Elsevier

Signal Processing

Volume 193, April 2022, 108407
Signal Processing

Poisson image restoration using a novel directional TVp regularization

https://doi.org/10.1016/j.sigpro.2021.108407Get rights and content

Highlights

  • We model by using a rotation matrix, an adaptive weighted matrix and the TVp-quasinorm.

  • We solve the 2p subproblem by the half-quadratic algorithm with guaranteed convergence.

  • We give the complexity analysis of the proposed method.

  • Our model can suppress the staircase effects and avoid blurry edges.

  • More structural information is retained by using our AWDTVp.

Abstract

The degree of Poisson noise depends on the image intensity, which makes Poisson image restoration very challenging. Moreover, complex structures of images desire suitable regularizations to describe. In this paper, we propose a new image restoration model under Poisson noise based on an adaptive weighted directional TVp regularization. The rotation matrix can keep the diffusion of the corresponding Euler–Lagrange equation along with the tangential direction of the edge, and the adaptive weighted matrix can enhance the diffusion. Owing to its adaptivity, our proposed model can simultaneously handle several dominant directions. Besides, the p-quasinorm regularization promotes the image sparsity. To solve it efficiently, we design an alternating direction method of multipliers (ADMM). The related 2p subproblem is solved by using the half-quadratic algorithm with guaranteed convergence. Experimental results on natural and synthetic images show the effectiveness of the proposed method over the state-of-the-art variational methods.

Introduction

Poisson noise is inevitable in various applications such as electronic microscopy [11] and astronomical imaging [22]. So it is very important to reduce its influence on images through image restoration. The degree of Poisson noise significantly depends on the peak value of pixel intensity values. Specifically, the smaller the peak value of an original image is, the higher the intensity of Poisson noise is. Moreover, the Poisson noise magnitude in an image increases with the pixel intensity of the region of this image. These facts make image restoration under Poisson noise very challenging. Since Poisson noise is different from the Gaussian noise, the restoration models proposed for Gaussian noise are not effective in removing Poisson noise [43]. Based on the statistical property of the Poisson distribution and maximum likelihood estimation [23], the proposed Poisson restoration methods often prefer the following data fidelity term [2], [14], [15], [27]:Ω(KuflogKu)dx,where Ω is a bounded open domain in R2, K is a bounded linear operator representing the convolution kernel, f and u represent an observed image and the corresponding restored image, respectively.

Due to the lack of some prior information, the inverse problem of Poisson image restoration is ill-posed. Therefore, many regularization techniques have been widely investigated to deal with this problem in the past two decades, such as total variation (TV) regularization [30], high-order TV regularization [26], [42], [44], [46], total generalized variation (TGV) regularization [5], fractional-order total variation [9], [10] and directional TV regularization [3]. In particular, the TV-based Poisson restoration model [25], [33], [45] can be written asminuu1+λΩ(KuflogKu)dx,where λ>0 is a regularization parameter. A number of fast algorithms have been proposed to solve this model, such as the EM-TV methods [31], [32], the augmented Lagrangian method [15], the split Bregman method [33] and the primal-dual algorithms [36]. These algorithms have also been successfully applied to remove noisy images corrupted by Gaussian, impulse, and multiplicative noises [4], [37], [38]. If K=I, (1.2) becomes a denoising model, where f is the known noisy image. In [12], for modeling the restoration of images corrupted by Poisson noise, the authors proposed to minimize a generalized Kullback–Leiblerdivergence term plus a smoothed version of TV.

For the Poisson restoration problem, the TV regularization performs very well on piecewise constant images for preserving edges while removing noise. But it often causes staircase effects in flat regions. In particular, this phenomenon is more obvious for processing piecewise smooth images. In this case, some high-order total variation (HOTV) regularizations have been introduced [18], [19], [24], [43]. These regularizations have better restoration performance than the TV regularization for preserving the smooth regions. One example of the HOTV regularizations can be described asminu2u1+λΩ(KuflogKu)dx,where2u1=i=1mj=1l(uxx(i,j))2+(uxy(i,j))2+(uyx(i,j))2+(uyy(i,j))2.

However, while using this model for Poisson image restoration, the image edges will be blurred. To preserve edges while suppressing staircase effects, the variational model with TGV regularization was presented for dealing with Poisson image restoration problem [16]. Recently, researchers have proposed a directional TGV regularization for Poisson image restoration [13].

In some special images, the textures show obvious directionality. Therefore, the proposed model should be built to describe the geometric features of images. However, in the numerical calculation, the finite difference scheme of gradient for the TV-based models only depends on the horizontal and vertical directions. In this case, these models based on this scheme cannot couple with local structures of images efficiently. Fortunately, the directional total variation (DTV) method could overcome this shortcoming by adding a rotation matrix to rotate the gradient operator [3], [21], [29], [41]. Besides, this method adds a weighted matrix to enhance the diffusion of the corresponding Euler–Lagrange equation. Specifically, the DTV-based Poisson restoration model can be formulated as follows:minuΛβRθu1+λΩ(KuflogKu)dx,where the matrix Λβ and the rotation matrix Rθ are defined byΛβ=[β001]andRθ=[cosθsinθsinθcosθ].Here β>1 is a weight parameter and θ is the affine angle. As reported in Bayram and Kamasak [3], for restoring images with a dominant direction, the DTV-based model has better performance than the classical TV-based models. However, the DTV-based model does not perform very well when restoring images with complex structures [29]. In other words, when the image has several dominant directions, the DTV-based model will not achieve the ideal restoration effect. This fact motivates us to study some new schemes. In a recent work, Pang et al. proposed an adaptive weighted TVp regularization for Gaussian noise removal [29]. Inspired by the model proposed in Pang et al. [29], in this paper, we propose a Poisson restoration model based on an adaptive weighted directional TVp regularization (AWDTVp) to describe the local structures of the image. The major contributions of the paper are three-fold:

  • To the best of our knowledge, this is the first time that AWDTVp is considered for Poisson noise. Owing to the combination of the rotation matrix, the adaptive weighted matrix and the TVp-quasinorm, the proposed model has more robust adaptivity and more stronger restoration abilities.

