A discrete approximation of Blake & Zisserman energy in image denoising and optimal choice of regularization parameters

We consider a multi-scale approach for the discrete approximation of a functional proposed by Bake and Zisserman (BZ) for solving image denoising and segmentation problems. The proposed method is based on simple and effective higher order varia-tional model. It consists of building linear discrete energies family which Γ-converges to the non-linear BZ functional. The key point of the approach is the construction of the diffusion operators in the discrete energies within a finite element adaptive procedure which approximate in the Γ-convergence sense the initial energy including the singular parts. The resulting model preserves the singularities of the image and of its gradient while keeping a simple structure of the underlying PDEs, hence efficient numerical method for solving the problem under consideration. A new point to make this approach work is to deal with constrained optimization problems that we circumvent through a Lagrangian formulation. We present some numerical experiments to show that the proposed approach allows us to detect first and second-order singularities. We also consider and implement to enhance the algorithms and convergence properties, an augmented Lagrangian method using the alternating direction method of Multipliers (ADMM)

An adaptive Cahn-Hilliard equation for enhanced edges in binary image inpainting

We consider the Cahn-Hilliard equation for solving the binary image inpainting problem with emphasis on the recovery of low-order sets (edges, corners) and enhanced edges. The model consists in solving a modified Cahn-Hilliard equation by weighting the diffusion operator with a function which will be selected locally and adaptively. The diffusivity selection is dynamically adopted at the discrete level using the residual error indicator. We combine the adaptive approach with a standard mesh adaptation technique in order to well approximate and recover the singular set of the solution. We give some numerical examples and comparisons with the classical Cahn-Hillard equation for different scenarios. The numerical results illustrate the effectiveness of the proposed model. </jats:p

A Nash Game Based Variational Model For Joint Image Intensity Correction And Registration To Deal With Varying Illumination

Registration aligns features of two related images so that information can be compared and/or fused in order to highlight differences and complement information. In real life images where bias field is present, this undesirable artefact causes inhomogeneity of image intensities and hence leads to failure or loss of accuracy of registration models based on minimization of the differences of the two image intensities. Here, we propose a non-linear variational model for joint image intensity correction (illumination and translation) and registration and reformulate it in a game framework. While a non-potential game offers flexible reformulation and can lead to better fitting errors, proving the solution existence for a non-convex model is non-trivial. Here we establish an existence result using the Schauder's fixed point theorem. To solve the model numerically, we use an alternating minimization algorithm in the discrete setting. Finally numerical results can show that the new model outperforms existing models

An Augmented Lagrangian Method for Solving a New Variational Model based on Gradients Similarity Measures and High Order Regularization for Multimodality Registration

In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages

A nonstandard higher-order variational model to speckle noise removal and thin-structure detection

In this work, we propose a multiscale approach for a nonstandard higher-order PDE based on the $p(\cdot)$-Kirchhoff energy. First, we consider a topological gradient approach for a semilinear case in order to detect important object of image. Then, we consider a fully nonlinear $p(\cdot)$-Kirchhoff equation with variables exponent functions that are chosen adaptively based on the map furnished by the topological gradient in order to preserve important features of the image. Then, we consider the split Bregman method for the numerical implementation of our proposed model. We compare our model with other classical variational approaches such that the TVL and biharmonic restoration models. Finally, we present some numerical results to illustrate the effectiveness of our approach

Non-standard fourth-order PDE related to the image denoising Multi-scale non-standard fourth-orderPDE in image denoising and its fixed point algorithm

International audienc

Non-standard fourth-order PDE related to the image denoising Multi-scale non-standard fourth-orderPDE in image denoising and its fixed point algorithm

International audienc

Restauration d'images par des méthodes d'équations aux dérivées partielles et des techniques de régularisation

Image inpainting refers to the process of restoring a damaged image with missing information. Different mathematical approaches were suggested to deal with this problem. In particular, partial differential diffusion equations are extensively used. The underlying idea of PDE-based approaches is to fill-in damaged regions with available information from their surroundings. The first purpose of this Thesis is to treat the case where this information is not available in a part of the boundary of the damaged region. We formulate the inpainting problem as a nonlinear boundary inverse problem for incomplete images. Then, we give a Nash-game formulation of this Cauchy problem and we present different numerical which show the efficiency of the proposed approach as an inpainting method.Typically, inpainting is an ill-posed inverse problem for it most of PDEs approaches are obtained from minimization of regularized energies, in the context of Tikhonov regularization. The second part of the thesis is devoted to the choice of regularization parameters in second-and fourth-order energy-based models with the aim of obtaining as far as possible fine features of the initial image, e.g., (corners, edges, … ) in the inpainted region. We introduce a family of regularized functionals with regularization parameters to be selected locally, adaptively and in a posteriori way allowing to change locally the initial model. We also draw connections between the proposed method and the Mumford-Shah functional. An important feature of the proposed method is that the investigated PDEs are easy to discretize and the overall adaptive approach is easy to implement numerically.Cette thèse concerne le problème de désocclusion d'images, au moyen des équations aux dérivées partielles. Dans la première partie de la thèse, la désocclusion est modélisée par un problème de Cauchy qui consiste à déterminer une solution d'une équation aux dérivées partielles avec des données aux bords accessibles seulement sur une partie du bord de la partie à recouvrir. Ensuite, on a utilisé des algorithmes de minimisation issus de la théorie des jeux, pour résoudre ce problème de Cauchy. La deuxième partie de la thèse est consacrée au choix des paramètres de régularisation pour des EDP d'ordre deux et d'ordre quatre. L'approche développée consiste à construire une famille de problèmes d'optimisation bien posés où les paramètres sont choisis comme étant une fonction variable en espace. Ceci permet de prendre en compte les différents détails, à différents échelles dans l'image. L'apport de la méthode est de résoudre de façon satisfaisante et objective, le choix du paramètre de régularisation en se basant sur des indicateurs d'erreur et donc le caractère à posteriori de la méthode (i.e. indépendant de la solution exacte, en générale inconnue). En outre, elle fait appel à des techniques classiques d'adaptation de maillage, qui rendent peu coûteuses les calculs numériques. En plus, un des aspects attractif de cette méthode, en traitement d'images est la récupération et la détection de contours et de structures fines

Diffeomorphic unsupervised deep learning model for mono- and multi-modality registration

Different from image segmentation, developing a deep learning network for image registration is less straightforward because training data cannot be prepared or supervised by humans unless they are trivial (e.g. pre-designed affine transforms). One approach for an unsupervised deep leaning model is to self-train the deformation fields by a network based on a loss function with an image similarity metric and a regularisation term, just with traditional variational methods. Such a function consists in a smoothing constraint on the derivatives and a constraint on the determinant of the transformation in order to obtain a spatially smooth and plausible solution. Although any variational model may be used to work with a deep learning algorithm, the challenge lies in achieving robustness. The proposed algorithm is first trained based on a new and robust variational model and tested on synthetic and real mono-modal images. The results show how it deals with large deformation registration problems and leads to a real time solution with no folding. It is then generalised to multi-modal images. Experiments and comparisons with learning and non-learning models demonstrate that this approach can deliver good performances and simultaneously generate an accurate diffeomorphic transformation. </jats:p