The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform f;. The method involves a window function w; and proceeds in three steps. First, the convolution g;=w;*f; is computed numerically on a Cartesian grid, using the available samples of f;. Then, g=wf is computed via the inverse discrete Fourier transform, and finally f is obtained as g/w. Due to the smoothing effect of the convolution, evaluating w;*f; is much less error prone than merely interpolating f;. The method was originally devised for image reconstruction in radio astronomy, but is actually applicable to a broad range of reconstructive imaging methods, including magnetic resonance imaging and computed tomography. In particular, it provides a fast and accurate alternative to the filtered backprojection. The basic method has several variants with other applications, such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the Radon (Hough) transform.

The recent developments in light generation and detection techniques have opened new possibilities for optical medical imaging, tomography, and diagnosis at tissue penetration depths of ~10 cm. However, because light scattering and diffusion in biological tissue are rather strong, the reconstruction of object images from optical projections needs special attention. We describe a simple reconstruction method for diffuse optical imaging, based on a modified backprojection approach for medical tomography. Specifically, we have modified the standard backprojection method commonly used in x-ray tomographic imaging to include the effects of both the diffusion and the scattering of light and the associated nonlinearities in projection image formation. These modifications are based primarily on the deconvolution of the broadened image by a spatially variant point-spread function that is dependent on the scattering of light in tissue. The spatial dependence of the deconvolution and nonlinearity corrections for the curved propagating ray paths in heterogeneous tissue are handled semiempirically by coordinate transformations. We have applied this method to both theoretical and experimental projections taken by parallel- and fan-beam tomography geometries. The experimental objects were biomedical phantoms with multiple objects, including in vitro animal tissue. The overall results presented demonstrate that image-resolution improvements by nearly an order of magnitude can be obtained. We believe that the tomographic method presented here can provide a basis for rapid, real-time medical monitoring by the use of optical projections. It is expected that such optical tomography techniques can be combined with the optical tissue diagnosis methods based on spectroscopic molecular signatures to result in a versatile optical diagnosis and imaging technology.

In magnetic resonance imaging (MRI), the spatial inhomogeneity of the static magnetic field can cause degraded images if the reconstruction is based on inverse Fourier transformation. This paper presents and discusses a range of fast reconstruction algorithms that attempt to avoid such degradation by taking the field inhomogeneity into account. Some of these algorithms are new, others are modified versions of known algorithms. Speed and accuracy of all these algorithms are demonstrated using spiral MRI.

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We describe a fast and accurate direct Fourier method for reconstructing a function f of three variables from a number of its parallel beam projections. The main application of our method is in single particle analysis, where the goal is to reconstruct the mass density of a biological macromolecule. Typically, the number of projections is extremely large, and each projection is extremely noisy. The projection directions are random and initially unknown. However, it is possible to determine both the directions and f by an iterative procedure; during each stage of the iteration, one has to solve a reconstruction problem of the type considered here. Our reconstruction algorithm is distinguished from other direct Fourier methods by the use of gridding techniques that provide an efficient means to compute a uniformly sampled version of a function g from a nonuniformly sampled version of Fg, the Fourier transform of g, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of f, the function to be reconstructed, in order to obtain Ff on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct f from this sampled version of Ff. This stage requires a proper weighting of the samples of Ff to compensate for their nonuniform distribution. We use a fast method for computing appropriate weights that exploits the special properties of the spherical sampling grid for Ff and involves the computation of a Voronoi diagram on the unit sphere. We demonstrate the excellent speed and accuracy of our method by using simulated data.

Analogous to reconstructive X-ray tomography one can determine the spatial distribution of the acoustic absorption coefficient or the acoustic refractive index within an object from many 'projections' of the object taken with ultrasound beams instead of X-rays. However, the reconstruction quality obtained so far is insufficient, mainly since the mathematical models employed assume the beams, propagating along straight lines which is not sufficiently satisfied. An improved mathematical model and a pertinent reconstruction algorithm are presented which take the bending of the beams into account and yield substantially improved reconstructions.

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