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Optimal Algorithm for Shape from Shading and Path Planning Ron Kimmel ([email protected])

Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel

James A. Sethian ([email protected])

Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA

December 2000 Abstract. An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian's fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a dierent partial dierential equation. A modi cation of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.

Keywords: Fast Marching, Eikonal Equations, Shape from Shading, Navigation

1. Introduction One of the earliest problems in the eld of computer vision is the reconstruction of a three dimensional object from its single gray level image. The problem, for the case of a diusive re ectance model of the surface, also known as Lambertian re ectance, is recognized as the `shape from shading problem' (Horn, 75; Horn, 90). Various numerical schemes were proposed over the years, most of these methods were based on variational principles that require additional regularization terms that introduce second order derivatives into the minimization process. These terms yield over-smoothed reconstructions, see e.g. (Horn and Brooks, 89). Only two early direct models for the shape from shading did not incorporate extra smoothness terms, the rst is the characteristic strips expansion method that Horn used when he rst introduced the problem (Horn, 75), the second is Bruckstein's equal height contours tracking

c 2000 Kluwer Academic Publishers.

Printed in the Netherlands.

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2 Kimmel & Sethian model (Bruckstein, 88). Unfortunately, the rst implementations of these algorithms suered from numerical instabilities. New numerical algorithms based on recent results in curve evolution theory, control theory, and the viscosity framework (Crandall, Ishii and Lions, 92), were applied to the shape from shading problem in (Rouy and Tourin, 92; Dupuis and Oliensis, 94; Kimmel and Bruckstein, 92; Kimmel and Bruckstein, 95). In these advanced numerical algorithms the smoothness assumption is embedded within the scheme without the need for an extra smoothness as a penalty. Recently, Sethian (Sethian, 96; Sethian, 96) introduced an O(N log N ) computational steps sequential algorithm for solving the Eikonal equation on a rectangular grid, where N is the total number of grid points. This algorithm, known as the `Fast Marching Method,' relies on a systematic causality relationship based on upwinding, coupled with a heap structure for eciently ordering the updated points. The method was applied to segmentation in 3D in (Malladi and Sethian, 96) and to edge integration in (Cohen and Kimmel, 97). An important property of Sethian's Fast Marching Method that distinguishes it from graph search based methods is that it converges to the continuous physical (viscosity) solution as the rectangular numerical grid is re ned. Sethian's method is a nite dierence scheme, based on upwind monotone schemes, and has been extended to higher order in Sethian SIAM Review article (Sethian, 99). In (Tsitsiklis, 95), Tsitsiklis gives a dierent, rst order Dijkstra-like algorithm which obtains the viscosity solution through a control-theoretic discretization using a causality relationship based on the optimality criterion. In this note we rst modify Sethian's fast marching method to construct a numerical solution for the oblique light source shape from shading problem. Next, the method is used for path planning in robotic navigation with small number of degrees of freedom.

2. Shape from Shading Let us rst review the shading image formation model for a 3D Lambertian object. Assume, that the object we try to reconstruct is given as a function z (x; y ) : R2 ! R, whose surface normal at each point is given by ~n(x; y ) : R2 ! S 2. Next, let the light source direction be given by ~l 2 S 2. Then, the intensity image, I (x; y ) : R2 ! R, for an orthographic projection of the object is given by the inner product I (x; y) = ~l
~n(x; y): For the simple vertical light source case ~l = (0; 0; 1), in which the light source is located near the viewer, the shading image is given

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3 by I (x; y ) = p1+1z2 +z2 : The problem in hand is the reconstruction of z (x; y ) from magnitude at each point that is given by q its gradient ? 2 jrz(x; y)j = (I (x; y)) ? 1: This equation is known as the Eikonal equation. See (Verbeek and Verwer, 90) for a `shading from shape' Eikonal based technique. The fast marching method is an O(N log N ) numerical algorithm for solving the Eikonal equation, e.g. jrz (x; y )j = f (x; y ). The rst version of the algorithm is based on the following numerical approximation of the Eikonal equation Optimal Algorithm for Shape from Shading and Path Planning x



