Shape representation and indexing based on region connection calculus and oriented matroid theory

International Conference on Discrete Geometry for Computer Imagery (DGCI), 2003, Naples (Italy)In this paper a novel method for indexing views of 3D objects is presented. The topological properties of the regions of the views of a set of objects are used to define an index based on the region connection calculus and oriented matroid theory. Both are formalisms for qualitative spatial representation and reasoning and are complementary in the sense that whereas the region connection calculus encodes information about connectivity of pairs of connected regions of the view, oriented matroids encode relative position of the disjoint regions of the view and give local and global topological information about their spatial distribution. This indexing technique is applied to 3D object hypothesis generation from single views to reduce candidates in object recognition processes.Peer Reviewe

Shape representation and indexing based on region connection calculus and oriented matroid theory

International Conference on Discrete Geometry for Computer Imagery (DGCI), 2003, Naples (Italy)In this paper a novel method for indexing views of 3D objects is presented. The topological properties of the regions of the views of a set of objects are used to define an index based on the region connection calculus and oriented matroid theory. Both are formalisms for qualitative spatial representation and reasoning and are complementary in the sense that whereas the region connection calculus encodes information about connectivity of pairs of connected regions of the view, oriented matroids encode relative position of the disjoint regions of the view and give local and global topological information about their spatial distribution. This indexing technique is applied to 3D object hypothesis generation from single views to reduce candidates in object recognition processes.Peer Reviewe

A diszkrét tomográfia új irányzatai és alkalmazása a neutron radiográfiában = New directions in discrete tomography and its application in neutron radiography

A projekt során alapvetően a diszkrét tomográfia alábbi területein végeztük eredményes kutatásokat: rekonstrukcó legyezőnyaláb-vetületekből; geometriai tulajdonságokon alapuló rekonsrukciós és egyértelműségi eredmények kiterjeszthetőségének vizsgálata; újfajta geometriai jellemzők bevezetése; egzisztenica, unicitás és rekonstrukció vizsgálata abszorpciós vetületek esetén; 2D és 3D rekonstrukciós algoritmusok fejlesztése neutron tomográfiás alkalmazásokhoz; bináris rekonstrukciós algoritmusok tesztelése, benchmark halmazok és kiértékelések; a rekonstruálandó kép geometriai és egyéb strukturális információinak kinyerése közvetlenül a vetületekből. A kidolgozott eljárásaink egy részét az általunk fejlesztett DIRECT elnevezésű diszkrét tomográfiai keretrendszerben implementáltuk, így lehetőség nyílt az ismertetett eljárások tesztelésére és a különböző megközelítések hatékonyságának összevetésére is. Kutatási eredményeinket több, mint 40 nemzetközi tudományos közleményben jelentettük meg, a projekt futamideje alatt két résztvevő kutató is doktori fokozatot szerzett a kutatási témából. A projekt során több olyan kutatási irányvonalat fedtünk fel, ahol elképzeléseink szerint további jelentős elméleti eredményeket lehet elérni, és ezzel egyidőben a gyakorlat számára is új jellegű és hatékonyabb diszkrét képalkotó eljárások tervezhetők és kivitelezhetők. | In the project entitled ""New Directions in Discrete Tomography and Its Applications in Neutron Radiography"" we did successful research mainly on the following topics on Discrete Tomography (DT): reconstruction from fan-beam projections; extension of uniqueness and reconstruction results of DT based on geometrical priors, introduction of new geometrical properties to facilitate the reconstruction; uniqueness and reconstruction in case of absorbed projections; 2D and 3D reconstruction algorithms for applications in neutron tomography; testing binary reconstruction algorithms, developing benchmark sets and evaluations; exploiting structural features of images from their projections. As a part of the project we implemented some of our reconstruction methods in the DIRECT framework (also developed at our department), thus making it possible to test and compare our algorithms. We published more than 40 articles in international conference proceedings and journals. Two of our project members obtained PhD degree during the period of the project (mostly based on their contributions to the work). We also discovered several research areas where further work can yield important theoretical results as well as more effective discrete reconstruction methods for the applications

Topological characterization of simple points by complex collapsibility

International audienceThinning is an image operation whose goal is to reduce object points in a "topology-preserving" way. Such points whose removal does not change the topology are called simple points and they play an important role in any thinning process. For efficient computation, local characterizations have been already studied based on the concept of point connectivity for two-and three-dimensional digital images. In this paper, we introduce a new topological characterization of simple points based on collapsibility of polyhedral complexes. We also study their topological characteristics and propose a linear thinning algorithm

Digital pseudomanifolds, digital weakmanifolds and Jordan–Brouwer separation theorem

AbstractIn this paper we introduce the new notion of n-pseudomanifold and n-weakmanifold in an (n+1)-digital image using (2(n+1),3(n+1)−1)-adjacency. For these classes, we prove the digital version of the Jordan–Brouwer separation theorem. To accomplish this objective, we construct a polyhedral representation of the (n+1)-digital image based on a cubical complex decomposition which enables us to translate some results from polyhedral topology into the digital space. Our main result extends the class of “thin” objects that are defined locally and verifying the Jordan–Brouwer separation theorem

Coding cells of digital spaces: a framework to write generic digital topology algorithms

This paper proposes a concise coding of the cells of n-dimensional finite regular grids. It induces a simple, generic and efficient framework for implementing classical digital topology data structures and algorithms. Discrete subsets of multidimensional images (e.g. regions, digital surfaces, cubical cell complexes) have then a common and compact representation. Moreover, algorithms have a straightforward and efficient implementation, which is independent from the dimension or sizes of digital images. We illustrate that point with generic hypersurface boundary extraction algorithms by scanning or tracking. This framework has been implemented and basic operations as well as the presented applications have been benchmarked

Combinatorial aspects of Escher tilings

International audienceIn the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell.Lorsque Maurits Cornelis Escher commença à la fin des années 30 à produire des pavages du plan avec des tuiles, il étonna le monde artistique par la singularité de ses dessins. En particulier, les pavages du plan obtenus avec des copies d'une seule tuile apparaissent souvent dans son œuvre et ont attiré peu à peu l'attention de la communauté mathématique. Puisqu'une tuile dans le monde continu peut être approximée par un chemin sur un réseau carré suffisamment fin - une méthode universellement utilisée dans les applications utilisant des écrans graphiques - l'objet combinatoire qui modèle adéquatement la tuile est le polyomino. Comme ceux-ci sont naturellement codés par des chemins sur un alphabet de quatre lettres, l'utilisation de la combinatoire des mots devient pertinente pour l'étude des propriétés des tuiles pavantes. Nous présentons dans ce papier plusieurs résultats, allant de la reconnaissance de ces tuiles à leur génération, conduisant à des liens surprenants avec les célèbres suites de Fibonacci et de Pell