Global Medical Shape Analysis Using the Laplace-Beltrami Spectrum

This paper proposes to use the Laplace-Beltrami spectrum (LBS) as a global shape descriptor for medical shape analysis, allowing for shape comparisons using minimal shape preprocessing: no registration, mapping, or remeshing is necessary. The discriminatory power of the method is tested on a population of female caudate shapes of normal control subjects and of subjects with schizotypal personality disorder

Modelling of realistic Blood Vessel Geometry

Projecte realitzat mitjançant programa de mobilitat. GOTTFRIED WILHELM LEIBNIZ UNIVERSITÄT HANNOVER. FAKULTÄT FÜR ELEKTROTECHNIK UND INFORMATIK. INSTITUT FÜR MENSCH-MASCHINE-KOMMUNIKATION FACHGEBIET GRAPHISCHE DATENVERARBEITUNGModel·lat realístic de vasos sanguinis en 3D, per ús en impresores 3D de teixits orgànics humans. A partir de certs paràmetres com el diàmetre i gruix dels dos extrems dels vasos, es crea una malla de triangles en 3D

Laplace–Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis

This paper proposes the use of the surface based Laplace-Beltrami and the volumetric Laplace eigenvalues and -functions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre-processing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated on a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the Laplace-Beltrami eigenvalues and -functions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the Morse-Smale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the Laplace-Beltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel

Laplace–Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis

This paper proposes the use of the surface based Laplace-Beltrami and the volumetric Laplace eigenvalues and -functions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre-processing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated on a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the Laplace-Beltrami eigenvalues and -functions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the Morse-Smale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the Laplace-Beltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel

Computational Topology Methods for Shape Modelling Applications

This thesis deals with computational topology, a recent branch of research that involves both mathematics and computer science, and tackles the problem of discretizing the Morse theory to functions defined on a triangle mesh. The application context of Morse theory in general, and Reeb graphs in particular, deals with the analysis of geometric shapes and the extraction of skeletal structures that synthetically represents shape, preserving the topological properties and the main morphological characteristics. Regarding Computer Graphics, shapes, that is a one-, two- or higher- dimensional connected, compact space having a visual appearance, are typically approximated by digital models. Since topology focuses on the qualitative properties of spaces, such as the connectedness and how many and what type of holes it has, topology is the best tool to describe the shape of a mathematical model at a high level of abstraction. Geometry, conversely, is mainly related to the quantitative characteristics of a shape. Thus, the combination of topology and geometry creates a new generation of tools that provide a computational description of the most representative features of the shape along with their relationship. Extracting qualitative information, that is the information related to semantic of the shape and its morphological structure, from discrete models is a central goal in shape modeling. In this thesis a conceptual model is proposed which represents a given surface based on topological coding that defines a sketch of the surface, discarding irrelevant details and classifying its topological type. The approach is based on Morse theory and Reeb graphs, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the geometry of the discrete shape model. To fully develop the method, both theoretical and computational aspects have been considered, related to the definition and the extension of the Reeb graph to the discrete domain. For the definition and automatic construction of the conceptual model, a new method has been developed that analyzes and characterizes a triangle mesh with respect to the behavior of a real and at least continuous function defined on the mesh. The proposed solution handles also degenerate critical points, such as non-isolated critical points. To do that, the surface model is characterized using a contour-based strategy, recognizing critical areas instead of critical points and coding the evolution of the contour levels in a graph-like structure, named Extended Reeb Graph, (ERG), which is a high-level abstract model suitable for representing and manipulating piece-wise linear surfaces. The descriptive power of the (ERG) has been also augmented with the introduction of geometric information together with the topological ones, and it has been also studied the relation between the extracted topological and morphological features with respect to the real characteristics of the surface, giving and evaluation of the dimension of the discarded details. Finally, the effectiveness of our description framework has been evaluated in several application contexts