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Accurate image restoration is of paramount importance for low-level vision, computer vision, and various other fields. Numerous complex restoration algorithms have been stated in the literature. Performance of these restoration algorithms differs with the nature of the image and distortion. These algorithms are evaluated either qualitatively or quantitatively by comparing the restored image with the original image. The practical drawback of this quantitative comparison is the requirement for the original image. In this paper, we propose a measure, with theoretical grounding, to objectively evaluate the restoration algorithms with no knowledge about the original image. This measure analyses the deblurring as well as the denoising nature of the restoration method. The main utility of this measure will be in designing automatic restoration systems. Effectiveness of this measure is substantiated with experimental results.

Images become blurred when there is a smearing of the point spread function caused by certain imaging conditions such as camera motion, object motion, atmospheric turbulence, and so forth. In addition, sensor noise leads to additional degradation of the image. Image restoration [

Numerous image restoration algorithms [

The improvement in the quality of the estimated image

In this paper, we introduce a new information theoretic approach (based on entropy) to evaluate restoration. Our new method provides a score to evaluate the parameterizations of a single restoration algorithm as well as fundamentally different restoration algorithms. This measure does not demand any knowledge about the original image thus making it feasible to be used to automate the restoration process in the real field. The proposed measure quantifies the noise reducing as well as the deblurring capability of restoration algorithms.

The advantages of the proposed objective evaluation are the following. (i) It automates the restoration process in real time as it does not demand any knowledge of the original image, (ii) it ranks the algorithms based on the data fidelity as well the denoising capability, and (iii) The proposed measure does not assume the statistics of the noise or the image.

Section

This section brings out our information theoretic approach that could be used for restoration evaluation and justifies the use of each and every term of the proposed measure in the evaluation process.

A good restoration method should concentrate on estimating images as close as possible to the original image. The deviation of the estimate from the original image is a quantitative measure for the data fidelity of the restoration algorithm. Thus, the data fidelity of restoration algorithms can be quantified with a function

Maximum likelihood (ML) estimation or least square (LS) estimation could be used as

Due to the inherent disadvantages of ML and LS, we use entropy (

Equation (

This section proves the significance of the DF term in quantifying the deblurring effect of the given algorithm. The residue

Let us consider the mutual information between

When the image and the blur are estimated perfectly,

There are restoration algorithms, which are good at deblurring but not at denoising (such as inverse filter). Consider two different algorithms (algo 1 and algo 2) producing estimates of the image and blur (

The entropy is a measure of disorder in the image. The randomness of noise leads to higher entropy, whereas the natural images which are not as random as that of noise, possess smaller entropy [

An alternate insight of our evaluation method can be obtained by applying MDL [

This section brings out the results of the experiments conducted on the restoration results, to test and validate the efficacy of the proposed measure. We have studied the effectiveness of the proposed measure by verifying its consistency with the conventional Distortion Measure (DM – (

Table

Details of the experiments conducted.

Experiment | Images | Blur | Noise | De-convolution methods |
---|---|---|---|---|

Exp 1a | single chip image | Box blur ( | No noise | Wiener [ |

Exp 1b | single Lena image | Box blur ( | Poisson noise | Wiener, CLS, Inverse |

Exp 2a | Two Rice images | Gaussian blur | Gaussian of variance (in dB) | Wiener, CLS, inverse filter, Total variation [ |

Exp 2b | 20 Mountain images | Each blurred by Motion blurs of different trajectory. | Gaussian noise | Wiener |

Exp 2c | Single Cameraman image | Box blur ( | Gaussian noise | CLS [ |

The entropies of the residue and blur kernel are estimated as given in (

_{1} term, _{1} (

where

This section gives the results of Exp 1a and 1b (Table

Restoration results in no noise scenario (a) Blurred Image, (b) Wiener, (c) Inverse, (d) CLS restored images.

Restoration results in Poisson noise scenario (a) Blurred Image, (b) Wiener, (c) Inverse, (d) CLS restored images.

Table

Comparison of Proposed and Distortion measure of Experiments

Algo | No noise | With noise | ||
---|---|---|---|---|

Proposed | DM | Proposed | DM | |

Wiener | ||||

Inverse | 12.96 | 30.9 | 15.4 | 18.8 |

CLS | 12.85 | 28.3 | 11.1 | 13.1 |

This section furnishes the results of the three experiments Exp 2a, 2b, and 2c (refer Table

Comparison of Proposed and Distortion measure of Experiment

Algo | Noise | Noise | ||
---|---|---|---|---|

Proposed | DM | Proposed | DM | |

Wiener | 11.8 | 10.5 | ||

CLS | 12.1 | 11.3 | 11.6 | 10.8 |

Inverse | 15.5 | 14.2 | 14.3 | 13.3 |

Lucy | 13.2 | 12.1 | 12.5 | 11.5 |

TV | 11.1 | 10 |

Figure

Cross correlation between Proposed and DM.

Figure

Results of Exp 2c. (a) Proposed versus lambda, (b) DM versus lambda.

We have proposed a measure to analyse the performance of restoration algorithms for the given blurred image, which (i) does not demand any knowledge about the original image and (ii) derived without any assumption of the noise or image statistics. This helps in automating the restoration process without the intervention of the user. The given measure has a data fidelity term and noise assessing terms, thus analysing the deblurring as well as the denoising nature of the restoration method. The authenticity of the proposed measure was substantiated with different sets of empirical results.