Fundamentals linking discrete and continuous
approaches to computer vision - A topological view

Date: Monday June 18, 2007
Time: 1:30 - 5:30p
Location: Greenway Room D, Hyatt Regency Hotel
Presenter: Leo Grady
In association with CVPR 2007


Movie - We show how the methods developed here can be used to translate the Mumford-Shah functional to a graph, which can be minimized to generate a clustering algorithm

Description:
Continuous and combinatorial methods of image analysis have developed in computer vision largely along separate lines of research. The primary difference between a combinatorial and continuous algorithm for computer vision is whether or not an image is treated as a large set of regular samples approximating a continuum domain or as a set of discrete objects. Although the former approach leads to mathematical exposition and analysis in terms of partial differential equations and the latter to graph theoretic methods, both toolsets for analysis may be derived from the common language of topology. In personal experience with members of the computer vision field, there appears to be a misconception that combinatorial and continuum mechanics are wholly separate from each other and that there can therefore be no meaningful cross-pollination of ideas between continuum and combinatorial algorithms. The primary goal of this short course will be to clarify this confusion and rebuild a common framework from primary principles that accommodates both approaches. The structure of this short course will be to start from the standpoint of topology (specifically algebraic topology) and show that the mechanisms underlying the structure of continuous and combinatorial PDEs is entirely analogous. In addition, the differing perspectives often give many new interpretations of older concepts. The goal is to give attendees working with disparate mathematical tools the ability to translate continuum methods to a combinatorial formulation and vice versa in order to facilitate idea exchange. The latter part of the course will take practical examples from the computer vision literature and show how they fit into this common setting. A running theme of the short course is that neither continuum nor combinatorial methods has any inherit primacy in computer vision (or physics) and both mathematical traditions are rich enough to support most concepts. Therefore, more justification should be given to why a new idea is better expressed in a continuum or combinatorial language. Most of the examples given throughout the short course will be in the area of image segmentation, since this is the presenter's particular area of focus.

Outline:
The course consists of three sections: Theory, Example and Practice.
1) Introduction
    Course goals are outlined and a history is provided of the mathematical treatment of space.
2) Theory
    We begin by revisiting integrals and continuous PDEs, adopting a particular topological viewpoint of the underlying mechanisms.  We then review the theory of combinatorial PDEs (i.e., cochains defined on chains) and show that the underlying structures in continuous and combinatorial methods have the same behavior and properties.  The conclusion is that in order for combinatorial PDEs to behave like their continuous counterparts, care must be taken to identify continuous quantities with combinatorial cells of appropriate dimension.  The theory section is divided into the following subsections
    a) Integrals
    b) Classification of physical laws
    c) Combinatorial spaces
    d) Combinatorial functions
    e) Conclusions
3) Example
    This section is devoted to detailing the depth of connection in structure of continuous and combinatorial theories.  We explore several, disparate ideas in continuous PDE theory and show with concrete examples how the behavioral properties are preserved exactly with the corresponding combinatorial analogues. Additionally, we give examples for which the combinatorial analogue provides new interpretations of traditionally continuous concepts.   The example section is divided into the following subsections
    a) Helmholtz decomposition
    b) Stokes Theorem
    c) Methods of solution
    d) Diffusion equation
    e) The Laplacian
4) Practice
    We now show that there is a practical benefit to understanding the above theory in the applications of computer vision and pattern recognition.  We begin by interpreting older algorithms from the viewpoint given above, drawing a common theme that image content is employed as material properties of the domain.  We then apply the knowledge from the previous two sections in developing the Random Walker segmentation method and in showing how to generalize the shortest path problem to 3D (and consequently, intelligent scissors/live wire).  Finally, we take the viewpoint of "translating" a continuous PDE to a combinatorial PDE and show how to solve the classical Mumford-Shah PDE on a general graph.  The practice sections is divided into the following subsections
    a) Images as constitutive
    b) Random walker image segmentation
    c) Generalized shortest paths
    d) Combinatorial Mumford-Shah
5) Conclusion