32 research outputs found
On the Determination of Lagrange Multipliers for a Weighted LASSO Problem Using Geometric and Convex Analysis Techniques
No full textCompressed Sensing (CS) encompasses a broad array of theoretical and applied techniques for recovering signals, given partial knowledge of their coefficients, cf. Candes (C. R. Acad. Sci. Paris, Ser. I 346, 589-592 (2008)), Candes et al. (IEEE Trans. Inf. Theo (2006)), Donoho (IEEE Trans. Inf. Theo. 52(4), (2006)), Donoho et al. (IEEE Trans. Inf. Theo. 52(1), (2006)). Its applications span various fields, including mathematics, physics, engineering, and several medical sciences, cf. Adcock and Hansen (Compressive Imaging: Structure, Sampling, Learning, p. 2021), Berk et al. (2019 13th International conference on Sampling Theory and Applications (SampTA) pp. 1-5. IEEE (2019)), Brady et al. (Opt. Express 17(15), 13040-13049 (2009)), Chan (Terahertz imaging with compressive sensing. Rice University, USA (2010)), Correa et al. (2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) pp. 7789-7793 (2014, May) IEEE), Gao et al. (Nature 516(7529), 74-77 (2014)), Liu and Kang (Opt. Express 18(21), 22010-22019 (2010)), McEwen and Wiaux (Mon. Notices Royal Astron. Soc. 413(2), 1318-1332 (2011)), Marim et al. (Opt. Lett. 35(6), 871-873 (2010)), Yu and Wang (Phys. Med. Biol. 54(9), 2791 (2009)), Motivated by our interest in the mathematics behind Magnetic Resonance Imaging (MRI) and CS, we employ convex analysis techniques to analytically determine equivalents of Lagrange multipliers for optimization problems with inequality constraints, specifically a weighted LASSO with voxel-wise weighting. We investigate this problem under assumptions on the fidelity term || Ax - b || (2)(2), either concerning the sign of its gradient or orthogonality-like conditions of its matrix. To be more precise, we either require the sign of each coordinate of 2( Ax - b)(T) A to be fixed within a rectangular neighborhood of the origin, with the side lengths of the rectangle dependent on the constraints, or we assume A(T) A to be diagonal. The objective of this work is to explore the relationship between Lagrange multipliers and the constraints of a weighted variant of LASSO, specifically in the mentioned cases where this relationship can be computed explicitly. As they scale the regularization terms of the weighted LASSO, Lagrange multipliers serve as tuning parameters for the weighted LASSO, prompting the question of their potential effective use as tuning parameters in applications like MR image reconstruction and denoising. This work represents an initial step in this direction
A computational model for grid maps in neural populations
No full textAbstract
Grid cells in the entorhinal cortex, together with head direction, place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this work we introduce a novel theoretical and algorithmic framework able to explain the optimality of hexagonal grid-like response patterns. We show that this pattern is a result of minimal variance encoding of neurons together with maximal robustness to neurons’ noise and minimal number of encoding neurons. The novelty lies in the formulation of the encoding problem considering neurons as an overcomplete basis (a frame) where the position information is encoded. Through the modern Frame Theory language, specifically that of tight and equiangular frames, we provide new insights about the optimality of hexagonal grid receptive fields. The proposed model 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 velocity-driven oscillations (oscillatory model) or translational symmetries in the synaptic connections (attractor model). We moreover demonstrate that the proposed encoding mechanism naturally explains axis alignment of neighbor grid cells and 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 Neuromathematical Model for Geometrical Optical Illusions
No full textGeometrical optical illusions have been object of many studies due to the possibility they offer to understand the behavior of low-level visual processing. They consist in situations in which the perceived geometrical properties of an object differ from those of the object in the visual stimulus. Starting from the geometrical model introduced by Citti and Sarti (J Math Imaging Vis 24(3):307â326, 2006), we provide a mathematical model and a computational algorithm which allows to interpret these phenomena and to qualitatively reproduce the perceived misperception
Topological Features of Electroencephalography are Reference-Invariant
