An adaptive approach for texture enhancement based on a fractional differential operator with non-integer step and order
Introduction
Image texture enhancement aims to improve the quality of an image by modifying its attributes. A number of cutting-edge techniques have been proposed which can be divided into two categories: transform-based [1] and spatial domain-based [2]. Transform-based methods regulate coefficients associated with the frequency domain, followed by an inverse transform to obtain the resulting image, based on which image enhancement can be achieved. However, these methods may introduce ringing effect and additional noise. On the other hand, spatial domain-based methods can avoid these problems without the need to perform frequency domain transform, resulting in less computation. Among the different enhancement approaches, the differential mask operator stands out as a particularly important example. Differential operator masks can be further categorized as integral differential and fractional differential operators. As for image improvement, most integral-differential operators (e.g., Sobel, Prewitt, and Laplacian of Gaussian operators) behave well when used for enhancing high-frequency features. Nevertheless, their performance deteriorates significantly when applied to smooth regions.
Pu et al. [3] apply the theory of fractional differential operator to address these problems. Since a fractional differential operator is capable of characterizing fractal-like structures [4] which are often found in the texture regions, this class of operator is considered as an effective tool for texture enhancement in images. Through analyzing the geometric and physical properties of fractional differential operators, Pu et al. [3] have developed an n×n fractional differential operator mask, and it was noticed that the adoption of the mask results in better enhancement of texture details compared to traditional integral-based differential operators [5]. It was further observed that the fractional differential operator has the capability of not only preserving high-frequency contour features, but enhancing the low-frequency texture details in smooth areas as well. Gao et al. [6] applied the fractional differential operator to quaternions, and designed a set of masks which are referred to as quaternion fractional differential (QFD) operators, which generalize the previous fractional differential operators.
However, for image enhancement, some problems still exist with the fractional differential operator. To begin with, traditional fractional differential operators usually consider fixed-size mask templates, leading to ineffective processing of pixels corrupted by noise and with low correlations. Moreover, the spatial step in the numerical implementation of the fractional differential operator based on the definition of Grümwal–Letnikov usually advances by one. In other words, the default minimum distance is assumed to be one pixel. As a result, the high degree of self-similarity that many images exhibit is not well characterized. In addition, it is not convenient to manually search for the optimal fixed fractional derivative order which matches the local texture details. In view of these problems, we propose a novel fractional differential operator mask with adaptive non-integral step and order in this paper for the enhancement of texture details. The main contributions of this paper are as follows:
- (1)
We identify local non-regular self-similar support regions by analyzing texture features, such that highly correlated pixels can be focused on while noisy pixels are excluded.
- (2)
We select non-integral steps and fractional orders for the support region in an adaptive way, such that the degree of self-similarity in complex textures can be well characterized.
- (3)
We design a non-regular fractional differential operator mask with fractional order and adaptive non-integral steps, such that the texture enhancement performance can be optimized regardless of whether the regions consist of high or low frequency patterns.
The paper is organized as follows. The proposed algorithm is introduced in Section 2, followed by the analysis of experimental results in Section 3. Finally, our conclusions are summarized in Section 4.
Section snippets
Fractional differential operator mask with adaptive non-integral step and order
As complex textures are characterized by irregular and disorderly patterns, a novel approach based on adaptive fractional differential operator is proposed in this paper to enhance these patterns. Fig. 1 provides an overview of the algorithm. As can be seen, with a suitable texture similarity measure, a local support cross skeleton domain, i.e. {hpi}, where denotes the four directions, can be defined. Once such a support skeleton domain, which can be partitioned into two sets H(p)
Experiments and analysis
We use the measures average gradient (AG) [8], average information entropy (AE) [9], and average peak signal to noise ratio (APSNR) to evaluate the performance of the proposed approach.
AG is a measure of image contrast, and larger values correspond to stronger texture patterns. The measure is defined as follows:whereg denotes intensity level, and M×N is the size of an image.Algorithm 1 : Fractional Differential Operator Mask with
Conclusion
In this paper, we have proposed an adaptive fractional differential operator mask for image enhancement. Through the identification of the main limitations of the previous fractional derivative-based methods, we obtain better performance by introducing a non-regular support region, and adaptively determining the associated fractional order such that the high degree of self-similarity in most images could be judiciously taken into consideration. In view of the capability of our proposed approach
Acknowledgment
This research is based upon work supported by the National Natural Science Foundation of China under Grant no. 61472267, the Nature Foundation of Jiangsu Province under Grant no. BK2012166, the Natural Science Foundation of Jiangsu University under Grant no. 12KJB10031, the Science and Technology Project of Jiangsu Province under Grant no. JH21, a Grant from the City University of Hong Kong (Project no. 7004220), the Open Foundation of Modern Enterprise Information Application Supporting
Fuyuan Hu was a postdoctoral researcher at Vrije Universiteit Brussel,Belgium, a Ph.D. student at Northwestern Polytechnical University, and a visiting Ph.D. student at the City University of Hong Kong. He is an associate professor in computer vision and machine learning at Suzhou University of Science and Technology. His research interests include graphical models, structured learning, and tracking.
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Cited by (0)
Fuyuan Hu was a postdoctoral researcher at Vrije Universiteit Brussel,Belgium, a Ph.D. student at Northwestern Polytechnical University, and a visiting Ph.D. student at the City University of Hong Kong. He is an associate professor in computer vision and machine learning at Suzhou University of Science and Technology. His research interests include graphical models, structured learning, and tracking.
Hau San Wong received the B.S. and M.Phil. degrees in Electronic Engineering from the Chinese University of Hong Kong, and the Ph.D. degree in Electrical and Information Engineering from the University of Sydney. He is an associate professor in the Department of Computer Science, City University of Hong Kong. His research interests include multimedia signal processing, neural networks and evolutionary computation.
Maoxin Si received his B.S. degree in applied math from Xinjiang Agricultural University, Xinjiang, China, in 1982. He is an associate professor in computer science at Suzhou Vocational University. His research interests include image processing and algorithm optimization.
Shaohui Si received his B.S. degree in electronic and information engineering from Suzhou University of Science and Technology, Suzhou, China, in 2012. He is currently pursuing his M.S. degree in Suzhou University of Science and Technology. His interests include image enhancement, and fractional derivative theory.
Baochuan Fu received his M.S. degree in electronic measurement and instrumentation from Harbin University of Science and Technology, Harbin, China, in 1988 and Ph.D. degree in control theory and control engineering from Tongji University, Shanghai, China, in 2008. He is a professor in computer science at Suzhou University of Science and Technology. His research interests include cybernetics and system optimization.
Heng Luo received his M.S. degree from Tongji University, Shanghai, China, in 2007 and his Ph.D. degree from University of Edinburgh, UK, in 2012 both in telecommunication engineering. His current research interests include information processing and system optimization.