## 1 Introduction

*image restoration problem*.

*u*is corrupted by some random noise

*η*, resulting in a noisy image

*f*. Our task is to reconstruct

*u*, knowing both

*f*and the distribution of

*η*. Of course, there is in general no way to find the

*exact*image

*u*; image restoration algorithms rather yield a good approximation of

*u*, usually noted ${u^{\ast }}$. To do so, they exploit

*a priori*knowledge on the image

*u*.

*et al.*, 1992; Pham and Kopylov, 2015), Poisson (Chan and Shen, 2005; Le

*et al.*, 2007), Cauchy (Sciacchitano

*et al.*, 2015), as well as some mixed noise models, e.g. mixed Gaussian-Impulse noise (Yan, 2013), mixed Gaussian–Salt and Pepper noise (Liu

*et al.*, 2017), mixed Poisson–Gaussian (Calatroni

*et al.*, 2017; Pham

*et al.*, 2018; Tran

*et al.*, 2019).

*et al.*, 2015). The mixture of Poisson and Gaussian noise occurs in several practical setups (e.g. microscopy, astronomy), where the sensors used to capture images have two sources of noise: a signal-dependent source which comes from the way light intensity is measured; and a signal-independent source which is simply thermal and electronic noise. Gaussian noise is just additive, so it cannot properly approximate the Poisson–Gaussian distributions observed in practice, which are strongly signal-dependent.

*f*is observed image,

*u*is the unknown image, $\mathcal{P}(u)$ means that the image

*u*is corrupted by Poisson noise, and $W\sim \mathcal{N}(0,{\sigma ^{2}})$ is a Gaussian noise with zero mean and variance

*σ*.

*et al.*, 2008; Jezierska

*et al.*, 2011; Lanza

*et al.*, 2014; Le Montagner

*et al.*, 2014). Many algorithms for denoising images corrupted by mixed Poisson–Gaussian noise have been investigated using approximations based on variance stabilization transforms (Zhang

*et al.*, 2007; Makitalo and Foi, 2013) or PURE-LET based approaches (Luisier

*et al.*, 2011; Li

*et al.*, 2018). Variational models based on the Bayesian framework have been also proposed for removing and denoising and deconvolution of mixed Poisson–Gaussian noise (Calatroni

*et al.*, 2017). This framework is perhaps a popular approach to mixed Poisson–Gaussian noise model. Authors in De Los Reyes and Schönlieb (2013) proposed a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation image denoising. Authors in Lanza

*et al.*(2014) focused on the maximum a posteriori approach to derive a variational formulation composed of the total variation (TV) regularization term and two fidelities. A weighted squared ${L_{2}}$ norm noise approximation was proposed for mixed Poisson–Gaussian noise in Li

*et al.*(2015), or an efficient primal-dual algorithm was also proposed in Chouzenoux

*et al.*(2015) by investigating the properties of the Poisson–Gaussian negative log-likelihood as a convex Lipschitz differentiable function. Recently, authors in Marnissi

*et al.*(2016) proposed a variational Bayesian method for Poisson–Gaussian noise, using an exact Poisson–Gaussian likelihood. Similarily, authors in Calatroni

*et al.*(2017) proposed a variational approach which includes an infimal convolution combination of standard data delities classically associated to one single-noise distribution, and a TV regularization as regularizing energy. Generally, image restoration by variational models based on TV can be a good solution to the mixed Poisson–Gaussian noise removal with the following formula (Calatroni

*et al.*, 2017; Pham

*et al.*, 2019):

##### (2)

\[ {u^{\ast }}=\underset{u\in S(\Omega )}{\operatorname{arg\,min}}{\int _{\Omega }}|\nabla u|+\frac{{\lambda _{1}}}{2}{\int _{\Omega }}{(u-f)^{2}}+{\lambda _{2}}{\int _{\Omega }}(u-f\log u),\]*f*is the observed image, $\Omega \subset {\mathbb{R}^{2}}$ is a bounded domain, and $S(\Omega )$ is the set of positive functions from Ω to $\mathbb{R}$; finally, ${\lambda _{1}},{\lambda _{2}}$ are positive regularization parameters (see Chan and Shen, 2005, for details on this method).

