We present a novel backprojection algorithm for three-dimensional (3-D) radon transform data that requires O(N3 log2 N) operations for reconstruction of an N x N x N volume from O(N2) plane-integral projections. Our algorithm uses a hierarchical decomposition of the 3-D radon transform to recursively decompose the backprojection operation. Simulations are presented demonstrating reconstruction quality comparable to the standard filtered backprojection, which requires O(N5) computations under the same circumstances.

We recently proposed a novel fast backprojection algorithm for reconstruction of an N x N pixel object from O(N) projections in O(N(2)log(2)N) operations. In this paper, we analyze a simplified version of that algorithm, to determine the effects of various parameter choices on the algorithm's theoretical performance. We derive a bound on the variance of the per-pixel error introduced by using hierarchical backprojection. This bound is with respect to an ensemble of input sinograms, and allows us to construct confidence intervals (for any specified level) for the per-pixel errors. The bound has a simple form, and we show how to use it to select algorithm parameters for different cost versus error tradeoffs. Simulation results show that the bound accurately predicts the performance of the algorithm over a wide range of parameter choices. These results are verified for different images, including a tomographic reconstruction from the visual human dataset (VHD). The analysis therefore provides an effective tool for the selection of parameters and operating point for the fast hierarchical backprojection algorithm.

In the standard two-dimensional (2-D) parallel beam tomographic formulation, it is assumed that the angles at which the projections were acquired are known. In certain situations, however, these angles are known only approximately (as in the case of magnetic resonance imaging (MRI) of a moving patient), or are completely unknown. The latter occurs in a three-dimensional (3-D) version of the problem in the electron microscopy-based imaging of viral particles. We address the problem of determining the view angles directly from the projection data itself in the 2-D parallel beam case. We prove the surprising result that under some fairly mild conditions, the view angles are uniquely determined by the projection data. We present conditions for the unique recovery of these view angles based on the Helgasson-Ludwig consistency conditions for the Radon transform, we also show that when the projections are shifted by some random amount which must be jointly estimated with the view angles, unique recovery of both the shifts and view angles is possible.

In the standard two-dimensional (2-D) parallel beam tomographic formulation, it is generally assumed that the angles at which the projections were acquired are known. We have previously demonstrated, however, that under fairly mild conditions these view angles can be uniquely recovered from the projections themselves. We address the question of reliability of such solutions to the angle recovery problem using moments of the projections. We demonstrate that under mild conditions, the angle recovery problem has unique solutions and is stable with respect to perturbations in the data. Furthermore, we determine the Cramer-Rao lower bounds on the variance of the estimates of the angles when the projection are corrupted by additive Gaussian noise. We also treat the case in which each projection is shifted by some unknown amount which must be jointly estimated with the view angles. Motivated by the stability results and relatively small values of the error bounds, we construct a simple algorithm to approximate the ML estimator and demonstrate that the problem can be feasibly solved in the presence of noise. Simulations using this simple estimator on a variety of phantoms show excellent performance at low to moderate noise levels, essentially achieving the Cramer-Rao bounds.

We present a new fast reconstruction algorithm for parallel beam tomography. The new algorithm is an accelerated version of the standard filtered backprojection (FBP) reconstruction, and uses a hierarchical decomposition of the backprojection operation to reduce the computational cost from O(N(3)) to O(N(2)log(2 )N). We discuss the choice of the various parameters that affect the behavior of the algorithm, and present numerical studies that demonstrate the cost versus distortion tradeoff. Comparisons with Fourier reconstruction algorithms and a multilevel inversion algorithm by Brandt et al., both of which also have O(N(2)log(2)N) cost, suggest that the proposed hierarchical scheme has a superior cost versus distortion performance. It offers RMS reconstruction errors comparable to the FBP with considerable speedup. For an example with a 512 x 512-pixel image and 1024 views, the speedup achieved with a particular implementation is over 40 fold, with reconstructions visually indistinguishable from the FBP.

We present a global Ziv-Zakai (1969) type lower bound on the mean square error for estimation of signal parameter vectors, where some components of the distortion function may be periodic. Periodic distortion functions arise naturally in the context of direction of arrival or phase estimation problems. The bound is applied to an image registration problem, and compared to the performance of the maximum-likelihood estimator (MLE).

The authors recently demonstrated that the view angles of tomographic projections were uniquely determined by the projection data alone for almost all objects, given enough projections. Here, they present results on the stability of the solution, as well as bounds on the variance of Convolution Back Projection (CBP) based reconstructions. To demonstrate that the problem can be feasibly solved in the presence of noise, the authors present a maximum likelihood estimator with a heuristic initialization algorithm which estimates the orientation of each projection.

We address the problem of parallel beam tomographic reconstruction when the angles at which the projections are taken are unknown. The problem arises in medical imaging owing to patient motion, and in imaging of viruses from a single projection of many identical units at random orientations. We determine conditions for unique identifiability of the angles from the projection data alone, and derive bounds on the variance of estimators of those angles in the presence of noise. Finally, we present a maximum likelihood estimator, along with a heuristic initialization procedure. Numerical simulations on a test phantom show excellent agreement with the bounds, and nearly perfect reconstructions at moderate noise levels.