A Factorization Approach to Grouping a Factorization Approach to Grouping

The foreground group in a scene may be `discovered' and computed as a factorized approximation to the pairwise a nity of the elements in the scene. A pointwise approximation of the pairwise a nity information may in fact be interpreted as a `saliency' index, and the foreground of the scene may be obtained by thresholding it. An algorithm called `a nity factorization' is thus obtained which may be used for grouping. The a nity factorization algorithm is demonstrated on displays composed of points, of lines and of brightness values. Its relationship to the Shi-Malik normalized cuts algorithms is explored both analytically and experimentally. The a nity factorization algorithm is shown to be computationally e cient (O(n) oating-point operations for a scene composed of n elements) and to perform well on displays where the background is unstructured. Generalizations to solve more complex problems are also discussed. Appeared in Proceedings, European Conference on Computer Vision, 1998 This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonpro t educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Information Technology Center America; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Information Technology Center America. All rights reserved. Copyright c Mitsubishi Electric Information Technology Center America, 1999 201 Broadway, Cambridge, Massachusetts 02139 A factorization approach to grouping P. Perona and W. Freeman 1 California Institute of Technology MS 136-93, Pasadena CA 91125, USA 2 Universit a di Padova, Italy 3 Mitsubishi Electric Research Lab. , 201 Broadway, Cambridge MA 02139 Abstract. The foreground group in a scene may be `discovered' and computed as a factorized approximation to the pairwise a nity of the elements in the scene. A pointwise approximation of the pairwise a nity information may in fact be interpreted as a `saliency' index, and the foreground of the scene may be obtained by thresholding it. An algorithm called `a nity factorization' is thus obtained which may be used for grouping. The a nity factorization algorithm is demonstrated on displays composed of points, of lines and of brightness values. Its relationship to the Shi-Malik normalized cuts algorithms is explored both analytically and experimentally. The a nity factorization algorithm is shown to be computationally e cient (O(n) oating-point operations for a scene composed of n elements) and to perform well on displays where the background is unstructured. Generalizations to solve more complex problems are also discussed. The foreground group in a scene may be `discovered' and computed as a factorized approximation to the pairwise a nity of the elements in the scene. A pointwise approximation of the pairwise a nity information may in fact be interpreted as a `saliency' index, and the foreground of the scene may be obtained by thresholding it. An algorithm called `a nity factorization' is thus obtained which may be used for grouping. The a nity factorization algorithm is demonstrated on displays composed of points, of lines and of brightness values. Its relationship to the Shi-Malik normalized cuts algorithms is explored both analytically and experimentally. The a nity factorization algorithm is shown to be computationally e cient (O(n) oating-point operations for a scene composed of n elements) and to perform well on displays where the background is unstructured. Generalizations to solve more complex problems are also discussed.