domain, ten, assuming no work for locating te smallest element in te set Close, we ave a computational complexity of O M. Because te eap can be re-ordered in O log M steps, tis yields a computational complexity of O M log M. For more details, see refs. 1 and 3. Te above tecnique is strongly reminiscent of Dijkstra s metod, wic allows one to compute minimum costs of pats on networks. Tat tecnique, wic is also O M log M), is numerically inconsistent; given two points on te grap, it produces te network minimum (Manattan) lengt, wic may not be optimal. Te Fast Marcing Metod is a consistent approximation to te continuous partial differential gradient operator, and tus it provides sub-grid resolution and ence finds te true sortest pat. Before proceeding, it is instructive to say a few words about te actual update procedure involved in solving Eq. 2. Imagine a uniform square grid and suppose tat te goal is to update te value of T at te center point i j. We label te values of T at te surrounding grid points T A = T i 1 j, T B = T i+1 j, T C = T i j 1, and T D = T i j+1 (see Fig. 2). Some of te values may be infinite, corresponding to Far values. Standing at te center point i j, we attempt to solve te quadratic given by eac possibility. For example, we refer to possible contributors A and C and assume, witout loss of generality, tat T A T C. We attempt to solve T T A 2 + T T C 2 = 2 F 2 i j were is te uniform grid spacing. Two possibilities are as follows: Tere is a real solution T, wit T, T A and T, T C, to te quadratic T T A 2 + T T C 2 = 2 F 2 i j 3. Fast Marcing on a Particular Triangulated Planar Domain Our goal now is to extend tis metod to triangular domains. To do so, we sall build a monotone update procedure on te triangulated mes. As motivation, in tis section we consider te obvious triangulation of a square grid in te plane. Imagine te triangulation given in Fig. 3. If we consider te possible contributors T A and T C, and look at te triangle formed by te points i j, i 1 j and i j 1, we can easily write down te equation of te plane determined by te known values T A and T C, as well as te unknown value T, namely [ T TA ] x + [ T TC ] y + T = z Computing te gradient, we ten want to select a value of T suc tat [ ] T 2 [ ] TA T 2 TC + = F 2 ij In oter words, we are tilting te plane by a value T at te center point i j to ave a gradient magnitude equal to F. We note tat tis is te exact same construction as te one produced by te ortogonal construction. Note also tat in tis case te gradient vector, wit origin at te center point, always points into te triangle from wic it is updated. Tis is not necessarily te case for an arbitrary triangle. To establis monotonicity, we will need to verify tis condition. Te inability to solve te above quadratic corresponds to an inability to tilt at an appropriate angle, and te requirement tat te solution T be greater tan te contributors means tat te solution is always constructed in an upwind manner. By using te same update rules as above, and te eap structure to maintain a list of grid points of Close points, tis provides a metod for executing te Fast Marcing Metod on suc a simple triangulation. 4. Fast Marcing Metods on Triangulated Domains Our goal now is to extend tis idea to an arbitrary triangulation. Here, we are essentially following te construction of upwind approximations to te gradient on triangulated meses developed by Bart and Setian in (4). Fig. 2. Construction of upwind solution on ortogonal mes. Simple triangulation for building a monotone update op- Fig. 3. erator.

