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# Second-order TGV model for Poisson noise image restoration

- Hou-biao Li
^{1}Email author, - Jun-yan Wang
^{1}and - Hong-xia Dou
^{1}

**Received:**25 April 2016**Accepted:**27 July 2016**Published:**5 August 2016

## Abstract

Restoring Poissonian noise images have drawn a lot of attention in recent years. There are many regularization methods to solve this problem and one of the most famous methods is the total variation model. In this paper, by adding a quadratic regularization on TGV regularization part, a new image restoration model is proposed based on second-order total generalized variation regularization. Then the split Bregman iteration algorithm was used to solve this new model. The experimental results show that the proposed model and algorithm can deal with Poisson image restoration problem well. What’s more, the restoration model performance is significantly improved both in visual effect and objective evaluation indexes.

## Keywords

- Image restoration
- Poisson noise
- Total generalized variation
- Split Bregman iteration
- Optimization problem

## Background

*u*to be defined on \(\Omega \subset {{\mathbb {R}}^2}\) and obtained a solution from a minimization problem

The rest of this article is organized as follows. In “Total generalized variation (TGV)” section, we briefly review the total generalized variation (TGV). The proposed model and algorithm are presented in “The proposed Poisson noise recovering model and algorithm” section. In “Experimental results and discussions” section, experimental results are illustrated to show the consistent performance of the proposed method. Finally, conclusions are given in “Conclusions” section.

## Total generalized variation (TGV)

Bredies et al. (2010) proposed the concept of total generalized variation (TGV), which is considered to be the generalization of TV. For convenience, some concepts of TGV are given as follows.

###
**Definition 1**

*k*with weight \(\alpha\) for \(u\in L_{loc}^1(\Omega )\) is defined as the value of the function

*k*with arguments in \({\mathbb {R}}^d\), and \({\alpha _l}\) are fixed positive parameters.

###
**Definition 2**

###
**Definition 3**

*l*. Moreover, the operator \(\varepsilon \left( {{u_{l - 1}}} \right)\) denotes the symmetrized gradient operator

## The proposed Poisson noise recovering model and algorithm

### The proposed model for Poisson noise image

*P*denotes the Poisson distribution function.

*P*(

*u*) is TGV and \(\left\| u \right\| ^2\), which can be written as

*S*

### The split Bregman algorithm for Poisson noise removal

*w*,

*x*,

*y*and

*z*, the problem (19) can be reformulated as the following constrained optimization problem

*w*-

*subproblem*, note that it is separable with respect to each component. It is easy to solve and the solution of

*w*may be written as

*x*,

*y*-

*subproblem*, we can directly obtain the solutions by using shrinkage operator:

*x*-

*subproblem*can be solved by

*y*-

*subproblem*is similarly obtained

*u*,

*p*)-subproblem is a saddle-point problem, which can be divided into the following two subproblems:

