Pedro Real

Region-based segmentation of 2D and 3D images with tissue-like P systems

Highlights? Membrane Computing is inspired by the structure and functioning of cells. ? We take a non-synchronous, distributed and parallel model. ? We present a membrane solution to segment 2D and 3D digital images. ? Our solution is logarithmic with respect to the input data. Membrane Computing is a biologically inspired computational model. Its devices are called P systems and they perform computations by applying a finite set of rules in a synchronous, maximally parallel way. In this paper, we develop a variant of P-system, called tissue-like P system in order to design in this computational setting, a region-based segmentation algorithm of 2D pixel-based and 3D voxel-based digital images. Concretely, we use 4-adjacency neighborhood relation between pixels in 2D and 6-adjacency neighborhood relation between voxel in 3D for segmenting digital images in a constant number of steps. Finally, specific software is used to check the validity of these systems with some simple examples.

Integral Operators for Computing Homology Generators at Any Dimension

Starting from an nDgeometrical object, a cellular subdivision of such an object provides an algebraic counterpart from which homology information can be computed. In this paper, we develop a process to drastically reduce the amount of data that represent the original object, with the purpose of a subsequent homology computation. The technique applied is based on the construction of a sequence of elementary chain homotopies (integral operators) which algebraically connect the initial object with a simplified one with the same homological information than the former.

Towards Digital Cohomology

We propose a method for computing the Z 2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particular 14–neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z 2–cohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed.

P systems and computational algebraic topology

Membrane Computing is a paradigm inspired from biological cellular communication. Membrane computing devices are called P systems. In this paper we calculate some algebraic-topological information of 2D and 3D images in a general and parallel manner using P systems. First, we present a new way to obtain the homology groups of 2D digital images in time logarithmic with respect to the input data involving an improvement with respect to the algorithms development by S. Peltier et al. Second, we obtain an edge-segmentation of 2D and 3D digital images in constant time with respect to the input data.

The homological reduction method for computing cocyclic Hadamard matrices

An alternate method for constructing (Hadamard) cocyclic matrices over a finite group G is described. Provided that a homological model for G is known, the homological reduction method automatically generates a full basis for 2-cocycles over G (including 2-coboundaries). From these data, either an exhaustive or a heuristic search for Hadamard cocyclic matrices is then developed. The knowledge of an explicit basis for 2-cocycles which includes 2-coboundaries is a key point for the designing of the heuristic search. It is worth noting that some Hadamard cocyclic matrices have been obtained over groups G for which the exhaustive searching techniques are not feasible. From the computational-cost point of view, even in the case that the calculation of such a homological model is also included, comparison with other methods in the literature shows that the homological reduction method drastically reduces the required computing time of the operations involved, so that even exhaustive searches succeeded at orders for which previous calculations could not be completed. With aid of an implementation of the method in MATHEMATICA, some examples are discussed, including the case of very well-known groups (finite abelian groups, dihedral groups) for clarity.

A combinatorial method for computing Steenrod squares

We present here a combinatorial method for computing Steenrod squares of a simplicial set X. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X. A generalization of this method to Steenrod reduced powers is sketched.

An algorithm computing homotopy groups

An algorithm computing homotopy groups of a reduced simplicial set with effective homology is described, using the Whitehead tower method.

Homological spanning forest framework for 2D image analysis

A 2D topology-based digital image processing framework is presented here. This framework consists of the computation of a flexible geometric graph-based structure, starting from a raster representation of a digital image I. This structure is called Homological Spanning Forest (HSF for short), and it is built on a cell complex associated to I. The HSF framework allows an efficient and accurate topological analysis of regions of interest (ROIs) by using a four-level architecture. By topological analysis, we mean not only the computation of Euler characteristic, genus or Betti numbers, but also advanced computational algebraic topological information derived from homological classification of cycles. An initial HSF representation can be modified to obtain a different one, in which ROIs are almost isolated and ready to be topologically analyzed. The HSF framework is susceptible of being parallelized and generalized to higher dimensions.

A bio-inspired software for segmenting digital images

Segmentation in computer vision refers to the process of partitioning a digital image into multiple segments (sets of pixels). It has several features which make it suitable for techniques inspired by nature. It can be parallelized, locally solved and the input data can be easily encoded by bio-inspired representations. In this paper, we present a new software for performing a segmentation of 2D digital images based on Membrane Computing techniques.

A genetic algorithm for cocyclic hadamard matrices

A genetic algorithm for finding cocyclic Hadamard matrices is described. Though we focus on the case of dihedral groups, the algorithm may be easily extended to cover any group. Some executions and examples are also included, with aid of Mathematica 4.0.