ALGEBRAIC MINING OF SOLID MODELS FROM IMAGES Francesco Furiani † , Claudio Paoluzzi ‡* and Alberto Paoluzzi † † Mathematics & Physics Dept, University “Roma Tre”, Rome, Italy ‡ Next S.p.A., Rome, Italy ABSTRACT We introduce a map algebra based on a cochain extension of the Lin- ear Algebraic Representation (LAR), used to efficiently represent and query geometric and physical information through sparse ma- trix 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 prop- erties, 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. Index Terms— Map algebra, image information mining, solid modelling, cochain complex, algebraic topology 1. INTRODUCTION Computational problems in science and technology must deal with increasingly complex geometric information and appli- cations. The complexity of geometric information stems from dramatic increase in size, diversity, and complexity of geo- metric data, including digital images, point clouds, bound- ary schemes, NURBs representations, finite element meshes, 3D medical images. This increasing complexity of geomet- ric information and applications, and the goals of unification, scalability, and support of massively parallel distributed com- puting, strongly push for rethinking the foundations of geo- metric and topological computing. In particular the emerging applications from space, nano & bio technology, and medi- cal 3D, require a novel convergence of shape synthesis and analysis methods from computer imaging, computer graph- ics, computer-aided geometric design, with meshing of com- putational domains and physical simulations. Objects and relations distributed in time and space are all cell complexes. Examples: digital images, finite element C.P. performed the work while on sabbatical from consulting at Thales Alenia Space S.p.A, Rome. The research of A.P and F.F. was supported by a POC grant 2012/13 by SOGEI, the ICT company of the Italian Ministry of Economy and Finance. meshes, B-reps of solids, assemblies, networks, and so on, are all cell complexes of various dimension [1]. Such decom- positions of space into k-cells (0 ≤ k ≤ d) may generate k-chain spaces, linear spaces constituted by any combination of k-cells, where (a) singletons of k-cells give a basis, i.e. a minimal set of generators, and (b) linear boundary operators compute the boundary chain of any given chain, by mean of a single SpMV (sparse matrix-vector) multiplication. Basically, k-cochains are discrete densities of quantities contained in the k-cells of a cell complex (such as a digi- tal image or a finite element mesh), k being the dimension. The LAR (Linear Algebraic Representation) scheme [1] is a simple, general and effective representation of (co)chain complexes [2], based on a CSR (Compressed Sparse Row) representation [3] for characteristic matrices of linear spaces of (co)chains. LAR supports all topological incidence struc- tures, is dimension-independent and not restricted to regular, i.e., dimensionally uniform complexes. It allows for fast va- lidity checks of the topology of geometric models, possibly generated from 3D scanner data or extracted from 3D images, using only elementary linear algebra, namely, sparse matrix- vector multiplication. Any query about incidence relations between chains or cochains of same or different dimensions are answered by a single SpMV product. 2. LINEAR ALGEBRAIC REPRESENTATION (LAR) 2.1. Cellular Complexes All geometrical objects considered in this paper are chains in a cellular complex Λ(X) partitioning a (topological) space X ⊂ E d . Informally, a cellular complex is made of basic building blocks called cells, suitably glued together [4]. More formally, a cellular complex is a Hausdorff space X, i.e. a topological space in which distinct points have disjoint neighbourhoods, together with a partition Λ=Λ 0 ∪· · ·∪Λ d of X into open cells (of varying dimension) that satisfies some additional properties. Some definitions useful in the remain- der follow. A compact topological subspace is a convex cell if it is the set of solutions of affine equalities and inequalities. A face of a cell is the convex cell obtained by replacing some of the inequalities by equalities. A facet of a cell is a face defined