An overview of the topological gradient approach in image processing: advantages and inconveniences.
Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting
This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in . A family of fully discrete approximation...
Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting
This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞. A family of fully...
Analysis of morphology-like operators used in color image contrast enhancement.
Analysis of total variation flow and its finite element approximations
We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter , and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our...
Analysis of total variation flow and its finite element approximations
We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since...
Application of the partitioning method to specific Toeplitz matrices
We propose an adaptation of the partitioning method for determination of the Moore-Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant...
Application of the random field theory in PET imaging - injection dose optimization
This work presents new application of the random field theory in medical imaging. Results from both integral geometry and random field theory can be used to detect locations with significantly increased radiotracer uptake in images from positron emission tomography (PET). The assumptions needed to use these results are verified on a set of real and simulated phantom images. The proposed method of detecting activation (locations with increased radiotracer concentration) is used to quantify the quality...
Applications of nonlinear diffusion in image processing and computer vision.
Applied mathematics meets signal processing.
Bivariate hahn moments for image reconstruction
Border figure detection using a phase oscillator network with dynamical coupling.
Color-texture-based image retrieval system using Gaussian Markov random field model.
Compact simple Lie groups and their -, -, and -transforms.
Connections between object classification criteria using an ultrasonic bi-sonar system
The paper presents connections between the criteria which make three types of objects possible to be recognized, namely, edges, planes and corners. These criteria can be applied while a binaural sonar system is used. It is shown that the criteria are specific forms of a general equation. The form of the equation depends on a single coefficient. In the paper, the meaning of this coefficient is discussed. The constructions of the arrangement of objects are presented and are bound with values of the...
Continuous limits of discrete perimeters
We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on submodular interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge...
Cross-task code reuse in genetic programming applied to visual learning
We propose a method that enables effective code reuse between evolutionary runs that solve a set of related visual learning tasks. We start with introducing a visual learning approach that uses genetic programming individuals to recognize objects. The process of recognition is generative, i.e., requires the learner to restore the shape of the processed object. This method is extended with a code reuse mechanism by introducing a crossbreeding operator that allows importing the genetic material from...
Curvature in image and shape processing
The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian...
Curve cuspless reconstruction via sub-riemannian geometry
We consider the problem of minimizing ∫ 0 ℓ ξ 2 + K 2 ( s ) d s for a planar curve having fixed initial and final positions and directions. The total lengthℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there...
Deconvolution of defocused image with multivariate local polynomial regression and iterative Wiener filtering in DWT domain.