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## Computing Exact Discrete Minimal Surfaces: Extending and Solving the Shortest Path Problem in 3D with Application to Segmentation

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Title: Computing Exact Discrete Minimal Surfaces: Extending and Solving the Shortest Path Problem in 3D with Application to Segmentation

1
Computing Exact Discrete Minimal Surfaces
Extending and Solving the Shortest Path Problem
in 3D with Application to Segmentation
• Department of Imaging and Visualization
• Siemens Corporate Research

2
Outline
• Introduction
• Extending the shortest path problem to 3D
• Method for computation
• Results
• Conclusion

3
Introduction Shortest path in 3D
A shortest path is easy to find in 3D, but it is
no longer a boundary
4
Introduction Minimal boundary segmentation
4?4 image
4?4 weighted graph
Minimal segmentation specified two ways
Cutting edges/specifying normals
Specifying boundary
Graph Cuts Max-flow/Min-cut
Intelligent scissors/Live wire Dijkstras
algorithm
5
Introduction Minimal boundary segmentation
Specifying boundary
Shortest path on dual graph

Sometimes called cracks or edgels
6
Introduction Minimal boundary segmentation
Dual - Surfaces
???
7
Introduction Minimal boundary segmentation
Max-flow/min-cut
Different algorithmic complexity
Both algorithms persist in 2D, but
max-flow/min-cut only option in 3D
8
Outline
• Introduction
• Extending the shortest path problem to 3D
• Method for computation
• Results
• Conclusion

9
Extending the Shortest Path Problem
Input Two points (0D boundary) Output Minimum
1D path having that boundary
How to extend problem to 3D?
10
Extending the Shortest Path Problem
Input Closed contour (1D boundary) Output
Minimum 2D surface having that boundary
minimal 2-dimensional object
1-dimensional boundary
Surface is minimal relative to weighting (metric)
11
Extending the Shortest Path Problem
2D intelligent scissors/live wire ubiquitous
segmentation method
12
Extending the Shortest Path Problem
13
Outline
• Introduction
• Extending the shortest path problem to 3D
• Method for computation
• Results
• Conclusion

14
Method
How to compute?
15
Method
Minimize path/surface
3D
16
Method - Digression
Continuum interpretation - I
Use generalized Stokes Theorem
Fundamental Theorem of calculus
Standard Stokes Theorem
Careful! Instead of bivectors, formulated in
primal space
17
Method - Digression
Continuum interpretation - II
3D
2D
- Vector field taking nonzeros along minimal
path
- Vector field taking nonzeros on the normals
of the surface
- Ambient vector field (e.g., derived from
• Ambient vector field
• (e.g., derived from image gradients)

Use boundary as constraint
Subject to
RHS consists of two delta functions at endpoints
RHS consists of a unit closed contour
18
Method
However Sometimes we can apply linear
programming to an integer programming problem
and guarantee an integer solution.
19
Method
Minimal surface problem
Subject to
If constraints are feasible, then
20
Method
Original minimal surface problem
Subject to
New minimal surface problem
21
Method
New minimal surface problem
Big question What is ?
The volume-face boundary operator
Followed by the
May also be stated as The boundary of the
boundary is zero
22
Method
Dual
Primal
Volume-face incidence in dual lattice is
node-edge incidence in primal lattice
23
Method
New minimal surface problem
is guaranteed to be integer.
24
Method
Conclusion
Minimal surface problem
Subject to
Solvable using generic linear programming code!
(and so is the other formulation)
25
Outline
• Introduction
• Extending the shortest path problem to 3D
• Method for computation
• Results
• Conclusion

26
Results - Correctness
27
Results 3D Segmentation
28
Outline
• Introduction
• Extending the shortest path problem to 3D
• Method for computation
• Results
• Conclusion

29
Conclusion
1. Natural extension of shortest path given two
points is minimal surface given a closed contour
2. Minimal surface problem solvable with generic
linear programming code
3. There are, in fact, two integral linear
programming problems that could be solved to
achieve the solution
30
Acknowledgements
Marie-Pierre Jolly Posing the problem Gareth
Funka-Lea Support and enthusiasm for the
work Yuri Boykov Enthusiasm and encouragement
of the topic Chenyang Xu Extensive comments on
analysis of LP problem
Writings and code
Combinatorial minimal surface MATLAB
tlab_code.zip
MATLAB toolbox for graph theoretic image
processing at http//eslab.bu.edu/software/grapha
nalysis/
31
Uniqueness
A minimal surface/path is not necessarily unique
3D
2D
32
Method
Is the face-edge boundary operator totally
unimodular?
Yes, provided that the complex is orientable.
Fortunately, the lattices used in vision are all
orientable.