y

2 

2

max(Dij?x z; ?Dij+x z; 0) + max(Dij?y z; ?Dij+y z; 0) = fij2 ; (1) where zij = z (i
x; j
y ), and Dij?x z = (zij ? zi?1;j )=
x is the standard backwards derivative approximation, Dij+x z = (zi+1;j ? zij )=
x is the standard forward derivative approximation in the x direction, and similarly for the y direction. This numerical approximation selects the correct viscosity solution for the shape from shading problem as proven by Rouy and Tourin (Rouy and Tourin, 92). One important observation is that information always ow form small to large values of the solution z. Therefore, the surface z may be reconstructed by rst setting all z values to 1, and the correct height value at the local minimum points. In case the height values at these locations are unknown, then a global topology solver can be applied (Kimmel and Bruckstein, 95; Brooks and Chojnacki, 94). Assume, for simplicity, that we deal with a single known minimum point. An alternate scanning directions of the numerical grid, while solving the quadratic equation (1) for zij at each visited grid point, would eventually converge. Yet, the rate of convergence depends on the complexity of the surface zij . If we reconstruct a connected spirals like surface, then there is a need for O(N ) scans in the worst case, that yields a complexity bounded by O(N 2), see e.g. (Bichsel and Pentland, 92; Dupuis and Oliensis, 94). Note that this is a worst case analysis. That is, for alternating scanning directions based methods the complexity is data dependent and ranges between O(N ) and O(N 2). For simple surfaces, convergence can be achieved in few iterations. Assume without loss of generality that
x =
y = 1, and initiate all zij = 1 besides the minimum point that is set to zero. Then, the update step for zij can be written as the following simple procedure ? Let z1 = minfzi?1;j ; zi+1;j g; and z2 = minfzi;j?1; zi;j+1g;

p 2 2 ? IF jz1 ? z2j < fij THEN zij = z1 +z2 + 2f2 ?(z1?z2 ) ; ELSE zij = minfz1 ; z2g + fij ; ij

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Kimmel & Sethian

The fast marching method introduces order to these update steps. Points are updated and accepted by their values from small to large. The selection of the smallest point among the set of candidate points and the update of its neighboring grid points involves an O(log N ) worst case complexity, that yields a total of O(N log N ) worst case computational complexity. The order of updates is similar to that of Dijkstra's graph search algorithm (Dijkstra, 59; Sedgewick, 88), and is based on a heap structure of the points at the front. The main dierence from Dijkstra's method is the numerical update step. Actually, one may use the nite numerical accuracy to avoid the ordering and reduce the total complexity to O(N ). Our shading image is usually given on a rectangular pixels grid. Therefore, the fast marching method can be directly applied to solve the shape from shading problem with a vertical light source. However, for the general oblique light source, the model to be solved reads jrz(x; y)j = f (x; y; z(x; y)). For this more general case, the right hand side depends on z (x; y ). The question is how to include this partial dierential equation, which is not an Eikonal equation anymore, within the fast marching framework. Let us focus on the oblique light source case in which the light source direction is dierent than that of the viewer. The shading image for this case is (?zx ; ?zy ; 1) : I (x; y) = ~l
~n = (l1; l2; l3)
q 1 + zx2 + zy2 W.l.o.g. we choose l2 = 0. The shading image is then given by I (x; y) = (l1; 0; l3)
~n; (2) where l12 + l32 = 1. Eq. (2) involves the term zx . It requires some additional thought to construct a monotonic approximation to this term and an appropriate update rule. If we could have the brightness image in the light source coordinates I~(~x), then the problem would have become the vertical light source case, which is given by the Eikonal equation z~x2~ + z~y2 = ~ 1 2 ? 1: (3) I (~x; y) Lee and Rosenfeld (Lee and Rosenfeld, 85) suggested the light source coordinates `to improve' early shape from shading algorithms. Adopting their suggestion, we view the re ectance map as an `almost' Eikonal equation which can be solved eciently. In the light source coordinate system, the right hand side depends on the surface itself via I~(~x; y) = I (l3x~ + l1z~; y): (4)

