Blind inpainting using and total variation regularization

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Our proposed system address the problem of image reconstruction with missing pixels or corrupted with impulse noise, when the locations of the corrupted pixels are not known. A logarithmic transformation is applied to convert the multiplication between the image and binary mask into an additive problem. The image and mask terms are then estimated iteratively with total variation regularization applied on the image, and regularization on the mask term which imposes sparseness on the support set of the missing pixels. The resulting alternating minimization scheme simultaneously estimates the image and mask, in the same iterative process. The logarithmic transformation also allows the method to be extended to the Rayleigh multiplicative and Poisson observation models. The method can also be extended to impulse noise removal by relaxing the regularizer from the norm to the norm. Experimental results show that the proposed method can deal with a larger fraction of missing pixels than two phase methods, which first estimate the mask and then reconstruct the image.

Early approaches for estimating the missing values used median filtering, which discards outliers. The Adaptive Median Filter (AMF) and Adaptive Center Weighted Median Filter were developed to detect the positions of noisy pixels with, respectively, salt-and-pepper and random valued impulse noise. The problems of reconstructing an image with missing data and removing impulse noise basically involve detecting outliers. Some methods do not use a separate mask detection stage, but estimate the mask or impulse noise field during the iterative process. A frame based method for image deblurring and decomposition into cartoon and texture components with impulse noise is presented. For the case of noise other than additive and Gaussian, there exist methods for image reconstruction from a partial set of pixels, such as and for Poisson noise, and for Rayleigh speckle noise. There are also the classical interpolation methods that have been used in ultrasound imaging.

In this paper, our propose a method to estimate the image x without knowing apriori the observation mask A, i.e., we simultaneously estimate the image and the mask. We formulate the masking operation as a summation after logarithmic compression, and apply a TV regularize on the term corresponding to the logarithm of the image, and an -norm regularizer on the term corresponding to the mask. The TV regularizer encourages the estimate of x to be piecewise smooth, while the -norm regularizer encourages the mask term to be sparse. The problem is solved iteratively using a Gauss-Seidel alternating minimization scheme. Experimental results show that our proposed method can deal with as many as 95% of the pixels missing, which is higher than reported in literature.
We extend the method to non-Gaussian noise models, namely multiplicative Rayleigh distributed speckle noise, and Poisson noise, by taking into account the data fidelity terms corresponding to their respective statistical models
We approach the problem of estimating x and A in a different manner. We use a logarithmic transform on both to convert the masking problem into an additive and separable one.
Title Name : An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems.

Author Name : Manya V. Afonso, José M. Bioucas-Dias, and Mário A. T. Figueiredo
These authors proposed the a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. an instance of the so-called alternating direction method of multipliers, for which convergence sufficient conditions are known.
Title Name: A fast and accurate first-order method for sparse recovery

Author Name: Stephen becker, jerome bobin and Emmanuel j. Candes
These authors is an approach is ideally suited for solving large-scale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is exible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods are known.

Title Name: A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems

Author Name: Amir Beck and Marc Teboulle
The authors are present a new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA is known.

Title Name: Fast Image Recovery Using Variable Splitting and Constrained Optimization .

Author Name: Manya V. Afonso, Jos´e M. Bioucas-Dias, A. T. Figueiredo
The author is a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an `2 data-fidelity term and a non smooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization and total-variation regularization. Based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so called alternating direction method of multipliers, for which convergence has been known.


  • Mat Lab R 2015a

  • Image Processing Toolbox 7.1

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