Robust corner detection with fractional calculus for magnetic resonance imaging

Highlights

• The algorithm scheme of the proposed detector is provided in the revision manuscript.

• We make comparative experiments between the proposed methods and integer-order one.

• We also present the experiments on the bone slices polluted by Gaussian noises.

• We make comparative experiments on the medical images polluted by Gaussian noises.

• The proposed detector is compared with two state-of-the-art methods: ROLG and SCD.

Abstract

In medical image processing and analyzing, corners are one of the most important features, which makes point-based registration and diagnosis studies essential. However, existing corner detectors are impressible to noise. This work proposes an order-steerable and robust image corner detector, while a fractional gradient operator consisting of fractional-order forward and backward differentiation and integration is derived. The fractional gradient operator achieving a 90° phase shift as the traditional first derivative does provides a new idea to calculate the gradient in an image. Qualitative and quantitative comparison experiments with exiting detection methods based on integer-order derivative are performed on simulated and real medical slices. The comparison experiments indicate that the present corner detector in this paper allows very strong response for corner detection and provides a better ability of detecting and locating image corners and robustness against noise.

Introduction

Medical image processing and diagnosis has been growing significant momentous in the medicinal field [1]. Image corners contain important information and can effectively reduce image data so that it can greatly reduce the amount of computation to improve the speed of operation. Corner detection is an important technique in medical image analysis and diagnosis, which refer to the process of identifying, locating, matching, registering, evaluating, and prognosis [2], [3], [4], [5], [6], [7]. Medical images distorted by noise are largely caused by the acquisition or transmission of digital images in a noisy channel [8]. Therefore, how to obtain corners and improve the accuracy of detection has become a research hot spot [9], [10], [11], [12].

There are many corner detection methods depending on the classical integer-order derivative, especially 1st-order [13], [14], [15], [9] or 2nd-order differentiation [16], [10]. Some popular corner detection methods (such as Harris, Shi–Tomasi, SUSAN and their advanced versions, etc.) are generally accomplished by calculating the gradient so that differential gradient operations naturally become the key technology of corner detection and location [17], [18], [4]. The popular categorization of corner detection methods based on integer-order differentiation operations (Roberts operator, Prewitt operator, Sobel operator, canny, Laplacian, LoG, etc.) has been widely used in image corner detection technology. The response of the first-order gradient operator to the gray scale region (gray slope or step) is stronger than that of the second-order Laplacian operator, but the gradient operator's response to the image corner information is weaker than the Laplacian operator. Hence, the above methods are sensitive to noise. In order to suppress the noise caused by the above-mentioned differential processing, it is common to smooth the image after the gradient operation. However, both noises and corner points are high-frequency components in image, the noise are eliminated noise while the corner points are also weaken. The sensitivity to noise and detection accuracy are contradictory, How to balance these two performance requirements is all the way a research topic [19], [20], [21].

The classical local first- or second-orders differentiation, whose magnitude-frequency characteristic is not steerable, is not stable in the presence of noise. Fractional calculus expand the order of calculus from integer to the range of real, and it achieves continuous order and global calculus. Nowadays, many domestic and foreign scholars have found that fractional calculus are very suitable for describing materials with memory and genetic properties in the real world. Therefore, the application of fractional order calculus theory in two-dimensional image signal has attracted the attention of researchers both at home and abroad in the last ten years, mainly in the fields of image enhancement, image denoising, image edge extraction and image singularity detection [22], [23], [24], [25], [26], [27]. Xiao et al. [25] proposed A general framework of fractional-order orthogonal moments, which are not only able to extract region-of-interest feature but also have potential for image reconstruction and have high noise robustness in invariant image recognition. Literature [26] integrated both multiple instance learning and self-paced learning into a unified learning framework and proposed a SP-MIL framework for co-saliency detection. Literature [27] proposed a 2D-LBP method, which counts the weighted occurrence number of the rotation invariant uniform LBP pattern pairs, obtain the spatial contextual information.

Although there are many proposed approaches for corner feature detection, but none of them are perfectly robust to noise especially for magnetic resonance images that need a higher accuracy in comparison with non-medical images. In this work, a comprehensive study is performed on the algorithm of an order-steerable and robust fractional gradient corner detector, including mathematical basis, characteristics analysis, and design motivation. The novel fractional gradient operator is implemented by the combination of forward and backward calculus, which effectively provides a capability balancing the immunity to noise and corner detection accuracy for magnetic resonance images.

Section 2 shows the development of the order-steerable corner detector based on fractional gradient. The fractional gradient corner detection, formed from the combination of forward and backward calculus (integration and derivative), is developed rigorously by Riemann–Liouville operator. The fractional gradient derivative, which provides a new approach to calculate the gradient of an image, performs 90° phase shift as the 1st differentiation does. The major differences from corner detection methods based on first-order derivative are that the fractional gradient corner detector is order-steerable, and one can tune magnitude characteristic by adjusting the orders of fractional gradient (α > β > 0, α denotes the order of fractional derivative, β denotes one of fractional integration) to reach a compromise between immunity to noise and corner detection accuracy. In Section 3, the quantitative and qualitative experiments are performed on images with various noises.