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Petter StrandmarkFredrik Kahl . Curvature Regularization for Curves and Surfaces in a Global Optimization Framework. Centre for Mathematical Sciences, Lund University. Length Regularization. Segmentation. Segmentation by minimizing an energy:. Data term. Length of boundary.

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Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

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Petter strandmark fredrik kahl

Petter StrandmarkFredrik Kahl

Curvature Regularization for Curves andSurfaces in a Global Optimization Framework

Centre for Mathematical Sciences, Lund University

Length regularization

Length Regularization


Segmentation by minimizing an energy:


Length of boundary

Long thin structures

Long, thin structures

Example from Schoenemann et al. 2009


Squared curvature

Length of boundary

Important papers

Important papers

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,


  • This paper:


Approximating c urves

Approximating Curves

Approximating curves

Approximating Curves

  • Start with a mesh of all possible line segments

variable for each region

variables for each pair of edges

Restricted to {0,1}

Linear objective function

Linear Objective Function

variable for each region; 1 meansforeground, 0 background

Incorporate curvature:

variables for each pair of edges

Linear constraints

Linear Constraints

Boundary constraints:


Surface continuation constraints:


New constraints

New Constraints

  • Problem with the existing formulation:

Nothing prevents a ”double boundary”

New constraints1

New Constraints

Existing formulation

Simple fix?

Global solution!

Require that

Not correct!

Not optimal (fractional)

New constraints2

New Constraints

  • Consistency:


New constraints3

New Constraints

New constraints

Existing formulation

Global solution!

Global + correct!

Not optimal (fractional)

Not correct!

Mesh types

Mesh Types








32 regions, 52 lines

12 regions, 18 lines

Mesh types1

Mesh Types

Adaptive meshes

Adaptive Meshes

Always split the most important region; use a priority queue

Adaptive meshes1

Adaptive Meshes

p. 69

Adaptive meshes2

Adaptive Meshes

Does it matter

Does It Matter?


Does it matter1

Does It Matter?


Curvature of surfaces

Curvature of Surfaces

Approximate surface with a mesh of faces

Want to measure how much the surface bends:

Willmore energy

3d mesh

3D Mesh

One unit cell

(5 tetrahedrons)

8 unit cells

3d results

3D Results

Problem: “Wrapping a surface around a cross”

Area regularization

Curvature regularization

Surface c ompletion results

Surface CompletionResults

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

The end

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