Pinv-Recon: Generalized MR Image Reconstruction via Pseudoinversion of the Encoding Matrix

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.

Distribution-informed and wavelength-flexible data-driven photoacoustic oximetry

Significance: Photoacoustic imaging (PAI) promises to measure spatially-resolved blood oxygen saturation, but suffers from a lack of accurate and robust spectral unmixing methods to deliver on this promise. Accurate blood oxygenation estimation could have important clinical applications, from cancer detection to quantifying inflammation. Aim: This study addresses the inflexibility of existing data-driven methods for estimating blood oxygenation in PAI by introducing a recurrent neural network architecture. Approach: We created 25 simulated training dataset variations to assess neural network performance. We used a long short-term memory network to implement a wavelength-flexible network architecture and proposed the Jensen-Shannon divergence to predict the most suitable training dataset. Results: The network architecture can handle arbitrary input wavelengths and outperforms linear unmixing and the previously proposed learned spectral decolouring method. Small changes in the training data significantly affect the accuracy of our method, but we find that the Jensen-Shannon divergence correlates with the estimation error and is thus suitable for predicting the most appropriate training datasets for any given application. Conclusions: A flexible data-driven network architecture combined with the Jensen-Shannon Divergence to predict the best training data set provides a promising direction that might enable robust data-driven photoacoustic oximetry for clinical use cases.