Fast Marching on Triangulated Domains

Fast Marching on Triangulated Domains

Ron Kimmel

www.cs.technion.ac.il/~ron

Computer Science Department

Geometric Image Processing Lab

Technion-Israel Institute of Technology

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

1D Distance: Example 1

Find distance T(x), given T(x0)=0.

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

except at x0. xx0

T(x)

1)( xTdx

d

1D Distance: Example 2

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. x

T(x)

x0 x1

1)( xTdx

d

1D Eikonal Equation

with boundary conditions T(x0)=T(x1)=0. Goal: Compute T that satisfies the equation

`the best'.

x

T(x)

x1x0

1)( xTdx

d

Numerical Approximation

Restrict , where h= grid spacing. Possible solutions for are

x

T(x)

x1x0

)(ihTTi

hTT

dxd iiT 1

1)( 1 h

TT ii

Approximation II

Updated i has always `upwind' from where the `wind blows'

x

T(x)

x1x0

1)( xTdx

d

0,)(,)(max)( 11 hTThTTxTdx

diiii

11,min iii TTT

Update Procedure

Set , and T(x0)=T(x1)=0.REPEAT UNTIL convergence,

FOR each i

T

i-1 i i+1

i+1

T

T

i-1

i

h

11,min ii TTm

hmTT ii ,min

iT

Update OrderWhat 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.

1

32

12

2

Weighted Domains Local weight , Arclength Goal: distance function characterized by: By the chain rule:

The Eikonal equation is

x

T(x)

x1x0

F(x)

)()(

1)())(( xT

dx

d

xFds

dxxT

dx

dsxT

ds

d

1Tds

d

)()( xFxTdx

d

RR:)(xF dxxFds )(

2D Rectangular Grids

Isotropic inhomogeneous domainsWeighted arclength:

the weight is Goal: Compute the distance T(x,y) from p0

where

),(),( yxFyxT

222),( TTyxT dy

ddxd

2222 ),( dydxyxFds RR 2:),( yxF

,}0},min{-max{

} 0}min{-max{

where

, }0 max{}0 max{

22211

211

1

222

iji,ji,jij

,ji,ji-ij

,ji-ijij

-x

ijijy

ij-y

ijx

ij-x

Fh,TTT

,,TTT

h

)-T (T T D

FT,T , - DDT,T, - DD

Upwind Approximation in 2D

T

i,j-1

i,j+1i-1,j

ij

i+1,j

i+1,j

i+1,j

TT

T

T

i-1,j

i,j-1

ij

2D Approximation Initialization: given initial value or Update:

Fitting a tilted plane with gradient , and two values anchored at the relevant neighboring grid points.

; },min{ ELSE2

)-(2

THEN ) |-(| IF

};,min{ };,min{

21

221

221

21

1,1,2,1,11

ijij

ij

ij

ij

jijijiji

FTTT

TTFTTT

FTT

TTTTT T

T

T

1

2i,j-1

i,j+1i-1,j

ij

i+1,j

ijTji :},{

ijF

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).

www.math.berkeley.edu/~sethian

Shortest Path on Flat Domains

Why do graph search based algorithms (like Dijkstra's) fail?

Edge IntegrationCohen-Kimmel, IJCV, 1997. Solve the 2D Eikonal equation

given T(p)=0 Minimal geodesic w.r.t.

222ijyx FTDTD

2222 )( dydxIFds

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.

222ijyx FTDTD

2222 )( dydxIFds

222 1 IIF

Path Planning 3 DOF

Solve the Eikonal Eq. in 3D {x,y,}-CS

given T(x0,y0,0)=0,Minimal geodesic w.r.t.

2222ijkyx FTDTDTD

22222 ),,( ddydxyxFds

Path Planning 3 DOF

2222ijkyx FTDTDTD

Path Planning 4 DOF

Solve the Eikonal Eq. in 4D

Minimal geodesic w.r.t.

22222

4321 ijklFTDTDTDTD

24

23

22

214321

22 ),,,( ddddFds

Update Acute AngleGiven ABC, update C. Consistency and monotonicity: Update only `from within the triangle' h in ABCFind t=EC that satisfies the gradient approximation

(t-u)/h= F.

c

c

We end up with:

t must satisfy u<t, and h in ABC. The update procedure is

IF (u<t) AND (a cos < b(t-u)/t < a/cos)THEN T(C) = min {T(C),t+T(A)};

ELSE T(C)= min {T(C),bF+T(A),aF+T(B)}.

Update Procedure

C B

A

u 0sin)cos(2 222222 Faubtbabutc

02 tt

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.

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!

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

max

max min

min

min2

2

min min min

max2

min min min2 3

1st Order Accuracy

The accuracy for acute triangles is O(e )Accuracy for the obtuse case

O(e /(- ))max max

max

Minimal Geodesics

Minimal Geodesics

Linear Interpolation

ODE ‘back tracking’

Quadratic Interpolation

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

jii

Geodesic Voronoi Diagrams and Geodesic Offsets

Geodesic Voronoi Diagrams and Geodesic Offsets

Marching Triangles

The intersection set of two functions is linearly interpolated via `marching triangle'

Voronoi Diagrams and Offsets on Weighted Curved Domains

Voronoi Diagrams and Offsets on Weighted Curved Domains

Cheap and Fast 3D Scanner

PC + video frame grabber. Video camera. Laser line pointer.

Joint with G. Zigelmanmotivated by simple shape from structure light methods, likeBouguet-Perona 99, Klette et al. 98

A frame grabber built at the Technion by Y Grinberg

Lego Mindstorms rotates the laser (E. Gordon)

Cheap and Fast 3D Scanner

Detection and Reconstruction

Examples of Decimation

Decimation - 3% of vertices Sub-grid sampling

Results

Results

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).

Difficulties

Need for user intervention. Local and global distortions. Restrictive boundary conditions. High computational complexity.

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).

000

Flattening

Flattening

Distances - comparison

Texture Mapping

Texture Mapping

Bending Invariant Signatures


Elad, Kimmel, CVPR’2001

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3Original surfaces Canonical surfaces in R


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0.40.6

0.81

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0.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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AAAA

D

DDD

BBBB

EEEE

FFFF

00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CE

CC

B

A

DE

EE

B

C

A

B

D

B

F

DAD

F

A

FF

Bending Invariant Clustering

2nd moments based MDS for clustering

Original surfaces Canonical forms

*A=human body*B=hand*C=paper*D=hat*E=dog*F=giraffe


More Applications

re-triangulation semi-manual segmentation halftoning in 3D

Adi, Kimmel 2002

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.

Documents

Fast Marching on Triangulated Domains