191 research outputs found

    Surface Denoising based on The Variation of Normals and Retinal Shape Analysis

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    Through the development of this thesis, starting from the curvature tensor, we have been able to understand the variation of tangent vectors to define a shape analysis operator and also a relationship between the classical shape operator and the curvature tensor on a triangular surface. In continuation, the first part of the thesis analyzed the variation of surface normals and introduced a shape analysis operator, which is further used for mesh and point set denoising. In the second part of the thesis, mathematical modeling and shape quantification algorithms are introduced for retinal shape analysis. In the first half, this thesis followed the concept of the variation of surface normals, which is termed as the normal voting tensor and derived a relation between the shape operator and the normal voting tensor. The concept of the directional and the mean curvatures is extended on the dual representation of a triangulated surface. A normal voting tensor is defined on each triangle of a geometry and termed as the element-based normal voting tensor (ENVT). Later, a deformation tensor is extracted from the ENVT and it consists of the anisotropy of a surface and the mean curvature vector is defined based on the ENVT deformation tensor. The ENVT-based mesh denoising algorithm is introduced, where the ENVT is used as a shape operator. A binary optimization technique is applied on the spectral components of the ENVT that helps the algorithm to retain sharp features in the concerned geometry and improves the convergence rate of the algorithm. Later, a stochastic analysis of the effect of noise on the triangular mesh based on the minimum edge length of the elements in the geometry is explained. It gives an upper bound to the noise standard deviation to have minimum probability for flipped element normals. The ENVT-based mesh denoising concept is extended for a point set denoising, where noisy vertex normals are filtered using the vertex-based NVT and the binary optimization. For vertex update stage in point set denoising, we added different constraints to the quadratic error metric based on features (edges and corners) or non-feature (planar) points. This thesis also investigated a robust statistics framework for face normal bilateral filtering and proposed a robust and high fidelity two-stage mesh denoising method using Tukey’s bi-weight function as a robust estimator, which stops the diffusion at sharp features and produces smooth umbilical regions. This algorithm introduced a novel vertex update scheme, which uses a differential coordinate-based Laplace operator along with an edge-face normal orthogonality constraint to produce a high-quality mesh without face normal flips and it also makes the algorithm more robust against high-intensity noise. The second half of thesis focused on the application of the proposed geometric processing algorithms on the OCT (optical coherence tomography) scan data for quantification of the human retinal shape. The retina is a part of the central nervous system and comprises a similar cellular composition as the brain. Therefore, many neurological disorders affect the retinal shape and these neuroinflammatory conditions are known to cause modifications to two important regions of the retina: the fovea and the optical nerve head (ONH). This thesis consists of an accurate and robust shape modeling of these regions to diagnose several neurological disorders by detecting the shape changes. For the fovea, a parametric modeling algorithm is introduced using Cubic Bezier and this algorithm derives several 3D shape parameters, which quantify the foveal shape with high accuracy. For the ONH, a 3D shape analysis algorithm is introduced to measure the shape variation regarding different neurological disorders. The proposed algorithm uses triangulated manifold surfaces of two different layers of the retina to derive several 3D shape parameters. The experimental results of the fovea and the ONH morphometry confirmed that these algorithms can provide an aid to diagnose several neurological disorders

    Surface Denoising based on Normal Filtering in a Robust Statistics Framework

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    During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as points-set or triangulated mesh). The noise-removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterwards. Therefore, in many available denoising algorithms, the computation of noise-free normals is a key factor. A variety of filters have been introduced for noise-removal from normals, with different focus points like robustness against outliers or large amplitude of noise. Although these filters are performing well in different aspects, a unified framework is missing to establish the relation between them and to provide a theoretical analysis beyond the performance of each method. In this paper, we introduce such a framework to establish relations between a number of widely-used nonlinear filters for face normals in mesh denoising and vertex normals in point set denoising. We cover robust statistical estimation with M-smoothers and their application to linear and non-linear normal filtering. Although these methods originate in different mathematical theories - which include diffusion-, bilateral-, and directional curvature-based algorithms - we demonstrate that all of them can be cast into a unified framework of robust statistics using robust error norms and their corresponding influence functions. This unification contributes to a better understanding of the individual methods and their relations with each other. Furthermore, the presented framework provides a platform for new techniques to combine the advantages of known filters and to compare them with available methods

