Divisive Normalization from Wilson-Cowan Dynamics

Divisive Normalization and the Wilson-Cowan equations are influential models of neural interaction and saturation [Carandini and Heeger Nat.Rev.Neurosci. 2012; Wilson and Cowan Kybernetik 1973]. However, they have not been analytically related yet. In this work we show that Divisive Normalization can be obtained from the Wilson-Cowan model. Specifically, assuming that Divisive Normalization is the steady state solution of the Wilson-Cowan differential equation, we find that the kernel that controls neural interactions in Divisive Normalization depends on the Wilson-Cowan kernel but also has a signal-dependent contribution. A standard stability analysis of a Wilson-Cowan model with the parameters obtained from our relation shows that the Divisive Normalization solution is a stable node. This stability demonstrates the consistency of our steady state assumption, and is in line with the straightforward use of Divisive Normalization with time-varying stimuli. The proposed theory provides a physiological foundation (a relation to a dynamical network with fixed wiring among neurons) for the functional suggestions that have been done on the need of signal-dependent Divisive Normalization [e.g. in Coen-Cagli et al., PLoS Comp.Biol. 2012]. Moreover, this theory explains the modifications that had to be introduced ad-hoc in Gaussian kernels of Divisive Normalization in [Martinez et al. Front. Neurosci. 2019] to reproduce contrast responses. The proposed relation implies that the Wilson-Cowan dynamics also reproduces visual masking and subjective image distortion metrics, which up to now had been mainly explained via Divisive Normalization. Finally, this relation allows to apply to Divisive Normalization the methods which up to now had been developed for dynamical systems such as Wilson-Cowan networks

Gamut mapping through perceptually-based contrast reduction

Comunicació presentada a: 6th Pacific-Rim Symposium on Image and Video Technology, (PSIVT 2013), celebrada del 28 d'octubre a l'1 de novembre a Guanajuato, Mèxic.In this paper we present a spatial gamut mapping algorithm that relies on a perceptually-based variational framework. Our method adapts a well-known image energy functional whose minimization leads to image enhancement and contrast modification. We show how by varying the importance of the contrast term in the image functional we are able to perform gamut reduction. We propose an iterative scheme that allows our algorithm to successfully map the colors from the gamut of the original image to a given destination gamut while preserving the colors’ perception and texture close to the original image. Both subjective and objective evaluation validate the promising results achieved via our proposed framework.This work was supported by the European Research Council, Starting Grant ref. 306337, and by Spanish grants ref. TIN2011-15954-E and ref. TIN2012-38112

A connection between image processing and artificial neural networks layers through a geometric model of visual perception

Comunicació presentada a: the 7th International Conference on Scale Space and Variational Methods in Computer Vision, celebrada del 30 de juny al 4 de juliol de 2019 a Hofgeismar, Alemanya.In this paper, we establish a connection between image processing, visual perception, and deep learning by introducing a mathematical model inspired by visual perception from which neural network layers and image processing models for color correction can be derived. Our model is inspired by the geometry of visual perception and couples a geometric model for the organization of some neurons in the visual cortex with a geometric model of color perception. More precisely, the model is a combination of a Wilson-Cowan equation describing the activity of neurons responding to edges and textures in the area V1 of the visual cortex and a Retinex model of color vision. For some particular activation functions, this yields a color correction model which processes simultaneously edges/textures, encoded into a Riemannian metric, and the color contrast, encoded into a nonlocal covariant derivative. Then, we show that the proposed model can be assimilated to a residual layer provided that the activation function is nonlinear and to a convolutional layer for a linear activation function. Finally, we show the accuracy of the model for deep learning by testing it on the MNIST dataset for digit classiffication

