Madjid Allili

Cubical homology and the topological classification of 2D and 3D imagery

There are a number of tasks in low level vision and image processing that involve computing certain topological characteristics of objects in a given image including connectivity and the number of holes. We combine a new combinatorial topology method to compute the number of connected components and holes of objects in a given image with fast segmentation methods to extract the objects.

Generating cubical complexes from image data and computation of the Euler number

Abstract A number of tasks in image processing and computer vision require the computation of certain topological characteristics of objects in a given image. In this paper, we introduce a new method based on the notion of the algebraic topology complex to compute the Euler number of a given object. First, we attach a cubical complex to the object of interest, then we associate an algebraic structure on which a number of simplifying operations preserving the topology but not necessarily the geometric nature of the complex are possible. This is a unifying dimension independent approach. We show that the Euler number can be obtained directly from the cubical structure or one can perform a collapsing operation that allows to reduce the given image to a lower dimension structure with equivalent topological properties. This reduced structure can be used in a further process, in particular, for the computation of the Euler number.

Morse homology descriptor for shape characterization

We propose a new topological method for shape description that is suitable for any multi-dimensional data set that can be modelled as a manifold. The description is obtained for all pairs (M, f), where M is a closed smooth manifold and f a Morse function defined on M. More precisely, we characterize the topology of all pairs of lower level sets (M/sub y/, M/sub x/) of f, where M/sub a/ = f/sup -1/((-/spl infin/,a]), for all a /spl isin/ R. Classical Morse theory is used to establish a link between the topology of a pair of lower level sets of f and its critical points lying between the two levels.

AN ALGORITHMIC APPROACH TO THE CONSTRUCTION OF HOMOMORPHISMS INDUCED BY MAPS IN HOMOLOGY

This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map f into ag raph of a multi-valued representable map F. The multi-valued representable maps have well developed continuity properties and admit a nite coding that per- mits treating them by combinatorial methods. We provide the construction of the homomorphism F induced by F such that F = f. The presented construction does not require subsequent barycentric subdivisions and simpli- cial approximations of f. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.

Reducing complexes in multidimensional persistent homology theory

Formans discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. An initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.

A computational algebraic topology approach for optical flow

This paper proposes an alternative to partial differential equations (PDEs) for the solution of the optical flow problem. The problem is modeled using the heat transfer process. Instead of using PDEs, we propose to use the global equation of heat conservation. We use a computational algebraic topology-based image model which allows as to encode some underlying physical laws by linking a global value on a domain with values on its boundary. The numerical scheme is derived in a straightforward way from the problem modeled and provides a physical explanation of each solving step. Experimental results are presented.

Image model: new perspective for image processing and computer vision

We propose a new image model in which the image support and image quantities are modeled using algebraic topology concepts. The image support is viewed as a collection of chains encoding combination of pixels grouped by dimension and linking different dimensions with the boundary operators. Image quantities are encoded using the notion of cochain which associates values for pixels of given dimension that can be scalar, vector, or tensor depending on the problem that is considered. This allows obtaining algebraic equations directly from the physical laws. The coboundary and codual operators, which are generic operations on cochains allow to formulate the classical differential operators as applied for field functions and differential forms in both global and local forms. This image model makes the association between the image support and the image quantities explicit which results in several advantages: it allows the derivation of efficient algorithms that operate in any dimension and the unification of mathematics and physics to solve classical problems in image processing and computer vision. We show the effectiveness of this model by considering the isotropic diffusion.

Image matching using algebraic topology

In this paper, two new approaches for the topological feature matching problem are proposed. The first one consists of estimating a combinatorial map between block structures (pixels, windows) of given binary images which is then analyzed for topological correspondence using the concept of homology of maps. The second approach establishes a matching by using a similarity measure between two sets of boundary representations of the connected components extracted from two given binary images. The similarity measure is applied on all oriented boundary components of given features. A number of experiments are carried out on both synthetic and real images to validate the two approaches.

Morse connections graph for shape representation

We present an algorithm for constructing efficient topological shape descriptors of three dimensional objects. Given a smooth surface S and a Morse function f defined on S, our algorithm encodes the relationship among the critical points of the function f by means of a connection graph, called the Morse Connections Graph, whose nodes represent the critical points of f. Two nodes are related by an edge if a connection is established between them. This graph structure is extremely suitable for shape comparison and shape matching and inherits the invariant properties of the given Morse function f.

A Computational Algebraic Topology Model for the Deformation of Curves

A new method for the deformation of curves is presented. It is based upon a decomposition of the linear elasticity problem into basic physical laws. Unlike other methods which solve the partial differential equation arising from the physical laws by numerical techniques, we encode the basic laws using computational algebraic topology. Conservative laws use exact global values while constitutive allow to make wise assumptions using some knowledge about the problem and the domain. The deformations computed with our approach have a physical interpretation. Furthermore, our algorithm performs with either 2D or 3D problems. We finally present an application of the model in updating road databases and results validating our approach.