    Courbure discrète : théorie et applications

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    International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

    Geodesic Active Fields:A Geometric Framework for Image Registration

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    Image registration is the concept of mapping homologous points in a pair of images. In other words, one is looking for an underlying deformation field that matches one image to a target image. The spectrum of applications of image registration is extremely large: It ranges from bio-medical imaging and computer vision, to remote sensing or geographic information systems, and even involves consumer electronics. Mathematically, image registration is an inverse problem that is ill-posed, which means that the exact solution might not exist or not be unique. In order to render the problem tractable, it is usual to write the problem as an energy minimization, and to introduce additional regularity constraints on the unknown data. In the case of image registration, one often minimizes an image mismatch energy, and adds an additive penalty on the deformation field regularity as smoothness prior. Here, we focus on the registration of the human cerebral cortex. Precise cortical registration is required, for example, in statistical group studies in functional MR imaging, or in the analysis of brain connectivity. In particular, we work with spherical inflations of the extracted hemispherical surface and associated features, such as cortical mean curvature. Spatial mapping between cortical surfaces can then be achieved by registering the respective spherical feature maps. Despite the simplified spherical geometry, inter-subject registration remains a challenging task, mainly due to the complexity and inter-subject variability of the involved brain structures. In this thesis, we therefore present a registration scheme, which takes the peculiarities of the spherical feature maps into particular consideration. First, we realize that we need an appropriate hierarchical representation, so as to coarsely align based on the important structures with greater inter-subject stability, before taking smaller and more variable details into account. Based on arguments from brain morphogenesis, we propose an anisotropic scale-space of mean-curvature maps, built around the Beltrami framework. Second, inspired by concepts from vision-related elements of psycho-physical Gestalt theory, we hypothesize that anisotropic Beltrami regularization better suits the requirements of image registration regularization, compared to traditional Gaussian filtering. Different objects in an image should be allowed to move separately, and regularization should be limited to within the individual Gestalts. We render the regularization feature-preserving by limiting diffusion across edges in the deformation field, which is in clear contrast to the indifferent linear smoothing. We do so by embedding the deformation field as a manifold in higher-dimensional space, and minimize the associated Beltrami energy which represents the hyperarea of this embedded manifold as measure of deformation field regularity. Further, instead of simply adding this regularity penalty to the image mismatch in lieu of the standard penalty, we propose to incorporate the local image mismatch as weighting function into the Beltrami energy. The image registration problem is thus reformulated as a weighted minimal surface problem. This approach has several appealing aspects, including (1) invariance to re-parametrization and ability to work with images defined on non-flat, Riemannian domains (e.g., curved surfaces, scalespaces), and (2) intrinsic modulation of the local regularization strength as a function of the local image mismatch and/or noise level. On a side note, we show that the proposed scheme can easily keep up with recent trends in image registration towards using diffeomorphic and inverse consistent deformation models. The proposed registration scheme, called Geodesic Active Fields (GAF), is non-linear and non-convex. Therefore we propose an efficient optimization scheme, based on splitting. Data-mismatch and deformation field regularity are optimized over two different deformation fields, which are constrained to be equal. The constraint is addressed using an augmented Lagrangian scheme, and the resulting optimization problem is solved efficiently using alternate minimization of simpler sub-problems. In particular, we show that the proposed method can easily compete with state-of-the-art registration methods, such as Demons. Finally, we provide an implementation of the fast GAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, so as to automatically delineate the corresponding regions on a subject brain given its feature map. In a leave-one-out cross-validation study on 39 brain surfaces with 35 manually delineated gyral regions, we show that the pairwise subject-atlas registration with the proposed spherical registration scheme significantly improves the individual alignment of cortical labels between subject and atlas brains, and, consequently, that the estimated automatic parcellations after label fusion are of better quality