No full textElectroencephalography (EEG) is among the most widely diffused, inexpensive, and applied neuroimaging techniques. Nonetheless, EEG requires measurements against a reference site(s), which is typically chosen by the experimenter, and specific pre-processing steps precede analysis. It is therefore valuable to obtain quantities that are reference-independent and minimally affected by pre-processing choices. Here, we show that the topological structure of embedding spaces, constructed either from multi-channel EEG timeseries or from their temporal structure, are subject-specific and robust to re-referencing and pre-processing pipelines. By contrast, the shape of correlation spaces, that is, discrete spaces where each point represents an electrode and the distance between them that is in turn related to the correlation between the respective timeseries, were neither significantly subject-specific nor robust to changes of reference. Our results suggest that the shape of spaces describing the observed configurations of EEG signals holds information about the individual specificity of the underlying individual’s brain dynamics, and that temporal correlations constrain to a large degree the set of possible dynamics. In turn, these encode the differences between subjects’ space of resting state EEG signals. Finally, our results and proposed methodology provide tools to explore the individual topographical landscapes and how they are explored dynamically. We propose therefore to augment conventional topographic analyses with an additional – topological – level of analysis, and to consider them jointly. More generally, these results provide a roadmap for the incorporation of topological analyses within EEG pipelines
Modèles mathématiques basé sur l'architecture fonctionnelle de la cortex pour les illusions d'optique géométrique
No full textThis thesis presents mathematical models for visual perception and deals with such phenomena in which there is a visible gap between what is represented and what we perceive. A phenomenon which drew the interest most is amodal completion, consisting in perceiving a completion of a partially occluded object, in contrast with the modal completion, where we perceive an object even though its boundaries are not present [Gestalt theory, 99]. Such boundaries reconstructed by our visual system are called illusory contours, and their neural processing is performed by the primary visual cortices (V1/V2), [93]. Geometric models of the functional architecture of primary visual areas date back to Hoffman [86]. In [139] Petitot proposed a model of single boundaries completion through constraint minimization, neural counterpart of the model of Mumford [125]. In this setting Citti and Sarti introduced a cortical based model [28], which justifies the illusions at a neural level and provides a neurogeometrical model for V1. Another class of phenomena are Geometric optical illusions (GOIs), discovered in the XIX century [83, 190], arising in presence of a mismatch of geometrical properties between an item in object space and its associated percept. The fundamental idea developed here is these phenomena arise due to a polarization of the connectivity of V1/V2, responsible for the misperception. Starting from [28] in which the connectivity building contours in V1 is modeled as a sub-Riemannian metric, we extend it claiming that in GOIs the cortical response to the stimulus modulates the connectivity of the cortex, becoming a coefficient for the metric. GOIs will be tested through this model.Cette thèse présente des modèles mathématiques pour la perception visuelle et s'occupe des phénomènes où on reconnait une brèche entre ce qui est représenté et ce qui est perçu. La complétion amodale consiste en percevoir un complètement d'un object qui est partiellement occlus, en opposition avec la complétion modale, dans laquelle on perçoit un object même si ses contours ne sont pas présents dans l'image [Gestalt, 99]. Ces contours, appelés illusoires, sont reconstruits par notre système visuelle et ils sont traités par les cortex visuels primaires (V1/V2) [93]. Des modèles géométriques de l'architecture fonctionnelle de V1 on le retrouve dans le travail de Hoffman [86]. Dans [139] Petitot propose un modèle pour le complètement de contours, équivalent neurale du modèle proposé par Mumford [125]. Dans cet environnement Citti et Sarti introduisent un modèle basé sur l'architecture fonctionnelle de la cortex visuel [28], qui justifie les illusions à un niveau neurale et envisage un modèle neuro-géometrique pour V1. Une autre classe sont les illusions d'optique géométriques (GOI), découvertes dans le XIX siècle [83, 190], qui apparaissent en présence d'une incompatibilité entre ce qui est présent dans l'espace object et le percept. L'idée fondamentale développée ici est que les GOIs se produisent suite à une polarisation de la connectivité de V1/V2, responsable de l'illusion. A partir de [28], où la connectivité qui construit les contours en V1 est modelée avec une métrique sub-Riemannian, on étend cela en disant que pour le GOIs la réponse corticale du stimule initial module la connectivité, en devenant un coefficient pour la métrique. GOIs seront testés avec ce modèle
Modèles mathématiques basé sur l'architecture fonctionnelle de la cortex pour les illusions d'optique géométrique