  • Since the proposed model is non-smooth and non-Lipschitz, an ADMM is presented for solving it, where the convergence of the relevant 2p subproblem is guaranteed. The complexity analysis of the proposed method is also given.

  • Experiments including natural images and synthetic images are conducted under different blurs and noise levels to demonstrate that our AWDTVp has superior performance over the state-of-the-art methods in Poisson image restoration.

This paper is organized as follows. In Section 2, the AWDTVp-based Poisson restoration model is proposed. Section 3 presents the alternating direction method of multipliers for solving this model. In Section 4, experiments are carried out to show the effectiveness of our method, compared with other three models for Poisson image restoration. Finally, Section 5 concludes the paper.

Section snippets

Our proposed model

In this section, we first introduce the definition of adaptive weighted DTV p-quasinorm (AWDTVp). Then we propose a Poisson restoration model based on the AWDTVp.

Generally speaking, different regions of an image have different structures. To describe them, it is very important to keep the diffusion of the corresponding Euler–Lagrange equation along with the tangential direction of the edge. Consequently, we need to adaptively rotate the gradient to fit in with the tangential direction. To this

Algorithm

In this section, we will present an ADMM for solving the proposed model (2.3). The solution of the original problem is transformed into alternating calculation of several related subproblems. Firstly, we need to focus on how to solve the 2p problem, since one of the subproblems has this form.

Numerical experiments

In this section, we give some experimental results to show the performance of AWDTVp. Compared with the TV model [33], the HOTV model [43] and the DTV model (1.5) in Poisson image restoration, numerical results and images will prove the superiority of our AWDTVp model. In fact, the DTV model is also our new model which has some advantages on restoring images with a dominate direction. All the numerical experiments are conducted in Matlab environment on a PC with 2.30 GHz Intel(R) Core(TM) i5

Conclusion

In this paper, we proposed a novel Poisson restoration model based on an adaptive weighted directional TVp regularization. Furthermore, we presented an efficient ADMM to solve the proposed model. One of the subproblems was solved by the half-quadratic algorithm with guaranteed convergence. Experimental results demonstrated the superior performance of the proposed approach over other three competing ones. In the future, we will explore some acceleration techniques to further reduce the

CRediT authorship contribution statement

Jun Zhang: Conceptualization, Methodology, Software, Formal analysis, Writing – review & editing. Pengcheng Li: Software, Investigation, Data curation, Writing – original draft. Junci Yang: Visualization. Mingxi Ma: Validation, Supervision. Chengzhi Deng: Writing – review & editing.

Declaration of Competing Interest

Authors declare that they have no conflict of interest.

Acknowledgments

This work was supported by the Science Foundation for Post Doctorate of China (2020M672484), the Natural Science Foundation of Jiangxi Province (20192BAB211005), and the NNSF of China (61865012), the Guangxi Natural Science Foundation Program (2018GXNSFAA138056).

References (46)

  • I. Bayram et al.

    Directional total variation

    IEEE Signal Process. Lett.

    (2012)
  • J.M. Bioucas-Dias et al.

    Multiplicative noise removal using variable splitting and constrained optimization

    IEEE Trans. Image Process.

    (2010)
  • K. Bredies et al.

    Total generalized variation

    SIAM J. Imaging Sci.

    (2010)
  • R.H. Chan et al.

    Half-quadratic algorithm for pq problems with applications to TV-1 image restoration and compressive sensing

    Global Optim. Methods

    (2014)
  • T.F. Chan et al.

    Continuation method for total variation denoising problems

    International Symposium on Optical Science, Engineering, and Instrumentation

    (1995)
  • X. Chen et al.

    An augmented Lagrangian method for non-Lipschitz nonconvex programming

    SIAM J. Numer. Anal.

    (2017)
  • M.R. Chowdhury et al.

    Poisson image denoising based on fractional-order total variation

    Inverse Probl. Imaging

    (2020)
  • M.R. Chowdhury et al.

    Non-blind and blind deconvolution under Poisson noise using fractional-order total variation

    J. Math. Imaging Vis.

    (2020)
  • N. Dey et al.

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

    Microsc. Res. Techniq.

    (2006)
  • D. di Serafino et al.

    Directional TGV-based image restoration under Poisson noise

    J. Imaging

    (2021)
  • F.X. Dupe et al.

    Image deconvolution under Poisson noise using sparse representations and proximal thresholding iteration

    IEEE International Conference on Acoustics, Speech and Signal Processing

    (2008)
  • M.A.T. Figueiredo et al.

    Restoration of Poissonian images using alternating direction optimization

    IEEE Trans. Image Process.

    (2010)
  • Y. Gao et al.

    Total generalized variation restoration with non-quadratic fidelity

    Multidimens. Syst. Signal Process.

    (2018)
  • Cited by (0)

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