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A class of nonlinear fourth-order telegraph-diffusion equations (TDE) for image restoration are proposed based on fourth-order TDE and bilateral filtering. The proposed model enjoys the benefits of both fourth-order TDE and bilateral filtering, which is not only edge preserving and robust to noise but also avoids the staircase effects. The existence, uniqueness, and stability of the solution for our model are proved. Experiment results show the effectiveness of the proposed model and demonstrate its superiority to the existing models.

The use of second-order partial differential equations (PDE) has been studied as a useful tool for image restoration (noise removal). They include anisotropic diffusion equations [

A rather detailed analysis of blocky effects associated with anisotropic diffusion which was carried out in [

In 2000, You and Kaveh [

Recently, Ratner and Zeevi [

Inspired by the ideas of [

The ability of edge preservation in the fourth-order TDE-based image restoration method strongly depends on the conductance coefficient

We will recall that the main purpose of the function

The remainder of this paper is organized as follows. In Section

In this section, we establish the existence and uniqueness of the following problem:

The following standard notations are used throughout. We denote

We denote by

Let

Now, we consider the existence and uniqueness of weak solutions of the following linear TD problem

A function

Suppose that

In order to prove Theorem

Suppose that

By the definition of

If there exists

Multiplying (

Note that

Consequently, (

Now, write

Consequently,

By Lemma

Suppose that

Suppose that

In the following, we prove the theorem in two parts.

In this first section, we show the existence of a weak solution of (

In order to apply the Schauder fixed point theorem, we need to prove that

Letting

Now, we turn to the proof of the uniqueness, following the idea in [

Finally, we apply the same argument on the intervals

In this section, we construct an explicit discrete scheme to numerically solve differential equation (

where

(a) calculating the Laplace of the image intensity functions

In this section, we present numerical results obtained by applying our proposed fourth-order TDE to image denosing. We test the proposed method on “Barbara” image with size

Comparison of different methods on “Barbara” image. (a) Original image. (b) Noised image. (c) Fourth-order TDE with

We first study the effects of damping coefficient

Graph of ISNR versus different damping coefficient

Comparison of different methods on “License plate” image. (a) Original image. (b) Noisy image. (c) PM smodel with

Next, we test the proposed method for image restoration on 50 synthetic degraded images generated using random white Gaussian noise of variance

ISNR (in dB) values for “license plate” image using random white Gaussian noise of variance

method | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

PM model | 5.7246 | 5.7309 | 5.7447 | 5.7127 | 5.7502 | 5.7354 | 5.7447 | 5.7049 | 5.7254 | 5.5188 |

TDE model | 5.8335 | 5.8536 | 5.8443 | 5.8751 | 5.8271 | 5.6747 | 5.8327 | 5.8573 | 5.8767 | 5.8187 |

ITDE model | 5.9147 | 5.9152 | 5.9217 | 5.9109 | 5.9198 | 5.9107 | 5.9122 | 5.9098 | 5.9150 | 5.9112 |

Fourth-order TDE | 6.1719 | 6.1382 | 6.1641 | 6.1268 | 6.1626 | 6.1431 | 6.1543 | 6.1797 | 6.1162 | 6.1883 |

Finally, we designed to further evaluate the good behavior of our proposed fourth-order TDE with white Gaussian noise across 5 noise levels. We added Gaussian white noise across five different variances

ISNR (in dB) values for “license plate” image using different methods across five noise levels.

Method | With Gaussian white noise | ||||

PM model | 6.9930 | 6.5647 | 5.7246 | 4.9859 | 4.3677 |

TDE model | 7.1032 | 6.6143 | 5.8335 | 5.1291 | 4.5038 |

ITDE model | 7.2165 | 6.8372 | 5.9147 | 5.3022 | 4.6195 |

Fourth-order TDE | 7.3648 | 6.9296 | 6.1719 | 5.5536 | 4.8564 |

The mean of ISNR (in dB) values for eight standard test images using random white Gaussian noise of variance 20.

Methods | Noise images | |||||

Lena | Boats | Cameraman | Peppers | House | Elaine | |

PM model | 4.3275 | 4.1487 | 4.0326 | 5.1120 | 4.2726 | 5.1737 |

TDE model | 4.4662 | 4.2745 | 4.1689 | 5.2022 | 4.3996 | 5.2461 |

ITDE model | 4.5743 | 4.3948 | 4.2917 | 5.2872 | 4.5127 | 5.3830 |

Fourth-order TDE | 4.7125 | 4.5222 | 4.4134 | 5.4431 | 4.6475 | 5.5705 |

A class of nonlinear fourth-order telegraph-diffusion equation (TDE) for image restoration is presented in this paper. The proposed model first extends the second order TDE for image restoration to fourth-order TDE. Moreover, our proposed model combines nonlinear fourth-order TDE with bilateral filtering, which is not only edge preserving and robust to noise but also avoids the staircase effects. Finally, we study the existence, uniqueness, and stability of the proposed model. A set of numerical experiments is presented to show the good performance of our proposed model. Numerical results indicate that the proposed model recovers well edges and reduces noise. In [

This work was supported by National Natural Science Foundation of China under Grant no. 60972001 and National Key Technologies R & D Program of China under Grant no. 2009BAG13A06. The authors would like to acknowledge Professor Qilin Liu and Yuxiang Li from Department of Mathematics for many fruitful discussions. The authors also thank the anonymous reviewer for his or her constructive and valuable comments, which helped in improving the presentation of our work.

^{1}solutions of a class of fourth order nonlinear equations for image processing