Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

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Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

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Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

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Petter StrandmarkFredrik Kahl

Curvature Regularization for Curves andSurfaces in a Global Optimization Framework

Centre for Mathematical Sciences, Lund University

Segmentation

Segmentation by minimizing an energy:

Dataterm

Length of boundary

Example from Schoenemann et al. 2009

Dataterm

Squared curvature

Length of boundary

Motivation from a psychological/biological standpoint

Improved multi-label formulation

- Schoenemann, Kahl and Cremers, ICCV 2009
- Schoenemann, Kahl, Masnou and Cremers, arXiv 2011
- Schoenemann, Masnou and Cremers, arXiv 2011

Continuous formulation

Global optimization of curvature

- Schoenemann, Kuang and Kahl, EMMCVPR 2011

- Goldluecke and Cremers, ICCV 2011

- Kanizsa, Italian Journal of Psychology 1971
- Dobbins, Zucker and Cynader, Nature 1987

Correct formulation,

efficiency,

- This paper:

3D

- Start with a mesh of all possible line segments

variable for each region

variables for each pair of edges

Restricted to {0,1}

variable for each region; 1 meansforeground, 0 background

Incorporate curvature:

variables for each pair of edges

Boundary constraints:

then

Surface continuation constraints:

then

- Problem with the existing formulation:

Nothing prevents a ”double boundary”

Existing formulation

Simple fix?

Global solution!

Require that

Not correct!

Not optimal (fractional)

- Consistency:

then

New constraints

Existing formulation

Global solution!

Global + correct!

Not optimal (fractional)

Not correct!

Too

coarse!

90°

60°

45°

27°

30°

32 regions, 52 lines

12 regions, 18 lines

Always split the most important region; use a priority queue

p. 69

16-connectivity

8-connectivity

Approximate surface with a mesh of faces

Want to measure how much the surface bends:

Willmore energy

One unit cell

(5 tetrahedrons)

8 unit cells

Problem: “Wrapping a surface around a cross”

Area regularization

Curvature regularization

Problem: “Connecting two discs”

Area regularization

Curvature regularization

491,000 variables

637,000 variables

128 seconds

- Curvature regularization is now more practical
- Adaptive meshes
- Hexagonal meshes

- New constraints give correct formulation
- Surface completion

Source code available online (2D and 3D)

The end