3 Applied Matematics: Kimmel and Setian Proc. Natl. Acad. Sci. USA 95 (1998) A Construction for Acute Triangulations. We start wit an acute triangulation and consider te triangulation around te grid point given in Fig. 4. A large number of triangles may sare te center vertex. Our procedure, motivated by te simple triangulation in te previous section, is to compute a possible value for T from eac triangle tat includes te center point as a vertex. Because several triangles can produce admissible values for T, we must select an appropriate value. Tere are several possibilities. In our examples we ave cosen te one tat produces te smallest new value for T ; tis will correspond to an algoritm similar to te one taken above. More elaborate upwind constructions on triangulated meses are given in ref. 4. We now construct a simple update procedure for one nonobtuse triangle ABC in wic te point to update is C. We sould verify tat te update is from witin te triangle, i.e., te altitude sould be inside te nonobtuse triangle ABC. See Fig. 5. Tis means tat we searc for t = EC suc tat t u = F Denote a = BC and b = AC; we ave by similarity tat t/b = DF/AD = u/ad, tus CD = b AD = b bu/t = b t u /u. Next, by te Law of Cosines: BD 2 = a 2 + CD 2 2a CD cos θ and by te Law of Sines: sin φ = CD sin θ Now, BD by using te rigt angle triangle CBG, we ave = a sin φ = a CD BD sin θ = a CD sin θ a 2 + CD 2 2a CD cos θ We end up wit te quadratic equation for t: a 2 + b 2 2ab cos θ t 2 + 2bu a cos θ b t [3] + b 2 u 2 F 2 a 2 sin 2 θ =0 [4] Te solution t must satisfy u + t, and sould be updated from witin te triangle, namely: a cos θ + b t u t Tus, te update procedure is given as follows: If u + t and a cos θ + b t u t ten T C =min T C t+ T A + a [5] cos θ + a cos θ Fig. 5. Given te triangle ABC, suc tat u = T B T A, find T C =T A +t suc tat t u / = F. (Left) A perspective view of te triangle stencil supporting te T values tat form a tilted plane wit a gradient magnitude equal to F. (Rigt) Te trigonometry on te plane defined by te triangle stencil Extension to General Triangulations. Finally, we note tat we ave required an acute triangulation. Tis is so tat any front entering te side of a triangle as two points to provide values before te tird is computed. In oter words, for monotonicity, we restrict te update to come from witin te triangle, i.e., te gradient of te solution at a grid point sould point into te triangle from wic it is updated. Te most straigtforward way to enforce tis requirement is to build a nonobtuse triangulation, tus making sure tat te grid captures all incoming fronts. In te case of an obtuse triangulation, te support may only include a limited (acute) section of incoming wavefronts. One approac to andle nonacute triangulations, wic we now describe, is to locally build numerical support at obtuse angles by splitting tese angles in a special way. An obtuse angle at vertex A can be updated by its neigboring points in a consistent way only at a limited section of upcoming fronts. Connecting te vertex to any point in tis section splits te obtuse angle into two acute ones. Te idea is to extend tis section by recursively unfolding te adjacent triangle(s), until a new vertex B is included in te extended section. Ten, te vertices are connected by a virtual directional edge from B to A (i.e., A may be updated by B). Te lengt of te edge AB is equal to te distance between A and B on te unfolded triangles plane (Fig. 6). We can perform a complexity analysis to ceck on te cost of tis virtual node capturing. Let max min be te maximal and minimal altitudes, respectively, i.e., triangles altitudes wit maximal and minimal lengt. Let θ max be te maximal obtuse angle, and denote α = π θ max te angle of te extended section, let θ min be te minimal (acute) angle for all triangles, let e max be te lengt of te longest edge, and let l be te lengt else T C =min T C bf + T A cf + T B It is easy to verify tat tis equation is a finite difference approximation to te Eikonal equation. It is monotone by construction, consistent, converges to te viscosity solution, and can be used to extend te Fast Marcing Metod to acute triangulated domains. Fig. 4. Acute triangulation around center grid point. Fig. 6. (Left) Te initialization of te construction for te splitting section. (Bottom) A triangulated surface patc. (Rigt) Te unfolded patc and te splitting section expansion up to te first vertex B, and te virtual edge connecting te two vertices AB.

