An automated image analysis system for determining myosin filament azimuthal rotations, or orientations, in electron micrographs of muscle cross sections is described. The micrographs of thin sections intersect the myosin filaments which lie on a triangular lattice. The myosin filament profiles are variable and noisy, and the images exhibit a variable contrast and background. Filament positions are determined by filtering with a point spread function that incorporates the local symmetry of the lattice. Filament orientations are determined by correlation with a template that incorporates the salient filament characteristics, and the orientations are classified using a Gaussian mixture model. The precision of the technique is assessed by application to a variety of micrographs and comparison with manual classification of the orientations. The system provides a convenient, robust, and rapid means of analysing micrographs containing many filaments to study the distribution of filament orientations.

The envelope problem in macromolecular x-ray crystallography involves determining the boundary of a molecule from measurements of amplitudes of x-rays diffracted from a crystalline specimen. This represents a highly underdetermined image reconstruction problem with a large number of degrees of freedom. We regularize the problem by applying binary and connectivity constraints to the image, and seek the solution using the method of iterated projections. However, since the constraints are highly non-convex, the usual methods of generalized projections are not effective. We use the difference map projection algorithm and show that this is effective with simulated diffraction data from a protein envelope.

In vivo optical molecular imaging involves the use of light emitting tracers combined with sophisticated sensing modalities to perform in vivo imaging of genetic and molecular information. In contrast to the classical diagnostic imaging tools which image the end effects of the diseases, optical molecular imaging could enhance our knowledge of biological phenomena, monitor genetic expression and the alteration of cells, and lead to earlier detection of diseases. With the development of exotic molecular probes with easily detectable bioluminescence and fluorescence labels, optical molecular imaging has emerged as an important new field within biomedical imaging. This paper reviews this state-of-the-art imaging technology and signal processing issues to monitor molecular and cellular events in living organisms.

Recovery of magnetic resonance images from irregular sampling sets is investigated from the point of view of moment discretization of the Fredholm equation of the first kind. The limited spatial extent of the object is known a priori and the sampling schemes considered each have mean density lower than that imposed by the Nyquist limit. The recovery formula obtained has the same form as a standard irregular sampling technique. A practical means of performing the recovery for very large data sets by utilising the block Toeplitz structure of the matrix involved is presented.

Optical diffusion tomography is a technique for imaging a highly scattering medium using measurements of transmitted modulated light. Reconstruction of the spatial distribution of the optical properties of the medium from such data is a difficult nonlinear inverse problem. Bayesian approaches are effective, but are computationally expensive, especially for three-dimensional (3-D) imaging. This paper presents a general nonlinear multigrid optimization technique suitable for reducing the computational burden in a range of nonquadratic optimization problems. This multigrid method is applied to compute the maximum a posteriori (MAP) estimate of the reconstructed image in the optical diffusion tomography problem. The proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality.

Optical diffusion tomography attempts to reconstruct an object cross section (a highly scattering media such as tissue) from measurements of scattered and attenuated light. While Bayesian approaches are well suited to this difficult nonlinear inverse problem, the resulting optimization problem is very computationally expensive. In this paper, we propose a nonlinear multigrid technique for computing the maximum a posteriori (MAP) reconstruction in the optical diffusion tomography problem. The multigrid approach improves reconstruction quality by avoiding a local minimum. In addition, it dramatically reduces computation. Each iteration of the algorithm alternates a Born approximation step with a single cycle of a nonlinear multigrid algorithm.

An image reconstruction problem motivated by X-ray fiber diffraction analysis is considered. The experimental data are sums of the squares of the amplitudes of particular sets of Fourier coefficients of the electron density, and a part of the electron density is known. The image reconstruction problem is to estimate the unknown part of the electron density, the "image." A Bayesian approach is taken in which a prior model for the image is based on the fact that it consists of atoms, i.e., the unknown electron density consists of separated, sharp peaks. Currently used heuristic methods are shown to correspond to certain maximum a posteriori estimates of the Fourier coefficients. An analytical solution for the Bayesian minimum mean-square-error estimate is derived. Simulations show that the minimum mean-square-error estimate gives good results, even when there is considerable data loss, and out-performs the maximum a posteriori estimates.

The restoration of an incomplete image from a known part and experimental data in the form of the Fourier amplitude squared sums is formulated as a Bayesian estimation problem. This problem is motivated by the structure completion problem in X-ray fiber diffraction analysis. An appropriate prior of uniformly distributed impulses is used. The Bayesian MMSE and MAP estimates are obtained. Simulations are used to compare the performance of the estimates. The results show that the MMSE estimate significantly outperforms the other estimates. The restored images exhibit some bias towards the known part of the image. This can be partly reduced by an unbiasing procedure.

There are a number of technical situations where inversion of measured data involves blind deconvolution (i.e. the point spread function is unknown a priori) in three (or more) dimensions. We show that a unique solution exists for three-dimensional (3D) blind deconvolution for a particular sampling of the Fourier transform of a blurred image whose average density is less than the Nyquist density. This suggests that 3D blind deconvolution problems may be easier to solve than 2D problems. Experiments using an iterative blind deconvolution algorithm indicate that convergence is faster for the 3D case than for the 2D case, and that reducing the sampling density has a rather small effect on the quality of reconstructions.>

The Hartley transform is an integral transform closely related to the Fourier transform. It has some advantages over the Fourier transform in the analysis of real signals as it avoids the use of complex arithmetic. However, the Hartley transform has other applications in signal and image reconstruction related to traditional phase retrieval problems. These can he understood by examining the analytic properties of the Hartley transform in the complex plane. In this paper, the analytic continuation of the Hartley transform into the complex plane is derived and its properties discussed. It is shown that for signals or images of finite extent, the Hartley transform is analytic in the entire finite complex plane, and this is used to derive properties of its complex zeros. Hilbert transform-type relationships for the Hartley transform, related to causal and analytic-signals, are also derived. The analytic properties derived are used to study the problem of image reconstruction from the Hartley transform intensity. Uniqueness and reconstruction algorithms for one- and two-dimensional problems are discussed, and examples are presented. Generation of image moments from the Hartley transform intensity is also described.>