Convexity Shape Prior for Segmentation

Convexity is known as an important cue in human vision. We propose shape convexity as a new high-order regularization constraint for binary image segmentation. In the context of discrete optimization, object convexity is represented as a sum of 3-clique potentials penalizing any 1-0-1 configuration on all straight lines. We show that these non-submodular interactions can be efficiently optimized using a trust region approach. While the quadratic number of all 3-cliques is prohibitively high, we designed a dynamic programming technique for evaluating and approximating these cliques in linear time. Our experiments demonstrate general usefulness of the proposed convexity constraint on synthetic and real image segmentation examples. Unlike standard second-order length regularization, our convexity prior is scale invariant, does not have shrinking bias, and is virtually parameter-free.

[1]  Pascal Mamassian,et al.  Observer biases in the 3D interpretation of line drawings , 1998, Vision Research.

[2]  Marie-Pierre Jolly,et al.  Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in N-D Images , 2001, ICCV.

[3]  Satoru Fujishige,et al.  Submodular functions and optimization , 1991 .

[4]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[5]  Vladimir Kolmogorov,et al.  Computing geodesics and minimal surfaces via graph cuts , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[6]  Daniel Cremers,et al.  Global Solutions of Variational Models with Convex Regularization , 2010, SIAM J. Imaging Sci..

[7]  Olga Veksler,et al.  Star Shape Prior for Graph-Cut Image Segmentation , 2008, ECCV.

[8]  Olga Veksler,et al.  Order-Preserving Moves for Graph-Cut-Based Optimization , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Lena Gorelick,et al.  GrabCut in One Cut , 2013, 2013 IEEE International Conference on Computer Vision.

[10]  Vladimir Kolmogorov,et al.  Partial Enumeration and Curvature Regularization , 2013, 2013 IEEE International Conference on Computer Vision.

[11]  IEEE conference on computer vision and pattern recognition , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[12]  Martial Hebert,et al.  An Integer Projected Fixed Point Method for Graph Matching and MAP Inference , 2009, NIPS.

[13]  Marie-Pierre Jolly,et al.  Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[14]  Vladimir Kolmogorov,et al.  Graph cut based image segmentation with connectivity priors , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[15]  Vladimir Kolmogorov,et al.  What energy functions can be minimized via graph cuts? , 2002, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Lena Gorelick,et al.  Efficient Squared Curvature , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[17]  Andrew Blake,et al.  Geodesic star convexity for interactive image segmentation , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[18]  Lena Gorelick,et al.  Efficient Regularization of Squared Curvature , 2013, ArXiv.

[19]  Thomas Pock,et al.  Convex Relaxation of a Class of Vertex Penalizing Functionals , 2013, Journal of Mathematical Imaging and Vision.

[20]  Thomas Schoenemann,et al.  Generalized sequential tree-reweighted message passing , 2012, ArXiv.

[21]  Sebastian Nowozin,et al.  Global Interactions in Random Field Models: A Potential Function Ensuring Connectedness , 2010, SIAM J. Imaging Sci..

[22]  Daniel Cremers,et al.  Generalized ordering constraints for multilabel optimization , 2011, 2011 International Conference on Computer Vision.

[23]  Andrew J. Davison,et al.  Active Matching , 2008, ECCV.

[24]  Daniel Cremers,et al.  A Linear Framework for Region-Based Image Segmentation and Inpainting Involving Curvature Penalization , 2011, International Journal of Computer Vision.

[25]  Olga Veksler,et al.  Tiered scene labeling with dynamic programming , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[26]  R. Basri,et al.  The role of convexity in perceptual completion: beyond good continuation , 1999, Vision Research.

[27]  Lena Gorelick,et al.  Fast Trust Region for Segmentation , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[28]  Jamie Shotton,et al.  The Layout Consistent Random Field for Recognizing and Segmenting Partially Occluded Objects , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).