We introduce a generalization of a deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We prove that our more general algorithm is globally convergent (i.e., converges to a local minimum from any initialization) under less restrictive conditions, even when the original cost functional is nonconvex. We apply this algorithm to tomographic reconstruction from limited-angle data by formulating the problem as one of regularized least-squares optimization. The results demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent limited-angle tomographic reconstructions. Two edge-preserving regularizers-one convex, the other nonconvex-are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited-angle scenarios, and to explore how factors, such as the choice of error norm, angular sampling rate and amount of noise, affect the reconstruction quality and algorithm performance. These simulation results show that for this application, the nonconvex regularizer produces consistently superior results.

We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N(2)logN) floating-point operations per iteration to reconstruct an NxN image from P view angles, as compared to O(N (2)P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary.

Uses a series-expansion approach and an operator framework to derive a new, fast and accurate, iterative tomographic reconstruction algorithm applicable for parallel-ray projections that have been collected at a finite number of arbitrary view angles and have been radially sampled at a rate high enough so that aliasing errors are small. The authors use the conjugate gradient algorithm to minimize a regularized least squares criterion, and prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the FFT. The proposed algorithm requires O(N/sup 2/ log N) multiplies per iteration to reconstruct an N/spl times/N image from P view angles, and requires the storage of half of a 2N/spl times/2N PSF.

We demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent tomographic reconstructions from limited-angle data. The tomography problem is formulated as a regularized least-squares optimization problem, and is then solved using a generalization of a recently proposed deterministic relaxation algorithm. This algorithm has been shown to converge under certain conditions when the original cost functional being minimized is convex. We have proven that our more general algorithm is globally convergent under less restrictive conditions, even when the original cost functional is nonconvex. Simulation results demonstrate the effectiveness of the algorithm, and show that for moderate to high photon counts, spectrally weighted error norms perform as well as, or better than a standard error norm that is commonly used for Poisson-distributed data. This suggests that a recently proposed fast Fourier algorithm, which is restricted to using a spectrally weighted error norm, can be used in many practical limited-angle problems to perform the minimization needed by the deterministic relaxation algorithm.

Shows how the separable two-dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm. This algorithm is similar to the conventional filtered backprojection algorithm, except that the filters are now angle dependent, and the backprojection gives the wavelet coefficients of the reconstruction, which are then used to synthesize the reconstruction at various resolution levels. By reconstructing only a small localized region at high resolution, the authors show how radiation exposure and computation can be significantly reduced, compared to a standard reconstruction.

We show how the separable two-dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm. This algorithm is similar to the conventional filtered backprojection algorithm, except that the filters are now angle dependent, and the backprojection gives us the wavelet coefficients of the reconstruction, which are then used to synthesize the reconstruction at various resolution levels. By reconstructing only a small localized region at high resolution, radiation exposure and computation can be significantly reduced, compared to a standard reconstruction.>

An algorithm for detecting and estimating the parameters of an unknown number of closely spaced, superimposed noisy echoes of a signal of unknown and arbitrary shape is derived. The algorithm exploits an approximate invariance structure in the frequency domain that allows the ESPRIT algorithm to be used for parameter estimation. An extension of the algorithm is applicable to the resolution of superimposed pulses of unknown and possibly different shapes. Computer simulation results for the algorithm are compared with the corresponding Cramer-Rao performance bounds.>