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**Fast Marching on Triangulated Domains**

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

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**Brief Historical Review**

Upwind schemes: Godunov 59 Level sets: Osher & Sethian 88 Viscosity SFS: Rouy & Tourin 92, (Osher & Rudin) Level sets SFS: Kimmel & Bruckstein 92 Continuous morphology: Brockett & Maragos 92,Sapiro et al. 93 Minimal geodesics: Kimmel, Amir & Bruckstein 93 Fast marching method: Sethian 95 Fast optimal path: Tsitsiklis 95 Level sets on triangulated domains:Barth & Sethian 98 Fast marching on triangulated domains: Kimmel & Sethian 98 Applications based on joint works with: Elad, Kiryati, Zigelman

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**1D Distance: Example 1 T(x) x x0 Find distance T(x), given T(x0)=0.**

Solution: T(x)=|x-x0|. except at x0. T(x) x x0

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**1D Distance: Example 2 T(x) x x0 x1**

Find the distance T(x), given T(x0)=T(x1)=0 Solution: T(x) = min{|x-x0|,|x-x1|}, Again, , except x0,x1 and (x0+x1)/2. T(x) x x0 x1

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**1D Eikonal Equation T(x) x x0 x1**

with boundary conditions T(x0)=T(x1)=0. Goal: Compute T that satisfies the equation `the best'. T(x) x x0 x1

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**Numerical Approximation**

Restrict , where h= grid spacing. Possible solutions for are T(x) x x0 x1

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**Approximation II T(x) x x0 x1 Updated i has always**

`upwind' from where the `wind blows' T(x) x x0 x1

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**Update Procedure T T T Set , and T(x0)=T(x1)=0.**

REPEAT UNTIL convergence, FOR each i T i T i+1 T i-1 i-1 i i+1 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. 2 1 2 3 1 2

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**Weighted Domains x x0 x1 Local weight , Arclength**

Goal: distance function characterized by: By the chain rule: The Eikonal equation is T(x) F(x) x x0 x1

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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**

ij T i+1,j T i,j-1 i+1,j T i,j-1 i+1,j T i-1,j ij i-1,j i,j+1

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**2D Approximation T T Initialization: given initial value or Update:**

Fitting a tilted plane with gradient , and two values anchored at the relevant neighboring grid points. T 1 i+1,j T 2 i,j-1 ij i-1,j i,j+1

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**Computational Complexity**

T is systematically constructed from smaller to larger T values. Update of a heap element is O(log N). Thus, 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, 1997.**

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 2001. 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,j}-CS**

given T(x0,y0,j0)=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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**Update Procedure We end up with: t must satisfy u<t, and h in ABC.**

The update procedure is IF (u<t) AND (a cos q < b(t-u)/t < a/cosq) THEN T(C) = min {T(C),t+T(A)}; ELSE T(C)= min {T(C),bF+T(A),aF+T(B)}. u C B A

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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 Initialization step!**

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 /(2a ). The minimal area of any unfolded triangle is bounded below a >= (h a ) q /2, The number of unfolded triangles before Q is found is bounded by m<= a /a = e /(q h a ). max 2 max min 2 min min min min 2 2 3 max min max min min min

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**1st Order Accuracy The accuracy for acute triangles is O(e )**

Accuracy for the obtuse case O(e /(p-q )) max max max

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Minimal Geodesics

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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**

Given n points, { p D, j 0,..,n-1} Voronoi region: G = {p D| d(p,p ) < d(p,p ), V j = i}. j i i j

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**Geodesic Voronoi Diagrams and Geodesic Offsets**

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**Geodesic Voronoi Diagrams and Geodesic Offsets**

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Marching Triangles The 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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**Voronoi Diagrams and Offsets on Weighted Curved Domains**

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**Cheap and Fast 3D Scanner**

PC + video frame grabber. Video camera. Laser line pointer. Joint with G. Zigelman motivated by simple shape from structure light methods, like Bouguet-Perona 99, Klette et al. 98 Lego Mindstorms rotates the laser (E. Gordon) A frame grabber built at the Technion by Y Grinberg

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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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Results

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**Texture Mapping Environment mapping: Blinn, Newell (76).**

Environment mapping: Greene, Bier and Sloan (86). Free-form surfaces: Arad and Elber (97). Polyhedral surfaces: Floater (96, 98), Levy and Mallet (98). Multi-dimensional scaling: Schwartz, Shaw and Wolfson (89).

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**Difficulties Need for user intervention. Local and global distortions.**

Restrictive boundary conditions. High computational complexity.

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**Flattening via MDS Compute geodesic distances between pairs of points.**

Construct a square distance matrix of geodesic distances^2. Find the coordinates in the plane via multi-dimensional scaling. The simplest is `classical scaling’. Use 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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Flattening

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**Distances - comparison**

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Texture Mapping

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Texture Mapping

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**Bending Invariant Signatures**

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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**

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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**Bending Invariant Clustering**

2nd moments based MDS for clustering Original surfaces Canonical forms D D D 0.8 A A E 0.8 E 0.7 A 0.7 D E 0.6 E B 0.6 E E E 0.5 C C C 0.5 C D C C 0.4 B C 0.4 D D E D 0.3 F A 0.3 B B B B 0.2 F B B C 0.2 A A A A 0.1 F 0.1 F 1 1 1 0.8 1 0.8 0.8 0.6 *A=human body 0.8 F F F 0.6 0.6 0.4 0.6 F 0.4 0.4 0.4 0.2 0.2 *B=hand 0.2 0.2 *C=paper *D=hat *E=dog *F=giraffe Elad, Kimmel, CVPR’2001

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**More Applications semi-manual re-triangulation segmentation**

halftoning in 3D Adi, Kimmel 2002

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Conclusions Applications of Fast Marching Method on rectangular grids: Path planning, edge integration, shape from shading. O(N) consistent method for weighted geodesic distance: ‘Fast marching on triangulated domains’. Applications: 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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