Variation totale d'une fonction

En mathématiques, la variation totale est liée à la structure (locale ou globale) du codomaine d'une fonction. Pour une fonction continue à valeurs réelles f, définie sur un intervalle [a, b] ⊂ ℝ, sa variation totale sur l'intervalle de définition est une mesure de la longueur d'arc de la projection sur l'axe des ordonnées de la courbe paramétrée (x, f(x)), pour x ∈ [a, b]. Note historique L'idée de variation totale pour les fonctions d'une variable réelle a d'abord été introduite par Camille Jordan, afin de démontrer un théorème de convergence pour les séries de Fourier de fonctions discontinues périodiques à variation bornée. L'extension du concept aux fonctions de plusieurs variables n'est pas si simple. Définition Fonctions d'une variable réelle La variation totale d'une fonction d'une variable réelle (ou complexe) f, définie sur un intervalle [a , b] \subset \mathbb{R} est donnée par : : V^a_b(f)=\sup_{\mathcal{P}} \sum_

Coarse graining and large-N behavior of the d-dimensional N-clock model

Matthias Ruf

We study the asymptotic behavior of the N-clock model, a nearest neighbors ferromagnetic spin model on the d-dimensional cubic epsilon-lattice in which the spin field is constrained to take values in a discretization S-N of the unit circle S-1 consisting of N equispaced points. Our Gamma-convergence analysis consists of two steps: we first fix N and let the lattice spacing epsilon -> 0, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in S-N; at a second stage, we let N -> +infinity. The final result of this two-step limit process is an anisotropic total variation of S-1-valued vector fields of bounded variation.

2021

Sparsity-Based Data Reconstruction Models for Biomedical Imaging

Emrah Bostan

We propose new regularization models to solve inverse problems encountered in biomedical imaging applications. In formulating mathematical schemes, we base our approach on the sparse signal processing principles that have emerged as a central paradigm in the field. We adopt a variational perspective and specify the proposed sparsity-promoting data reconstruction models as energy minimization problems. To design practical algorithms, we develop novel iterative methods to efficiently perform the consequent optimization task. The thesis is organized in three main parts. In the first part, our main contribution is the introduction of a proper statistically-based discretization paradigm for inverse problems. In particular, our framework considers a continuous-domain stochastic signal model and characterizes a specific class of inference algorithms. We show that derived inference-based methods cover the classical Tikhonov-type techniques as well as a wide range of the sparsity-promoting schemes including the well-known l1-norm regularization and its nonconvex variants. This provides a unifying stochastic perception of the resolution of inverse problems. In the second part, we propose a novel phase retrieval algorithm for imaging unstained biological samples that are optically thin. In specific terms, we use the transport-of-intensity equation (TIE) relating the spatial phase map of a field to the derivative of its intensity along the propagation direction. We analyze the implications of using the TIE formalism with finer and coarser approximations of said derivative. Based on this analysis, our contribution is a practical phase reconstruction algorithm that incorporates a sparsity-based regularization. The developed technique operates with a standard bright-field microscope. Experiments on real data illustrate that our phase reconstruction algorithm is viable and can be a low-cost alternative to dedicated phase microscopes. In the last part, we develop regularization schemes for vector fields, which have an increasing prevalence in medical imaging. In this context, our first contribution is a new regularizer that imposes sparsity on the singular values of the Jacobian of a given vector field. We show that the proposed regularization functional is a valid extension of total variation (TV) regularization to vector-valued functions. We utilize the developed framework for enhancing the streamline visualizations of experimentally acquired 4D flow MRI data. Since vector field regularization requires processing large volumes of multidimensional data, our second contribution is the development of a non-iterative denoising algorithm. In particular, we design model-based tight wavelet frames that are able to remove the spurious divergence content from the vector field, which is of interest in aortic blood flow imaging.

EPFL2016

Structurally-Informed Deconvolution Of Functional Magnetic Resonance Imaging Data

Thomas William Arthur Bolton, Younes Farouj, Mert Inan, Dimitri Nestor Alice Van De Ville

Neural activity occurs in the shape of spatially organized patterns: networks of brain regions activate in synchrony. Many of these functional networks also happen to be strongly structurally connected. We use this information to revisit the fundamental problem of functional magnetic resonance imaging (fMRI) data deconvolution. Using tools from graph signal processing (GSP), we extend total activation, a spatio-temporal deconvolution technique, to data defined on graph domains. The resulting approach simultaneously cancels out the effect of the haemodynamics, and promotes spatial patterns that are in harmony with predefined structural wirings. More precisely, we minimize a functional involving one data fidelity and two regularization terms. The first regularizer uses the concept of generalized total variation to promote sparsity in the activity transients domain. The second term controls the overall spatial variation over the graph structure. We demonstrate the relevance of this structurally-driven regularization on synthetic and experimental data.

IEEE2019

Sparsity-Based Data Reconstruction Models for Biomedical Imaging

Emrah Bostan

EPFL2016