Algebraic Mining of Solid Models from Images
Francesco Furiani, C Paoluzzi, Alberto Paoluzzi · 2014
We introduce a map algebra based on a cochain extension of the Linear Algebraic Representation (LAR), used to efficiently represent and query geometric and physical information through sparse matrix algebra. LAR, based on standard algebraic topology methods, supports all incidence structures, including enumerative (images), decompositive (meshes) and boundary (CAD) representations, is dimension-independent and not restricted to regular complexes. This algebraic representation enjoys a neat mathematical format— being based on chains, the domains of discrete integration, and cochains, the discrete prototype of differential forms, so naturally integrating the geometric shape with the supported physical properties, and provides a mechanism for strongly typed representation of all physical quantities associated with images. It is easy to show that k-cochains form a linear vector space over k-cells, which means that they can used as basic objects in a rich and virtually unlimited calculus of physical properties.