Abstract
The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.
Highlights
Fractional calculus has been an important branch of mathematical analysis over the last 300 years [1,2,3,4]; it is still little known by many mathematicians and physical scientists in both the domestic and overseas engineering fields
We propose to introduce a new mathematical method—fractional calculus to the denoising field for texture image and implementing a fractional partial differential equation to solve the above problems by the integer-order partial differential equation-based denoising algorithms [23, 33,34,35,36,37,38]
Equation (1) shows that GrumwaldLetnikov definition in the Euclidean measure extends the step from integer to fractional, and it extends the order from integer differential to fractional differential
Summary
Fractional calculus has been an important branch of mathematical analysis over the last 300 years [1,2,3,4]; it is still little known by many mathematicians and physical scientists in both the domestic and overseas engineering fields. We propose to introduce a new mathematical method—fractional calculus to the denoising field for texture image and implementing a fractional partial differential equation to solve the above problems by the integer-order partial differential equation-based denoising algorithms [23, 33,34,35,36,37,38]. It took the gradient operator of the energy norm from first order to fractional order and still cannot essentially solve the problem of how to nonlinearly maintain the texture details via the anisotropic diffusion. The fractional calculus along the slope is neither zero nor constant but is a nonlinear curve, while integer-order differential along slope is the constant From this discussion, we can see that fractional calculus can nonlinearly enhance the complex texture details during the digital image processing. We show the denoising capabilities of the proposed model by comparing with Gaussian denoising, fourth-order TV denoising, bilateral filtering denoising, contourlet denoising, wavelet denoising, nonlocal means noise filtering (NLMF) denoising, and fractional-order anisotropic diffusion denoising
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