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5 That is, we need to evaluate the value of the surface at a point in order to nd the `brightness' and only then plug it to Eq. (4) and use the fast marching method to solve Eq. (3). In order to overcome this dependence, we use the directional propagation and `adopt' the smallest z~ value from all the neighbors of the updated grid point. The new update step then reads ? Let z~1 = minfz~i?1;j ; z~i+1;j g and z~2 = minfz~i;j?1; z~i;j+1g; ? Let k = l3i + l1 minfz~1; z~2g; Optimal Algorithm for Shape from Shading and Path Planning

p 2 2 ? IF jz~1 ? z~2j < fkj THEN z~ij = z~1 +~z2 + 2f2 ?(~z1 ?z~2 ) ; ELSE z~ij = minfz~1 ; z~2g + fkj ; Where z~ij = z (i
~x; j
y ), and fkj = f (k
x; j
y ). Again, w.l.o.g. we assume
~x =
y = 1, and f (x; y ) = I (x; y )?2 ? 1. The numerical kj

algorithm in this case is still consistent, one pass since the smallest

z~ neighbor will never change its value, and is thus within the fast

marching framework. The map between the light source coordinates (~x; y; z~) and the image coordinates (x; y; z ) is a rotation given by

0x1 0 l 0 3 @yA=@ 0 1 z ?l1 0

l1 1 0 x~ 1 0 A@ y A: l3 z~

We have thereby extended the fast marching method to the case of jrz j = F (z ) relevant to the oblique light source shape from shading problem. A consistent solution can be computed with O(N log N ), where N is the total number of pixels (grid points). Original Surface

Shading Image

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Figure 1. The reconstruction of the surface at the left, from its shading image, with ~l = (0:2; 0; 0:96). Right is the dierence between the original surface and its reconstruction.

We tested the algorithm on a synthetic shading image, see Figure 1. Observe that we do not deal here with self casting shadows (see (Lions, Rouy and Tourin, 93)), nor with solving the global topological structure

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6 Kimmel & Sethian (see (Kimmel and Bruckstein, 95; Dupuis and Oliensis, 94; Brooks and Chojnacki, 94)).

3. Fast Marching Methods for Path Planning In this section, we apply the fast marching method to construct optimal paths for navigation problems with relatively few degrees of freedom. We begin with a two-dimensional path planning problem with constraints; and consider a point robot in a uniform domain with obstacles. Here, we treat obstacles by simply setting the possible speed of the robot g (x; y ) to be in nity inside the obstacles. Thus, given a starting point A and end point B , the robot is allowed to come arbitrarily close to the obstacles. The optimal path is de ned by the weighted arclength ds~2 = g (x; y )2ds2 , where ds2 = dx2 + dy 2 is the Euclidean arclength, and g (x; y ) is the weight over the domain. We search for the path C (s) = fx(s); y (s)g, with C (0) = A Rand C (L) = B , where L is the total arclength, that minimizes minC g (C (s))ds, . For an arbitrary parameterization p of the curve, the above geometric functional reads

Z1 0

g(C (p))jCp(p)jdp;

(5)

for which the Euler-Lagrange geometric equation is g = hrg; N~ i:

Where  = hCss ; N~ i is the curvature of the curve C , and N~ its normal. In order to link between the shape from shading and the path planning problem we show that the optimal paths are the gradient descent contours of a speci c Eikonal equation. LEMMA 1. The optimal paths between two points A and B , i.e. the curves that minimize Eq. (5) are the gradient descent contours of the function u that satis es the Eikonal equation jruj = g; and u(A) = 0:

Proof: Let us prove that the EL equation g = hrg; N~ i holds for the ` ow lines' of the function u that satis es jruj = g . The curve tangent of the ow lines is de ned by T~ = Cs = jrruuj , u ;u g . The curvature is de ned as thus N~ = jrruuj = f?jr uj y

x

 = hCss ; N~ i = h@sT~ ; N~ i

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Optimal Algorithm for Shape from Shading and Path Planning

u + r r u = @s ;   g ugx 1 u y = g hr( g ); Csi; hr( g ); Csi ; f?uy ; uxg   = g13 ux uy (uyy ? uxx ) + uxy (u2x ? u2y ) :

