Understanding the reconstruction of non-Cartesian sampled magnetic resonance imaging data via the Schwartz spaces

Gordon E. Sarty

doi:10.1117/12.349571

28 May 1999 Understanding the reconstruction of non-Cartesian sampled magnetic resonance imaging data via the Schwartz spaces

Gordon E. Sarty

Author Affiliations +

Proceedings Volume 3659, Medical Imaging 1999: Physics of Medical Imaging; (1999) https://doi.org/10.1117/12.349571
Event: Medical Imaging '99, 1999, San Diego, CA, United States

Abstract

Purpose: To provide a mathematically rigorous basis for the study of data sampling issues in fast Magnetic Resonance Imaging (MRI). Methods: The problem of reconstructing MRI data that have been sampled along arbitrary curves or trajectories in k-space was formulated in the language of distribution theory. With this formulation the nature of the associated point-spread function could then be related to conventional band-pass operators. Results: The definition of band-pass operators can be extended to curve-pass and sample-pass operators in a rigorous way. Each of these operators can be associated with point-spread functions that are distributional Fourier transforms of generalized functions defined by the given sampling region in k-space. For band-pass operators, the generalized function is related to the functional of line integration. For sample-pass operators, the generalized function is a sum of weighted Dirac delta functions. Breakthroughs: The problem of reconstructing an approximation for a two-dimensional function (an image) from two-, one- and zero-dimensional spatial frequency data sets has been shown to be well-posed. A rigorous relationship, exposing the nature of the approximation in terms of distributional Fourier transforms, between the different approaches to reconstruction has been found. Conclusions: The principle unsolved problem in reconstructing MRI data collected on non-Cartesian grids in k- space is the formulation of a sampling theorem similar to the Nyquist sampling theorem. The distributional formulation clearly reveals the connections between continuous and discrete distributional Fourier transforms and promises to be a useful tool for the formulation of a sampling theorem.

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Gordon E. Sarty "Understanding the reconstruction of non-Cartesian sampled magnetic resonance imaging data via the Schwartz spaces", Proc. SPIE 3659, Medical Imaging 1999: Physics of Medical Imaging, (28 May 1999); https://doi.org/10.1117/12.349571

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KEYWORDS

Fourier transforms

Point spread functions

Magnetic resonance imaging

Destructive interference

Heat treatments

Bessel functions

Data acquisition

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