Post on 23-Mar-2016
description
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