1. Unitialized, Globally Optimal, Graph-Based Rectilinear Shape Segmentation - The Opposing Metrics Method
Computer Science Department –
Carnegie Mellon University, Pittsburgh
Department of Imaging and Visualization –
Siemens Corporate Research, Princeton
Ali Kemal Sinop and Leo Grady
SIEMENS
Main Idea
Weak boundary completion
Optimization
Globally optimal variational segmentation requires two propreties:
1) Ability to measure “shapeness” of a segmentation
2) Ability to find a segmentation that optimizes the shapeness measure
Rectilinearity measure
For a given boundary P, measure rectilinearity as the ratio
x – binary indicator vector on nodes
L1 – Laplacian matrix of L1 graph
L2 – Laplacian matrix of L2 graph
Kaniza square
gen. eigenvector
segmentation
Relax binary formulation to allow real values for x
Generalized eigenvector problem!
Merged circle/square
gen. eigenvector
segmentation
Threshold solution x at value producing maximal
ratio
for an intrinsic parameterization in terms of u and v
For a segmentation on lattice, L1 and L2 boundary metrics
representable as cuts on weighted graph
Natural image results
Effect of resolution on measure
Per1 (P) = 16
Per2 (P) = 13.9
Q(P) = 1
Graph formulation allows us to measure exact dependence
of shape descriptor on the number of pixels comprising object
Q(P) = .8536
Per1 (P) = 16
Per2 (P) = 11.8
Correctness
Input
1st
2nd
3rd
4th