*et al.*(2018) proposed a modified scheme of gradient descent (MSGD) that guarantee positive values for each pixel in the image domain.

##### (3)

\[\begin{aligned}{}{u^{\ast }}& =\underset{u\in S(\Omega )}{\operatorname{arg\,min}}E(u),\\ {} E(u)& ={\int _{\Omega }}\alpha (x)\big|\nabla u(x)\big|dx+\frac{{\lambda _{1}}}{2}{\int _{\Omega }}{\big(u(x)-f(x)\big)^{2}}dx\\ {} & \hspace{1em}+{\lambda _{2}}{\int _{\Omega }}\big(u(x)-f(x)\log u(x)\big)dx,\end{aligned}\]*f*is the observed image, ${\lambda _{1}}$ and ${\lambda _{2}}$ are positive regularization parameters, $S(\Omega )=\{u\in \text{BV}(\Omega ):u>0\}$ is closed and convex, with $\text{BV}(\Omega )$ being the space of functions $\Omega \to \mathbb{R}$ with bounded variation; and finally $\alpha (x)$ is a continuous function in $S(\Omega )$.

*l*is a threshold value and $v(x)=|\nabla {G_{\sigma }}(x)\ast f|$, in which ∗ denotes the convolution with ${G_{\sigma }}(x)=\frac{1}{2\pi {\sigma ^{2}}}\exp \big(-\frac{{x^{2}}}{2{\sigma ^{2}}}\big)$, i.e. the Gaussian filter with standard deviation

*σ*.

*convex*, which enables us to use larger time-step parameters during gradient descent when solving (3). We introduce the influence function $\alpha (x)$, which acts as an edge-detection function, to get the model (3) in order to improve the ability of edge preservation and to control the speed of smoothing. In addition, we propose a new method to solve (3) that perceptibly improves the quality of the denoised images. By changing the time-step parameter, users can either get faster denoising with comparable results to previous methods, or better quality denoising with comparable running times. Our method is a technical improvement over the split-Bregman algorithm. We report experimental results for the aforementioned method, for various parameters in the noise distribution. The quality of denoising is measured with the SSIM and PSNR metrics. If we tune the time-step parameter to get similar quality result as the original split-Bregman method, we get faster running times.

## 2 Preliminaries

*u*, knowing the noisy image

*f*. Our strategy is to find the image

*u*which maximizes the conditional probability $P(u|f)$. Bayes’s rule gives: The probability density function of the observed image

*f*corrupted by Gaussian noise ${P_{\mathcal{N}}}$ (respectively, by Poisson noise ${P_{\mathcal{P}}}$) is:

*σ*is the variance of the Gaussian noise. As we explained in the introduction, the two sources of noise are independent of each other, so the distribution of the mixed noise may be expressed as:

*not*assume that the noise and the original image are independent of each other.) Suppose that

*f*has size $M\times N$, and let $I=\{1,\dots ,M\}\times \{1,\dots ,N\}$ denote the domain of

*f*. For

*i*in

*I*, we write ${f_{i}}$ the pixel of

*f*at position

*i*(and similarly ${u_{i}}$ the pixel of

*u*at position

*i*). Then,

##### (5)

\[ \sum \limits_{i\in I}{u_{i}}-{f_{i}}\log ({u_{i}})+\log ({f_{i}}!)+y{({u_{i}}-{f_{i}})^{2}},\]*u*varies but

*f*is constant. Since our goal is to minimize the whole expression, we can ignore the term $\log ({f_{i}}!)$ altogether.

*et al.*, 2007): where

*z*is a normalization factor. We need to make a couple of comments here. First,

*u*is not a function ${\mathbb{R}^{2}}\to \mathbb{R}$, but rather a discrete array of pixels; thus the integral in that expression is going to be translated to a sum, while $\nabla u$ will be translated as a linear approximation. Second, this assumption appears to contradict the previous one, that the pixels of the original image are independent of one another. However, the assumption on $P(u)$ is local: each pixel depends (weakly) on the neighbouring pixels only, so we do not lose much by assuming independence. This turns out to yield good results in practice (Chan and Shen, 2005).

