Cartoon-texture composite regularization based non-blind deblurring method for partly-textured blurred images with Poisson noise

doi:10.1016/j.sigpro.2015.04.020

Signal Processing

Volume 116, November 2015, Pages 127-140

https://doi.org/10.1016/j.sigpro.2015.04.020 Get rights and content

Highlights

•
Our method utilizes different regularization terms for cartoon and texture components.
•
Our method can eliminate the staircase effect without distinguishing the regions of image.
•
Our method can eliminate the artifact in non-textured region caused by the nonlocal TV.
•
Our ADMM based algorithm can solve the new model efficiently.
•
Our method can improve subjective vision, PSNR and SSIM efficiently.

Abstract

In recent years, deblurring image with Poisson noise has attracted more and more attention in many areas such as astronomy and biological imaging. This is an ill-posed problem and can be regularized to improve the quality of the solution. Fractional-order total variation regularization can eliminate the staircase effect caused by the total variation regularization and avoid over-smoothing at the edges caused by the high-order total variation regularization, but it cannot preserve textures well. Non-local regularization can preserve textures well but often causes extra artifacts especially in the non-texture regions. In this paper, we deal with the cartoon component and the texture component differently and propose a cartoon-texture composite regularization based non-blind deblurring method. We utilize fractional-order total variation regularization for the cartoon component to eliminate the staircase effect and avoid over-smoothing at the edges, and use the non-local total variation regularization for the texture component to preserve the details and eliminate the artifacts in the non-textured regions. We develop an alternating direction method of multipliers based algorithm to solve the proposed model. Experiments results show that our method can improve the result both visually and in terms of the peak signal to noise ratio and structural similarity index efficiently.

Graphical abstract

Introduction

Image deblurring aims to retrieve a clean sharp image from a noisy blurred observation. In general, we assume that the noise is signal-independent additive white Gaussian noise because this makes both analysis and estimation significantly more convenient. However, in many areas such as astronomy imaging and some biological imaging systems (e.g. microscopy and X-ray imaging), noise is signal-dependent and follows Poisson distribution [1], [2], [3]. Out-of-focus blur is an important degradation factor in microscopy imaging caused by poor localization of the point spread function, while motion blur is most common in X-ray imaging because of the movement of the patient. Restoration. The restoration of a blurred image with Poisson noise has received considerable attention in recent years. From the mathematical point of view, this problem is ill-posed. One of the most well-known techniques to deal with ill-posed problem is by energy minimization and regularization. To improve the quality of the image, numerous regularization methods have been proposed, and therein methods regarded to be particularly efficient include total variation regularization [4], [5], [6], non-local regularization [7], [8], [9] and sparsity regularization [10], [11], [12].

Total variation regularization favors piecewise constant solution, and therefore it performs well in preserving edges but often causes staircase effect. High order partial differential equations and high order total variation regularization [13], [14] can eliminate the staircase effect efficiently, but often cause edge blurring. To remedy this, one feasible method is to choose the order of the model adaptively. Following this idea, Jiang et al. [15] proposed a composite regularization method by utilizing the total variation regularization at the edges and high order total variation regularization in the flat regions. However, this method is not practical as for blurred images it is difficult to locate the position of edges exactly and achieve a proper estimate of the order of the model. Another interesting compromise between the total variation regularization and high order total variation regularization is the fractional-order total variation regularization [16]. The numerical results show that when the order is larger slightly than one, the fractional-order models can eliminate the staircase effect and avoid over-smoothing at the edges [17], [18], [19]. It is for this advantage that the fractional-order total variation regularization has received more and more attention in recent years [17], [18], [19], [20], [21].

However, the fractional-order total variation regularization cannot preserve textures well, though it performs better than the total variation regularization in this respect. In terms of texture preservation, non-local methods perform very well [7], [8], [9]. However, these non-local methods often cause extra artifacts. These artifacts have little effect in the textured regions because of the visual masking of texture, but they are undesirable especially in the non-textured regions with gray changes. It should be noted that these extra artifacts mainly result from the weighted averaging of the patches used in the non-local methods. If the change of the gray values is very small, then the error of the weighted averaging of the patches is small, and therefore these extra artifacts can be significantly reduced.

In this paper, we consider the non-blind deblurring of the partly-textured blurred image with Poisson noise. In order to preserve textures, eliminate the staircase effect and avoid over-smoothing of the edges simultaneously, we propose a cartoon-texture composite regularization based model to deal with this problem. We have two main contributions:

(i)
Assuming that the image consists of cartoon component and texture component, we utilize fractional-order total variation regularization for the cartoon component and non-local total variation regularization for the texture component. By dealing with these two components separately, we can retain the advantages of the fractional-order total variation regularization and the non-local total variation regularization, but avoid their disadvantages.
(ii)
We develop an alternating direction method of multipliers (ADMM) based algorithm to solve the new composite regularization model efficiently.