No full textThis thesis presents mathematical models for visual perception and deals with such phenomena in which there is a visible gap between what is represented and what we perceive. A phenomenon which drew the interest most is amodal completion, consisting in perceiving a completion of a partially occluded object, in contrast with the modal completion, where we perceive an object even though its boundaries are not present [Gestalt theory, 99]. Such boundaries reconstructed by our visual system are called illusory contours, and their neural processing is performed by the primary visual cortices (V1/V2), [93]. Geometric models of the functional architecture of primary visual areas date back to Hoffman [86]. In [139] Petitot proposed a model of single boundaries completion through constraint minimization, neural counterpart of the model of Mumford [125]. In this setting Citti and Sarti introduced a cortical based model [28], which justifies the illusions at a neural level and provides a neurogeometrical model for V1. Another class of phenomena are Geometric optical illusions (GOIs), discovered in the XIX century [83, 190], arising in presence of a mismatch of geometrical properties between an item in object space and its associated percept. The fundamental idea developed here is these phenomena arise due to a polarization of the connectivity of V1/V2, responsible for the misperception. Starting from [28] in which the connectivity building contours in V1 is modeled as a sub-Riemannian metric, we extend it claiming that in GOIs the cortical response to the stimulus modulates the connectivity of the cortex, becoming a coefficient for the metric. GOIs will be tested through this model.Cette thèse présente des modèles mathématiques pour la perception visuelle et s'occupe des phénomènes où on reconnait une brèche entre ce qui est représenté et ce qui est perçu. La complétion amodale consiste en percevoir un complètement d'un object qui est partiellement occlus, en opposition avec la complétion modale, dans laquelle on perçoit un object même si ses contours ne sont pas présents dans l'image [Gestalt, 99]. Ces contours, appelés illusoires, sont reconstruits par notre système visuelle et ils sont traités par les cortex visuels primaires (V1/V2) [93]. Des modèles géométriques de l'architecture fonctionnelle de V1 on le retrouve dans le travail de Hoffman [86]. Dans [139] Petitot propose un modèle pour le complètement de contours, équivalent neurale du modèle proposé par Mumford [125]. Dans cet environnement Citti et Sarti introduisent un modèle basé sur l'architecture fonctionnelle de la cortex visuel [28], qui justifie les illusions à un niveau neurale et envisage un modèle neuro-géometrique pour V1. Une autre classe sont les illusions d'optique géométriques (GOI), découvertes dans le XIX siècle [83, 190], qui apparaissent en présence d'une incompatibilité entre ce qui est présent dans l'espace object et le percept. L'idée fondamentale développée ici est que les GOIs se produisent suite à une polarisation de la connectivité de V1/V2, responsable de l'illusion. A partir de [28], où la connectivité qui construit les contours en V1 est modelée avec une métrique sub-Riemannian, on étend cela en disant que pour le GOIs la réponse corticale du stimule initial module la connectivité, en devenant un coefficient pour la métrique. GOIs seront testés avec ce modèle
Mean curvature flow in SE (2) and applications to visual perception
No full textOur goal in this thesis is to provide a result of existence of the degenerate non-linear, non-divergence PDE which describes the mean curvature flow in the Lie group SE(2) equipped with a sub-Riemannian metric. The research is motivated by problems of visual completion and models of the visual cortex
Learning and navigating digitally rendered haptic spatial layouts.
No full textLearning spatial layouts and navigating through them rely not simply on sight but rather on multisensory processes, including touch. Digital haptics based on ultrasounds are effective for creating and manipulating mental images of individual objects in sighted and visually impaired participants. Here, we tested if this extends to scenes and navigation within them. Using only tactile stimuli conveyed via ultrasonic feedback on a digital touchscreen (i.e., a digital interactive map), 25 sighted, blindfolded participants first learned the basic layout of an apartment based on digital haptics only and then one of two trajectories through it. While still blindfolded, participants successfully reconstructed the haptically learned 2D spaces and navigated these spaces. Digital haptics were thus an effective means to learn and translate, on the one hand, 2D images into 3D reconstructions of layouts and, on the other hand, navigate actions within real spaces. Digital haptics based on ultrasounds represent an alternative learning tool for complex scenes as well as for successful navigation in previously unfamiliar layouts, which can likely be further applied in the rehabilitation of spatial functions and mitigation of visual impairments
Several aspects of modern signal analysis: optimal tuning parameters in MRI inverse problems and symplectic time-frequency analysis on modulation spaces
No full text