##### (7)

\[ {u^{\ast }}=\underset{u}{\operatorname{arg\,min}}\sum \limits_{i\in I}\frac{1}{z}|\nabla {u_{i}}|+y{({u_{i}}-{f_{i}})^{2}}+\big({u_{i}}-{f_{i}}\log ({u_{i}})\big),\]##### (8)

\[ E(u)={\int _{\Omega }}|\nabla u|dx+\frac{{\lambda _{1}}}{2}{\int _{\Omega }}{(u-f)^{2}}dx+{\lambda _{2}}{\int _{\Omega }}(u-f\log u)dx,\]*z*, which is positive and constant, so the minimum is the same.) Intuitively, the last two terms are

*data fidelity*terms, which ensure that the restored image

*u*is not “too far” from the original image

*u*(taking the distribution of the noise into account). By contrast, $|\nabla u|$ is a

*smoothness*term, which guarantees that the reconstructed image is not too irregular (this is where our

*a priori*knowledge on the original picture lies). The parameters ${\lambda _{1}}$ and ${\lambda _{2}}$ will have to be determined experimentally later on.

## 3 Existence and Unicity of the Solution

##### Proof.

*h*are:

*f*is a positive, and $u\in S(\Omega )$, we have: ${h^{\prime\prime }}(u)>0$, i.e. $h(u)$ is strictly convex. Moreover, the TV regularization is convex, thence $E(u)$ is also strictly convex. □

##### Theorem 2.

*Let*$f\in S(\Omega )\cap {L^{\infty }}(\Omega )$

*, then the problem*(3)

*has an exactly one solution*$u\in BV(\Omega )$

*and satisfying*:

##### Proof.

*h*on ${\mathbb{R}^{+}}$, where

*g*satisfies:

*g*decreases if $t\in (0,f(x))$ and increases if $t\in (f(x),+\infty )$. Therefore, for every $V\geqslant f(x)$, we have

*et al.*(1999), we have: ${\textstyle\int _{\Omega }}|\nabla \inf (u,b)|\leqslant {\textstyle\int _{\Omega }}|\nabla u|$. Hence, $E(\inf (u,b))\leqslant E(u)$. In the same way, we have: $E(\sup (u,a))\leqslant E(u)$, where $a=\inf (f)$. Thence, we can assume $a\leqslant {u_{n}}\leqslant b$, the sequence $\{{u_{n}}\}$ is bounded in ${L^{1}}(\Omega )$.

## 4 Numerical Method

### 4.1 Discretization Scheme

##### (9)

\[\begin{aligned}{}& {u^{\ast }}=\underset{u\in S(\Omega )}{\operatorname{arg\,min}}E(u),\\ {} & E(u)={\int _{\Omega }}\alpha (x)|\nabla u|dx+\frac{{\lambda _{1}}}{2}{\int _{\Omega }}{(Ku-f)^{2}}dx+{\lambda _{2}}{\int _{\Omega }}(Ku-f\log Ku)dx,\end{aligned}\]*K*is a blurring operator (convex),

*f*is the observed image, $S(\Omega )$ is the set of positive functions defined over Ω with bounded total variation, and ${\lambda _{1}},{\lambda _{2}}$ are positive regularization parameters. This functional $u\mapsto E(u)$ is still strictly convex, because

*K*is assumed to be convex.

*u*is a image, we write ${u_{j,k}}$ for the pixel at coordinates $(j,k)$ in

*u*. We define the following quantities:

*ε*is a small positive number, added to avoid divisions by 0 in the implementation of the algorithms. Finding a minimum for the problem (2) can be achieved via the steepest gradient descent method

*et al.*, 2011; Boyd

*et al.*, 2010) which can be used for the minimization problem in (2). In this paper, we extend the split-Bregman algorithm (Goldstein and Osher, 2009) to solve the minimization problem.