- 1.For
*u*, we havewhich can be solved by considering the following normal equation$$\begin{aligned} \begin{aligned} {u^{k + 1}}&= \mathop {\arg \min }\limits _u \frac{\lambda }{2}\left\| u \right\| _2^2 + \frac{{\beta {\mu _1}}}{2}\left\| {Ku - {w^{k + 1}} - b_1^k} \right\| _2^2\\&\quad + \frac{{{\alpha _1}{\mu _2}}}{2}\left\| {Du - p - {x^{k + 1}} - b_2^k} \right\| _2^2 + \frac{{{\mu _4}}}{2}\left\| {u - {z^{k + 1}} - b_4^k} \right\| _2^2, \end{aligned} \end{aligned}$$(28)Finally,$$\begin{aligned} \begin{aligned}&\lambda u + {\alpha _1}{\mu _2}\sum \limits _{j = 1}^2 {D_j^T\left( {{D_j}u - {p_j} - x_j^{k + 1} - b_{{2_j}}^k} \right) } \\& \quad + \beta {\mu _1}{K^T}\left( {Ku - {w^{k + 1}} - b_1^k} \right) + {\mu _4}\left( {u - {z^{k + 1}} - b_4^k} \right) = 0. \end{aligned} \end{aligned}$$(29)*u*is solved by$$\begin{aligned} \begin{aligned} {u^{k + 1}}&= {\left( \lambda I + \beta {\mu _1}{K^T}K + {\alpha _1}{\mu _2}\sum \limits _{j = 1}^2 {D_j^T{D_j}} + {\mu _4}I \right) ^{ - 1}}\\&\quad \times \left( \beta {\mu _1}{K^T}\left( {w^{k + 1}} + b_1^k \right) \right. + {\alpha _1}{\mu _2}\sum \limits _{j = 1}^2 {D_j^T} \left( {p_j} + x_j^{k + 1} + b_{{2_j}}^k \right) \\&\quad \left. {+ {\mu _4}({z^{k + 1}} + b_4^k)} \right) . \end{aligned} \end{aligned}$$ - 2.For the sub-problem
*p*, it can be written as the following minimization problemwhere \(p=(p_1,p_2)^T\) is a \(2\times 1\) vector, \(\varepsilon (p)\) is a \(2\times 2\) matrix.$$\begin{aligned} \begin{aligned} {p^{k + 1}}& = {} \mathop {\arg \min }\limits _p \frac{{{\alpha _1}{\mu _2}}}{2}\left\| {Du - p - {x^{k + 1}} - b_2^k} \right\| _2^2\\& \quad + \frac{{{\alpha _0}{\mu _3}}}{2}\left\| {\varepsilon (p) - {y^{k + 1}} - b_3^k} \right\| _2^2, \end{aligned} \end{aligned}$$(30)For \(p_1\), it can be solved by considering the following linear systemTherefore,$$\begin{aligned}&{\alpha _1}{\mu _2}\left( {{p_1} - {D_1}u + x_1^{k + 1} + b_{{2_1}}^k} \right) + {\alpha _0}{\mu _3}D_1^T\left( {{D_1}{p_1} - y_1^{k + 1} - b_{{3_1}}^k} \right) \nonumber \\&\quad +\frac{{{\alpha _0}{\mu _3}}}{2}D_2^T\left( {{D_2}{p_1} + {D_1}{p_2} - 2y_3^{k + 1} - 2b_{{3_3}}^k} \right) = 0. \end{aligned}$$(31)Similarly, we can obtain the solution of \(p_2\) as$$\begin{aligned} p_1^{k + 1}& = {({\alpha _1}{\mu _2}I + {\alpha _0}{\mu _3}D_1^T{D_1} + \frac{{{\alpha _0}{\mu _3}}}{2}D_2^T{D_2})^{ - 1}} \nonumber \\& \quad \times ({\alpha _1}{\mu _2}({D_1}u - x_1^{k + 1} - b_{{2_1}}^k) + {\alpha _0}{\mu _3}D_1^T(y_1^{k + 1} + b_{{3_1}}^k)\nonumber \\& \quad +\frac{{{\alpha _0}{\mu _3}}}{2}D_2^T(2y_3^{k + 1} + 2b_{{3_3}}^k - {D_1}{p_2})). \end{aligned}$$(32)$$\begin{aligned} p_2^{k + 1}& = {\left( {\alpha _1}{\mu _2}I + {\alpha _0}{\mu _3}D_2^T{D_2} + \frac{{{\alpha _0}{\mu _3}}}{2}D_1^T{D_1}\right) ^{ - 1}} \nonumber \\& \quad \times {\alpha _1}{\mu _2}\left( {D_2}u - x_2^{k + 1} - b_{{2_3}}^k \right) + {\alpha _0}{\mu _3}D_2^T\left( y_2^{k + 1} + b_{{3_2}}^k \right) \nonumber \\& \quad + \frac{{{\alpha _0}{\mu _3}}}{2}D_1^T\left( 2y_3^{k + 1} + 2b_{{3_3}}^k - {D_2}{p_1}\right) . \end{aligned}$$(33)

## Experimental results and discussions

In this section, we illustrate some numerical results of the proposed model for the Poisson noise removal problem. We compare our method with the one proposed in Figueiredo and Bioucas-Dias (2010) (PIDAL) and the other proposed in Liu and Huang (2012) (PID-Split). In order to prove the superiority of the proposed model, we compare our model with TGV regularization model.To show the effectivity of the proposed model, we choose four pictures possed abundant detail information.

*u*and \(\widehat{u}\) are the ideal image and the restored image, respectively.

*u*and \(\widehat{u}\), respectively. \(\sigma _u\) and \(\sigma _{\widehat{u}}\) are the variance of

*u*and \(\widehat{u}\), respectively. \(\sigma _{u{\widehat{u}}}\) is the covariance of

*u*and \(\widehat{u}\). The positive constants \(C_1\) and \(C_2\) can be thought of as stabilizing constants for near-zero denominator values. Generally speaking, the more bigger value of SNR, PSNR or the smaller value of RelErr is, the better quality of the reconstructed image is.

