An algorithm for total variation inpainting based on nonlinear multi-grid methods

Image inpainting refers to restoring a damaged image with missing information. The total variation (TV) inpainting model is one such method that simultaneously fills in the regions with available information from their surroundings and eliminates noises. The method works well with small narrow inpainting domains. However there remains an urgent need to develop fast iterative solvers, as the underlying problem sizes are large. In addition one needs to tackle the imbalance of results between inpainting and denoising. When the inpainting regions are thick and large, the procedure of inpainting works quite slowly and usually requires a significant number of iterations and leads inevitably to oversmoothing in the outside of the inpainting domain. To overcome these difficulties, we propose a solution for TV inpainting method based on the nonlinear multi-grid algorithm.

Denoising di immagini mediante tecniche basate sulla Total Variation.

Il lavoro svolto in questa tesi è legato allo studio ed alla formulazione di metodi computazionali volti all’eliminazione del noise (rumore) presente nelle immagini, cioè il processo di “denoising” che è quello di ricostruire un’immagine corrotta da rumore avendo a disposizione una conoscenza a priori del fenomeno di degrado. Il problema del denoising è formulato come un problema di minimo di un funzionale dato dalla somma di una funzione che rappresenta l’adattamento dei dati e la Variazione Totale. I metodi di denoising di immagini saranno affrontati attraverso tecniche basate sullo split Bregman e la Total Variation (TV) pesata che è un problema mal condizionato, cioè un problema sensibile a piccole perturbazioni sui dati. Queste tecniche permettono di ottimizzare dal punto di vista della visualizzazione le immagini in esame.

Gibbs point process approximation: Total variation bounds using Stein's method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

Total Variation Blind Deconvolution: The Devil Is in the Details

In this paper we study the problem of blind deconvolution. Our analysis is based on the algorithm of Chan and Wong [2] which popularized the use of sparse gradient priors via total variation. We use this algorithm because many methods in the literature are essentially adaptations of this framework. Such algorithm is an iterative alternating energy minimization where at each step either the sharp image or the blur function are reconstructed. Recent work of Levin et al. [14] showed that any algorithm that tries to minimize that same energy would fail, as the desired solution has a higher energy than the no-blur solution, where the sharp image is the blurry input and the blur is a Dirac delta. However, experimentally one can observe that Chan and Wong's algorithm converges to the desired solution even when initialized with the no-blur one. We provide both analysis and experiments to resolve this paradoxical conundrum. We find that both claims are right. The key to understanding how this is possible lies in the details of Chan and Wong's implementation and in how seemingly harmless choices result in dramatic effects. Our analysis reveals that the delayed scaling (normalization) in the iterative step of the blur kernel is fundamental to the convergence of the algorithm. This then results in a procedure that eludes the no-blur solution, despite it being a global minimum of the original energy. We introduce an adaptation of this algorithm and show that, in spite of its extreme simplicity, it is very robust and achieves a performance comparable to the state of the art.

A total variation discontinuous Galerkin approach for image restoration

The focal point of this paper is to propose and analyze a P 0 discontinuous Galerkin (DG) formulation for image denoising. The scheme is based on a total variation approach which has been applied successfully in previous papers on image processing. The main idea of the new scheme is to model the restoration process in terms of a discrete energy minimization problem and to derive a corresponding DG variational formulation. Furthermore, we will prove that the method exhibits a unique solution and that a natural maximum principle holds. In addition, a number of examples illustrate the effectiveness of the method.

A Clearer Picture of Total Variation Blind Deconvolution

Blind deconvolution is the problem of recovering a sharp image and a blur kernel from a noisy blurry image. Recently, there has been a significant effort on understanding the basic mechanisms to solve blind deconvolution. While this effort resulted in the deployment of effective algorithms, the theoretical findings generated contrasting views on why these approaches worked. On the one hand, one could observe experimentally that alternating energy minimization algorithms converge to the desired solution. On the other hand, it has been shown that such alternating minimization algorithms should fail to converge and one should instead use a so-called Variational Bayes approach. To clarify this conundrum, recent work showed that a good image and blur prior is instead what makes a blind deconvolution algorithm work. Unfortunately, this analysis did not apply to algorithms based on total variation regularization. In this manuscript, we provide both analysis and experiments to get a clearer picture of blind deconvolution. Our analysis reveals the very reason why an algorithm based on total variation works. We also introduce an implementation of this algorithm and show that, in spite of its extreme simplicity, it is very robust and achieves a performance comparable to the top performing algorithms.