### 4.2 Proposed Algorithm

*γ*and two convex functionals $\Psi (\cdot )$ and $G(\cdot )$; and that we need to solve the following constrained optimization problem:

##### (11)

\[ \text{find}\hspace{2.5pt}\underset{u,d}{\operatorname{arg\,min}}\| d{\| _{1}}+\frac{\gamma }{2}G(u)+\frac{\rho }{2}\| d-\Psi (u)-b{\| _{2}^{2}},\]*ρ*is a penalty parameter (a positive constant) and

*b*is a variable related to the split-Bregman iteration algorithm (to be explicited later). The solution to problem (11) can be approximated by the split-Bregman Algorithm (Goldstein and Osher, 2009):

##### (12)

\[ \underset{u,d}{\operatorname{arg\,min}}\bigg(\hspace{-0.1667em}\| d{\| _{1}}+\frac{\gamma }{2}G(\nu )+\frac{{\rho _{1}}}{2}\| \nu -Ku-c{\| _{2}^{2}}+\frac{{\rho _{2}}}{2}\hspace{-0.1667em}\sum \limits_{i=1,2}\| {d_{i}}-\alpha {\nabla _{i}}u-{b_{i}}{\| _{2}^{2}}\hspace{-0.1667em}\bigg),\]*γ*are positive, $d=({d_{1}},{d_{2}})$, $b=({b_{1}},{b_{2}})$ and $\nabla u=({\nabla _{1}}u,{\nabla _{2}}u)$.

*u*,

*ν*and

*d*.

*Subproblem 1.*The

*u*subproblem is a quadratic optimization problem, whose optimality condition reads:

##### (13)

\[\begin{aligned}{}& \bigg({\rho _{1}}{K^{T}}\cdot K+{\rho _{2}}\alpha \sum \limits_{i=1,2}{\nabla _{i}^{T}}{\nabla _{i}}\bigg){u^{(k+1)}}\\ {} & \hspace{1em}={\rho _{1}}{K^{T}}\big({\nu ^{(k)}}-{c^{(k)}}\big)+{\rho _{2}}\sum \limits_{i=1,2}{\nabla _{i}^{T}}\big({d_{i}^{(k)}}-{b_{i}^{(k)}}\big),\end{aligned}\]*et al.*(2008), equation (13) can be solved efficiently with one fast Fourier transform (FFT) operation and one inverse FFT operation as:

##### (14)

\[ u={\mathcal{F}^{-1}}\bigg(\frac{\mathcal{F}(r)}{{\rho _{1}}\mathcal{F}({K^{T}})\cdot \mathcal{F}(K)-{\rho _{2}}\mathcal{F}(\alpha )\cdot \mathcal{F}(\Delta )}\bigg),\]*Subproblem 2.*The optimality condition for the

*ν*subproblem is given by

*Subproblem 3.*The solution of the

*d*subproblem can readily be obtained by applying the soft thresholding operator (see Micchelli

*et al.*, 2011). We can use shrinkage operators to compute the optimal values of ${d_{1}}$ and ${d_{2}}$ separately:

##### (16)

\[ {d_{i}^{(k+1)}}=\text{shrink}\bigg(\alpha {\nabla _{i}}{u^{(k+1)}}+{b_{i}^{(k)}},\frac{1}{{\rho _{2}}}\bigg).\]*The algorithm.*The complete method is summarized in Algorithm 1. We need a stopping criterion for the iteration; we end the loop if the maximum number of allowed outer iterations

*N*has been carried out (to guarantee an upper bound on running time) or the following condition is satisfied for some prescribed tolerance

*ς*: where

*ς*is a small positive parameter. For our experiments, we set tolerance $\varsigma =0.0005$ and $N=500$.

##### Algorithm 1

## 5 Numerical Simulations

### 5.1 Implementation Issues

*et al.*, 2010), the Modified scheme for Mixed Poisson–Gaussian model (MS-MPG) (Pham

*et al.*, 2018) and the Bregman method (Goldstein and Osher, 2009). All of the compared methods perform image denoising with their optimal parameters. For a fair comparison, the regularization parameters of both MS-MPG and our proposed are the same: ${\lambda _{1}}=0.4$, ${\lambda _{2}}=0.6$. We set ${\rho _{1}}=1$, ${\rho _{2}}=1$. The parameter

*σ*in $\alpha (x)$ is set to 1. The threshold value

*l*in the function $\alpha (x)$ and the parameters

*γ*are chosen to keep the poise between noise removal and detail preservation capabilities.

*M*and

*N*are the number of image pixels in rows and columns.

