--Quantitative magnetic resonance imaging (qMRI) provides tissue-specific parameters vital for clinical diagnosis. Although simultaneous multi-parametric qMRI (MP-qMRI) technologies enhance imaging efficiency, robustly reconstructing qMRI from highly undersampled, high-dimensional measurements remains a significant challenge. This difficulty arises primarily because current reconstruction methods that rely solely on a single prior or physics-informed model to solve the highly ill-posed inverse problem, which often leads to suboptimal results. T o overcome this limitation, we propose LoREIN, a novel unsupervised and dual-prior-integrated framework for accelerated 3D MP-qMRI reconstruction. T echnically, LoREIN incorporates both low-rank prior and continuity prior via low-rank representation (LRR) and implicit neural representation (INR), respectively, to enhance reconstruction fidelity. The powerful continuous representation of INR enables the estimation of optimal spatial bases within the low-rank subspace, facilitating high-fidelity reconstruction of weighted images. Simultaneously, the predicted multi-contrast weighted images provide essential structural and quantitative guidance, further enhancing the reconstruction accuracy of quantitative parameter maps. Furthermore, our work introduces a zero-shot learning paradigm with broad potential in complex spatiotemporal and high-dimensional image reconstruction tasks, further advancing the field of medical imaging.

Accurate estimation of the speed-of-sound (SoS) is important for ultrasound (US) image reconstruction techniques and tissue characterization. Various approaches have been proposed to calculate SoS, ranging from tomography-inspired algorithms like CUTE to convolutional networks, and more recently, physics-informed optimization frameworks based on differentiable beamforming. In this work, we utilize implicit neural representations (INRs) for SoS estimation in US. INRs are a type of neural network architecture that encodes continuous functions, such as images or physical quantities, through the weights of a network. Implicit networks may overcome the current limitations of SoS estimation techniques, which mainly arise from the use of non-adaptable and oversimplified physical models of tissue. Moreover, convolutional networks for SoS estimation, usually trained using simulated data, often fail when applied to real tissues due to out-of-distribution and data-shift issues. In contrast, implicit networks do not require extensive training datasets since each implicit network is optimized for an individual data case. This adaptability makes them suitable for processing US data collected from varied tissues and across different imaging protocols. We evaluated the proposed SoS estimation method based on INRs using data collected from a tissue-mimicking phantom containing four cylindrical inclusions, with SoS values ranging from 1480 m/s to 1600 m/s. The inclusions were immersed in a material with an SoS value of 1540 m/s. In experiments, the proposed method achieved strong performance, clearly demonstrating the usefulness of implicit networks for quantitative US applications.

We propose the POD-DNN, a novel algorithm leveraging deep neural networks (DNNs) along with radial basis functions (RBFs) in the context of the proper orthogonal decomposition (POD) reduced basis method (RBM), aimed at approximating the parametric mapping of parametric partial differential equations on irregular domains. The POD-DNN algorithm capitalizes on the low-dimensional characteristics of the solution manifold for parametric equations, alongside the inherent offline-online computational strategy of RBM and DNNs. In numerical experiments, POD-DNN demonstrates significantly accelerated computation speeds during the online phase. Compared to other algorithms that utilize RBF without integrating DNNs, POD-DNN substantially improves the computational speed in the online inference process. Furthermore, under reasonable assumptions, we have rigorously derived upper bounds on the complexity of approximating parametric mappings with POD-DNN, thereby providing a theoretical analysis of the algorithm's empirical performance.

We propose a categorical semantics for machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, and different architectures, shedding new light on their similarities and differences. Furthermore, our approach to learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realised in the discrete setting of Boolean and polynomial circuits. We demonstrate the practical significance of our framework with an implementation in Python.

In this work, we analyze the suitability of deep neural networks (DNNs) for the numerical solution of parametric problems. Such problems connect a parameter space with a solution state space via a so-called parametric map, [53]. One special case of such a parametric problem arises when the parametric map results from solving a partial differential equation (PDE) and the parameters describe physical or geometrical constraints of the PDE such as, for example, the shape of the physical domain, boundary conditions, or a source term. Applications that lead to these problems include modeling unsteady and steady heat and mass transfer, acoustics, fluid mechanics, or electromagnetics, [34]. Solving a parametric PDE for every point in the parameter space of interest individually typically leads to two types of problems.

In the context of dynamic emission tomography, the conventional processing pipeline consists of independent image reconstruction of single time frames, followed by the application of a suitable kinetic model to time activity curves (TACs) at the voxel or region-of-interest level. The relatively new field of 4D PET direct reconstruction, by contrast, seeks to move beyond this scheme and incorporate information from multiple time frames within the reconstruction task. Existing 4D direct models are based on a deterministic description of voxels' TACs, captured by the chosen kinetic model, considering the photon counting process the only source of uncertainty. In this work, we introduce a new probabilistic modeling strategy based on the key assumption that activity time course would be subject to uncertainty even if the parameters of the underlying dynamic process were known. This leads to a hierarchical Bayesian model, which we formulate using the formalism of Probabilistic Graphical Modeling (PGM). The inference of the joint probability density function arising from PGM is addressed using a new gradient-based iterative algorithm, which presents several advantages compared to existing direct methods: it is flexible to an arbitrary choice of linear and nonlinear kinetic model; it enables the inclusion of arbitrary (sub)differentiable priors for parametric maps; it is simpler to implement and suitable to integration in computing frameworks for machine learning. Computer simulations and an application to real patient scan showed how the proposed approach allows us to weight the importance of the kinetic model, providing a bridge between indirect and deterministic direct methods.

Parametric images provide insight into the spatial distribution of physiological parameters, but they are often extremely noisy, due to low SNR of tomographic data. Direct estimation from projections allows accurate noise modeling, improving the results of post-reconstruction fitting. We propose a method, which we name kinetic compressive sensing (KCS), based on a hierarchical Bayesian model and on a novel reconstruction algorithm, that encodes sparsity of kinetic parameters. Parametric maps are reconstructed by maximizing the joint probability, with an Iterated Conditional Modes (ICM) approach, alternating the optimization of activity time series (OS-MAP-OSL), and kinetic parameters (MAP-LM). We evaluated the proposed algorithm on a simulated dynamic phantom: a bias/variance study confirmed how direct estimates can improve the quality of parametric maps over a post-reconstruction fitting, and showed how the novel sparsity prior can further reduce their variance, without affecting bias. Real FDG PET human brain data (Siemens mMR, 40min) images were also processed. Results enforced how the proposed KCS-regularized direct method can produce spatially coherent images and parametric maps, with lower spatial noise and better tissue contrast. A GPU-based open source implementation of the algorithm is provided.