The conventional wisdom states that the interior problem (reconstruction of an interior region from projection data along
lines only through that region) is NOT uniquely solvable. While it remains correct, our recent theoretical and numerical results demonstrated that this interior problem CAN be solved in a theoretically exact and numerically stable fashion if a sub-region within the interior region is known. In contrast to the well-established lambda tomography, the studies on this type of exact interior reconstruction are referred to as "interior tomography". In this paper, we will overview the development of interior tomography, involving theory, algorithms and applications. The essence of interior tomography is to find the unique solution from highly truncated projection data via analytic continuation. Such an extension can be done either in the filtered backprojection or backprojection filtration formats. The key issue for the exact interior reconstruction is how to invert the truncated Hilbert transform. We have developed a projection onto convex set (POCS)
algorithm and a singular value decomposition (SVD) method and produced excellent results in numerical simulations
and practical applications.
The projection equations in the research of discrete tomography are obtained by either counting the number of points that each line passes or computing the fractional areas of the intersection of each strip and the grid. In this work, a system of linear equations for strip-based projections with rational slopes is obtained. The linear dependency number of these equations is derived.
In this paper, we for the first time define the concept of 3D skew lambda tomography (SLT) based on the 3D Calderon
Operator, and formulate an approximate local reconstruction algorithm for cone beam data collected along an arbitrary
scanning curve. The main idea is to rewrite the filtering operator in an exact filtered-backprojection reconstruction
formula as a local projection. While the practical cone-beam lambda tomography works well for spirals with small
pitches, the proposed SLT is more suitable to the spirals with large pitch. Simulation using the 3D differentiable Shepp-
Logan phantom is performed to demonstrate the utility of this new technique.
This paper investigates the feasibility of reconstructing a Computed Tomography (CT) image from truncated Lambda
Tomography (LT), a gradient-like image of it's original. An LT image can be regarded as a convolution of the object
image and the point spread function (PSF) of the Calderon operator. The PSF's infinite support provides the LT image
infinite support; even the original CT image is of compact support. When the support of a truncated LT image fully
covers the compact support of the corresponding CT image, we develop an extrapolation method to recover the CT
image more precisely. When the support of the CT image fully covers the support of the truncated LT image, we design
a template-based scheme to compensate the cupping effects and reconstruct a satisfactory image. Our algorithms are
evaluated in numerical simulations and the results demonstrate the feasibilities of our methods. Our approaches provide a
new way to reconstruct high-quality CT images.
Because of the importance of the so-called long object problem, spiral cone-beam computed tomography (CT) has
become a hot area in the CT field since it was first proposed in 1991. As a main stream in the development of the next
generation medical CT, spiral cone-beam CT has been greatly improved, especially in the aspect of image reconstruction
methods. Now, the state-of-the-art cone-beam algorithms can reconstruct images exactly from longitudinally truncated
data collected along a rather general scanning trajectory. Here we present a brief overview of this area with an emphasis
on the results achieved by our team.
As a potentially important technology for medical x-ray Computed Tomography (CT), Lambda tomography (LT) is to
reconstruct a gradient-like image only from local projection data. Based on our recently derived exact fan-beam LT
formula, here we propose a practical cone-beam LT algorithm for LT reconstruction from local data collected along
an arbitrary smooth 3D curve. A key step in our algorithm is to determine an appropriate vector perpendicular to the line
connecting the x-ray source and an image point. The algorithm is implemented assuming an equi-spatial planar detector
and a nonstandard spiral trajectory. The numerical simulation results demonstrate the merits of our method.
In this article we consider cone-beam CT projections along a nonstandard 3-D spiral with variable radius and variable pitch. Specifically, we generalize an exact image reconstruction formula by Zou and Pan (2004a) and (2004b) to the case of nonstandard spirals, by giving a new, analytic proof of the reconstruction formula. Our proof is independent of the shape of the spiral, as long as the object is contained in a region inside the spiral, where there is a PI line passing through any interior point. Our generalized reconstruction formula can also be applied to much more general situations, including cone-beam scanning along standard (Pack, et al. 2004) and nonstandard saddle curves, and any smooth curve from one endpoint of a line segment to the other endpoint, for image reconstruction of that line segment. In other words, our results can be regarded as a generalization of Orlov's classical papers (1975) to cone-beam scanning.
In this paper, we perform numerical studies on Feldkamp-type and Katsevich-type algorithms for cone-beam reconstruction with a nonstandard spiral locus to develop an electron-beam micro-CT scanner. Numerical results are obtained using both the approximate and exact algorithms in terms of image quality. It is observed that the two algorithms produce similar quality if the cone angle is not large and/or there is no sharp density change along the z-direction. The Katsevich-type algorithm is generally preferred due to its nature of exactness.
To solve the long object problem, an exact and efficient algorithm has been recently developed by Katsevich. While the Katsevich algorithm only works with standard helical cone-beam scanning, there is an important need for nonstandard spiral cone-beam scanning. Specifically, we need a scanning spiral of variable radius for our newly proposed electron-beam CT/micro-CT prototype. In this paper, for variable radius spiral cone-beam CT we construct two Katsevich-type cone-beam reconstruction algorithms in the filtered backprojection (FBP) and backprojected filtration (BPF) formats, respectively. The FBP algorithm is developed based on the standard Katsevich algorithm, and consists of four steps: data differentiation, PI-line determination, slant filtration and weighted backprojection. The BPF algorithm is designed based on the scheme by Zou and Pan, and also consists four steps: data differentiation, PI-line determination, weighted backprojection and inverse Hilbert transform. Numerical experiments are conducted with mathematical phantoms.