*

We also have that

D

E

*

7

+

 rg; N~ = rg; rgu   = 1 fuxuxx + uy uxy ; uxuxy + uy uyy g ; f?uy ; uxg g  g  1 = g 2 ux uy (uyy ? uxx ) + uxy (u2x ? u2y ) :

2

We just showed that by solving the Eikonal equation for the function u anchored at the initial location, we can backtrack the optimal path by owing along the gradient descent of that function from any given ` nal' location. Note that for g = 1, the function u is the distance map and the ow lines are straight lines, which are indeed the optimal paths between two points in a Euclidean domain. This backtracking idea was used in (Cohen and Kimmel, 97; Deschamps and Cohen, 00; Kimmel, Amir and Bruckstein, 95) for nding the optimal paths in 2D and 3D domains. Next, we add one more dimension to the problem and allow the robot to rotate. Instead of the point robot, we now consider a two-dimensional rectangle with a given width and length. Thus, the initial position A of the robot in the con guration space is speci ed by the position of the center of the rectangle, plus an orientation angle  between 0 and 2 . The nal con guration B is similarly speci ed, and the goal is to construct the optimal path from A to B , that minimizes the integration along the arclength ds2 = dx2 + dy 2 + d 2. In the absence of obstacles, a completely straightforward application of the fast marching method is possible; one discretizes the con guration space into a three-dimensional grid, that is, we similarly grid both IR2 and  between 0 and 2 , employing periodic boundary conditions in . Then, we solve the Eikonal equation

h

u2x + u2y + u2

i1=2

= 1;

(6)

with u(A) = 0. Back tracking along the gradient descent of u yields the optimal path. In the presence of obstacles, we rst apply the classical

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8 Kimmel & Sethian shrinking approach, where the robot is shrank into a point. For every discretized angle i , we dilate the shape of the obstacles with a `structuring element' corresponding to the robot at the given angle. This 3D con guration space construction can be done in O(N ) using the fast morphological operations methods in (Gil and Kimmel, 00). In Figure 2, we show several examples of a two-dimensional robot with rotational angle, navigating in a plane with obstacles. 80 80

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Figure 2. Navigating a rectangle with rotation in 2D (3DOF).

We now turn to examples in which the goal is to compute the optimal path, and imagine trying to not simply avoid obstacles, but to navigate under a potential function which weights the free work space (Latombe, 91; Khatib, 86), or even the whole free C-space. Now, the \uneven" terrain weighting is determined by the distance from the obstacles. Figure 3 presents the eect of the potential de ned in C-space on the course of the optimal path. We have tested several potentials that are functions of the distance map: a constant potential P1 (x; y; ) = 1, and P1 (x; y; ) = d8(x; y; ), where d(x; y; ) is the distance from the point (x; y; ) to the closest obstacle. The solution to the Eikonal equation jru(x; y; )j = 1=(1 + P (x; y; )) via the fast marching method yields the desired input for a back propagation (via an inverse gradient descent) procedure. We note, that in this case, we used the fast marching method itself to determine the value of the distance map d(x; y; ) by computing the distance map d(x; y; ) in the R2  S 1 C-space. Figure 4 presents the optimal path for a four joints arm robot (4 DOF). The optimal path is de ned via the arclength ds2 = d 12 + d 22 + d 32 + d 42, where i is the angle between the stick i, and stick i ? 1 along the robot arm. In the appendix we derive a useful algorithm for the update stage of the fast marching numerical method for an n dimensional Eikonal equation. Finally, we present results of an extension of the fast marching method to non-rectangular grids that was presented in (Kimmel and Sethian, 98). The method extends the numerical monotonic approximation to the Eikonal equation through a geometric interpretation of

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Optimal Algorithm for Shape from Shading and Path Planning 80

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Figure 3. Path planning under dierent potentials of the free con guration space

Figure 4. Navigation with 4 DOF: Left to right top to bottom.

the update procedure. We applied this method to path planning on curved domains with possible weights, and to computation of Voronoi diagrams, and oset curves on non- at triangulated domains (Kimmel and Sethian, 99). Figure 5 presents minimal geodesics as optimal paths, Voronoi diagram and oset curves on a curved triangulated surface. Figure 6 presents minimal geodesics as optimal paths on weighted (right) and unweighted curved surfaces (left). The darker the texture on the left, the lower the cost or weight.

