ALMA: a mathematics-driven approach for determining tuning parameters in generalized LASSO problems, with applications to MRI

Magnetic Resonance Imaging (MRI) is a powerful technique employed for non-invasive in vivo visualization of internal structures. Sparsity is often deployed to accelerate the signal acquisition or overcome the presence of motion artifacts, improving the quality of image reconstruction. Image reconstruction algorithms use TV-regularized LASSO (Total Variation-regularized LASSO) to retrieve the missing information of undersampled signals, by cleaning the data of noise and while optimizing sparsity. A tuning parameter moderates the balance between these two aspects; its choice affecting the quality of the reconstructions. Currently, there is a lack of general deterministic techniques to choose these parameters, which are oftentimes manually selected and thus hinder the reliability of the reconstructions. Here, we present ALMA (Algorithm for Lagrange Multipliers Approximation), an iterative mathematics-inspired technique that computes tuning parameters for generalized LASSO problems during MRI reconstruction. We analyze quantitatively the performance of these parameters for imaging reconstructions via TV-LASSO in an MRI context on phantoms. Although our study concentrates on TV-LASSO, the techniques developed here hold significant promise for a wide array of applications. ALMA is not only adaptable to more generalized LASSO problems but is also robust to accommodate other forms of regularization beyond total variation. Moreover, it extends effectively to handle non-Cartesian sampling trajectories, broadening its utility in complex data reconstruction scenarios. More generally, ALMA provides a powerful tool for numerically solving constrained optimization problems across various disciplines, offering a versatile and impactful solution for advanced computational challenges.

Cortical-inspired Wilson-Cowan-type equations for orientation-dependent contrast perception modelling

We consider the evolution model proposed in [9, 6] to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with widely used Wilson-Cowan equations [48], mainly in terms of efficient representation properties. Then, in order to explicitly encode local directional information, we exploit the model of the primary visual cortex V1 proposed in [20] and largely used over the last years for several image processing problems [24,38,28]. The resulting model is capable to describe assimilation and contrast visual bias at the same time, the main novelty being its explicit dependence on local image orientation. We report several numerical tests showing the ability of the model to explain, in particular, orientation-dependent phenomena such as grating induction and a modified version of the Poggendorff illusion. For this latter example, we empirically show the existence of a set of threshold parameters differentiating from inpainting to perception-type reconstructions, describing long-range connectivity between different hypercolumns in the primary visual cortex.

A computational model for grid maps in neural populations

Grid cells in the entorhinal cortex, together with place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this contribution we introduce a novel theoretical and algorithmic framework able to explain the emergence of hexagonal grid-like response patterns from the statistics of the input stimuli. We show that this pattern is a result of minimal variance encoding of neurons. The novelty lies into the formulation of the encoding problem through the modern Frame Theory language, specifically that of equiangular Frames, providing new insights about the optimality of hexagonal grid receptive fields. The model proposed overcomes some crucial limitations of the current attractor and oscillatory models. It is based on the well-accepted and tested hypothesis of Hebbian learning, providing a simplified cortical-based framework that does not require the presence of theta velocity-driven oscillations (oscillatory model) or translational symmetries in the synaptic connections (attractor model). We moreover demonstrate that the proposed encoding mechanism naturally maps shifts, rotations and scaling of the stimuli onto the shape of grid cells' receptive fields, giving a straightforward explanation of the experimental evidence of grid cells remapping under transformations of environmental cues.

A cortical-inspired model for orientation-dependent contrast perception: a link with Wilson-Cowan equations

We consider a differential model describing neuro-physiological contrast perception phenomena induced by surrounding orientations. The mathematical formulation relies on a cortical-inspired modelling [10] largely used over the last years to describe neuron interactions in the primary visual cortex (V1) and applied to several image processing problems [12,19,13]. Our model connects to Wilson-Cowan-type equations [23] and it is analogous to the one used in [3,2,14] to describe assimilation and contrast phenomena, the main novelty being its explicit dependence on local image orientation. To confirm the validity of the model, we report some numerical tests showing its ability to explain orientation-dependent phenomena (such as grating induction) and geometric-optical illusions [21,16] classically explained only by filtering-based techniques [6,18].