Reconstruction of magnetic resonance images using one-dimensional techniques

Whenever DFT (discrete Fourier transform) processing of a multidimensional discrete signal is required, one can apply either a multidimensional FFT (fast Fourier transform) algorithm, or a single-dimension FFT algorithm, both using the same number of points. That is, the dimensions of a "multidimensional" signal, and of its spectrum, are a matter of choice. Every multidimensional sequence is completely equivalent to a one-dimensional function in both "time" and "frequency" domains. This statement applied to MRI (magnetic resonance imaging) explains why one can reconstruct the slice by using either one-dimensional or two-dimensional methods, as it is already done in echo planar methods. In the commonly used spin warp methods, the image can be also reconstructed by either one- or two-dimensional processing. However, some artifacts in the images reconstructed from the original "zig-zag" echo planar trajectory, are shown to be due to the wrong dimensionality of the FFT applied.

Uniqueness condition for the phase problem in one dimension

A result indicating the uniqueness of the phase problem for one-dimensional sequences of disconnected support is presented. This result rests on a mapping that converts a one-dimensional convolution into a two-dimensional one.