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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.

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. [5] 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.