    Large-scale Machine Learning in High-dimensional Datasets

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    New contributions in overcomplete image representations inspired from the functional architecture of the primary visual cortex = Nuevas contribuciones en representaciones sobrecompletas de imágenes inspiradas por la arquitectura funcional de la corteza visual primaria

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    The present thesis aims at investigating parallelisms between the functional architecture of primary visual areas and image processing methods. A first objective is to refine existing models of biological vision on the base of information theory statements and a second is to develop original solutions for image processing inspired from natural vision. The available data on visual systems contains physiological and psychophysical studies, Gestalt psychology and statistics on natural images The thesis is mostly centered in overcomplete representations (i.e. representations increasing the dimensionality of the data) for multiple reasons. First because they allow to overcome existing drawbacks of critically sampled transforms, second because biological vision models appear overcomplete and third because building efficient overcomplete representations raises challenging and actual mathematical problems, in particular the problem of sparse approximation. The thesis proposes first a self-invertible log-Gabor wavelet transformation inspired from the receptive field and multiresolution arrangement of the simple cells in the primary visual cortex (V1). This transform shows promising abilities for noise elimination. Second, interactions observed between V1 cells consisting in lateral inhibition and in facilitation between aligned cells are shown efficient for extracting edges of natural images. As a third point, the redundancy introduced by the overcompleteness is reduced by a dedicated sparse approximation algorithm which builds a sparse representation of the images based on their edge content. For an additional decorrelation of the image information and for improving the image compression performances, edges arranged along continuous contours are coded in a predictive manner through chains of coefficients. This offers then an efficient representation of contours. Fourth, a study on contour completion using the tensor voting framework based on Gestalt psychology is presented. There, the use of iterations and of the curvature information allow to improve the robustness and the perceptual quality of the existing method. La presente tesis doctoral tiene como objetivo indagar en algunos paralelismos entre la arquitectura funcional de las áreas visuales primarias y el tratamiento de imágenes. Un primer objetivo consiste en mejorar los modelos existentes de visión biológica basándose en la teoría de la información. Un segundo es el desarrollo de nuevos algoritmos de tratamiento de imágenes inspirados de la visión natural. Los datos disponibles sobre el sistema visual abarcan estudios fisiológicos y psicofísicos, psicología Gestalt y estadísticas de las imágenes naturales. La tesis se centra principalmente en las representaciones sobrecompletas (i.e. representaciones que incrementan la dimensionalidad de los datos) por las siguientes razones. Primero porque permiten sobrepasar importantes desventajas de las transformaciones ortogonales; segundo porque los modelos de visión biológica necesitan a menudo ser sobrecompletos y tercero porque construir representaciones sobrecompletas eficientes involucra problemas matemáticos relevantes y novedosos, en particular el problema de las aproximaciones dispersas. La tesis propone primero una transformación en ondículas log-Gabor auto-inversible inspirada del campo receptivo y la organización en multiresolución de las células simples del cortex visual primario (V1). Esta transformación ofrece resultados prometedores para la eliminación del ruido. En segundo lugar, las interacciones observadas entre las células de V1 que consisten en la inhibición lateral y en la facilitación entre células alineadas se han mostrado eficientes para extraer los bordes de las imágenes naturales. En tercer lugar, la redundancia introducida por la transformación sobrecompleta se reduce gracias a un algoritmo dedicado de aproximación dispersa el cual construye una representación dispersa de las imágenes sobre la base de sus bordes. Para una decorrelación adicional y para conseguir más altas tasas de compresión, los bordes alineados a lo largo de contornos continuos están codificado de manera predictiva por cadenas de coeficientes, lo que ofrece una representacion eficiente de los contornos. Finalmente se presenta un estudio sobre el cierre de contornos utilizando la metodología de tensor voting. Proponemos el uso de iteraciones y de la información de curvatura para mejorar la robustez y la calidad perceptual de los métodos existentes
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