Image reconstruction by an alternating minimisation

This paper focuses on the problem of incomplete data in the applications of the circular cone-beam computed tomography. This problem is frequently encountered in medical imaging sciences and some other industrial imaging systems. For example, it is crucial when the high density region of objects can only be penetrated by X-rays in a limited angular range. As the projection data are only available in an angular range, the above mentioned incomplete data problem can be attributed to the limited angle problem, which is an ill-posed inverse problem. This paper reports a modified total variation minimisation method to reduce the data insufficiency in tomographic imaging. This proposed method is robust and efficient in the task of reconstruction by showing the convergence of the alternating minimisation method. The results demonstrate that this new reconstruction method brings reasonable performance. (C) 2010 Elsevier B.V. All rights reserved.

Identification of fluorescent beads using a coded aperture snapshot spectral imager.

We apply a coded aperture snapshot spectral imager (CASSI) to fluorescence microscopy. CASSI records a two-dimensional (2D) spectrally filtered projection of a three-dimensional (3D) spectral data cube. We minimize a convex quadratic function with total variation (TV) constraints for data cube estimation from the 2D snapshot. We adapt the TV minimization algorithm for direct fluorescent bead identification from CASSI measurements by combining a priori knowledge of the spectra associated with each bead type. Our proposed method creates a 2D bead identity image. Simulated fluorescence CASSI measurements are used to evaluate the behavior of the algorithm. We also record real CASSI measurements of a ten bead type fluorescence scene and create a 2D bead identity map. A baseline image from filtered-array imaging system verifies CASSI's 2D bead identity map.

Millimeter-wave compressive holography.

We describe an active millimeter-wave holographic imaging system that uses compressive measurements for three-dimensional (3D) tomographic object estimation. Our system records a two-dimensional (2D) digitized Gabor hologram by translating a single pixel incoherent receiver. Two approaches for compressive measurement are undertaken: nonlinear inversion of a 2D Gabor hologram for 3D object estimation and nonlinear inversion of a randomly subsampled Gabor hologram for 3D object estimation. The object estimation algorithm minimizes a convex quadratic problem using total variation (TV) regularization for 3D object estimation. We compare object reconstructions using linear backpropagation and TV minimization, and we present simulated and experimental reconstructions from both compressive measurement strategies. In contrast with backpropagation, which estimates the 3D electromagnetic field, TV minimization estimates the 3D object that produces the field. Despite undersampling, range resolution is consistent with the extent of the 3D object band volume.

Resolution of sharp fronts in the presence of model error in variational data assimilation

We show that the four-dimensional variational data assimilation method (4DVar) can be interpreted as a form of Tikhonov regularization, a very familiar method for solving ill-posed inverse problems. It is known from image restoration problems that L1-norm penalty regularization recovers sharp edges in the image more accurately than Tikhonov, or L2-norm, penalty regularization. We apply this idea from stationary inverse problems to 4DVar, a dynamical inverse problem, and give examples for an L1-norm penalty approach and a mixed total variation (TV) L1–L2-norm penalty approach. For problems with model error where sharp fronts are present and the background and observation error covariances are known, the mixed TV L1–L2-norm penalty performs better than either the L1-norm method or the strong constraint 4DVar (L2-norm)method. A strength of the mixed TV L1–L2-norm regularization is that in the case where a simplified form of the background error covariance matrix is used it produces a much more accurate analysis than 4DVar. The method thus has the potential in numerical weather prediction to overcome operational problems with poorly tuned background error covariance matrices.

Métodos numéricos para o retoque digital

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Robust optical flow estimation

[EN] In this work, we describe an implementation of the variational method proposed by Brox et al. in 2004, which yields accurate optical flows with low running times. It has several benefits with respect to the method of Horn and Schunck: it is more robust to the presence of outliers, produces piecewise-smooth flow fields and can cope with constant brightness changes. This method relies on the brightness and gradient constancy assumptions, using the information of the image intensities and the image gradients to find correspondences. It also generalizes the use of continuous L1 functionals, which help mitigate the efect of outliers and create a Total Variation (TV) regularization. Additionally, it introduces a simple temporal regularization scheme that enforces a continuous temporal coherence of the flow fields.