*u*, ${u^{\ast }}$, respectively; ${\sigma _{u}},{\sigma _{{u^{\ast }}}}$, their standard deviations; ${\sigma _{u,{u^{\ast }}}}$, the covariance of two images

*u*and ${u^{\ast }}$; ${c_{1}}={({K_{1}}L)^{2}}$; ${c_{2}}={({K_{2}}L)^{2}}$;

*L*is the dynamic range of the pixel values (255 for 8-bit grayscale images); and finally ${K_{1}}\ll 1$, ${K_{2}}\ll 1$ are small constants.

### 5.2 Numerical Results and Discussion

#### 5.2.1 Image Denoising

##### Fig. 2

##### Fig. 3

##### Fig. 4

##### Fig. 5

##### Fig. 6

##### Fig. 7

##### Table 1

Image | Method | CPU time (s) | |

${I_{\max }}=120$ | ${I_{\max }}=60$ | ||

TV ${L_{1}}$ | 4.3449 | 5.6730 | |

Clock | Bregman | 0.9460 | 0.8212 |

MS-MPG | 4.1465 | 4.8734 | |

Ours | 1.0945 | 1.1081 | |

TV ${L_{1}}$ | 5.6229 | 7.4171 | |

Coco | Bregman | 1.0265 | 0.8414 |

MS-MPG | 4.0844 | 5.0879 | |

Ours | 1.1239 | 1.2251 | |

TV ${L_{1}}$ | 4.3096 | 6.4129 | |

Lamp | Bregman | 0.9225 | 0.9473 |

MS-MPG | 4.1810 | 4.8758 | |

Ours | 0.9431 | 1.1266 |

##### Table 2

Image | PSNR | MSSIM | ||||||||

Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | |

${I_{\max }}=120$, $\sigma =10$ | ||||||||||

Jetplane | 18.9416 | 22.7203 | 24.1190 | 24.7848 | 25.3251 | 0.4045 | 0.7061 | 0.7514 | 0.7511 | 0.7748 |

Lake | 19.6413 | 21.3675 | 22.5906 | 22.9972 | 24.4798 | 0.5235 | 0.6360 | 0.6812 | 0.7069 | 0.7603 |

Aerial | 17.4471 | 18.9550 | 19.5840 | 19.3051 | 19.8806 | 0.5582 | 0.5083 | 0.5808 | 0.5711 | 0.7130 |

Clock | 18.3852 | 24.6040 | 25.7945 | 24.8844 | 26.1201 | 0.2997 | 0.8339 | 0.8822 | 0.7796 | 0.8970 |

Car | 19.1385 | 21.4694 | 22.1559 | 22.8793 | 24.0620 | 0.4848 | 0.6106 | 0.6542 | 0.6804 | 0.7256 |

Coco | 16.9119 | 20.4242 | 20.4215 | 20.3426 | 20.6539 | 0.2755 | 0.8551 | 0.8798 | 0.8296 | 0.8950 |

Lamp | 17.8770 | 24.2808 | 24.3594 | 24.1062 | 24.6339 | 0.2446 | 0.8522 | 0.8891 | 0.7889 | 0.8985 |

Poulina | 18.8381 | 25.2567 | 25.7203 | 25.9781 | 26.0653 | 0.3250 | 0.7648 | 0.7934 | 0.7982 | 0.8074 |

Spine | 21.0004 | 25.2561 | 24.6855 | 25.5349 | 26.1010 | 0.6180 | 0.7925 | 0.7763 | 0.7967 | 0.8206 |

Head | 21.7787 | 24.3567 | 26.2348 | 26.9061 | 27.0979 | 0.6324 | 0.8033 | 0.8043 | 0.8273 | 0.8400 |

Average | 18.9960 | 22.8691 | 23.5666 | 23.7719 | 24.4420 | 0.4366 | 0.7363 | 0.7693 | 0.7530 | 0.8132 |

${I_{\max }}=120$, $\sigma =15$ | ||||||||||

Jetplane | 16.7150 | 22.2033 | 23.4915 | 23.6918 | 24.1415 | 0.3320 | 0.6761 | 0.7248 | 0.6959 | 0.7320 |

Lake | 17.2574 | 20.8215 | 22.0827 | 22.2260 | 23.0442 | 0.4384 | 0.6021 | 0.6732 | 0.6709 | 0.7040 |

Aerial | 15.8006 | 18.7671 | 19.2795 | 19.1060 | 19.4706 | 0.4622 | 0.4594 | 0.5740 | 0.5139 | 0.6472 |