4 8434 Applied Matematics: Kimmel and Setian Proc. Natl. Acad. Sci. USA 95 (1998) Fig. 7. Te smallest possible area of one triangle covered by te longest possible extended section bounds te number of unfolded triangles. Te triangle wit a min is te brigt saded one. of te virtual directional edge (AB in te above example). Furtermore, assume tat α θ min are small enoug angles so tat sin α 8 tan α 8 α. Ten, te angular widt of te narrower section is α = π θ max, so tat we ave sin α e max Tis relation, for small α 2 2l angles, yields l e max Denote by l max = e max /α. α Te maximal area of te extended sections is bounded from above by a max = e2 max and te minimal area of an unfolded tri- 2α angle is bounded from below by a min = minα 2 θ min 2 (see Fig. 7). Terefore, te number of triangles tat are needed to be unfolded before a vertex is found in te extended section is bounded by te ratio of tese areas m = a max a min = e 2 max [6] θ min 2 minα3 Te accuracy of te first order sceme for acute triangles is of O max 8 O e max. Te accuracy for te obtuse case wit te above construction becomes O l max =O e max / π θ max, as expected. In te worst case scenario, te sceme accuracy depends on te largest edge and te widest angle. Te complexity of tis virtual construction does not cange substantially cange te operation count of te underlying algoritm. Te construction of te virtual directional edges includes unfolding triangles for eac obtuse angle until a vertex is detected in te extended splitting section. Because te number of unfolded triangles is bounded by a constant, te construction of te virtual directional edges takes O M, and running time complexity is still optimal O M log M. 5. Construction of Minimal Geodesics We can use tis algoritm to compute distances on triangulated manifolds, and ence construct minimal geodesics. First, we solve te Eikonal equation wit speed F = 1 on te triangulated surface to compute te distance from a source point; we ten backtrack along te gradient of te arrival time field by solving te ordinary differential equation dx s ds = T were X s traces out te geodesic pat. We use second order Huen s integration metod on te triangulated surface wit a switc to a first order sceme at sonic points in te gradient. Wen integrating witin a given triangle, its tree neigboring triangles support te computation and are used to interpolate T as a second order polynomial wose six coefficients are computed from tose given T values at te vertices. Fig. 8 presents a perspective view of te triangulation and sortest pats for two surfaces. Fig. 8a sows sortest pats on regular triangulation of te surface given by te function z x y =0 45 sin 2πx sin 2πy on , for grid size of Te minimal geodesics are painted on te triangulated surface and projected to te xy plane. Fig. 8b gives Fig. 8. surfaces. Computing minimal weigted geodesics on triangulated a polyedron example, in wic a different speed function F was assigned to eac side, causing a Snell Law effect along te edges. Te speed F is 2 at te close side wit te start point, 1 at te second top side, and 4 at te side of te destination point. Finally, we sow two additional computations to illustrate te algoritm performance on manifolds wit an underlying nonacute (obtuse) triangulation. Fig. 9 sows te computation of minimal geodesics on a torus genus one object, in wic some sortest pats in fact cut troug te middle. Te equidistance curves are sown, as well as te sortest pats. In Fig. 10, we compute geodesic distances on a syntetic ead. We tank T. Bart, L. C. Evans, and A. Vladimirsky for elpful discussions. All calculations were performed at te University of California at Berkeley and te Lawrence Berkeley Laboratory. Tis work was supported in part by te Applied Matematical Science subprogram of te Office of Energy Researc, U.S. Department of Energy, under Contract DE-AC03-76SF00098, and te Office of Naval Researc under under Grant FDN

5 Applied Matematics: Kimmel and Setian Proc. Natl. Acad. Sci. USA 95 (1998) 8435 Fig. 9. Sortest pats on a bead (a genus one two-dimensional manifold). Fig. 10. Sortest pats on syntetic ead. 1. Setian, J. A. (1996) Proc. Natl. Acad. Sci. USA 93, Dijkstra, E. W. (1959) Numerisce Matematik 1, Setian, J. A. (1996) Level Set Metods: Evolving Interfaces in Geometry, Fluid Mecanics, Computer Vision and Material Science (Cambridge Univ. Press, Cambridge, U.K.). 4. Bart, T. & Setian, J. A. (1998) J. Comp. Pys., in press. 5. Setian, J. A. (1985) Commun. Mat. Pys. 101, Rouy, E. & Tourin, A. (1992) SIAM J. Numer. Anal. 29 (3), Copp, D. L. (1993) J. Comp. Pys. 106, Malladi, R., Setian, J. A. & Vemuri, B. C. (1995) IEEE Trans. Pattern Anal. Macine Intelligence 17 (2). 9. Adalsteinsson, D. & Setian, J. A. (1995) J. Comp. Pys. 118,