The Poissonian images used for our experiments are generated as follows: the original images are convoluted with a blur kernel and additionally contaminated by Poisson noise, here we use the \(\mathbf {poissrnd}\) function in MATLAB’s Statistics Toolbox after blurring the true images with the given point spread functions to generate the blurred and noise images.

The selection of the regularization parameters highly affects the image restoration results, and related to make the fair comparison with different denoising models. The penalty parameters \(\mu\) which relies on unknown noise level highly influences the speed of the algorithms. In experiments, we set \(\mu =[0.01, 0.001]\) in the PIDAL algorithm. In the PID-Split algorithms, we choose \(\mu =[0.0004; 0.1, 0.0001]\). In the TGV model, we set \(\mu =[0.1; 10, 5, 3]\). The penalty parameter in the proposed method is empirically set \(\mu =[0.1; 0.6; 0.1; 0.02]\). Thus, we may have a good restoration results.

*psfGauss*(5, 2) proposed in Nagy et al. (2004) on the original image and add the Poisson noise to the blurred data to generate the degraded image in Fig. 1b. The parameter of this test, we set \(\beta =120\) in PIDAL algorithm, \(\beta =6,\lambda =0.01\) for PID-Split algorithms,due to the TGV model we set \(\beta =450, \alpha =[8, 10]\), set \(\beta =54, \lambda =0.001, \alpha =[16, 9]\) for the proposed model. The pictures of Fig. 1c–f are the restoration images, which represent the difference between the three methods. From these pictures, we can see the proposed model have more advantages. In order to more effectively reflect the experiment result, Fig. 1g–j present the residual images refer to the difference of the original image and the restoration image. From these pictures, we can see that the proposed model can preserve more details than other methods. In the Table 1, the

*SNR*,

*PSNR*,

*RelErr*and

*SSIM*values of the restored images by the proposed model are better than other methods.

*PIDAL*method and PID-Split algorithms have better restoration results. In Fig. 2g–j, we have enlarged some details of the images, which can be clearly see the advantages of the proposed model for the recovery of edge details. The

*SNR*,

*RelRrr*and

*SSIM*values in Table 1 showed that the proposed model have a better restoration result.

*r*pixels, with an angle of \(\theta\) degrees in a counter-clockwise direction. In this example, \(r = 2\) and \(\theta = 45\), then add the Poisson noise to the blurred data to generate the degraded image in Fig. 3b. The parameters choose as the same as those in second experiment and also may be adjusted.

Summarized all of the experiment restoration results

Method | SNR | PSNR | RelErr | SSIM |
---|---|---|---|---|

Test1 | ||||

PIDAL (Figueiredo and Bioucas-Dias 2010) | 25.33 | 30.30 | 0.054 | 0.904 |

PID-Split (Liu and Huang 2012) | 25.18 | 30.16 | 0.055 | 0.917 |

TGV | 25.31 | 30.32 | 0.053 | 0.908 |

Proposed | 25.48 | 30.45 | 0.053 | 0.922 |

Test2 | ||||

PIDAL (Figueiredo and Bioucas-Dias 2010) | 21.17 | 25.03 | 0.087 | 0.828 |

PID-Split (Liu and Huang 2012) | 21.38 | 25.36 | 0.084 | 0.837 |

TGV | 21.24 | 25.10 | 0.086 | 0.835 |

Proposed | 21.74 | 25.60 | 0.080 | 0.843 |

Test3 | ||||

PIDAL (Figueiredo and Bioucas-Dias 2010) | 21.50 | 27.87 | 0.0841 | 0.811 |

PID-Split (Liu and Huang 2012) | 21.49 | 27.86 | 0.0357 | 0.811 |

TGV | 20.64 | 27.01 | 0.092 | 0.801 |

Proposed | 21.01 | 27.23 | 0.0803 | 0.820 |

## Conclusions

In this paper, we investigate the second-order total generalized variation with a quadratic regularization to deal with the Poissonian images restoration problem. The proposed model is solved efficiently by split Bregman iterative algorithm in this way the calculation speed is fast. Numerical results show that our proposed method is particularly advantageous for restoration the Poisson images in terms of *SNR*, *SSIM* and *RelErr* quality compared to other methods. In the model, the parameters selection is a difficult problem which needs further study.

## Declarations

### Authors' contributions

All authors completed the paper together. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments. This work was supported by National Natural Science foundation of China (51175443, 11101071).

### Competing interests

The authors declare that they have no competing interests.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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