Clock | 16.4619 | 24.2165 | 25.3645 | 24.2371 | 25.7740 | 0.2440 | 0.8105 | 0.8601 | 0.8186 | 0.8805 |

Car | 16.8589 | 20.9512 | 21.7735 | 22.1269 | 22.7608 | 0.4015 | 0.5809 | 0.6402 | 0.6338 | 0.6727 |

Coco | 15.4193 | 20.3398 | 20.4109 | 20.1332 | 20.5488 | 0.2181 | 0.8265 | 0.8599 | 0.7741 | 0.8789 |

Lamp | 16.0461 | 23.8972 | 23.9090 | 23.5169 | 24.3063 | 0.1964 | 0.8225 | 0.8695 | 0.7210 | 0.8799 |

Poulina | 16.6627 | 24.9195 | 25.2709 | 25.2753 | 25.4142 | 0.2452 | 0.7346 | 0.7659 | 0.7491 | 0.7739 |

Spine | 18.5582 | 23.7301 | 24.4015 | 24.3122 | 24.9272 | 0.5378 | 0.7418 | 0.7689 | 0.7521 | 0.7794 |

Head | 19.3512 | 24.549 | 25.4199 | 25.8356 | 25.9893 | 0.5588 | 0.7567 | 0.7836 | 0.7854 | 0.7991 |

Average | 16.9131 | 22.4395 | 23.1404 | 23.0461 | 23.6377 | 0.3634 | 0.7011 | 0.7520 | 0.7115 | 0.7748 |

##### Table 3

Image | PSNR | MSSIM | ||||||||

Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | |

${I_{\max }}=60$, $\sigma =10$ | ||||||||||

Jetplane | 14.0929 | 21.4515 | 22.5116 | 22.3705 | 22.8057 | 0.2570 | 0.6396 | 0.6482 | 0.6730 | 0.6854 |

Lake | 14.7190 | 20.1480 | 20.9885 | 20.7335 | 21.5586 | 0.3488 | 0.5567 | 0.5987 | 0.5945 | 0.6325 |

Aerial | 13.9091 | 18.7122 | 18.9386 | 18.6929 | 19.2898 | 0.3465 | 0.4036 | 0.5296 | 0.3801 | 0.5635 |

Clock | 13.9941 | 23.7554 | 24.7607 | 24.3166 | 25.0682 | 0.1866 | 0.7759 | 0.7865 | 0.7931 | 0.8439 |

Car | 14.2393 | 20.3390 | 20.8993 | 20.8988 | 21.4920 | 0.3124 | 0.5417 | 0.5723 | 0.5709 | 0.5864 |

Coco | 13.4373 | 19.9609 | 19.9082 | 20.0459 | 20.2665 | 0.1573 | 0.7969 | 0.7815 | 0.8218 | 0.8535 |

Lamp | 13.6235 | 23.3118 | 23.2870 | 23.4892 | 23.6101 | 0.1466 | 0.7898 | 0.7823 | 0.8067 | 0.8568 |

Poulina | 14.1692 | 24.2429 | 24.8768 | 24.8704 | 24.9272 | 0.1804 | 0.6901 | 0.7169 | 0.7252 | 0.7316 |

Spine | 16.0910 | 22.749 | 23.3981 | 23.3266 | 23.5011 | 0.4597 | 0.6821 | 0.7286 | 0.7153 | 0.7308 |

Head | 16.9718 | 23.667 | 24.2991 | 24.2780 | 24.4763 | 0.4970 | 0.7059 | 0.7494 | 0.7284 | 0.7550 |

Average | 14.5247 | 21.8338 | 22.3868 | 22.3022 | 22.6996 | 0.2892 | 0.6582 | 0.6894 | 0.6809 | 0.7240 |

${I_{\max }}=60$, $\sigma =15$ | ||||||||||

Image | Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours |

Jetplane | 11.4314 | 20.5604 | 21.0317 | 21.2727 | 21.3729 | 0.1911 | 0.5883 | 0.5208 | 0.6247 | 0.6319 |

Lake | 12.0450 | 19.3676 | 19.7789 | 19.9102 | 20.0911 | 0.2441 | 0.5053 | 0.5209 | 0.5526 | 0.5545 |

