Fourier-Dual Gaussian Representations for 3D Non-Cartesian Parallel MRI Reconstruction

Gaussian primitives are an attractive continuous representation choice for the MRI reconstruction inverse problem because their Fourier transform is available in closed form, enabling fitting directly from the measurement in the spatial Fourier domain ($k$-space). However, existing Gaussian-based MRI reconstruction methods fail to leverage this, as they either rasterize an image during the forward model or use expensive parameterizations and parallel imaging formulations that limit this analytical advantage. In addition, they parameterize each primitive by its amplitude and couples its energy to its spatial extent, which degrades the conditioning of the fit in $k$-space. We propose a Fourier-dual representation for 3D non-Cartesian parallel MRI that removes discretization entirely as follows: (i) Integral-Parameterized Gaussian (IPG) decouples the integral from spatial scale in $k$-space, and we prove better conditioning both theoretically and experimentally; (ii) Coil sensitivities are expressed in a compact Fourier basis for direct measurement at arbitrary $k$-space locations from a single shared Gaussian representation; (iii) A surrogate continuous total variation prior by Monte Carlo sampling of the analytical spatial gradients. We compare against compressed sensing, scan-specific implicit neural representations (INR), and existing Gaussian methods on 19 3D brain volumes, as well as a phantom acquisition. The proposed method obtains the highest PSNR and SSIM across this benchmark.