A model based on local graphs for colour images and its application for Gaussian noise smoothing

[EN] In this paper, a new model for processing colour images is presented. A graph is built for each image pixel taking into account some constraints on links. Each pixel is characterized depending on the features of its related graph, which allows to process it appropriately. As an example, we provide a characterization of each pixel based on the link cardinality of its connected component. This feature enables us to properly distinguish flat image regions respect to edge and detail regions. According to this, we have designed a hybrid filter for colour image smoothing. It combines a filter able to properly process flat image regions with another one that is more appropriate for details and texture. Experimental results show that our model performs appropriately. We also see that our proposed filter is competitive with respect to state-of-the-art methods. It is close closer to the corresponding optimal switching filter respect to other analogous hybrid method.Samuel Morillas acknowledges the support of grant MTM2015-64373-P (MINECO/FEDER, UE). Cristina Jordan acknowledges the support of grant TEC2016-79884-C2-2-R.Pérez-Benito, C.; Morillas, S.; Jordan-Lluch, C.; Conejero, JA. (2018). A model based on local graphs for colour images and its application for Gaussian noise smoothing. Journal of Computational and Applied Mathematics. 330:955-964. https://doi.org/10.1016/j.cam.2017.05.013S95596433

Livrable D3.3 of the PERSEE project : 2D coding tools

49Livrable D3.3 du projet ANR PERSEECe rapport a été réalisé dans le cadre du projet ANR PERSEE (n° ANR-09-BLAN-0170). Exactement il correspond au livrable D3.3 du projet. Son titre : 2D coding tool

An adaptive noise removal approach for restoration of digital images corrupted by multimodal noise

Data smoothing algorithms are commonly applied to reduce the level of noise and eliminate the weak textures contained in digital images. Anisotropic diffusion algorithms form a distinct category of noise removal approaches that implement the smoothing process locally in agreement with image features such as edges that are typically determined by applying diverse partial differential equation (PDE) models. While this approach is opportune since it allows the implementation of feature-preserving data smoothing strategies, the inclusion of the PDE models in the formulation of the data smoothing process compromises the performance of the anisotropic diffusion schemes when applied to data corrupted by non-Gaussian and multimodal image noise. In this paper we first evaluate the positive aspects related to the inclusion of a multi-scale edge detector based on the generalisation of the Di Zenzo operator into the formulation of the anisotropic diffusion process. Then, we introduce a new approach that embeds the vector median filtering into the discrete implementation of the anisotropic diffusion in order to improve the performance of the noise removal algorithm when applied to multimodal noise suppression. To evaluate the performance of the proposed data smoothing strategy, a large number of experiments on various types of digital images corrupted by multimodal noise were conducted.Keywords — Anisotropic diffusion, vector median filtering, feature preservation, multimodal noise, noise removal

A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices

A Second Order TV-type Approach for Inpainting and Denoising Higher Dimensional Combined Cyclic and Vector Space Data

In this paper we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space valued data. These kind of data appear when dealing with nonlinear color spaces such as HSV, and they can be obtained by changing the space domain of, e.g., an optical flow field to polar coordinates. For such nonlinear data spaces, we develop algorithms for the solution of the corresponding second order total variation (TV) type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and we apply the algorithms to concrete problems.Comment: revised submitted versio

Modern Regularization Methods for Inverse Problems

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research. In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.Leverhulme Trust Early Career Fellowship ‘Learning from mistakes: a supervised feedback-loop for imaging applications’ Isaac Newton Trust Cantab Capital Institute for the Mathematics of Information ERC Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse German Ministry for Science and Education (BMBF) project MED4D EPSRC grant EP/K032208/

Analysis of motion in scale space

This work includes some new aspects of motion estimation by the optic flow method in scale spaces. The usual techniques for motion estimation are limited to the application of coarse to fine strategies. The coarse to fine strategies can be successful only if there is enough information in every scale. In this work we investigate the motion estimation in the scale space more basically. The wavelet choice for scale space decomposition of image sequences is discussed in the first part of this work. We make use of the continuous wavelet transform with rotationally symmetric wavelets. Bandpass decomposed sequences allow the replacement of the structure tensor by the phase invariant energy operator. The structure tensor is computationally more expensive because of its spatial or spatio-temporal averaging. The energy operator needs in general no further averaging. The numerical accuracy of the motion estimation with the energy operator is compared to the results of usual techniques, based on the structure tensor. The comparison tests are performed on synthetic and real life sequences. Another practical contribution is the accuracy measurement for motion estimation by adaptive smoothed tensor fields. The adaptive smoothing relies on nonlinear anisotropic diffusion with discontinuity and curvature preservation. We reached an accuracy gain under properly chosen parameters for the diffusion filter. A theoretical contribution from mathematical point of view is a new discontinuity and curvature preserving regularization for motion estimation. The convergence of solutions for the isotropic case of the nonlocal partial differential equation is shown. For large displacements between two consecutive frames the optic flow method is systematically corrupted because of the violence of the sampling theorem. We developed a new method for motion analysis by scale decomposition, which allows to circumvent the systematic corruption without using the coarse to fine strategy. The underlying assumption is, that in a certain neighborhood the grey value undergoes the same displacement. If this is fulfilled, then the same optic flow should be measured in all scales. If there arise inconsistencies in a pixel across the scale space, so they can be detected and the scales containing this inconsistencies are not taken into account

PDE Based Enhancement of Color Images in RGB Space

International audienceA novel method for color image enhancement is proposed as an extension of scalar diffusion-shock filter coupling model, where noisy and blurred images are denoised and sharpened. The proposed model is based on using single vectors of the gradient magnitude and the second derivatives as a technique to relate different color components of the image. This model can be viewed as a generalization of Bettahar-Stambouli filter to multi-valued images. The proposed algorithm is more efficient than the mentioned filter and some previous works on color image denoising and deblurring without creating false colors