Jason Peter Stockmann1, Gigi Galiana2,
Vicente Parot3,4, Leo Tam1, Robert Todd Constable2
1Biomedical Engineering,
Yale University, New Haven, CT, United States; 2Diagnostic
Radiology, Yale University, New Haven, CT, United States; 3Biomedical
Imaging Center, Pontificia Universidad Catlica de Chile, Santiago, Chile; 4Department
of Electrical Engineering , Pontificia Universidad Catlica de Chile,
Santiago, Chile
Scalable image reconstruction from phase-scrambled data has previously been performed by expressing the signal equation as a Fresnel transform, which has the form of a chirp convolution. Scaling is achieved by varying the frequency of the chirp kernel that is multiplied by and deconvolved with the data. This approach has the form of a fractional Fourier transform (FrFT) but has not been previously described as such in the MR community. In this work, the relationship between the Fresnel transform and the FrFT is elucidated. The FrFT is then used to reconstruct alias-free scalable images from undersampled data that have been phase-scrambled using a powerful Z2 gradient coil.