Anisotropic Diffusion Partial Differential Equations in Multi-Channel Image Processing : Framework and Applications

We review recent methods based on diffusion PDE's (Partial Differential Equations) for the purpose of multi-channel image regularization. Such methods have the ability to smooth multi-channel images anisotropically and can preserve then image contours while removing noise or other undesired local artifacts. We point out the pros and cons of the existing equations, providing at each time a local geometric interpretation of the corresponding processes. We focus then on an alternate and generic tensor-driven formulation, able to regularize images while specifically taking the curvatures of local image structures into account. This particular diffusion PDE variant is actually well suited for the preservation of thin structures and gives regularization results where important image features can be particularly well preserved compared to its competitors. A direct link between this curvature-preserving equation and a continuous formulation of the Line Integral Convolution technique (Cabral and Leedom, 1993) is demonstrated. It allows the design of a very fast and stable numerical scheme which implements the multi-valued regularization method by successive integrations of the pixel values along curved integral lines. Besides, the proposed implementation, based on a fourth-order Runge Kutta numerical integration, can be applied with a subpixel accuracy and preserves then thin image structures much better than classical finite-differences discretizations, usually chosen to implement PDE-based diffusions. We finally illustrate the efficiency of this diffusion PDE's for multi-channel image regularization - in terms of speed and visual quality - with various applications and results on color images, including image denoising, inpainting and edge-preserving interpolation

Cortical Functional architectures as contact and sub-Riemannian geometry

In a joint paper, Jean Petitot together with the authors of the present paper described the functional geometry of the visual cortex as the symplectization of a contact form to describe the family of cells sensitive to position, orientation and scale. In the present paper, as a "homage" to the enormous contribution of Jean Petitot to neurogeometry, we will extend this approach to more complex functional architectures built as a sequence of contactization or a symplectization process, able to extend the dimension of the space. We will also outline a few examples where a sub-Riemannian lifting is needed

An energy-stable time-integrator for phase-field models

We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework

Stabilized pressure segregation methods and their application to fluid-stucture interaction problems

In this work we design and analyze pressure segregation methods in order to approximate the Navier-Stokes equations. Pressure correction methods are widely used because they allow the decoupling of velocity and pressure computation, decreasing the computational cost. We have analyzed some of these schemes, obtaining inherent pressure stability. However, for second order accurate methods (in time) this inherent stability is too weak, requiring the introduction of a stabilized finite element methodology for the space dis- cretization. Moreover, we have carried out a complete convergence analysis of a first order pressure segregation method. We have used a stabilization technique justified from a multiscale approach that allows the use of equal velocity-pressure interpolation spaces and convection dominated °ows. A new kind of methods has been motivated from an alternative version of the mono- lithic fluid solver where the continuity equation is replaced by a discrete pressure Poisson equation. These methods belong to the family of velocity correction schemes, where it is the velocity instead of the pressure the extrapolated unknown. Some stability bounds have been proved, revealing that their inherent pressure stability is too weak. Further, predictor corrector schemes easily arise from the new monolithic system. Numerical ex- perimentation shows the good behavior of these methods. We have introduced the ALE framework in order for the fluid governing equations to be formulated on moving domains. Taking as the model equation the convection-di®usion equation, we have analyzed the blend of the ALE framework and a stabilized finite element method. We suggest a coupling procedure for the fluid-structure problem taking benefit from the ingredients previously introduced: pressure segregation methods, a stabilized finite element formulation and the ALE framework. The final algorithm, using one loop, tends to the monolithic (fluid-structure) system. This method has been applied to the simulation of bridge aerodynamics, obtaining a good convergence behavior. We end with the simulation of wind turbines. The fact that we have a rotary body surrounded by the fluid (air) has motivated the introduction of a remeshing strategy. We consider a selective remeshing procedure that only affects a tiny portion of the domain, with little impact on the overall CPU time

Diskrete Spin-Geometrie für Flächen

This thesis proposes a discrete framework for spin geometry of surfaces. Specifically, we discretize the basic notions in spin geometry, such as the spin structure, spin connection and Dirac operator. In this framework, two types of Dirac operators are closely related as in smooth case. Moreover, they both induce the discrete conformal immersion with prescribed mean curvature half-density.In dieser Arbeit wird ein diskreter Zugang zur Spin-Geometrie vorgestellt. Insbesondere diskretisieren wir die grundlegende Begriffe, wie zum Beispiel die Spin-Struktur, den Spin-Zusammenhang und den Dirac Operator. In diesem Rahmen sind zwei Varianten für den Dirac Operator eng verwandt wie in der glatten Theorie. Darüber hinaus induzieren beide die diskret-konforme Immersion mit vorgeschriebener Halbdichte der mittleren Krümmung

Stability and inference in discrete diffusion scale-spaces

Taking averages of observations is the most basic method to make inferences in the presence of uncertainty. In late 1980's, this simple idea has been extended to the principle of successively average less where the change is faster, and applied to the problem of revealing a signal with jump discontinuities in additive noise. Successive averaging results in a family of signals with progressively decreasing amount of details, which is called the scale-space and further conveniently formalized by viewing it as a solution to a certain diffusion-inspired evolutionary partial differential equation (PDE). Such a model is known as the diffusion scale-space and it possesses two long-standing problems: (i) model analysis which aims at establishing stability and guarantees that averaging does not distort important information, and (ii) model selection, such as identification of the optimal scale (diffusion stopping time) given an initial noisy signal and an incomplete model. This thesis studies both problems in the discrete space and time. Such a setting has been strongly advocated by Lindeberg [1991] and Weickert [1996] among others. The focus of the model analysis part is on necessary and sufficient conditions which guarantee that a discrete diffusion possesses the scale-space property in the sense of sign variation diminishing. Connections with the total variation diminishing and the open problem in a multivariate case are discussed too. Considering the model selection, the thesis unifies two optimal diffusion stopping principles: (i) the time when the Shannon entropy-based Liapunov function of Sporring and Weickert [1999] reaches its steady state, and (ii) the time when the diffusion outcome has the least correlation with the noise estimate, contributed by Mrázek and Navara [2003]. Both ideas are shown to be particular cases of the marginal likelihood inference. Moreover, the suggested formalism provides first principles behind such criteria, and removes a variety of inconsistencies. It is suggested that the outcome of the diffusion should be interpreted as a certain expectation conditioned on the initial signal of observations instead of being treated as a random sample or probabilities. This removes the need to normalize signals in the approach of Sporring and Weickert [1999], and it also better justifies application of the correlation criterion of Mrázek and Navara [2003]. Throughout this work, the emphasis is given on methods that enable to reduce the problem to that of establishing the positivity of a quadratic form. The necessary and sufficient conditions can then be approached via positivity of matrix minors. A supplementary appendix is provided which summarizes a novel method of evaluating matrix minors. Intuitive examples of difficulties with statistical inference conclude the thesis.reviewe