Let $\mathcal R$ denote the generalized Radon transform (GRT), which integrates over a family of $N$-dimensional smooth submanifolds $\mathcal S_{\tilde y}\subset\mathcal U$, $1\le N\le n-1$, where an open set $\mathcal U\subset\mathbb R^n$ is the image domain. The submanifolds are parametrized by points $\tilde y\subset\tilde{\mathcal V}$, where an open set $\tilde{\mathcal V}\subset\mathbb R^n$ is the data domain. The continuous data are $g={\mathcal R} f$, and the reconstruction is $\check f=\mathcal R^*\mathcal B g$. Here $\mathcal R^*$ is a weighted adjoint of $\mathcal R$, and $\mathcal B$ is a pseudo-differential operator. We assume that $f$ is a conormal distribution, $\text{supp}(f)\subset\mathcal U$, and its singular support is a smooth hypersurface $\mathcal S\subset\mathcal U$. Discrete data consists of the values of $g$ on a lattice $\tilde y^j$ with the step size $O(ε)$. Let $\check f_ε=\mathcal R^*\mathcal B g_ε$ denote the reconstruction obtained by applying the inversion formula to an interpolated discrete data $g_ε(\tilde y)$. Pick a generic pair $(x_0,\tilde y_0)$, where $x_0\in\mathcal S$, and $\mathcal S_{\tilde y_0}$ is tangent to $\mathcal S$ at $x_0$. The main result of the paper is the computation of the limit $$ f_0(\check x):=\lim_{ε\to0}ε^κ\check f_ε(x_0+ε\check x). $$ Here $κ\ge 0$ is selected based on the strength of the reconstructed singularity, and $\check x$ is confined to a bounded set. The limiting function $f_0(\check x)$, which we call the discrete transition behavior, allows computing the resolution of reconstruction.
A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) $\mathcal R$ of a function $f$ in $\mathbb R^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of $f$ are reconstructed. The GRT integrates over a fairly general family of surfaces $\mathcal S_y$ in $\mathbb R^3$. Here $y$ is the parameter in the data space, which runs over an open set $\mathcal V\subset\mathbb R^3$. Assume that the data $g(y)=(…
Phantoms can serve as a gold standard for the validation of MRI numerical methods. In some special cases, it is possible to compute analytically the Radon transform, or sinogram, of a phantom. In this work, we present analytical formulae to compute the exact sinograms of three classes of phantoms. We compare the use of the discrete Radon transform, that yields an approximate sinogram, and the correspondent analytical sinogram for image reconstruction.
We study the inversion of the conical Radon which integrates a function in three-dimensional space from integrals over circular cones. The conical Radon recently got significant attention due to its relevance in various imaging applications such as Compton camera imaging and single scattering optical tomography. The unrestricted conical Radon transform is over-determined because the manifold of all cones depends on six variables: the center position, the axis orientation and the opening angle of the cone. In this work, we consider a particular restricted transform using triple line sensors whe…
A method of approximating the inverse Radon transform on the plane by integrating against a smooth kernel is investigated. For piecewise smooth integrable functions, convergence theorems are proven and Gibbs phenomena are ruled out. Geometric properties of the kernel and their implications for computer implementation are discussed. Suggestions are made for applications and an example is presented.
This paper presents a novel mathematical framework for understanding pixel-driven approaches for the parallel beam Radon transform as well as for the fanbeam transform, showing that with the correct discretization strategy, convergence - including rates - in the $L^2$ operator norm can be obtained. These rates inform about suitable strategies for discretization of the occurring domains/variables, and are first established for the Radon transform. In particular, discretizing the detector in the same magnitude as the image pixels (which is standard practice) might not be ideal and in fact, asymp…
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
We give an exact inversion formula for the approximate discrete Radon transform introduced in [Brady, SIAM J. Comput., 27(1), 107--119] that is of cost $O(N \log N)$ for a square 2D image with $N$ pixels and requires only partial data.
