We propose a fourth-order total bounded variation regularization model which could reduce undesirable effects effectively. Based on this model, we introduce an improved split Bregman iteration algorithm to obtain the optimum solution. The convergence property of our algorithm is provided. Numerical experiments show the more excellent visual quality of the proposed model compared with the second-order total bounded variation model which is proposed by Liu and Huang (2010).

Image restoration problem is one of the earliest and most classic linear inverse problems [

Equation (

In [

As mentioned in [

Let

The proof of the existence, uniqueness, convergence, and stability of our proposed model (

The organization of the rest of paper is as follows. In Section

Bregman iteration is a concept that originated in function analysis for finding extrema of convex function [

In [

The Bregman distance of function

Goldstein and Osher [

Then pay attention to the subproblem

Step 1:

Step 2:

Due to [

So the proposed model (

Then we perform the split Bregman iteration to solve problem (

For more precisely, we derive our algorithm as follows.

We have the following steps.

Set input value

Update

Update

Update

If stopping criterion holds, output the

In this section, we concentrate on the rigorous convergence proof of our iterative algorithm. Our analysis below is similar to that in [

We note that the subproblems (

Assume that there exists a unique solution

Reference [

Let

In this section, a number of experiments are performed to demonstrate the effectiveness and efficiency of our proposed split Bregman iteration (PSBI) algorithm for the fourth-order diffusive model (

The performance of all algorithms is measured by the improved signal-to-noise ratio (ISNR) and mean squared error (MSE) defined as

Three classical grayscale images with resolution of

Images used for synthetic degradations: (a) “Cameraman,” (b) “Boat,” (c) “Lena.”

In the first experiment, the original image “Cameraman” with complex background is blurred by Gaussian blur, out-of-focus blur, linear motion blur, and uniform blur, respectively. The blurred images are showed in Figures

Computational cost, ISNR value, and MSE value for different deblur cases.

Blur type | CPU time (s) | ISNR (dB) | MSE (dB) | |||||
---|---|---|---|---|---|---|---|---|

SBI | PSBI | Blurred | SBI | PSBI | Blurred | SBI | PSBI | |

Motion | 3.1512 | 4.8672 | 8.6206 | 21.4132 | 31.2089 | 443.8974 | 36.8159 | 2.9369 |

Out-of-focus | 2.9952 | 4.3992 | 8.0647 | 18.7751 | 28.8994 | 501.1467 | 50.5383 | 4.9883 |

Uniform | 3.0264 | 4.8828 | 8.5342 | 17.4389 | 25.1493 | 449.3335 | 68.5647 | 11.7966 |

Gaussian | 3.1200 | 7.3164 | 9.1581 | 16.5027 | 21.9641 | 394.0131 | 84.6321 | 24.4465 |

Deblurring of “Cameraman." (a)–(d) Gaussian blurred image, linear motion blurred image, out-of-focus blurred image and uniform blurred image, respectively. (e)–(h) Images deblurred by SBI. (i)–(l) Images deblurred by PSBI.

In the next test, we use Gaussian blur and out-of-focus blur to degrade the “Boat” image and then run the two algorithms many times to obtain the best results. In PSBI method, the selected parameters and iterations are

Image deblurring using SBI and PSBI for “Boat.”

Blur type | CPU time (s) | ISNR (dB) | MSE (dB) | |||||
---|---|---|---|---|---|---|---|---|

SBI | PSBI | Blurred | SBI | PSBI | Blurred | SBI | PSBI | |

Out-of-focus | 2.9796 | 4.6488 | 7.2471 | 17.8352 | 28.6621 | 381.1651 | 40.9274 | 3.4444 |

Gaussian | 3.2448 | 7.3008 | 8.3851 | 16.0370 | 22.4141 | 297.3680 | 61.4972 | 14.4226 |

Comparison with SBI method. First column: out-of-focus blurred image and Gaussian blurred image. Second column: images deblurred by SBI. Third column: images deblurred by PSBI.

Close-ups of selected section of Figure

Figures

Image deblurring using SBI and PSBI for “Lena.”

Blur type | CPU time (s) | ISNR (dB) | MSE (dB) | |||||
---|---|---|---|---|---|---|---|---|

SBI | PSBI | Blurred | SBI | PSBI | Blurred | SBI | PSBI | |

Motion | 3.1824 | 4.4460 | 8.5843 | 17.6570 | 27.3921 | 300.0612 | 46.4993 | 5.1319 |

Uniform | 2.9952 | 4.8672 | 8.7268 | 15.7435 | 25.7129 | 289.5325 | 71.4055 | 7.2947 |

Comparison with SBI method. First column: motion blurred image and uniform blurred image. Second column: images deblurred by SBI. Third column: images deblurred by PSBI.

Close-ups of selected section of Figure

In this paper, we propose the fourth-order total bounded variation regularization based image deblurring model and exploit the split Bregman iteration method to solve this new model. The convergence analysis of our algorithm is provided. Numerical experiments show that our algorithm works well for images with complex background or simple background. In our synthetic experiments, the fourth-order diffusive model yields better results than the second-order diffusive model. It is believed that the proposed model can be extended to further applications in image processing and computer vision.

The authors would like to thank the referees very much for their valuable comments and suggestions.