The rest of this paper is organized as follows: In Section 2, we describe the new cartoon-texture composite regularization based non-blind deblurring model; in Section 3, we present the details of the ADMM based algorithm for solving the new model. Some numerical experiments are given to illustrate the performance of the proposed method in Section 4. Finally, conclusions are provided in Section 5.

Section snippets

Cartoon-texture composite regularization based non-blind deblurring model

To simplify, we assume that $f \in ℝ^{N \times N}$ is the blurred image of size $N \times N$ corrupted by Poisson noise and $u \in ℝ^{N \times N}$ is the latent clean sharp image, and then we have $f = P (Hu),$ where $H : ℝ^{N \times N} \to ℝ^{N \times N}$ stands for the matrix notation of the convolution of a point spread function and $P$ is a Poisson process. Moreover, we assume that ${(Hu)}_{i, j} > 0$ for $i, j = 1, 2, \dots, N$ . Our goal is to retrieve the latent clean sharp image $u$ from the blurred noisy image $f .$ When $H$ is known, we call this problem non-blind deblurring; otherwise, we

ADMM based algorithm for solving the new model

Eq. (5) is a non-linear multi-parameter constrained optimization problem. The traditional gradient decent algorithm has slow convergence speed. In this paper, we propose to solve Eq. (5) by using the ADMM. The ADMM is an efficient technique to solve multi-parameter constrained optimization problems [25]. First, we give a short review of the ADMM, which solves the model in the form of $\begin{array}{l} \min_{x, y} f (x) + g (y), \\ s .t . A x + B y = b, \end{array}$ where $f$ and $g$ are convex functions. Depending on the actual situation, $x, y$ and $b$ can

Numerical experiments

In our experiments, we compare the proposed cartoon-texture composite regularization based non-blind deblurring method with four total variation regularization or high-order total variation regularization based methods as follows:

(1)
The Richardson–Lucy algorithm proposed in [4];
(2)
The ADMM based Poisson noisy image deblurring method proposed in [5];
(3)
The composite regularization based method proposed in [15] recently.

The models considered in [4], [5] are the same. They utilized the total variation

Conclusion

In this paper, instead of treating the image as a whole, we assume that it consists of the cartoon component and the texture component. Using different regularization for these two components, we propose a cartoon-texture composite regularization based non-blind deblurring method for partly-textured blurred images with Poisson noise. By dealing with these two components separately, our method retains the advantages of fractional-order total variation and non-local total variation, but avoids

Acknowledgements

This work is supported by the National Natural Science Fund of China (Nos. 61101198 and 11431015), the Natural Science Fund of Jiangsu Province of China (No. BK2012800), the China Postdoctoral Science Fund (No. 2012M511281), the Jiangsu Planned Projects for Postdoctoral Research Fund (No. 1102064C) and the Fundamental Research Funds for the Central Universities (No. 30915012204).

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View full text

Cartoon-texture composite regularization based non-blind deblurring method for partly-textured blurred images with Poisson noise

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Cartoon-texture composite regularization based non-blind deblurring model

ADMM based algorithm for solving the new model

Numerical experiments

Conclusion

Acknowledgements

A discrepancy principle for Poisson data

Inverse Prob.

A proximal iteration for deconvolving Poisson noisy images using sparse representations

IEEE Trans. Image Process.

Restoration of Poissonian images using alternating direction optimization

IEEE Trans. Image Process.

Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution

Microsc. Res. Tech.

Removing multiplicative noise by Douglas-Rachford splitting methods

J. Math. Imaging Vis.

Bregmanized nonlocal regularization for deconvolution and sparse reconstruction

SIAM J. Imag. Sci.

Poisson noise reduction with non-local PCA

J. Math. Imaging Vis.

Sparse Poisson noisy image deblurring

IEEE Trans. Image Process.

This is SPIRAL-TAP: sparse Poisson intensity reconstruction algorithms—theory and practice

IEEE Trans. Image Process.

Sparsity regularization for image reconstruction with Poisson data

S&T/SPIE Electron. Imaging Int. Soc. Opt. Photonics

Fourth-order partial differential equations for noise removal

IEEE Trans. Image Process.

Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time

IEEE Trans. Image Process.

Alternating direction method for the high-order total variation-based Poisson noise removal problem

Numer. Algorithms

A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising

Appl. Math. Modell.