corp. research princeton, nj computing geodesics and minimal surfaces via graph cuts yuri boykov,...
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![Page 1: Corp. Research Princeton, NJ Computing geodesics and minimal surfaces via graph cuts Yuri Boykov, Siemens Research, Princeton, NJ joint work with Vladimir](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d1c5503460f949f189b/html5/thumbnails/1.jpg)
Corp. ResearchPrinceton, NJ
Computing geodesics and minimal surfaces via graph cuts
Yuri Boykov, Siemens Research, Princeton, NJ
joint work with
Vladimir Kolmogorov, Cornell University, Ithaca, NY
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Corp. ResearchPrinceton, NJ
Two standard object extraction methods
Interactive Graph cuts [Boykov&Jolly ‘01]
• Discrete formulation
• Computes min-cuts on N-D grid-graphs
Geodesic active contours [Caselles et.al. ‘97, Yezzi et.al ‘97]
• Continuous formulation
• Computes geodesics in image- based N-D Riemannian spaces
Geo-cuts
• Minimal geometric artifacts
• Solved via local variational technique (level sets)
• Possible metrication errors
• Global minima
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Corp. ResearchPrinceton, NJGeodesics and minimal surfaces
The shortest curve between two points is a geodesic
Riemannian metric(space varying, tensor D(p))
Geodesic contours use image-based Riemannian metric
Euclidian metric (constant)
A
B
A
B
Generalizes to 3D (minimal surfaces)
distance map distance
map
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Corp. ResearchPrinceton, NJ
Graph cuts (simple example à la Boykov&Jolly, ICCV’01)
n-links
s
t a cuthard constraint
hard constraint
Minimum cost cut can be computed in polynomial time
(max-flow/min-cut algorithms)
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Corp. ResearchPrinceton, NJMetrication errors on graphs
discrete metric ???
Minimum cost cut (standard 4-neighborhoods)
Continuous metric space
(no geometric artifacts!)
Minimum length geodesic contour (image-based Riemannian metric)
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Corp. ResearchPrinceton, NJ
Cut Metrics :cuts impose metric properties on graphs
C
Cut metric is determined by the graph topology and by edge weights. Can a cut metric approximate a given Riemannian metric?
Ce
eC ||||||
Cost of a cut can be interpreted as a geometric “length” (in 2D) or “area” (in 3D) of the corresponding contour/surface.
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Corp. ResearchPrinceton, NJOur key technical result
The main technical problem is solved via Cauchy-Crofton formula from integral geometry.
We show how to build a grid-graph such that its
cut metric approximates any given Riemannian metric
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Corp. ResearchPrinceton, NJ
Integral Geometry andCauchy-Crofton formula
C
ddnC L21||||Euclidean length of C :
the number of times line L intersects C
2
0
a set of all lines L
CL
a subset of lines L intersecting contour C
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Corp. ResearchPrinceton, NJ
Cut Metric on gridscan approximate Euclidean Metric
C
k
kkknC 21||||
Euclidean length
2kk
kw
gcC ||||graph cut cost
for edge weights:the number of edges of family k intersecting C
Edges of any regular neighborhood system
generate families of lines
{ , , , }
Graph nodes are imbeddedin R2 in a grid-like fashion
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Corp. ResearchPrinceton, NJCut metric in Euclidean case
2kk
kw
“standard” 4-neighborhoods
(Manhattan metric)
256-neighborhoods8-neighborhoods
“Distance maps” (graph nodes “equidistant” from a given node) :
(Positive!) weights depend only on edge direction k.
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Corp. ResearchPrinceton, NJReducing Metrication Artifacts
originalnoisy image
Image restoration [BVZ 1999]
restoration with “standard” 4-neighborhoods
restoration with 8-neighborhoods
using edge weights
2kk
kw
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Corp. ResearchPrinceton, NJCut Metric in Riemannian case
The same technique can used to compute edge weights that approximate arbitrary Riemannian metric defined by tensor D(p)
• Idea: generalize Cauchy-Crofton formula
4-neighborhoods 8-neighborhoods 256-neighborhoods
Local “distance maps” assuming anisotropic D(p) = const
))(,()( pDkfpwk (Positive!) weights depend on
edge direction k and on location/pixel p.
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Corp. ResearchPrinceton, NJ
||||||||0,0
CC gc
Convergence theorem
Theorem: For edge weights set by tensor D(p)
0|| e
C
)( pwk
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Corp. ResearchPrinceton, NJ“Geo-Cuts” algorithm
image-derivedRiemannian metric
D(p) )( pwk
regular grid edge weights
Boundary conditions(hard/soft
constraints)
Global optimization
Graph-cuts[Boykov&Jolly, ICCV’01]
||ˆ||||ˆ|| CC gc
min-cut = geodesic
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Corp. ResearchPrinceton, NJ
Minimal surfaces in image inducedRiemannian metric spaces (3D)
3D bone segmentation (real time screen capture)
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Corp. ResearchPrinceton, NJ
Our results reveal a relation between…
Level Sets Graph Cuts [Osher&Sethian’88,…] [Greig et. al.’89, Ishikawa et. al.’98, BVZ’98,…]
Gradient descent method VS. Global minimization tool
variational optimization method for combinatorial optimization for fairly general continuous energies a restricted class of energies [e.g. KZ’02]
finds a local minimum finds a global minimum near given initial solution for a given set of boundary conditions
anisotropic metrics are harder anisotropic Riemannian metrics to deal with (e.g. slower) are as easy as isotropic ones
numerical stability has to be carefully
addressed [Osher&Sethian’88]:continuous formulation -> “finite
differences”
numerical stability is not an issue
discrete formulation ->min-cut algorithms
(restricted class of energies)
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Corp. ResearchPrinceton, NJConclusions
“Geo-cuts” combines geodesic contours and graph cuts. • The method can be used as a “global” alternative to variational level-sets.
Reduction of metrication errors in existing graph cut methods• stereo [Roy&Cox’98, Ishikawa&Geiger’98, Boykov&Veksler&Zabih’98, ….]• image restoration/segmentation [Greig’86, Wu&Leahy’97,Shi&Malik’98,…]• texture synthesis [Kwatra/et.al’03]
Theoretical connection between discrete geometry of graph cuts and concepts of integral & differential geometry
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Corp. ResearchPrinceton, NJ
Geo-cuts (more examples)
3D segmentation (time-lapsed)