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 H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization of H R , H L has been previously obtained, but only asymptotically when b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

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. The interior problem does not have a unique solution, and a direct inversion on the truncated projection data often leads to large reconstruction errors in ROI. To reduce the truncation artifacts, conventional attenuation-based interior reconstruction problems rely on a known region or piecewise constant constraint within the ROI. Here we propose a quasi-interior XDT scheme that incorporates a small fraction of projection information from the exterior region to assist ROI reconstruction. In the phantom simulation, a small amount (17% of exterior region) of added exterior projection data improves the reconstruction quality by ~50%. The addition of exterior samplings in the experiment demonstrates improved spatial and XRD profile reconstructions compared to total-variation-based reconstruction or sinogram extrapolation. We expect our quasi-interior XDT to obviate the requirement on prior knowledge of the object or its support, and to allow the ROI reconstruction to be performed with the fast, widely-used filtered back-projection algorithm for easy integration into real-time XDT imaging modules.

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. Imaging applications in medical and industrial settings usually do not require the examination of the entire object. Therefore, a diffraction tomography modality covering only the region of interest (ROI) and subsequent image reconstruction techniques with truncated projections are highly desirable. Here we propose a table-top diffraction tomography system that can resolve the spatially-variant diffraction form factor from internal regions within extended samples. We demonstrate that the interior reconstruction maintains the material contrast while reducing the imaging time by 6 folds. The presented method could accelerate the acquisition of XDT and be applied in portable imaging applications with a reduced radiation dose.

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. However, its solution is non-unique in a general unconstrained setting. To solve the interior problem uniquely and stably, in recent years both the prior knowledge- and compressive sensing (CS)-based methods have been developed. Those theoretically exact solutions for the interior problem are called interior tomography. Along this direction, we propose here a new CS-based method for the interior problem based on the curvelet transform. A curvelet is localized in both radial and angular directions in the frequency domain. A two-dimensional (2D) image can be represented in a curvelet frame. We employ the curvelet transform coefficients to regularize the interior problem and obtain a curvelet frame based regularization method (CFRM) for interior tomography. The curvelet coefficients of the reconstructed image are split into two sets according to their visibility from the interior data, and different regularization parameters are used for these two sets. We also presents the results of numerical experiments, which demonstrate the feasibility of the proposed approach.

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. In this paper, we improve a novel continuous singular value decomposition (SVD) method for interior reconstruction assuming a known sub-region. First, two sets of orthogonal eigen-functions are calculated for the Hilbert and image spaces respectively. Then, after the interior Hilbert data are calculated from projection data through the ROI, they are projected onto the eigen-functions in the Hilbert space, and an interior image is recovered by a linear combination of the eigen-functions with the resulting coefficients. Finally, the interior image is compensated for the ambiguity due to the null space utilizing the prior sub-region knowledge. Experiments with simulated and real data demonstrate the advantages of our approach relative to the POCS type interior reconstructions.

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. The first algorithm utilizes two families of filtering lines. These lines are parallel to the tangent of the scanning trajectory and the so-called L lines. The second algorithm utilizes two families of filtering lines tangent to the boundaries of the Zhao window and L lines, respectively, but it eliminates the filtering paths along the tangent of the scanning trajectory, thus reducing the required detector size greatly. The first algorithm is theoretically exact for r < 0.265R and quasi-exact for 0.265R

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). It is based on optimizing the contribution weights of convolution families used in exact Katsevich 3PI algorithms, so that the total weight of each Radon plane is as close to 1 as possible. Optimization is based on solving a least squares problem subject to linear constrains. Numerical evaluation shows good noise and artifact reduction properties of the proposed algorithm.

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. We discuss full-scan and short-scan versions of the algorithm. Evaluation of our algorithm uses simulated and real 256-slice data.

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. Second, axial truncation of the cone beam data is allowed. Third, the length of the arc is determined only by the region of interest and is independent of the size of the entire object. The algorithm is quite flexible and can be used for even more general trajectories that consist of several circular segments and arcs. The algorithm applies also in the case when the circle (or, circles) is complete. A numerical experiment with the clock phantom demonstrated good image quality.

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. The algorithm is quite flexible and can also be used for more general trajectories consisting of several (complete or incomplete) circles and line segments. Results of numerical experiments demonstrate good image quality.

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. As a special case, we show that gantry tilt can also be handled by our dynamic pitch formula. Simulation results demonstrate the resulting algorithm.

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. Another numerical experiment shows that the terms depending on the non-standard distance weighting l/[x - y (s)] can no longer be ignored.

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. Finally, we show that according to the formula the processed cone beam projections should be backprojected using both the inverse distance squared law and the inverse distance law.