Curvature Regularization for Curves andSurfaces in a Global Optimization Framework

Petter Strandmark Fredrik Kahl

Centre for Mathematical Sciences, Lund University

Length RegularizationSegmentation

𝐸=∫Γ

❑

𝑔 (𝑥 )𝑑𝑥+|𝛾|

Data termLength of boundary

Segmentation by minimizing an energy:

Γ

𝛾

Long, thin structures

𝐸=∫Γ

❑

𝑔 (𝑥 )𝑑𝑥

Data term

Example from Schoenemann et al. 2009

Length of boundary

+|𝛾|

Squared curvature

+∫𝛾

❑

|𝜅 (𝑠 )|2𝑑𝑠

Important papers

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

Global optimization of curvature

• Schoenemann, Kuang and Kahl, EMMCVPR 2011 Improved multi-label formulation

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

Motivation from a psychological/biological standpoint

• This paper: Correct formulation,

efficiency,

3D

• Goldluecke and Cremers, ICCV 2011 Continuous formulation

Approximating Curves

𝛼1 𝛼3

𝛼2𝛼4

𝛼5

𝛼6𝛼7

∫𝛾

❑

(𝜅 (𝑠 ) )2𝑑𝑠❑→∑𝑖=1

𝑚

𝑏𝑖(𝛼𝑖) ,𝑚❑→∞

Approximating Curves

Start with a mesh of all possible line segments

𝑥𝑖variable for each region𝑦 𝑖 , 𝑗 , 𝑦 𝑗 ,𝑖variables for each pair of edges Restricted to {0,1}

Linear Objective FunctionIncorporate curvature: 𝐸=∫

Γ

❑

𝑔 (𝑥 )𝑑𝑥+|𝛾|+∫𝛾

❑

|𝜅 (𝑠 )|2𝑑𝑠

∑𝑖=1

𝑛

𝑔𝑖 𝑥𝑖+12 ∑

𝑖 , 𝑗

❑

ℓ𝑖𝑗 𝑦 𝑖𝑗 +∑𝑖 , 𝑗

❑

𝑏𝑖(𝛼 𝑖)𝑦 𝑖𝑗𝐸≈

𝑥𝑖variable for each region; 1 means foreground, 0 background

𝑦 𝑖 , 𝑗variables for each pair of edges

Linear Constraints𝑦 𝑖𝑗

𝑦 𝑖𝑘

𝑦 𝑖𝑙

𝑥𝑎

𝑥𝑏

𝑥𝑐

𝑥𝑑

𝑥𝑒

𝑥 𝑓

Boundary constraints:

then

∑𝑚𝑦 𝑖𝑚=1

Surface continuation constraints:

then

∑𝑚𝑦 𝑗𝑚=1𝑦 𝑖𝑗 𝑦 𝑗𝑙

𝑦 𝑗𝑘

New Constraints

Problem with the existing formulation:

Nothing prevents a ”double boundary”

New Constraints

Simple fix?Require that 𝑦 𝑖𝑗+𝑦 𝑗𝑖≤1

Not optimal (fractional)

Existing formulationGlobal solution!Not correct!

𝑦 𝑖𝑗

𝑦 𝑗𝑖

New Constraints

Consistency:

𝑦 𝑖𝑗

𝑥𝑎

𝑥𝑏

then

New Constraints

Existing formulationGlobal solution!Not correct!

𝑦 𝑖𝑗+𝑦 𝑗𝑖≤1Not optimal (fractional)

New constraintsGlobal + correct!

Mesh Types

90° 60° 45°

27° 30°32 regions, 52 lines 12 regions, 18 lines

Too coarse!

Mesh Types

Adaptive Meshes

Always split the most important region; use a priority queue

Adaptive Meshes

p. 69

Adaptive Meshes

Does It Matter?16-connectivity

2.470×1082.458×108

Does It Matter?8-connectivity

Curvature of Surfaces

𝛼

Approximate surface with a mesh of faces

Want to measure how much the surface bends:∫𝐻 2𝑑𝐴∫(𝐻 ¿¿ 2−𝐾 )𝑑𝐴 ¿ Willmor

e energy

3D Mesh

One unit cell

8 unit cells

(5 tetrahedrons)

3D Results

Area regularization Curvature regularization

Problem: “Wrapping a surface around a cross”

Surface Completion Results

Area regularization Curvature regularization

491,000 variables 637,000 variables128 seconds

Problem: “Connecting two discs”

Conclusions

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