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Fast Marching on Triangulated Domains Ron Kimmel Computer Science Department Geometric Image Processing Lab Technion-Israel Institute of Technology

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Brief Historical Review qUpwind schemes: Godunov 59 qLevel sets: Osher & Sethian 88 qViscosity SFS: Rouy & Tourin 92, (Osher & Rudin) qLevel sets SFS: Kimmel & Bruckstein 92 qContinuous morphology: Brockett & Maragos 92,Sapiro et al. 93 qMinimal geodesics: Kimmel, Amir & Bruckstein 93 qFast marching method: Sethian 95 qFast optimal path: Tsitsiklis 95 qLevel sets on triangulated domains:Barth & Sethian 98 qFast marching on triangulated domains: Kimmel & Sethian 98 qApplications based on joint works with: Elad, Kiryati, Zigelman

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1D Distance: Example 1 qFind distance T(x), given T(x0)=0. qSolution: T(x)=|x-x0|. q except at x0. x x0 T(x)

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1D Distance: Example 2 qFind the distance T(x), given T(x0)=T(x1)=0 qSolution: T(x) = min{|x-x0|,|x-x1|}, qAgain,, except x0,x1 and (x0+x1)/2. x T(x) x0 x1

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1D Eikonal Equation with boundary conditions T(x0)=T(x1)=0. qGoal: Compute T that satisfies the equation `the best'. x T(x) x1 x0

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Numerical Approximation qRestrict, where h= grid spacing. q qPossible solutions for are x T(x) x1 x0

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Approximation II q qUpdated i has always q`upwind' from where the `wind blows' x T(x) x1 x0

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Update Procedure Set, and T(x0)=T(x1)=0. REPEAT UNTIL convergence, nFOR each i ® T i-1 i i+1 T T i-1 i h

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Update Order What is the optimal order of updates? Solution I: Scan the line successively left to right. N scans, i.e. O(N ) Solution II: Left to right followed by right to left. Two scans are sufficient. (Danielson`s distance map 1980) Solution III: Start from x0, update its neighboring points, accept updated values, and update their neighbors, etc

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Weighted Domains qLocal weight, Arclength qGoal: distance function characterized by: qBy the chain rule: qThe Eikonal equation is x T(x) x1 x0 F(x)

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2D Rectangular Grids Isotropic inhomogeneous domains Weighted arclength: the weight is Goal: Compute the distance T(x,y) from p0 where

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Upwind Approximation in 2D T i,j-1 i,j+1 i-1,j ij i+1,j T T T T i-1,j i,j-1 ij

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2D Approximation Initialization: q given initial value or Update: q Fitting a tilted plane with gradient, and two values anchored at the relevant neighboring grid points. T T 1 2 i,j-1 i,j+1 i-1,j ij i+1,j

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Computational Complexity qT is systematically constructed from smaller to larger T values. qUpdate of a heap element is O(log N). qThus, upper bound of the total is O(N log N).

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Shortest Path on Flat Domains Why do graph search based algorithms (like Dijkstra's) fail?

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Edge Integration Cohen-Kimmel, IJCV, Solve the 2D Eikonal equation given T(p)=0 Minimal geodesic w.r.t.

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Shape from Shading Rouy-Tourin SIAM-NU 1992, Kimmel-Bruckstein CVIU 1994, Kimmel-Sethian JMIV Solve the 2D Eikonal equation where Minimal geodesic w.r.t.

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Path Planning 3 DOF Solve the Eikonal Eq. in 3D {x,y, }-CS given T(x0,y0, 0)=0, Minimal geodesic w.r.t.

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Path Planning 3 DOF

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Path Planning 4 DOF Solve the Eikonal Eq. in 4D Minimal geodesic w.r.t.

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Update Acute Angle Given ABC, update C. Consistency and monotonicity: Update only `from within the triangle' h in ABC Find t=EC that satisfies the gradient approximation (t-u)/h= F. c c

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We end up with: t must satisfy u

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Obtuse Problems This front first meets B, next A, and only then C. A is `supported’ by a single point. The supported section of incoming fronts is a limited section.

