Algebraic Methods and Computational Strategies for Pseudoinverse-Based MR Image Reconstruction (Pinv-Recon)

Image reconstruction in Magnetic Resonance Imaging (MRI) is fundamentally a linear inverse problem, such that the image can be recovered via explicit pseudoinversion of the encoding matrix by solving $\textbf{data} = \textbf{Encode} \times \textbf{image}$ - a method referred to here as Pinv-Recon. While the benefits of this approach were acknowledged in early studies, the field has historically favored fast Fourier transforms (FFT) and iterative techniques due to perceived computational limitations of the pseudoinversion approach. This work revisits Pinv-Recon in the context of modern hardware, software, and optimized linear algebra routines. We compare various matrix inversion strategies, assess regularization effects, and demonstrate incorporation of advanced encoding physics into a unified reconstruction framework. While hardware advances have already significantly reduced computation time compared to earlier studies, our work further demonstrates that leveraging Cholesky decomposition leads to a two-order-of-magnitude improvement in computational efficiency over previous Singular Value Decomposition-based implementations. Moreover, we demonstrate the versatility of Pinv-Recon on diverse $\textit{in vivo}$ datasets encompassing a range of encoding schemes, starting with low- to medium-resolution functional and metabolic imaging and extending to high-resolution cases. Our findings establish Pinv-Recon as a versatile and robust reconstruction framework that aligns with the increasing emphasis on open-source and reproducible MRI research.