Inverse reconstruction of nonuniform data and the stability of RBF method with uniform centers
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Given equispaced Fourier data, the image reconstruction suffers from spurious oscillations, i.e. the Gibbs phenomenon, in the vicinity of the discontinuities using the Fourier partial sum. Many techniques, including the Gegenbauer reconstruction method, and the inverse method have been proposed to resolve the Gibbs phenomenon. In most practical applications, the given Fourier samples are not uniform, and the finite section method, convolution gridding, re-sampling and Gegenbauer reprojection method have been developed to diminish the amplitude of the oscillations near the discontinuous locations. New methodologies are proposed in this dissertation to obtain exponential accuracy of the reconstruction from nonharmonic Fourier data. The inverse reconstruction method was first developed for the resolution of the Gibbs phenomenon in the reconstruction of the piecewise analytic functions using the spectral data, such as the uniform Fourier data. We show that the reconstruction with the inverse method successfully recovers high-order accuracy of the reconstruction in polynomials from nonuniform Fourier data, as well as reduces the Gibbs oscillations. The inverse reconstruction formula is derived for the truncated operator S N and the finite approximation R N . In the goal of circumventing the ill-conditioning issue of the transformation matrix for the inverse method for R N , the hybrid method is adopted, in which a covariance matrix is introduced and the pseudo-inversion is utilized to solve the linear system. For the inverse method for S N , the method of least squares is used to solve the over-determined linear system. In order for the reconstruction algorithm to be stable, the relation between (2 N + 1), the total number of Fourier frame coefficients, and ( m +1), the number of polynomials used for the reconstruction, is obtained such the transformation matrix is well conditioned. Moreover, given nonharmonic Fourier data of a $2$D image, the utilization of the inverse reconstruction method for S N,M , slice-by-slice, also yields great accuracy, provided that the edge locations of the images are known. Since the radial basis function (RBF) method has attracted lots of attentions for being regarded as mesh-free, in this work, we also study the inverse method using the multiquadric (MQ) radial basis functions (RBFs) as the reconstruction base. Furthermore, the inverse method is successfully applied to considerably resolve the Gibbs phenomenon in RBF interpolation of discontinuous functions. It is achieved by applying the inverse method to each smooth region separately. Since the RBF methods have been widely applied to solve partial differential equations, the stability condition of the application of RBF method to solve the hyperbolic equations is investigated by using the matrix stability analysis for various RBF methods, including the single and multi-domain method and the local RBF method. The CFL condition for each RBF method is obtained numerically. It is proven that the obtained CFL condition is only a necessary condition. That is, the numerical solution may grow for a finite time. It is also explained that the boundary condition is crucial for stability, however, it may degrade accuracy if it is imposed.