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Petter Strandmark Fredrik 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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Petter strandmark fredrik kahl

Petter Strandmark Fredrik 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 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

Adaptive meshes
Adaptive Meshes

Always split the most important region; use a priority queue

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)