Adaptive fractional differential approach and its application to medical image enhancement

doi:10.1016/j.compeleceng.2015.02.013

Computers & Electrical Engineering

Volume 45, July 2015, Pages 324-335

https://doi.org/10.1016/j.compeleceng.2015.02.013 Get rights and content

Highlights

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We use the improved Otsu algorithm to segment medical images.
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The function of order v is constructed based on the features of image areas.
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The fractional differential order of each pixel is obtained by the function.
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Each pixel of medical image is processed by adaptive fractional differential mask.
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The edge of medical image is enhanced while the weak texture is preserved.

Abstract

This paper presents a new medical image enhancement method that adjusts the fractional order according to the dynamic gradient feature of the entire image. The presented method can extract the edges of an image accurately and enhance them while preserving smooth areas and weak textures; these improvements can be particularly helpful to doctors’ diagnoses. The primary contribution of this paper is the Adaptive Fractional Differential Algorithm (AFDA), which uses the improved Otsu algorithm to segment the edges, textures and smooth areas of images. This algorithm allows the optimal fractional order of each pixel to be obtained using an adaptive fractional differential function constructed based on the area feature of image. As a result, the image can be enhanced adaptively. Experimental results show that for medical images, AFDA shows better image enhancement than other methods by making edges clearer and textures richer.

Graphical abstract

Introduction

Medical image quality has become an indispensable part of modern medicine and directly influences the accuracy of doctors’ diagnoses and treatments. Low resolution and contrast in medical images have made correct diagnosis difficult; this directly influences the speed and accuracy of doctors’ diagnoses. Therefore, it is necessary to improve medical image enhancement to reflect the information of an illness more clearly and accurately [1], [2], [3].

Fractional differentials, which is a theory of arbitrary order, are similar to integral differentials [4], [5]. Compared with traditional integral differential approaches, fractional differentials applied to image processing can enhance edges, make texture details clearer, and preserve smooth areas [6], [7], [8]. Therefore, medical images processed by fractional differentials are clearer and have higher contrast. Traditional fractional differentials use the same fractional order to process edges, textures and smooth areas of image; however, while edges would be enhanced by high fractional orders, weak textures and smooth areas would be ignored, and while weaker textures and smoother areas would be preserved by low fractional orders, edges would be weakened. Thus, image enhancement is difficult to attain in practice. To manage these issues, traditional and improved fractional differential algorithms have been developed for digital image processing in [9], [10], [11], [12]. Adaptive fractional derivatives for image denoising problems have also been considered [13], [14], [15]. In particular, Ref. [6] proposed six fractional differential masks and the operator YiFeiPu-2, which has the best performance in precision and convergence because the fractional orders in the differential operators mentioned above are constant values and determined by a human; thus, image enhancement cannot be optimized if area features of the image are modified.

To enhance medical images effectively, this paper presents a new AFDA method based on an improved Otsu algorithm [16]. First, the improved Otsu algorithm is used to segment edges, textures and smooth areas accurately. Second, the optimal fractional order of each pixel is obtained by an adaptive fractional differential function constructed based on the features of different image areas. Finally, an optimal fractional order is substituted into the fractional differential mask to process the corresponding pixel of the image. In the experimental comparison and analysis below, five typical medical images are processed by AFDA as examples, and evaluation parameters of image texture such as entropy, contrast, average gray, average gradient and the proportion of edges are used for quantitative analysis and experimental verification. The experimental results show that the proposed method produces better image enhancements than traditional methods by extracting and enhancing the edges of an image more accurately while preserving more smooth areas and weak textures.

The remainder of this paper is organized as follows. Section 2 describes the related fractional differential theories and the realization of fractional differential masks. Section 3 presents the adaptive fractional differential function of the proposed approach in detail. Section 4 discusses the experiments and comparisons. Lastly, Section 5 presents the conclusions.

Section snippets

Definition of fractional differential

Until now, no unified formula has been developed to define fractional calculus. Mathematicians have analyzed the problem from different points of view and obtained different definitions of fractional calculus. Currently, there are three classical definitions of fractional calculus: the Grünwald–Letnikov (G–L) definition, the Riemann–Liouville (R–L) definition and the Capotu definition [17], [18], [19]. Because the G–L definition is less complex than the others and only uses one coefficient, the

Adaptive fractional differential function

The best effect of image enhancement is to make edges clearer and preserve weak textures and smooth areas in an image. According to the characteristics of fractional differentials, we construct the corresponding adaptive fractional differential function, which has a high order in edge pixels and a relatively small order in the weak texture pixels. The piecewise function of v-order fractional differential is given as: $v = \{\begin{matrix} \frac{M (i, j) - t}{M (i, j)}, & M (i, j) ⩾ t and \frac{M (i, j) - t}{M (i, j)} ⩾ v_{1} \\ v_{1}, & M (i, j) ⩾ t and \frac{M (i, j) - t}{M (i, j)} < v_{1} \\ v_{2}, & 2 \end{matrix}$

Comparison and analysis of experiment

As introduced in Section 3.3, five typical types of medical images shown in Fig. 4 were chosen as examples. The effect of image enhancement of AFDA is compared with that of the Histogram, Sobel, Laplacian and Traditional fractional differential methods, which are 0.5-order, 0.8-order and 1-order, respectively. The effect of image enhancement is evaluated by visual analysis, segment results and some metrics.

The experimental results are shown as Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11,

Conclusions

Fractional differentials applied to medical image processing are a new research topic. The methods of image enhancement can improve the quality of medical images and can thus be helpful to doctors’ diagnoses. The proposed AFDA method comprehensively considers global and local information in medical images and yields better enhancement effects than traditional image processing methods. The AFDA method uses the improved Otsu algorithm to segment every area of an image accurately and processes

Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (61174098), Guangdong Natural Science Foundation (S2013010012537) and Project on the Integration of Industry, Education and Research of Guangdong Province (2013B090600062).

Bo Li received his M.E. in Communication and Information System (2010) from Center China Normal University. He is currently pursuing his PhD degree at South China University of Technology. His current research interests include Computer vision and Image processing.

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Wei Xie received his PhD in System Science (2003) from Kitami Institute of Technology, Japan. Now he is full professor of Control theory and control engineering at College of Automation Science and Engineering, South China University of Technology, China. His current research interests include computer vision and adaptive control.

^☆: Reviews processed and recommended for publication to the Editor-in-Chief by Associate Editor Dr. M.R. Daliri.

View full text

Adaptive fractional differential approach and its application to medical image enhancement☆

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Definition of fractional differential

Adaptive fractional differential function

Comparison and analysis of experiment

Conclusions

Acknowledgements

Signal Process

J Vis Commun Image Represent

Elsevier Sci.

Speckle reduction and contrast enhancement of echocardiograms via multiscale nonlinear processing

IEEE Trans Med Imag

Contrast enhancement of medical images using multi-scale edge representation

Opt Eng

Despeckling of medical ultrasound images

IEEE Trans Ultrason Ferroelectr Frequency Control

Fractional calculus: theory and applications, differentiation and integration to arbitrary order

Theory and applications of fractional differential equations

Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement

IEEE Trans Image Process

Edge detection of colour image based on quaternion fractional differential

IET Image Process

Image thresholding framework based on two-dimensional digital fractional integration and Legendre moments

IET Image Process