Aerial | 11.6216 | 18.4021 | 18.9001 | 18.5632 | 19.1482 | 0.2425 | 0.3435 | 0.4216 | 0.3518 | 0.4362 |

Clock | 11.4506 | 22.9914 | 23.6737 | 23.4250 | 24.3187 | 0.1365 | 0.7297 | 0.6387 | 0.7298 | 0.8123 |

Car | 11.5354 | 19.6031 | 19.8705 | 20.1081 | 20.2498 | 0.2163 | 0.4898 | 0.4776 | 0.5254 | 0.5311 |

Coco | 11.1477 | 19.6694 | 19.1809 | 19.7580 | 19.8315 | 0.1149 | 0.7432 | 0.6010 | 0.7628 | 0.8227 |

Lamp | 11.1182 | 22.6734 | 22.1005 | 22.8145 | 22.9551 | 0.1014 | 0.7341 | 0.6151 | 0.7430 | 0.8263 |

Poulina | 11.5927 | 23.4904 | 23.8398 | 23.8808 | 24.0040 | 0.1257 | 0.6353 | 0.6106 | 0.6818 | 0.6960 |

Spine | 13.4551 | 20.8085 | 21.9682 | 22.0129 | 22.1122 | 0.3844 | 0.6115 | 0.6493 | 0.6548 | 0.6658 |

Head | 14.3442 | 22.4799 | 22.4954 | 22.5105 | 22.9698 | 0.4370 | 0.6445 | 0.6890 | 0.6853 | 0.6991 |

Average | 11.9742 | 21.0046 | 21.2840 | 21.4256 | 21.7053 | 0.2194 | 0.6025 | 0.5745 | 0.6312 | 0.6676 |

#### 5.2.2 Image Deblurring and Denoising

##### Fig. 8

##### Fig. 9

##### Fig. 10

##### Fig. 11

##### Table 4

Image | PSNR | MSSIM | ||||||||

Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | Noisy | TV ${L_{1}}$ | Bregman | MS-MPG | Ours | |

${I_{\max }}=120$, $\sigma =15$ | ||||||||||

Jetplane | 14.9522 | 18.7079 | 18.72012 | 18.0420 | 19.0029 | 0.2282 | 0.6384 | 0.6600 | 0.5883 | 0.6860 |

Lake | 16.0535 | 19.6419 | 19.6100 | 19.6472 | 20.2675 | 0.2876 | 0.5506 | 0.5449 | 0.5654 | 0.6090 |

Aerial | 15.3701 | 18.3325 | 18.8549 | 18.7495 | 18.9921 | 0.3107 | 0.4647 | 0.4902 | 0.4960 | 0.5030 |

Clock | 16.1891 | 23.2348 | 23.5898 | 22.5893 | 23.7133 | 0.1761 | 0.7758 | 0.8196 | 0.6575 | 0.8313 |

Car | 15.5905 | 19.6202 | 19.6600 | 19.3774 | 20.1486 | 0.2408 | 0.5375 | 0.5486 | 0.5286 | 0.5863 |

Coco | 15.3829 | 20.1479 | 20.1762 | 19.0025 | 20.3572 | 0.1410 | 0.8082 | 0.8468 | 0.7524 | 0.8608 |

Lamp | 15.9477 | 23.4005 | 23.6315 | 21.6657 | 23.7635 | 0.1296 | 0.8001 | 0.8597 | 0.6919 | 0.8679 |

Poulina | 15.3475 | 19.5518 | 19.6801 | 20.2705 | 20.4392 | 0.1710 | 0.6871 | 0.7099 | 0.6923 | 0.7205 |

Spine | 16.0476 | 19.1907 | 18.6797 | 18.8847 | 19.3544 | 0.3865 | 0.5694 | 0.5832 | 0.6051 | 0.6286 |

Head | 14.9812 | 16.7888 | 16.6991 | 16.6044 | 18.3481 | 0.4590 | 0.6352 | 0.6562 | 0.6711 | 0.7061 |

Average | 15.5862 | 19.8617 | 19.9301 | 19.4833 | 20.4387 | 0.2531 | 0.6467 | 0.6719 | 0.6249 | 0.6999 |