In this paper we explore how to quantify the complexity of discrete objects (images, meshes, networks) which can be encoded in terms of graphs, digraphs or hypergraphs. Herein we present our Heat Flow-Thermodynamic Depth approach which combines ingredients of spectral graph theory, information theory and information projection. We illustrate the approach with several applications: image exploration (image complexity), mesh regularization and selection of optimal map functions in Extended Reeb Graphs (graph and digraph complexity) and structural categorization (hypergraph complexity). 1 Randomness vs. Statistical Complexity Given a discrete mathematical object (image, mesh, network) which can be encoded by a graph, digraph or hypergraph, the quantification of its intrinsic complexity plays a key role in understanding the underpinning structural principles shaping it. Such principles include: the information content of the encoding, the set of constraints over the information flowing through it, and the combinatorial exploration of the hypothesis that best explain its genesis. Information content may be posed in terms of estimating a given entropy. Information flow constraints may be discovered by probing the structure through heat kernels and wave equations. Generative hypothesis may rely on depth-based representations like logical depth or thermodynamic depth, hierarchical representations like grammars and compositional models, or dynamic rules like preferential attachment. The weight of each of the latter principles in the characterization of the encoding defines a computable complexity measure for it. For instance, strategies considering information content through Shannon or cross entropy are referred to as randomness complexity measures for they quantify the degree of disorganization (see a recent survey on graph entropy measures in [1]) . On the other hand, methods relying on spectral measures, like the von Neumann entropy [2], or on Dirichlet sums, like Estrada’s heterogeneity index are [3] more focused on quantifying the degree of regularity of the structure. These are good examples of the so called statistical complexity measures that vanished both for completely regular and completely random structures. Thermodynamic depth (TD) is a physics based approach [4] also belonging to the latter category. When dealing with graphs, termodynamic depth aims to quantify how hard is to build a given graph (the macroscopic state) from scracth (microscopic states): if there

Discrete Geometry for Computer Imagery

Invited Paper.- Digital Circularity and Its Applications.- Digital Geometry - Curves, Straightness, and Convexity.- On the Convex Hull of the Integer Points in a Bi-circular Region.- Multi-primitive Analysis of Digital Curves.- What Does Digital Straightness Tell about Digital Convexity?.- Digital Geometry - Geometric Transformations, Metrics, Distance Transforms, and Skeletons.- Hierarchical Discrete Medial Axis for Sphere-Tree Construction.- Quasi-Affine Transformation in 3-D: Theory and Algorithms.- Farey Sequences and the Planar Euclidean Medial Axis Test Mask.- Neighborhood Sequences on nD Hexagonal/Face-Centered-Cubic Grids.- Neighborhood Sequences in the Diamond Grid - Algorithms with Four Neighbors.- Hinge Angles for 3D Discrete Rotations.- Segmentation, Thinning and Skeletonization.- Surface Thinning in 3D Cubical Complexes.- A General Bayesian Markov Random Field Model for Probabilistic Image Segmentation.- An Order-Independent Sequential Thinning Algorithm.- Blurred Segments in Gray Level Images for Interactive Line Extraction.- Multi-scale Analysis of Discrete Contours for Unsupervised Noise Detection.- Sub-pixel Segmentation with the Image Foresting Transform.- Image Representation, Processing, Analysis, Reconstruction and Recognition - Algorithms and Applications.- Phase-Correlation Guided Search for Realtime Stereo Vision.- PCIF: An Algorithm for Lossless True Color Image Compression.- Ego-Vehicle Corridors for Vision-Based Driver Assistance.- Adaptive Pixel Resizing for Multiscale Recognition and Reconstruction.- Contour Reconstruction for Multiple 2D Regions Based on Adaptive Boundary Samples.- Digital Tomography.- Reconstruction of Canonical hv-Convex Discrete Sets from Horizontal and Vertical Projections.- About the Complexity of Timetables and 3-Dimensional Discrete Tomography: A Short Proof of NP-Hardness.- Image Models Based on Geometry, Combinatorics, Arithmetics, Algebra, Mathematical Morphology, Topology and Grammars.- Theoretical Issues of Cluster Pattern Interfaces.- ?-Arithmetization: A Discrete Multi-resolution Representation of Real Functions.- Tree-Based Encoding for Cancellations on Morse Complexes.- Parallel Contextual Hexagonal Array Grammars and Languages.- Connected Viscous Filters.- Signatures of Combinatorial Maps.- Digital Topology and its Applications to Image Modeling and Analysis.- Using Membrane Computing for Obtaining Homology Groups of Binary 2D Digital Images.- Collapses and Watersheds in Pseudomanifolds.- The Inscribed Square Conjecture in the Digital Plane.- Convenient Closure Operators on .

Combinatorial image analysis : 13th International Workshop, IWCIA 2009, Playa del Carmen, Mexico, November 24-27, 2009 : proceedings

Le contexte est la geometrie discrete dans Zn. Il s'agit de decrire les courbes et surfaces discretes composees de voxels: les definitions usuelles de droites et plans discrets epais se comportent mal quand on passe a des ensembles courbes. Comment garantir un bon comportement topologique, les connexites requises, dans une situation qui generalise les droites et plans discrets?Le calcul de donnees sur ces courbes, normales, tangentes, courbure, ou des fonctions plus generales, fait appel a des moyennes utilisant des masques. Une question est la pertinence theorique et pratique de ces masques. Une voie exploree, est le calcul de masques fondes sur la marche aleatoire. Une marche aleatoire partant d'un centre donne sur une courbe ou une surface discrete, permet d'affecter a chaque autre voxel un poids, le temps moyen de visite. Ce noyau permet de calculer des moyennes et par la, des derivees. L'etude du comportement de ce processus de diffusion, a permis de retrouver des outils classiques de geometrie sur des surfaces maillees, et de fournir des estimateurs de tangente et de courbure performants. La diversite du champs d'applications de ce processus de diffusion a ete mise en avant, retrouvant ainsi des methodes classiques mais avec une base theorique identique.

Processus de Diffusion Discret Opérateur Laplacien appliqué à l'étude de surfaces

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Processus de Diffusion Discret Opérateur Laplacien appliqué à l'étude de surfaces