A Generalized Inverse Fourier Transformation (GIFT) Approach for Direct Image Reconstruction from Arbitrary K-space Trajectories

Motivation: K-space re-gridding or sampling density compensation is required for image reconstruction with arbitrary K-space trajectory, e.g., in FFT, NUFFT, GFFT, etc. Goal(s): We propose a generalized inverse Fourier transform (GIFT) approach to direct image reconstruction. The reconstruction is continuous in image space. Approach: We generalize continuous Fourier transform to any coordinate systems, arbitrary K-space sampling/trajectory, and arbitrary K-point size and shape. Results: Images were calculated from a spiral k-space trajectory in the 2D Cartesian coordinate system, which we use as examples to demonstrate GIFT's reconstruction flexibility for different resolutions, also within any small, focused region of interest (ROI). Impact: The generalized image reconstruction algorithm apply to both Cartesian and Polar, can be readily used for non-uniform K-space with arbitrary trajectory. Images can be reconstructed with arbitrary resolution also within any small ROIs.