PMID: 9660548Jul 11, 1998Paper

An optimal and efficient new gridding algorithm using singular value decomposition

Magnetic Resonance in Medicine : Official Journal of the Society of Magnetic Resonance in Medicine
D Rosenfeld

Abstract

The problem of handling data that falls on a nonequally spaced grid occurs in numerous fields of science, ranging from radio-astronomy to medical imaging. In MRI, this condition arises when sampling under time-varying gradients in sequences such as echo-planar imaging (EPI), spiral scans, or radial scans. The technique currently being used to interpolate the nonuniform samples onto a Cartesian grid is called the gridding algorithm. In this paper, a new method for uniform resampling is presented that is both optimal and efficient. It is first shown that the resampling problem can be formulated as a problem of solving a set of linear equations Ax = b, where x and b are vectors of the uniform and nonuniform samples, respectively, and A is a matrix of the sinc interpolation coefficients. In a procedure called Uniform Re-Sampling (URS), this set of equations is given an optimal solution using the pseudoinverse matrix which is computed using singular value decomposition (SVD). In large problems, this solution is neither practical nor computationally efficient. Another method is presented, called the Block Uniform Re-Sampling (BURS) algorithm, which decomposes the problem into solving a small set of linear equations for each uniform g...Continue Reading

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