Henry Stark

High-resolution image recovery from image-plane arrays, using convex projections

We consider the problem of reconstructing remotely obtained images from image-plane detector arrays. Although the individual detectors may be larger than the blur spot of the imaging optics, high-resolution reconstructions can be obtained by scanning or rotating the image with respect to the detector. As an alternative to matrix inversion or least-squares estimation [Appl. Opt. 26, 3615 (1987)], the method of convex projections is proposed. We show that readily obtained prior knowledge can be used to obtain good-quality imagery with reduced data. The effect of noise on the reconstruction process is considered.

Tomographic Image Reconstruction from Incomplete View Data by Convex Projections and Direct Fourier Inversion

We consider the problem of reconstructing CAT imagery by the direct Fourier method (DFM) when not all view data are available. To restore the missing information we use the method of projections onto convex sets (POCS). POCS is a recursive image restoration technique that finds a solution consistent with the measured data and a priori known constraints in both the space and Fourier domain. Because DFM reconstruction is a frequency-domain technique it is ideally matched to POCS restoration when, for one reason or another, we are forced to generate an image from a less than complete set of view data. We design and apply an algorithm (PRDF) which interpolates/extrapolates the missing Fourier domain information by POCS and reconstructs an image by DFM. A simulated human thorax cross section is restored and reconstructed. The restorations using POCS are compared with the Gerchberg-Papoulis extrapolation method and shown to be superior. Applications of PRDF to other types of medical imaging modalities are discussed.

Projection-based blind deconvolution

We present a new projection-based algorithm for solving the classical blind-deconvolution problem. In our approach all known a priori information about both the unknown source and the blurring functions is expressed through constraint sets. In computer simulations the algorithm performed well even when the prior information was not accurate. To see how well our algorithm compares against others, we compared it with another recently published deconvolution method [ J. Opt. Soc. Am. A9, 932 ( 1992)]. The advantages of each method are discussed.

Iterative and one-step reconstruction from nonuniform samples by convex projections

We apply the method of projections onto convex sets (POCS) to recover a signal or image from nonuniform samples and prior knowledge. Both iterative and one-step algorithms to implement POCS are furnished. We discuss our results in relation to the recent work by Sauer and Allebach [ IEEE Trans. Circuits Syst.CAS-34, 1497 ( 1987)], who considered the same problem from a different point of view.

Spiral Sampling in Magnetic Resonance Imaging-The Effect of Inhomogeneities

A study of a spiral scan echo planar method in the presence of static field inhomogeneities is presented. The approach consists of obtaining the samples over the Fourier plane trajectory of the spiral scan, as modified by the inhomogeneities. The resulting images are compared to a Cartesian scan echo-planar method also subjected to inhomogeneities in the static field. The conclusion is that the spiral scan is more sensitive to this type of inhomogeneity. The possibility of compensating for the inhomogeneities during the reconstruction procedure is suggested by a preliminary experimental result.

Direct Fourier Reconstruction in Fan-Beam Tomography

We consider the problem of reconstructing tomographic imagery from fan-beam projections using the direct Fourier method (DFM). Previous DFM reconstructions from parallel-beam projections produced images of quality comparable to that of filtered convolution back-projection. Moreover, the number of operations using DFM in the parallel-beam case is proportional to N2 log N versus N3 for back projection [3]. The fan-beam case is more complicated because additional interpolation of the nonuniformly spaced rebinned data is required. We derive bounds on the detector spacing in fan-beam CT that enable direct Fourier reconstruction and describe the full algorithm necessary for processing the fan-beam data. The feasibility of the method is demonstrated with an example. A key result of this paper is that high-quality imagery can be reconstructed from fan-beam data using the DFM in 0 (N2 log N) operations.

Constrained FIR filter design by the method of vector space projections

A new technique for designing linear and arbitrary-phase finite-impulse response (FIR) filters with various types of constraints is proposed. The approach is based on the method of vector space projections. We describe the constraint sets and their associated projections that capture the properties of the desired filters. In filter design, as in many other engineering problems, one is primarily interested in meeting design constraints, i.e., finding a feasible solution, not necessarily an optimum one. Vector space projection methods are well-suited for this purpose. We furnish numerous examples of FIR filter design by vector space projections, including the important and difficult arbitrary phase/magnitude problem. Examples that demonstrate the advantages and flexibility of this method over other known methods are furnished.

Design of phase gratings by generalized projections

We address the problem of designing phase gratings that generate prescribed light intensities in the far field. The method of generalized projections is used to find a solution consistent with prior constraints and available data. While the method of generalized projections is known to converge to erroneous solutions occasionally, such solutions were not observed in this problem.

Design of self-healing arrays using vector-space projections

We consider the problem of reconfiguring array antennas whose performance has been degraded by the failure of one or more elements. We show that the method of vector-space projections is well suited for this task and enables the recovery of reasonable antenna performance when as many as 30% of the elements are inoperable.

Design of digital linear-phase FIR crossover systems for loudspeakers by the method of vector space projections

A new technique for designing digital linear-phase FIR crossover systems for loudspeakers is proposed. The approach is based on the principle of vector space projections. We describe the constraint sets and their projections that capture the properties of the desired crossover filters. The proposed approach is capable of designing crossover networks for multiple band-splitting as well as for equalization. Designs that demonstrate the advantages and flexibility of this method are furnished.