Learning Shapes by Convex Composition

We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that overlap and occlude one another in such a way that they "match" a given scene as closely as possible. This paper derives general sufficient conditions under which this convex shape composition identifies a target composition. From a computational standpoint, we present two different methods for solving the associated optimization programs. The first method simply recasts the problem as a linear program, while the second uses the alternating direction method of multipliers with a series of easily computed proximal operators.

[1]  Justin K. Romberg,et al.  Sparse Shape Reconstruction , 2013, SIAM J. Imaging Sci..

[2]  Justin K. Romberg,et al.  Convex Cardinal Shape Composition , 2015, SIAM J. Imaging Sci..

[3]  W. Eric L. Grimson,et al.  A shape-based approach to the segmentation of medical imagery using level sets , 2003, IEEE Transactions on Medical Imaging.

[4]  Rachid Deriche,et al.  A Review of Statistical Approaches to Level Set Segmentation: Integrating Color, Texture, Motion and Shape , 2007, International Journal of Computer Vision.

[5]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[6]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[7]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[8]  J. Romberg,et al.  Terahertz time-gated spectral imaging for content extraction through layered structures , 2016, Nature Communications.

[9]  Emmanuel Cand.  Simple Bounds for Recovering Low-complexity Models , 2012 .

[10]  Olivier D. Faugeras,et al.  Statistical shape influence in geodesic active contours , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[11]  Robert J. Fowler,et al.  Optimal Packing and Covering in the Plane are NP-Complete , 1981, Inf. Process. Lett..

[12]  R. Rockafellar.  Directionally Lipschitzian Functions and Subdifferential Calculus , 1979 .

[13]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[14]  Justin K. Romberg,et al.  Convex cardinal shape composition and object recognition in computer vision , 2015, 2015 49th Asilomar Conference on Signals, Systems and Computers.