1. Pinv-Recon: Generalized MR Image Reconstruction via Pseudoinversion of the Encoding Matrix
- Author
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Yeung, Kylie, Gleeson, Fergus V, Schulte, Rolf F, McIntyre, Anthony, Serres, Sebastien, Morris, Peter, Auer, Dorothee, Tyler, Damian J, Grist, James T, and Wiesinger, Florian
- Subjects
Physics - Medical Physics ,Electrical Engineering and Systems Science - Image and Video Processing - Abstract
Purpose: To present a novel generalized MR image reconstruction based on pseudoinversion of the encoding matrix (Pinv-Recon) as a simple yet powerful method, and demonstrate its computational feasibility for diverse MR imaging applications. Methods: MR image encoding constitutes a linear mapping of the unknown image to the measured k-space data mediated via an encoding matrix ($ data = Encode \times image$). Pinv-Recon addresses MR image reconstruction as a linear inverse problem ($image = Encode^{-1} \times data$), explicitly calculating the Moore-Penrose pseudoinverse of the encoding matrix using truncated singular value decomposition (tSVD). Using a discretized, algebraic notation, we demonstrate constructing a generalized encoding matrix by stacking relevant encoding mechanisms (e.g., gradient encoding, coil sensitivity encoding, chemical shift inversion) and encoding distortions (e.g., off-center positioning, B$_0$ inhomogeneity, spatiotemporal gradient imperfections, transient relaxation effects). Iterative reconstructions using the explicit generalized encoding matrix, and the computation of the spatial-response-function (SRF) and noise amplification, were demonstrated. Results: We evaluated the computation times and memory requirements (time ~ (size of the encoding matrix)$^{1.4}$). Using the Shepp-Logan phantom, we demonstrated the versatility of the method for various intertwined MR image encoding and distortion mechanisms, achieving better MSE, PSNR and SSIM metrics than conventional methods. A diversity of datasets, including the ISMRM CG-SENSE challenge, were used to validate Pinv-Recon. Conclusion: Although pseudo-inversion of large encoding matrices was once deemed computationally intractable, recent advances make Pinv-Recon feasible. It has great promise for both research and clinical applications, and for educational use., Comment: 26 pages, 8 figures, 2 tables (+ Supplementary Material). Submitted to Magnetic Resonance in Medicine
- Published
- 2024