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PPT – Computing Exact Discrete Minimal Surfaces: Extending and Solving the Shortest Path Problem in 3D with Application to Segmentation PowerPoint presentation | free to view - id: 21aa16-ZDc1Z

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Computing Exact Discrete Minimal Surfaces

Extending and Solving the Shortest Path Problem

in 3D with Application to Segmentation

- Leo Grady
- Department of Imaging and Visualization
- Siemens Corporate Research

Outline

- Introduction
- Extending the shortest path problem to 3D
- Method for computation
- Results
- Conclusion

Introduction Shortest path in 3D

A shortest path is easy to find in 3D, but it is

no longer a boundary

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

Introduction Minimal boundary segmentation

Specifying boundary

Shortest path on dual graph

Sometimes called cracks or edgels

Introduction Minimal boundary segmentation

Dual - Surfaces

???

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

Outline

- Introduction
- Extending the shortest path problem to 3D
- Method for computation
- Results
- Conclusion

Extending the Shortest Path Problem

Input Two points (0D boundary) Output Minimum

1D path having that boundary

How to extend problem to 3D?

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)

Extending the Shortest Path Problem

2D intelligent scissors/live wire ubiquitous

segmentation method

Extending the Shortest Path Problem

Outline

- Introduction
- Extending the shortest path problem to 3D
- Method for computation
- Results
- Conclusion

Method

How to compute?

Method

Minimize path/surface

3D

Method - Digression

Continuum interpretation - I

Use generalized Stokes Theorem

Fundamental Theorem of calculus

Standard Stokes Theorem

Careful! Instead of bivectors, formulated in

primal space

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

image gradients)

- 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

Method

Integer programming problem Bad!

However Sometimes we can apply linear

programming to an integer programming problem

and guarantee an integer solution.

Method

Minimal surface problem

Subject to

If constraints are feasible, then

Joint work with Vladimir Kolmogorov

Method

Original minimal surface problem

Subject to

New minimal surface problem

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

Method

Dual

Primal

Volume-face incidence in dual lattice is

node-edge incidence in primal lattice

Method

New minimal surface problem

is guaranteed to be integer.

Method

Conclusion

Minimal surface problem

Subject to

Solvable using generic linear programming code!

(and so is the other formulation)

Outline

- Introduction
- Extending the shortest path problem to 3D
- Method for computation
- Results
- Conclusion

Results - Correctness

Results 3D Segmentation

Outline

- Introduction
- Extending the shortest path problem to 3D
- Method for computation
- Results
- Conclusion

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

More Information

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

the paper Vladimir Kolmogorov Technical

analysis of LP problem

Writings and code

My webpage http//cns.bu.edu/lgrady

Combinatorial minimal surface MATLAB

code http//cns.bu.edu/lgrady/minimal_surface_ma

tlab_code.zip

MATLAB toolbox for graph theoretic image

processing at http//eslab.bu.edu/software/grapha

nalysis/

Uniqueness

A minimal surface/path is not necessarily unique

3D

2D

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.