Total Variation Regularization Algorithms for Images Corrupted with Different Noise Models: A Review

Total Variation (TV) regularization has evolved from an image denoising method for images corrupted with Gaussian noise into a more general technique for inverse problems such as deblurring, blind deconvolution, and inpainting, which also encompasses the Impulse, Poisson, Speckle, and mixed noise models. This paper focuses on giving a summary of the most relevant TV numerical algorithms for solving the restoration problem for grayscale/color images corrupted with several noise models, that is, Gaussian, Salt & Pepper, Poisson, and Speckle (Gamma) noise models as well as for the mixed noise scenarios, such the mixed Gaussian and impulse model. We also include the description of the maximum a posteriori (MAP) estimator for each model as well as a summary of general optimization procedures that are typically used to solve the TV problem. 1. Introduction Acquired digital images are subject to different kinds of noise [1, Chapter 7] depending on the hardware used for their acquisition which may involve additional degradations due to transmission errors or other external factors. The more common image noise models include the Gaussian, Impulse (e.g., Salt & Pepper), Poisson, and Speckle (e.g., Gamma) and the mixed Gaussian and impulse noise models. While there are several algorithms that can be considered state of the art for a particular noise model, typically the adaptation of such specialized algorithms to handle other noise models has been proven to be either severely difficult or just plain impossible. It is in this regard that regularization methods stand atop due to their flexibility to use any given noise model; while there are several examples of such methods (Tikhonov regularization, Wavelet image restoration, sparsity based denoising and inversion, etc.), here we focus on the Total Variation (TV) regularization [2] method. The original TV regularization method targeted image denoising under Gaussian noise [2]; nevertheless it has evolved into a more general technique for inverse problems (see [3] for more specific details) while retaining its edge preserving property ([4] gives an extended analysis of TV's properties). The TV regularized solution of the inverse problem involving data and observation operator (the identity in the case of denoising) is the minimum of the functional where is the data fidelity term, which depends on the noise model (for instance see (40), (42), (43), and (44)), the scalar is the regularization parameter, represents the total variation of solution , and the norm of vector is denoted by . Moreover, depending on the noise