Research on the Principle of Homology-Continuity in Image Degradation

Research on the Principle of Homology-Continuity in

Image Degradation

Liang Chen, Weijun Li*, and Chen Chen

Lab of Artificial Neural Networks,

Institute of Semiconductors, Chinese Academy of Sciences,

Beijing, China

Tel.: 010 8230 4336

{achenliang,wjli,eunicechen}@

Abstract. Image restoration is the inverse process of image degradation. Based

on the general imaging model to represent image degradation, various

restoration algorithms have been designed. However, none of these algorithms

have taken into account the inherent properties of Homology-Continuity in the

image degradation process. Such neglect leads to the ill-posedness and detail

loss that can not be completely overcome by the traditional image restoration.

In this paper, according to the Principle of Homology-Continuity (PHC)

proposed in High Dimensional Biomimetic Informatics, we will offer insight

into image degradation process and discuss the advantages of the corresponding

restoration algorithm.

Keywords: Image restoration, degradation process, general imaging model, the

Principle of Homology-Continuity, High Dimensional Biomimetic Informatics.

1 Introduction

Image restoration technologies date from the 50’s in the 20th century when space exploration projects were launched [1], and it has so far become a crucial part of digital image processing. Traditional image restoration methods are usually based on an objective imaging model to describe the image degradation process, and estimate the original image from the degraded one directly. The well-known Wiener Filtering [2], Tikhonov Regularization Method [4,5], ARMA Model Estimation [6,7] and Iterative Blind Deconvolution [8] are all derived from this idea. No matter whether they work in the spatial domain or the frequency domain, these approaches aimed directly at the original image, and extracted information merely from the single blurred image. However, they have ignored the inherent properties of the whole degradation process, and thus arouse two main problems in restoration: ill-posedness and non-unique solutions [9].

To conquer these hard-tackling problems, we turned the way of restoration. After in-depth study on the laws of image degradation, we have discovered the existence of * Corresponding author.

H. Deng et al. (Eds.): AICI 2011, Part II, LNAI 7003, pp. 255–263, 2011.

© Springer-Verlag Berlin Heidelberg 2011

256 L. Chen, W. Li, and C. Chen

Homology-Continuity properties in the degradation process, which are quite in conformity with subjective human perception. And thus we can form an entirely different way for restoration on the basis of the Principle of Homology-Continuity (PHC) [3,9].

In [12], Cao Yu proposed the concept of Signal-Noise Angle (SNA) to assess the definition of images. When she applied circular inverse deduction to locate the homologous points of given images, SNA is accordingly decreased as the iteration proceeded, which shows the experimental evidence of Homology-Continuity contained in “Blur — Clear” process. And Chen Yang has proposed another way of inverse deduction and a different biomimetic definition assessment metric in [13]. In both of their experiments, the advantages of PHC methods over traditional methods have been shown through contrast tests. PHC methods are mainly superior in the aspects of robustness to noise and less dependence of PSF estimation.

In this paper, we go on discussing the properties of Homology-Continuity existing in image degradation, and will give mathematical certification of its existence in detail. Based on the proofs, we offer insight into the real image degradation process and theoretical basis for the further algorithm designs.

2 Objective Imaging Model and the PHC Model

Fig. 1 shows a general continuous imaging model. To simplify the calculation, this process is usually modeled as a type of L inear Spatial-Invariant (L SI) System

[1,10,14].

The degradation process can be formulated as following:

),(),(*),(),(),(),(),(y x n y x h y x f y x n y x h f y x d +=

+−−=∫∫∞ηξηξ (1)

where the notation * denotes convolution, d denotes the degraded image, h denotes PSF, f denotes the original image, and n is the system noise.

Fig. 1. Continuous Imaging Model

According to the later proofs, there exist Homology-Continuity properties between the original image and the degraded image. On the basis of PHC, it is a gradual change from the original image to the degraded one, so is the reversed process. Therefore, we can model the PHC restoration as following:

In the high dimensional space n

R (that is, a whole image can be denoted by a single point in the space), a point set containing all homologous patterns of one certain image is named A, the points corresponding to the blurred image and the clear image in set A are x and y respectively, and ε∀, there definitely exists a set B that