12 research outputs found
Mathematical and Data-driven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining
Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this thesis, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns' properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we present suitable mathematical representations using techniques ranging from group theory to deep neural networks.Ph.D
Book of short Abstracts of the 11th International Symposium on Digital Earth
The Booklet is a collection of accepted short abstracts of the ISDE11 Symposium
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Mathematical Approaches for Image Enhancement Problems
This thesis develops novel techniques that can solve some image enhancement problems using theoretically and technically proven and very useful mathematical tools to image processing such as wavelet transforms, partial differential equations, and variational models. Three subtopics are mainly covered. First, color image denoising framework is introduced to achieve high quality denoising results by considering correlations between color components while existing denoising approaches can be plugged in flexibly. Second, a new and efficient framework for image contrast and color enhancement in the compressed wavelet domain is proposed. The proposed approach is capable of enhancing both global and local contrast and brightness as well as preserving color consistency. The framework does not require inverse transform for image enhancement since linear scale factors are directly applied to both scaling and wavelet coefficients in the compressed domain, which results in high computational efficiency. Also contaminated noise in the image can be efficiently reduced by introducing wavelet shrinkage terms adaptively in different scales. The proposed method is able to enhance a wavelet-coded image computationally efficiently with high image quality and less noise or other artifact. The experimental results show that the proposed method produces encouraging results both visually and numerically compared to some existing approaches. Finally, image inpainting problem is discussed. Literature review, psychological analysis, and challenges on image inpainting problem and related topics are described. An inpainting algorithm using energy minimization and texture mapping is proposed. Mumford-Shah energy minimization model detects and preserves edges in the inpainting domain by detecting both the main structure and the detailed edges. This approach utilizes faster hierarchical level set method and guarantees convergence independent of initial conditions. The estimated segmentation results in the inpainting domain are stored in segmentation map, which is referred by a texture mapping algorithm for filling textured regions. We also propose an inpainting algorithm using wavelet transform that can expect better global structure estimation of the unknown region in addition to shape and texture properties since wavelet transforms have been used for various image analysis problems due to its nice multi-resolution properties and decoupling characteristics
Estimation du flux optique en présence d'occultations par une approche TAC
Nous proposons une résolution du problème de flux optique en présence d'occultations par une approche multi-résolution basée sur la topologie algébrique calculatoire (TAC). Plusieurs méthodes d'estimation du flux optique sont basées sur les contraintes d'intensité et de continuité spatiale du champ de vitesse et conduisent à la résolution d'une équation aux dérivées partielles (EDP).Nous proposons de résoudre le problème de lissage indésirable des contours d'occultation, produit par la contrainte de continuité spatiale, en exploitant le principe de diffusion non-linéaire basé sur le gradient multispectral du flux optique. En effet, le calcul du flux optique peut être interprété par le phénomène de réaction- diffusion du flux optique et le lissage des contours d'occultation est la conséquence du fait que la diffusion se fait d'une manière linéaire dans toute l'image. Une manière d'éviter ce problème est la modification de la conductivité de la diffusion à chaque pixel selon son appartenance au voisinage d'un contour d'occultation. Cela requiert une mesure de détection précise des contours d'occultation afin de les préserver.Nous montrons que le gradient multispectral du flux optique est une mesure qui convient.Nous utilisons l'alternative aux EDPs que représente l'approche TAC qui exploite les lois globales de la diffusion et les principes d'algèbre topologique afin d'offrir plus de robustesse et de précision dans les calculs.Nous utilisons également le modèle TAC multi-résolution pour résoudre le problème de validité de la contrainte d'intensité limitée à de petits décalages.Nous appliquons ensuite notre algorithme au recalage d'images par le flux optique