Research On Image Processing Based On Fractional Calculus

The fractional order calculus research on non-integer order differential and integral is different from the traditional integer order calculus.Its memory and non-local characteristics are well suited to describe the materials with memory and genetic in the real world.At present,the fractional calculus theory is gradually applied to a wide range of fields such as physics,chemistry,biology,finance,and fluid.Since the order of its implementation are flexible and with a large degree of freedom,its application in the field of image processing has attracted many researchers' extensive attentions.Images are two-dimensional signal and have complex edges and texture details.The fractional calculus plays a special role in the edge and texture information processing of the image.Fractional differential can improve the high-frequency component of a signal and nonlinearly retain low-frequency component of the signal,whereas fractional integral can improve the low-frequency component of a signal and nonlinearly retain high-frequency component of the signal.Thus,research on image processing based on the fractional calculus is very meaningful.How to establish the relationship between fractional calculus and image processing,and how to construct the fractional order calculus equation model in image processing are the two major problems in the area of applications of fractional order calculus in the signal processing.In this paper,we establish the relationship between fractional calculus and image transformation by the specific definition of fractional calculus and the important tool of image processing– the Fourier transform and fractional Fourier transform of fractional order calculus and image transformation relationship.Fractional varying-order differential is also proposed.Then we designed an image denoising model based on the fractional varying-order differential,and an image restoration model based on the fractional varying-order differential.Finally,we obtain the cross application of the two fields of fractional order calculus and image processing.Fractional order calculus,Fourier transform,fractional Fourier transform each have good properties.The relationship formula is concise,and the definition of fractional varying-order differential breaks through the traditional thought.The new models are very effective for dealing with the details of signal and image processing.Numerical experiments and matlab simulations are provided to demonstrate the effectiveness of the proposed approach.The main contributions of this paper are summarized as follows:1.We study the relationship between fractional calculus and image processing.We derive and analyze the relationships between fractional calculus and two important tools of image processing – Fourier transform and fractional Fourier transform,and present the block diagram cross application of fractional calculus and image processing.Therefore,it lays the foundation for the application of fractional calculus in image processing field.2.We propose the concept of fractional varying-order differential.The differential operation of matrix functions can be performed on each element of the function of the differential treatment of different orders.It is different from the traditional constant order integral differential.Fractional differential has the characteristics for a signal that it can enhance high frequency component at the same time nonlinearly retain the lowfrequency components of the signal.We apply the characteristics in image denoising and establish a new model of image denoising based on fractional varying-order differential.The differential order used to process different parts of an image is different,and it is determined by the value of gradient modulus of the image.Therefore,the image can be better denoised and the ”staircase effect” in the non-texture region can be also suppressed,the texture area can better maintain the texture.3.We propose an image restoration method based on total variational and fractional varying-order differential.It combines fractional varying-order differential with the total variational image restoration concept,which neither one norm a ”smoothness” measure of the image,nor secondary norm of ”smoothness” measure are used.This method uses a Fractional order norm ”smoothness” of the measure of the image.This will not only overcome one norm to render the ”piecewise constant” phenomenon,but also solve quadratic norm edge blurring caused problems.So it can get a very satisfactory restoration effect.