A rapid hybrid algorithm for image restoration combining parametric Wiener filtering and wave atom transform

https://doi.org/10.1016/j.jvcir.2016.08.009Get rights and content

Highlights

  • Some shortcomings of some restoration algorithms were cited.

  • A rapid and fast hybrid algorithm for image restoration is proposed.

  • The efficiency of the new algorithm in the filtering process was shown.

  • The proposed technique was found to be very interesting and generates good results.

Abstract

Image restoration refers to removal or minimization of known degradations in an image. This includes de-blurring images degraded by the limitations of sensors or source of captures in addition to noise filtering and correction of geometric distortion due to sensors. There are several classical image restoration methods such as Wiener filtering. To find an estimate of the original image, Wiener filter requires the prior knowledge of the degradation phenomenon, the blurred image and the statistical properties of the noise process. In this work, we propose a new rapid and blind algorithm for image restoration that does not require a priori knowledge of the noise distribution. The degraded image is first de-convoluted in Fourier space by parametric Wiener filtering, and then, it is smoothed by the wave atom transform after setting the threshold to its coefficients. Experiment results are significant and show the efficiency of our algorithm compared with other techniques in use.

Introduction

Image restoration attempts to reconstruct or recover an image that has been degraded by using a priori knowledge of the degradation phenomenon. Thus, restoration techniques focus on modeling the degradation and applying the inverse model in order to recover a de-noised image. A fundamental method in the filtering theory used commonly for image restoration is the Wiener filter. This method’s objective was to restore image as close as possible to the original one. This is, indeed, a method of least squares estimation in which we consider that the images are realizations of a stationary and ergodic random process. The drawback of this method is the need for a priori knowledge of the degradation phenomenon, which is denoted as the degradation function of the imaging system, i.e. the point spread function (PSF), the blurred image and the statistical properties of the noise process. To restore distorted image and improve Wiener filtering without prior knowledge of the degradation phenomenon, several authors propose to estimate the PSF using the specification of the imaging system or ground characters in the acquired images. Then an appropriate method is selected to restore the images with the estimated PSF [1], [2]. In other works, authors proposed to restore images in the wavelet domain [3], [4], [5]. Donoho and Johnstone in [13] proposed an algorithm for solving the inverse problem known as the wavelet-vaguelletes algorithm. It consists of applying an inverse filter and then the wavelet transform. Recently, a fast algorithm (VSFIR) for image restoration is proposed. This approach is based on a variable splitting to obtain an equivalent constrained optimization formulation which is addressed with an augmented Lagrangian method [7]. In other work [8], authors proposed a TV blind method for single frame by employing split Bregman iteration called as TVBDSB. The proposed method can excellently recover the degraded images not only with simple background but also with complex background. In [9], a non-local regularization algorithm with a single forward backward operator splitting is proposed to solve the sub-problems of the Bregman iterations. In related work [10], authors proposed a non-blind image restoration method (NBID) that combines the TV and NLTV models. Firstly, original image is decomposed into three regions, salient edges, details and constant regions and then the TV model and the NLTV model are used to smooth the image and preserve the salient edges and details respectively. Other techniques propose the partition techniques applied for video coding such as [11], [12]. In the present study, and to benefit from the advantages of the Fourier transform and those of wave atom, we propose a scheme that consists of two steps: deconvolution in the Fourier domain followed by a de-noising in the wave atom domain. The simulations results and the comparative study with other recent techniques are conducted and showed that the proposed schema generates a wide improvement in the quality of the restored images. This improvement has been shown subjectively in terms of visual quality, and objectively with reference to the computation of some criteria.

Section snippets

Image degradation and Wiener filter theory

Assuming that the noise is additive white and that it is uncorrelated with the signal, the degradation process is modeled by a degradation function that, together with an additive noise term, operates on an input image f(x, y) to produce a degraded image g(x, y) [6].g(x,y)=H[f(x,y)]+η(x,y)Given g(x, y), some knowledge about the degradation function H and the additive noise term η(x, y), the objective of restoration was to obtain an estimate f̂(x,y) of the original image. The estimated image must be

Technical and mathematical development of the proposed method

In this work, we suppose that the PSF is known. Our algorithm consists of two steps: deconvolution in the Fourier domain and de-noising in the wave atom domain.

The first step consists of regularized deconvolution in the Fourier space by the parametric Wiener filter whose purpose is to offset in part the PSF and reduce degradations.

Leth(x,y)-1g(x,y)=f(x,y)originalimage+h(x,y)-1η(x,y)deconvolutednoisewhere the term n(x, y) = h−1(x, y)  η(x, y) is a term that corresponds to the noise. This

Mathematical Wave atom transform introduction

In conventional wavelet transform, the transition from a level to another causes the approximation decomposition. While in the wavelets packets, the decomposition could be pursued into the other sets (details and approximation), which is not optimal. The optimality is related to the maximum energy of the decomposition. The idea is then to search for the path leading to the maximum energy through the different sub-bands.

In 2007 Demanet introduced the wave atom transform WAT as an efficient

Experimental results

In order to test the effectiveness of our algorithm objectively and subjectively, experiments were performed on simulated low resolution images. Those images were initially degraded by variable spread functions (Gaussian blur, motion blur). In a second step, the obtained images were further degraded by adding an additive Gaussian noise. The resulting images were then restored by three methods: the classical Wiener method, deconvolution into wavelet domain method and the proposed method.

Conclusion

In this work, we presented a hybrid model that combines parametric Wiener filter with wave atoms transform for image restoration. The importance of our method is its ability to restore the image without a priori knowledge of the spectral density of the original image and the noise. The proposed method was compared to the classic Wiener filter and the deconvolution by wavelets. Experiment results show the efficiency of our algorithm compared to other algorithms. Indeed, the proposed method shows

Acknowledgment

We would like to express our gratitude to the editor and anonymous reviewers for their constructive comments that will lead to this manuscript’s improvement in quality and representation.

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    Blind image restoration based on Wiener filtering and defocus point spread function estimation

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    Image de-noising based on combination of Wiener filter and wavelet shrinkage

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