Short Note: The type 3 nonuniform FFT and its applications
Journal of Computational Physics
The fractional Fourier transform and time-frequency representations
IEEE Transactions on Signal Processing
Uncertainty Principle for Real Signals in the Linear Canonical Transform Domains
IEEE Transactions on Signal Processing - Part I
An uncertainty principle for real signals in the fractional Fouriertransform domain
IEEE Transactions on Signal Processing
Support vector machine for breast MR image classification
Computers & Mathematics with Applications
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The fractional Fourier transform (FrFT) is revisited in the framework of strongly continuous periodic semigroups to restate known results and to explore new properties of the FrFT. We then show how the FrFT can be used to reconstruct Magnetic Resonance (MR) images acquired under the presence of quadratic field inhomogeneity. Particularly, we prove that the order of the FrFT is a measure of the distortion in the reconstructed signal. Moreover, we give a dynamic interpretation to the order as time evolution of a function. We also introduce the notion of @r-@a space as an extension of the Fourier or k-space in MR, and we use it to study the distortions introduced in two common MR acquisition strategies. We formulate the reconstruction problem in the context of the FrFT and show how the semigroup theory allows us to find new reconstruction formulas for discrete sampled signals. Finally, the results are supplemented with numerical examples that show how it performs in a standard 1D MR signal reconstruction.