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 classication.

1 aBatard, Thomas1 aMaldonado, Eduard, Ramon1 aSteidl, Gabriele1 aBertalmío, Marcelo uhttp://ip4ec.upf.edu/geometricModelPerception01442nas a2200109 4500008004100000245011700041210006900158520101000227100001901237700002401256856005201280 2016 eng d00aA Class of Nonlocal Variational Problems on a Vector Bundle for Color Image Local Contrast Reduction/Enhancement0 aClass of Nonlocal Variational Problems on a Vector Bundle for Co3 a
We extend two existing variational models from the Euclidean space to a vector bundle over a Riemannian manifold. The Euclidean models, dedicated to regularize or enhance some color image features, are based on the concept of nonlocal gradient operator acting on a function of the Euclidean space. We then extend these models by generalizing this operator to a vector bundle over a Riemannian manifold with the help of the parallel transport map associated to some class of covariant derivatives. Through the dual formulations of the proposed models, we obtain the expressions of their solutions, which exhibit the functional spaces that describe the image features. Finally, for a well-chosen covariant derivative and its nonlocal extension, the proposed models perform local contrast modification (reduction or enhancement) and experiments show that they preserve more the aspect of the original image than the Euclidean models do while modifying equally its contrast.

1 aBatard, Thomas1 aBertalmío, Marcelo uhttp://ip4ec.upf.edu/VariationalProblemsGIC201600393nas a2200109 4500008004100000245007000041210006900111100002400180700001900204700001600223856004400239 2016 eng d00aCorrecting for Induction Phenomena on Displays of Differrent Size0 aCorrecting for Induction Phenomena on Displays of Differrent Siz1 aBertalmío, Marcelo1 aBatard, Thomas1 aKim, Jihyun uhttp://f1000research.com/posters/5-121501198nas a2200109 4500008004100000245007600041210006900117520081400186100001901000700002401019856004501043 2014 eng d00aOn Covariant Derivatives and Their Applications to Image Regularization0 aCovariant Derivatives and Their Applications to Image Regulariza3 aWe present a generalization of the Euclidean and Riemannian gradient operators to a vector bundle, a geometric structure generalizing the concept of manifold. One of the key ideas is to replace the standard differentiation of a function by the covariant differentiation of a section. Dealing with covariant derivatives satisfying the property of compatibility with vector bundle metrics, we construct generalizations of existing mathematical models for image regularization that involve the Euclidean gradient operator, namely the linear scale-space and the Rudin-Osher-Fatemi denoising model. For well-chosen covariant derivatives, we show that our denoising model outperforms state-of-the-art variational denoising methods of the same type both in terms of PSNR and Q-index [45].

1 aBatard, Thomas1 aBertalmío, Marcelo uhttp://ip4ec.upf.edu/ImageRegularization