Irregular graph pyramids and representative cocycles of cohomology generators

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns ‘quantities’ to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. Extension to nD and application in the context of pattern recognition are discussed

Algorithm to Compute a Minimal Length Basis of Representative Cocycles of Cohomology Generators

An algorithm to compute a minimal length basis of representative cocycles of cohomology generators for 2D images is proposed. We based the computations on combinatorial pyramids foreseeing its future extension to 3D objects. In our research we are looking for a more refined topological description of deformable 2D and 3D shapes, than they are the often used Betti numbers. We define contractions on the object edges toward the inner of the object until the boundaries touch each other, building an irregular pyramid with this purpose. We show the possible use of the algorithm seeking the minimal cocycles that connect the convex deficiencies on a human silhouette. We used minimality in the number of cocycle edges in the basis, which is a robust description to rotations and noise

Invariant Representative Cocycles of Cohomology Generators using Irregular Graph Pyramids

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns `quantities' to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given.Comment: Special issue on Graph-Based Representations in Computer Visio

Incremental-Decremental Technique for Delineating Tunnels and Pockets in 3D Digital Images

In this paper, we combine two complementary techniques for computing homol- ogy: Incremental Algorithm for computing AT-models (which consist of an algebraic set of data that provide, in particular, homological information of the given object) is suitable for homology computation in cases in which new cells are added to the existing complex, whereas Decremental Algorithm for computing AT-models is more appropriated in the case that some cells are removed from the complex. Using these algorithms, we are able to de- scribe tunnels and pockets of a 3D digital image (given as a sequence of 2D digital images) in terms of sets of equivalent 1-cycles.Junta de Andalucía FQM-296Junta de Andalucía TIC-02268Ministerio de Educación y Ciencia MTM2006-0372

Nanoinformatics

Machine learning; Big data; Atomic resolution characterization; First-principles calculations; Nanomaterials synthesi

Nanoinformatics

Machine learning; Big data; Atomic resolution characterization; First-principles calculations; Nanomaterials synthesi

Twenty-Second Annual Catalogue of John B. Stetson University DeLand, Florida

1906-1907 (22nd annual) John B. Stetson University Catalogue

Twenty-Third Annual Catalogue of John B. Stetson University DeLand, Florida

1907-1908 (23rd annual) John B. Stetson University Catalogue

Architecture and Development

Ayala Levin charts the settler colonial imagination and practices that undergirded Israeli architectural development aid in Africa

Architecture and Development

Ayala Levin charts the settler colonial imagination and practices that undergirded Israeli architectural development aid in Africa