HAViX is an interdisciplinary project on the interplay between frame theory, nonlinear approximation and group theory, aimed to obtain results relevant to the understanding of the functional architecture of brain's visual cortex and to applications to artificial vision.The objectives are highly relevant mathematical results with respect to deep current problems in harmonic analysis, and concern the characterization of noncommutative shift-invariant systems, the study of geometric properties of continuous wavelet transforms and the introduction of new techniques in nonlinear approximation. New approaches will be developed to address such topics, involving techniques from different areas of mathematical analysis such as abstract harmonic analysis and noncommutative Fourier duality, complex analysis, geometric and nonlinear analysis on groups.The expected results will permit to severely improve the present models of the functional architecture of the visual cortex and provide solid instruments for geometric data encoding and compression.The proposed mobility is from France, research center CAMS, mixed unit CNRS - EHESS, to Spain, Autonoma University of Madrid, Mathematics Department, and is motivated by the complementarity of competencies between the applicant, D. Barbieri, whose research activity in Italy and France allowed him to develop a strong interdisciplinary experience on different techniques of Analysis on Groups and on Modeling of the Visual Cortex, and the Scientist in Charge, E. Hernandez, who has an extremely high research expertise in Wavelet Theory, Nonlinear Approximation and Function Spaces, and belongs to a Host Institution that is a world excellence in Harmonic Analysis.