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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
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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
Find distance T(x), given T(x0)=0.
Solution: T(x)=|x-x0|.
except at x0. xx0
T(x)
1)( xTdx
d
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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
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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
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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
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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
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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
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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
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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 )(
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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
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,}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
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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
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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).
www.math.berkeley.edu/~sethian
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Shortest Path on Flat Domains
Why do graph search based algorithms (like Dijkstra's) fail?
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Edge IntegrationCohen-Kimmel, IJCV, 1997. Solve the 2D Eikonal equation
given T(p)=0 Minimal geodesic w.r.t.
222ijyx FTDTD
2222 )( dydxIFds
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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.
222ijyx FTDTD
2222 )( dydxIFds
222 1 IIF
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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.
2222ijkyx FTDTDTD
22222 ),,( ddydxyxFds
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Path Planning 3 DOF
2222ijkyx FTDTDTD
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Path Planning 4 DOF
Solve the Eikonal Eq. in 4D
Minimal geodesic w.r.t.
22222
4321 ijklFTDTDTDTD
24
23
22
214321
22 ),,,( ddddFds
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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
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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
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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
max
max min
min
min2
2
min min min
max2
min min min2 3
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1st Order Accuracy
The accuracy for acute triangles is O(e )Accuracy for the obtuse case
O(e /(- ))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
jii
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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. 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)
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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).
000
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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
3Original surfaces Canonical surfaces in R
Elad, Kimmel, CVPR’2001
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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
CCCC
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
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
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.