Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach.

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms and . These operators arise when one studies the interior problem of tomography. The diagonalization of has been previously obtained, but only asymptotically when .

Interior x-ray diffraction tomography with low-resolution exterior information.

X-ray diffraction tomography (XDT) resolves spatially-variant XRD profiles within macroscopic objects, and provides improved material contrast compared to the conventional transmission-based computed tomography (CT). However, due to the small diffraction cross-section, XDT suffers from long imaging acquisition time, which could take tens of hours for a full scan using a table-top x-ray tube. In medical and industrial imaging applications, oftentimes only the XRD measurement within a region-of-interest (ROI) is required, which, together with the demand to reduce imaging time and radiation dose to the sample, motivates the development of interior XDT systems that scan and reconstruct only an internal region within the sample.

X-ray diffraction tomography with limited projection information.

X-ray diffraction tomography (XDT) records the spatially-resolved X-ray diffraction profile of an extended object. Compared to conventional transmission-based tomography, XDT displays high intrinsic contrast among materials of similar electron density and improves the accuracy in material identification thanks to the molecular structural information carried by diffracted photons. However, due to the weak diffraction signal, a tomographic scan covering the entire object typically requires a synchrotron facility to make the acquisition time more manageable.

Interior tomography with curvelet-based regularization.

The interior problem, i.e. reconstruction from local truncated projections in computed tomography (CT), is common in practical applications.

Interior tomography with continuous singular value decomposition.

The long-standing interior problem has important mathematical and practical implications. The recently developed interior tomography methods have produced encouraging results. A particular scenario for theoretically exact interior reconstruction from truncated projections is that there is a known sub-region in the ROI.

Fast exact/quasi-exact FBP algorithms for triple-source helical cone-beam CT.

Cardiac computed tomography (CT) has been improved over past years, but it still needs improvement for higher temporal resolution in the cases of high or irregular cardiac rates. Given successful applications of dual-source cardiac CT scanners, triple-source cone-beam CT seems a promising mode for cardiac CT. In this paper, we propose two filtered-backprojection algorithms for triple-source helical cone-beam CT.

Optimized reconstruction algorithm for helical CT with fractional pitch between 1PI and 3PI.

We propose an approximate approach to use redundant data outside the 1PI window within the exact Katsevich reconstruction framework. The proposed algorithm allows a flexible selection of the helical pitch, which is useful for clinical applications. Our idea is an extension of the one proposed by KOhler, Bontus, and Koken (2006).

Exact image reconstruction for a circle and line trajectory with a gantry tilt.

We investigate image reconstruction with a circle and line trajectory with a tilted gantry. We derive new equations for reconstruction from the line data, such as equations of filtering lines, range of filtering lines and range of the line scan. We analyze the detector requirements and show that the line scan does not impose extra requirements on the cylindrical detector size with our algorithm, that is, the axial truncation of the filtering lines does not occur.

Formulation of four Katsevich algorithms in native geometry.

We derive formulations of the four exact helical Katsevich algorithms in the native cylindrical detector geometry, which allow efficient implementation in modern computed tomography scanners with wide cone beam aperture. Also, we discuss some aspects of numerical implementation.

Image reconstruction for the circle-and-arc trajectory.

Proposed is an exact shift-invariant filtered backprojection algorithm for the circle-and-arc trajectory. The algorithm has several important features. First, it allows for the circle to be incomplete.

Image reconstruction for the circle and line trajectory.

We propose an exact shift-invariant filtered backprojection algorithm for inversion of the cone beam data in the case when the source trajectory consists of an incomplete circle and a line segment. The algorithm allows for axial truncation of the cone beam data. The length of the line scan is determined only by the region of interest and is independent of the size of the entire object.

Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography.

We present an exact filtered backprojection reconstruction formula for helical cone beam computed tomography in which the pitch of the helix varies with time. We prove that the resulting algorithm, which is functionally identical to the constant pitch case, provides exact reconstruction provided that the projection of the helix onto the detector forms convex boundaries and that PI lines are unique. Furthermore, we demonstrate that both of these conditions are satisfied provided the sum of the translational velocity and the derivative of the translational acceleration does not change sign.

On two versions of a 3-pi algorithm for spiral CT.

A 3pi algorithm is obtained in which all the derivatives are confined to a detector array. Distance weighting of backprojection coefficients of the algorithm is studied. A numerical experiment indicates that avoiding differentiation along the source trajectory improves spatial resolution.

Analysis of an exact inversion algorithm for spiral cone-beam CT.

In this paper we continue studying a theoretically exact filtered backprojection inversion formula for cone beam spiral CT proposed earlier by the author. Our results show that if the phantom f is constant along the axial direction, the formula is equivalent to the 2D Radon transform inversion. Also, the inversion formula remains exact as spiral pitch goes to zero and in the limit becomes again the 2D Radon transform inversion formula.