Application of the fractional fourier transform to image reconstruction in MRI

The classic paradigm for MRI requires a homogeneous B₀ field in combination with linear encoding gradients. Distortions are produced when the B₀ is not homogeneous, and several post-processing techniques have been developed to correct them. Field homogeneity is difficult to achieve, particularly for short-bore magnets and higher B₀ fields. Nonlinear magnetic components can also arise from concomitant fields, particularly in low-field imaging, or intentionally used for nonlinear encoding. In any of these situations, the second-order component is key, because it constitutes the first step to approximate higher-order fields. We propose to use the fractional Fourier transform for analyzing and reconstructing the object's magnetization under the presence of quadratic fields. The fractional fourier transform provides a precise theoretical framework for this. We show how it can be used for reconstruction and for gaining a better understanding of the quadratic field-induced distortions, including examples of reconstruction for simulated and in vivo data. The obtained images have improved quality compared with standard Fourier reconstructions. The fractional fourier transform opens a new paradigm for understanding the MR signal generated by an object under a quadratic main field or nonlinear encoding.