Research On Fractional B-spline Wavelet And Its Application In Image Reconstruction

In 2000,Unser proposed a fractional-order B-spline function,which extended the original integer-order B-spline to a fractional order.Fractional B-spline has good properties such as attenuation,two-scale equation and fractional approximation.It can construct fractional B-spline wavelet and realize fractional B-spline wavelet transform.Wavelet transform is a multi-scale differential operator.A wavelet with n-th order vanishing moments can be written as the n-th order derivative of a function.The fractional differential operator corresponds to the wavelet function with fractional approximation,so the approximation performance of the wavelet function has been improved from integer order to fractional order.Fractional B-spline wavelet transform has a wide range of applications in signal processing.In addition,the fractional B-spline wavelet can also well describe the features of fractional singularity such as fine textures in images.In medical applications,Computed Tomography(CT)is widely used to evaluate various sample characteristics.Therefore,CT technology plays an important role in disease prevention and clinical diagnosis.However,the sample reconstructed by CT will not be able to accurately obtain the projection data due to physical measurement,resulting in noise artifacts in the reconstructed image and reducing the image quality.When the number of these artifacts is large and the amplitude is high,the image quality will be reduced,so that the image is neither recognizable nor credible.Especially for biological samples with complex texture structure,it is a challenging task to suppress the noise of the reconstructed image while maintaining the detailed structure.By enhancing the quality of the reconstructed image to obtain more reliable and realistic images,the denoising operation of the reconstructed image has become a necessary scientific work for image reconstruction.The noise processing methods of CT reconstructed images can be divided into three categories,namely projection domain,iterative reconstruction and image denoising based on post-processing.This paper studies the properties of fractional B-spline and the composition of fractional B-spline wavelet.Finally,one-dimensional and two-dimensional discrete Fourier transform formulas of fractional B-spline wavelet are obtained,and the wavelet transform is always realized.Aiming at the problem of noise in reconstructed CT images,we propose an anisotropic total variation(TV)minimization method based on online phase contrast tomography,which can achieve better image reconstruction results.In addition,based on the advantages of the fractional B-spline wavelet transform,we introduce an improved model of total variation(TV)that is truncated TV.Truncated TV model can solve the problem of large penalties for the TV model.Therefore,this paper proposes to further reduce image noise by truncated TV in the fractional B-spline wavelet domain.We validated this method by analyzing CT reconstruction images of actual biological pigeon samples.The results show that compared with some classic image denoising methods,the denoising algorithm proposed in the paper is more effective in suppressing the artifact noise of the reconstructed image while maintaining the detailed structure of the image.