4. Conclusion We presented an optimal algorithm for surface reconstruction from its shading image. The computational complexity upper bound, of O(N log N ), is data independent. It is the most ecient sequential algorithm for Horn's original formulation of the shape from shading

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Figure 5. Fast marching on triangulated surfaces: Minimal geodesics, Voronoi diagram, and geodesic osets.

Figure 6. Fast marching on triangulated surfaces: Minimal geodesics, and minimal weighted geodesics.

problem and a natural extension and application of the fast marching method. Next, we showed how the fast marching method can be applied to path planning problems as a consistent numerical method for locating optimal paths in con guration spaces with small number of dimensions. These two seemingly unrelated problems and their optimal solutions are samples of the wide range of possible applications for the fast marching method and its extensions.

Acknowledgment This research was supported in part by the fund for the promotion of research at the Technion.

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11

Appendix Let us derive a simple and useful procedure for the update stage of the fast marching numerical method for an n dimensional Eikonal equation. Consider the numerical approximation to the Eikonal equation jruj ? F = 0 in n dimensions that is given by k2f1;:::;ng max(D?k ; ?D+k ; 0)2 ? F 2 = 0: Where D?k and D+k denote the backwards and forward partial derivatives with respect to the k coordinate (e.g. D?1 = ui;j ? ui?1;j for the 2D case with
x = 1). We need to solve for the largest u1;:::;n = t, that is a root of the above quadratic equation. Let us rewrite the equation, and again search for the largest t that satis es k2f1;:::;ng max(t ? min(uk?1 ; uk+1); 0)2 ? F 2 = 0; where uk?1 indicates the value at the ?1 point along the k coordinate (e.g. u1?1  ui?1;j in the 2D case). Let ak = min(uk?1 ; uk+1 ), where the min is taken along the k coordinate (e.g. a2  min(ui;j ?1 ; ui;j +1) in the 2D case). In case there exists a root bigger than all ak , the solution is the root of the quadratic equation k2f1;:::;ng(t ? ak )2 ? F 2 = 0; and is given by   q (7) t = 1  a + nF 2 +   a a ? n a2 :

n

k k

k l k l

k k

Let aM = maxk (ak ). Inserting the demand t > aM in the above equation (t should be larger than all the values that generate it) we end up with the following simple condition

F 2 > k (aM ? ak )2: Then, we readily obtain the following lemma that help us derive a simple selection procedure for the update stage of the fast marching method. LEMMA 2. Under the condition F 2 > k (aM ? ak )2 the discriminant of Eq. (7) is greater than zero: nF 2 + k l ak al ? nk a2k > 0. I.e. the above t is a legitimate value that satis es t > aM .

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12 Kimmel & Sethian Proof: Let F 2 = k (aM ? ak )2. Then the discriminant of the cubic root of Eq. (7) is given by nF 2 + k lak al ? nk a2k = nk (aM ? ak )2 + k lak al ? nk a2k = nk (a2M ? 2aM ak + a2k ? a2k ) + k l ak al = nk (a2M ? 2aM ak ) + k l ak al = n2 a2M ? 2naM k ak + k lak al = (naM )2 ? 2(naM )(k ak ) + (k ak )2 = (k ak ? naM )2 > 0 2 Based on the above lemma, an ecient way of applying the n dimensional update step of the numerical approximation to Eikonal equation via the fast marching method is given as follows: 1. Compute fak g, and sort them in an increasing order, so that a1 is the smallest, and an is the largest value. Let m = n. 2. While F 2 < k2f1;::;mg(am ? ak )2 do m = m ? 1. 3. Compute the updated value   q t = 1  a + mF 2 +   a a ? m a2 ;

m

k k

k l k l

k k

where the  indexes are f1; ::; mg. Let the point value be t, if its less than its current value.

