An Alternative Variational Framework for Image Denoising

Elisha Achieng Ogada,Zhichang Guo,Boying Wu

doi:10.1155/2014/939131

Elisha Achieng Ogada, Zhichang Guo + Show 1 more

Open Access

https://doi.org/10.1155/2014/939131

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Abstract

We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method.

Highlights

The objective of any image restoration process should not focus only on the removal of noise, but it should observe, as Perona and Malik [1], Koenderink [2], and Witkin [3] determined, that no new spurious details are created in the restored image; at each scale-space representation, the boundaries/edges are sharp or preserved, and, at all scales, intraregion smoothing is preferred to interregion smoothing.In the light of the above considerations, researchers have observed that it is logical to obtain or develop edge indicators that would be adapted to the local image structure [4]
We have proposed an alternative framework for total variation based image denoising models
The model is based on minimization of total variation with a functional coefficient

Summary

Introduction

The objective of any image restoration process should not focus only on the removal of noise, but it should observe, as Perona and Malik [1], Koenderink [2], and Witkin [3] determined, that no new spurious details are created in the restored image; at each scale-space representation, the boundaries/edges are sharp or preserved, and, at all scales, intraregion smoothing is preferred to interregion smoothing. TV functionals are defined in the space of functions of bounded variation (BV) and do not necessarily require image functions to be continuous and smooth This fact makes them allow for “jumps” or discontinuities and be able to protect edges. We propose an alternative framework of variational model for image denoising, where the regularization potential is a product of a gradient based functional coefficient and the norm of gradient of image (potential function for TV); namely, F(u) = ∫Ω Ψ(|∇u|) ⋅ |∇u|dx.

Proposed Model

The Minimization Problem

Preliminaries

Weak Solution to the Flow Associated with the Minimization Problem

Numerical Experiments

Conclusion