High-order Caputo Fractional Order Differential Operator and Its Application in Image Enhancement

Unlike the traditional integer calculus, which usually has intuitive geometric and physical meaning, the definition of fractional calculus is generally complex and presents different forms. However, its characteristics such as memory and nonlocality have laid a good mathematical foundation for solving some complex problems in the engineering field. At the same time, signal and image processing based on fractional calculus have also attracted attention in recent years. At present, the common fractional differential operators used in image processing include Grünwald-Letnikov (G-L) fractional differential, Riemann-Liouville (R-L) fractional differential and Caputo fractional differential. Although G-L and R-L operators can enhance the image to a certain extent, their capabilites of the improvement of image contrast and definition is limited. At present, Caputo differential mask operators are mostly limited to low-order operators of order in (0,1), and the research and application of high-order operators are relatively few. In this paper, the high-order Caputo fractional differential operator and its application in image enhancement are studied. Firstly, a differential mask operator based on forward difference is constructed for Caputo fractional differential with order in (1,2) and (2,3), and its error is demonstrated; Further, the general form of high-order Caputo fractional differential operator is studied, and a representation based on matrix is given. On this basis, the proposed high-order Caputo fractional differential mask operator is applied to image enhancement, and the comparative experiments of image enhancement are carried out for mask operators of different orders and sizes. The experimental results show that the proposed high-order Caputo fractional differential operator achieves good image enhancement effect, especially for improving image contrast, clarity and average gradient have obvious advantages.