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

We consider the evolution model proposed in Bertalmío (Front Comput Neurosci 8:71, 2014), Bertalmío et al. (IEEE Trans Image Process 16(4):1058–1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson–Cowan equations (Wilson and Cowan in BioPhys J 12(1):1–24, 1972), 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 Citti and Sarti (J Math Imaging Vis 24(3):307–326, 2006) and largely used over the last years for several image processing problems (Duits and Franken in Q Appl Math 68(2):255–292, 2010; Prandi and Gauthier in A semidiscrete version of the Petitot model as a plausible model for anthropomorphic image reconstruction and pattern recognition. SpringerBriefs in Mathematics, Springer, Cham, 2017; Franceschiello et al. in J Math Imaging Vis 60(1):94–108, 2018). The resulting model is thus defined in the space of positions and orientation, and it is capable of describing assimilation and contrast visual bias at the same time. We report several numerical tests showing the ability of the model to reproduce, 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 and describing long-range connectivity between different hypercolumns in V1.The authors acknowledge the anonymous referees for their suggestions which improved significantly the quality of their manuscript. M. B. acknowledges the support of the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 761544 (Project HDR4EU) and under Grant Agreement No. 780470 (Project SAUCE), and of the Spanish government and FEDER Fund, Grant Ref. PGC2018-099651-B-I00 (MCIU/AEI/FEDER, UE). L. C., V. F. and D. P. acknowledge the support of a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d’avenir program, through the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02 and of the research project LiftME funded by INS2I, CNRS. V. F. acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant No. 794592 and from the INdAM project Problemi isoperimetrici in spazi Euclidei e non. V. F. and D. P. also acknowledge the support of ANR-15-CE40-0018 project SRGI - Sub-Riemannian Geometry and Interactions. B. F. acknowledges the support of the Fondation Asile des Aveugles

A geometric model of brightness perception and its application to color images correction

Human perception involves many features like contours, shapes, textures, and colors to name a few. Whereas several geometric models for contours, shapes and textures perception have been proposed, the geometry of color perception has received very little attention, possibly due to the fact that our perception of colors is still not fully understood. Nonetheless, there exists a class of mathematical models, gathered under the name Retinex, that aim at modeling the color perception of an image, that are inspired by psychophysical/physiological knowledge about color perception, and that can geometrically be viewed as the averaging of perceptual distances between image pixels. Some of the Retinex models turn out to be associated to an efficient image processing technique for the correction of camera output images. The aim of this paper is to show that this image processing technique can be improved by including more properties of the human visual system. To that purpose, we first present a generalization of the perceptual distance between image pixels by considering the parallel transport map associated to a covariant derivative on a vector bundle, and from which can be derived a new image processing model for color images correction. Then, we show that the family of covariant derivatives constructed in [T. Batard and N. Sochen, J. Math. Imaging Vision, 48(3) (2014), pp. 517-543] can model some color appearance phenomena related to brightness perception. Finally, we conduct experiments in which we show that the image processing techniques induced by these covariant derivatives outperform the original approach.This work was supported by the European Research Council, Starting Grant ref. 306337, by the Spanish government and FEDER Fund, grant ref. TIN2015-71537-P (MINECO/FEDER,UE), and by the Icrea Academia Award

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

Comunicació presentada a: The 7th International Conference on Scale Space and Variational Methods in Computer Vision, celebrada del 30 de juny al 4 de juliol de 2019 a Hofgeismar, Alemanya.We consider a differential model describing neuro-physiological contrast percep- tion phenomena induced by surrounding orientations. The mathematical formulation relies on a cortical-inspired modelling [10] largely used over the last years to describe neuron inter- actions 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 phenom- ena (such as grating induction) and geometric-optical illusions [21,16] classically explained only by filtering-based techniques [6,18]

Denoising an image by denoising its components in a moving frame

Comunicació presentada a: the 6th International Conference, (ICISP 2014), celebrada del 30. de juny al 2 de juliol de 2014 a Cherbourg, França.In this paper, we provide a new non-local method for image denoising. The key idea we develop is to denoise the components of the image in a well-chosen moving frame instead of the image itself.We prove the relevance of our approach by showing that the PSNR of a grayscale noisy image is lower than the PSNR of its components. Experiments show that applying the Non Local Means algorithm of Buades et al. on the components provides better results than applying it directly on the image.This work was supported by European Research Council, Starting Grant ref. 306337, and by Spanish grants AACC, ref. TIN2011-15954-E, and Plan Na- cional, ref. TIN2012-38112. S. Levine acknowledges partial support by NSF-DMS #0915219

Parameterization of Contractible Domains Using Sequences of Harmonic Maps

Abstract. In this paper, we propose a new method for parameterizing a contractible domain (called the computational domain) which is defined by its boundary. Using a sequence of harmonic maps, we first build a mapping from the computational domain to the parameter domain, i.e., the unit square or unit cube. Then we parameterize the original domain by spline approximation of the inverse mapping. Numerical simulations of our method were performed with several shapes in 2D and 3D to demonstrate that our method is suitable for various shapes. The method is particular useful for isogeometric analysis because it provides an extension from a boundary representation of a model to a volume representation.