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Solution by splitting Extend this section and link the vertex to one within the extended section. Recursive unfolding: Unfold until a new vertex Q is found. Initialization step!

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Recursive Unfolding: Complexity e = length of longest edge The extended section maximal area is bounded by a<= e /(2 ). The minimal area of any unfolded triangle is bounded below a >= (h ) /2, The number of unfolded triangles before Q is found is bounded by m<= a /a = e /( h . max min 2 2 max 2 min 2 3

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1st Order Accuracy The accuracy for acute triangles is O(e ) Accuracy for the obtuse case O(e /( - )) max

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Minimal Geodesics

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Linear Interpolation ODE ‘back tracking’

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Quadratic Interpolation

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Voronoi Diagrams and Offsets uGiven n points, { p D, j 0,..,n-1} uVoronoi region: uG = {p D| d(p,p ) < d(p,p ), V j = i}. j ji i

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Geodesic Voronoi Diagrams and Geodesic Offsets

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Marching Triangles uThe intersection set of two functions is linearly interpolated via `marching triangle'

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Voronoi Diagrams and Offsets on Weighted Curved Domains

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Cheap and Fast 3D Scanner qPC + video frame grabber. qVideo camera. qLaser line pointer. Joint with G. Zigelman motivated by simple shape from structure light methods, like Bouguet-Perona 99, Klette et al. 98 A frame grabber built at the Technion by Y Grinberg Lego Mindstorms rotates the laser (E. Gordon)

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Cheap and Fast 3D Scanner

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Detection and Reconstruction

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Examples of Decimation Decimation - 3% of vertices Sub-grid sampling

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Results

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Texture Mapping qEnvironment mapping: Blinn, Newell (76). qEnvironment mapping: Greene, Bier and Sloan (86). qFree-form surfaces: Arad and Elber (97). qPolyhedral surfaces: Floater (96, 98), Levy and Mallet (98). qMulti-dimensional scaling: Schwartz, Shaw and Wolfson (89).

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Difficulties qNeed for user intervention. qLocal and global distortions. qRestrictive boundary conditions. qHigh computational complexity.

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Flattening via MDS qCompute geodesic distances between pairs of points. qConstruct a square distance matrix of geodesic distances^2. qFind the coordinates in the plane via multi- dimensional scaling. The simplest is `classical scaling’. qUse the flattened coordinates for texturing the surface, while preserving the texture features. Zigelman, Kimmel, Kiryati, IEEE T. on Visualization and Computer Graphics (in press)

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Flattening

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Distances - comparison

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Texture Mapping

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Bending Invariant Signatures

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Elad, Kimmel, CVPR’2001 ?

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Bending Invariant Signatures ? Elad, Kimmel, CVPR’2001

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Bending Invariant Signatures ? Elad, Kimmel, CVPR’2001

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Bending Invariant Signatures Elad, Kimmel, CVPR’2001

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Bending Invariant Signatures Elad, Kimmel, CVPR’2001

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Bending Invariant Signatures 3 Original surfaces Canonical surfaces in R Elad, Kimmel, CVPR’2001

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CC C C A A A A D DD D B B B B E E E E F F F F C E C C B A D E E E B C A B D B F D A D F A F F Bending Invariant Clustering q2 nd moments based MDS for clustering Original surfaces Canonical forms *A=human body *B=hand *C=paper *D=hat *E=dog *F=giraffe Elad, Kimmel, CVPR’2001

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More Applications re-triangulation semi-manual segmentation halftoning in 3D Adi, Kimmel 2002

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Conclusions uApplications of Fast Marching Method on rectangular grids: Path planning, edge integration, shape from shading. uO(N) consistent method for weighted geodesic distance: ‘Fast marching on triangulated domains’. uApplications: Minimal geodesics, geodesic offsets, geodesic Voronoi diagrams, surface flattening, texture mapping, bending invariant signatures and clustering of surfaces, triangulation, and semi- manual segmentation.

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