References M. Bichsel and A. P. Pentland. A simple algorithm for shape from shading. In Proceedings IEEE CVPR, pages 459{465, Champaign, Illinois, May 1992. M. J. Brooks and W. Chojnacki. direct computation of shape from shading. In Proc. of ICPR, Int. Conf. of Pattern Recognition, pages 114{119, Israel, October 1994. A. M. Bruckstein. On shape from shading. Comput. Vision Graphics Image Process., 44:139{154, 1988. L. Cohen and R. Kimmel. Global minimum for active contours models: A minimal path approach. Int. J. of Comp. Vision. 24(1):57-78, 1997. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial linear dierential equations. Bulletin of the American Math. Society, 27:1{67, 1992. , T. Deschamps and L.D. Cohen. Minimal paths in 3D images and application to virtual endoscopy. In Proc. of 6th ECCV, Dublin, Ireland, June, 2000. E. W. Dijkstra. A note on two problems in connection with graphs. Numerische Mathematic, 1:269{271, 1959.

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P. Dupuis and J. Oliensis. An optimal control formulation and related numerical methods for a problem in shape reconstruction. Ann. Appl. Probab., 4(2):287{ 346, 1994. J. Gil and R. Kimmel. Ecient dilation, erosion, opening and closing algorithms. In Proc. of . Int. Symposium on Mathematical Morphology V, Palo-Alto, June, 2000. B. K. P. Horn. Obtaining shape from shading information. In P H Winston, editor, The Psychology of Computer Vision, pages 115{155. McGraw Hill, 1975. B. K. P. Horn. Height and gradient from shading. Int. J. of Comp. Vision, 5:37{75, 1990. B. K. P. Horn and M. J. Brooks, editors. Shape from Shading. MIT Press, Cambridge MA, 1989. O. Khatib. Real-time obstacle avoidance for manipulators and mobile robots. The International Journal of Robotics Research, 5(1):396{404, 1986. R. Kimmel, A. Amir, and A. M. Bruckstein. Finding shortest paths on surfaces using level sets propagation. IEEE Trans. on PAMI, 17(1):635-640, 1995. R. Kimmel and A. M. Bruckstein. Shape from shading via level sets. CIS Report #9209, Technion{Israel Institute of Technology, Israel, June 1992. R. Kimmel and A. M. Bruckstein. Global shape from shading. CVIU, 62(3):360{369, November 1995. R. Kimmel and A. M. Bruckstein. Tracking level sets by level sets: A method for solving the shape from shading problem. CVIU, 62(2):47{58, July 1995. R. Kimmel and J. A. Sethian. Computing geodesic paths on manifolds. Proceedings of National Academy of Sciences, USA, 95(15):8431{8435, 1998. R. Kimmel and J. A. Sethian. Fast Voronoi diagrams and osets on triangulated surface. In Curves and Surfaces. A K Peters, Ltd., Wellesley, Massachusetts, 1999. R. Malladi and J.A. Sethian An O(N log(N)) Algorithm for Shape Modeling, In Proc. of National Academy of Sciences, USA. 93:9389-9392, 1996. J. C. Latombe. Robot motion planning. Kluwer Academic Publishers, Boston, MA, 1991. C. H. Lee and A. Rosenfeld. Improved methods of estimating shape from shading using the light source coordinate system. Arti cial Intelligence, 26(1):125{143, May 1985. P. L. Lions, E. Rouy, and A. Tourin. Shape-from-shading, viscosity solutions and edges. Numerische Mathematik, 64:323{353, 1993. E. Rouy and A. Tourin. A viscosity solutions approach to shape-from-shading. SIAM. J. Numer. Analy., 29(3):867{884, June 1992. R. Sedgewick. Algorithms. Addison-Wesley, 1988. J. A. Sethian. Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge Univ. Press, 1996. J. A. Sethian. A marching level set method for monotonically advancing fronts. Proc. Nat. Acad. Sci., 93(4), 1996. J.A. Sethian. Fast Marching Methods, SIAM Review, 41, July, 1999. J. N. Tsitsiklis. Ecient algorithms for globally optimal trajectories. IEEE Trans. on Automatic Control, 40(9):1528{1538, 1995. P. W. Verbeek and B. J. H. Verwer. Shading from shape, the eikonal equation solved by gray-weighted distance transform. Pattern. Rec. Letters, 11:681{690, 1990.

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