Computing geodesics and minimal surfaces via graph cuts

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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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Two standard object extraction methods

• Geodesic active contours
• Caselles et.al. 97, Yezzi et.al 97
• Continuous formulation
• Computes geodesics in image- based N-D
  Riemannian spaces

• Interactive Graph cuts
• BoykovJolly 01
• Discrete formulation
• Computes min-cuts on N-D grid-graphs

Geo-cuts
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Geodesics and minimal surfaces

• The shortest curve between two points is a
  geodesic

B
B
A
A
Euclidian metric (constant)
Riemannian metric (space varying, tensor D(p))

• Geodesic contours use image-based Riemannian
  metric

• Generalizes to 3D (minimal surfaces)

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Graph cuts (simple example à la BoykovJolly,
ICCV01)
Minimum cost cut can be computed in polynomial
time (max-flow/min-cut algorithms)
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Metrication errors on graphs
discrete metric ???
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Cut Metrics cuts impose metric properties on
graphs

• Cut metric is determined by the graph topology
  and by edge weights.
• Can a cut metric approximate a given Riemannian
  metric?

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Our key technical result
We show how to build a grid-graph such that its
cut metric approximates any given Riemannian
metric

• The main technical problem is solved via
  Cauchy-Crofton formula from integral geometry.

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Integral Geometry andCauchy-Crofton formula
C
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Cut Metric on gridscan approximate Euclidean
Metric
Graph nodes are imbedded in R2 in a grid-like
fashion
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Cut metric in Euclidean case

• (Positive!) weights
  depend only on
• 
  edge direction k.

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Reducing Metrication Artifacts
Image restoration BVZ 1999
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Cut 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

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Convergence theorem
Theorem For edge weights set by
tensor D(p)
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Geo-Cuts algorithm
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Minimal surfaces in image inducedRiemannian
metric spaces (3D)
3D bone segmentation (real time screen capture)
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Our results reveal a relation between

• Level Sets
  Graph Cuts
• OsherSethian88,
  Greig et. al.89, Ishikawa et.
  al.98, BVZ98,

(restricted class of energies)
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Conclusions

• 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 RoyCox98, IshikawaGeiger98,
  BoykovVekslerZabih98, .
• image restoration/segmentation Greig86,
  WuLeahy97,ShiMalik98,
• texture synthesis Kwatra/et.al03
• Theoretical connection between discrete geometry
  of graph cuts and concepts of integral
  differential geometry

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Geo-cuts (more examples)
3D segmentation (time-lapsed)