Reconstruction from Radon projections and orthogonal expansion on a ball

The relation between Radon transform and orthogonal expansions of a function on the unit ball in \RR^d is exploited. A compact formula for the partial sums of the expansion is given in terms of the Radon transform, which leads to algorithms for image reconstruction from Radon data. The relation between orthogonal expansion and the singular value decomposition of the Radon transform is also exploited.Comment: 15 page

Fast OPED algorithm for reconstruction of images from Radon data

A fast implementation of the OPED algorithm, a reconstruction algorithm for Radon data introduced recently, is proposed and tested. The new implementation uses FFT for discrete sine transform and an interpolation step. The convergence of the fast implementation is proved under the condition that the function is mildly smooth. The numerical test shows that the accuracy of the OPED algorithm changes little when the fast implementation is used.Comment: 13 page

Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$H^{\alpha} f(n) : = \sum_{m \neq 0} \frac{e^{2 \pi i\alpha m^2} f(n - m)}{m}.$$ We prove that, uniformly in $\alpha \in \mathbb{T}$, there is a sparse bound for the bilinear form $\langle H^{\alpha} f , g \rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse H\"older classes.Comment: 9 page

Improving estimates for discrete polynomial averages

For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.Comment: 10 pages. This version combines arXiv:1910.12448 by J. Madrid with arXiv:1910.14630v1 by the remaining four authors. To appear in JFA

OPED reconstruction algorithm for limited angle problem

The structure of the reconstruction algorithm OPED permits a natural way to generate additional data, while still preserving the essential feature of the algorithm. This provides a method for image reconstruction for limited angel problems. In stead of completing the set of data, the set of discrete sine transforms of the data is completed. This is achieved by solving systems of linear equations that have, upon choosing appropriate parameters, positive definite coefficient matrices. Numerical examples are presented.Comment: 17 page