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Final Report Summary - HAVIX (Harmonic Analysis for optimal coding and the design principles of brain's Visual corteX)

The overall objectives of HAViX are crucial theoretical results concerning information coding with new mathematical tools, as well as relevant improvements to our present understanding of the data analysis performed by the visual cortex. The purpose of the project is the development of a new theoretical framework for problems of data approximation and processing arising from neural behaviors, with harmonic and geometric instruments of Fourier analysis and group theory.
The main object of the theoretical research is generalized wavelet analysis arising from representations of noncommutative groups, considered with respect to reproducing properties, dimensionality reduction and learning mechanisms. The main focus of applications concerns specific cellular mechanisms and morphologies in primary visual cortex V1, that is the largest and best studied of brain’s cortical areas dedicated to the early stages of the processing of vision.
The main idea is that the information content of data, in particular visual stimuli, can be well represented by identifying its approximated symmetries and embedding the data in the corresponding symmetric space. The structure of such an embedding is that of a generalized wavelet transform, obtained by the action of symmetries on basic generators which can eventually be learned from data. This approach is coherent with the contemporary research in applied harmonic analysis, and leads to natural open problems related to group representations, approximation theory and machine learning. These tools have a natural role for the development of functional models of neural behaviors at the level of brain's cortical tissues dedicated to the early analysis of sensory measurements.

The project HAViX produced 8 papers, 14 conferences, 1 organized workshop, and 3 organized outreach activities, together with the beginning of several new collaborations on the developments of new ideas as well as a large amount of material for works in preparation and the invited participation to several conferences after the end of the project. Moreover, the success of the developed research has contributed to the award to the fellow of a tenure track Assistant Professor position at the institution where the project was developed.
The project's scientific production can be broadly divided into three main classes of results:
1) A general theory of shift-invariant spaces associated to unitary group representations and the development of a general modular framework to deal with Hilbert spaces invariant under the action of discrete groups. The new approach introduced allows to characterize reproducing properties associated to a substantially larger class of generalized wavelet transform. The related papers produced are:
- Noncommutative Shift-Invariant Spaces, with E. Hernández and V. Paternostro. Preprint.
- The Zak transform and the structure of spaces invariant by the action of an LCA group, with E. Hernández and V. Paternostro. Published on Journal of Functional Analysis.
- Riesz and frame systems generated by unitary actions of discrete groups, with E. Hernández and J. Parcet. Published on Applied and Computational Harmonic Analysis.
2) Advances on computational models of feature extraction in primary visual cortex, neural frameworks for spatio-temporal segmentation and reinforcement learning in auditory perception, with the introduction of new harmonic, spectral and nonlinear techniques able to reproduce well known experiments in the neurophysiology and psychology of perception. The related papers produced are:
- Morphology of meaning in the sensory cortex, with A. Sarti. Preprint.
- Geometry and dimensionality reduction of feature spaces in primary visual cortex. Published on Proceedings of the SPIE.
- Cortical spatio-temporal dimensionality reduction for visual grouping, with G. Citti, G. Cocci and A. Sarti. Published on Neural Computation.
3) A new approach of Fourier analysis for the proof of equivalences between lattice coverings and reproducing properties of families of group characters. The related papers produced are:
- Lattice sub-tilings and frames in LCA groups, with E. Hernández and A. Mayeli. Preprint.
- Tiling by lattices for locally compact abelian groups, with E. Hernández and A. Mayeli. Preprint.

I) Supervised Master Theses
2015: Roberto González Martín. M. Sci. in Mathematics, Universidad Autónoma de Madrid. Thesis: Shearlets and edge detection. Co-direction with E. Hernández
2016: Simona Baccherini. M. Sci. in Mathematics, Università di Bologna. Thesis: Harmonic and geometric methods for robotic prosthetic control. (In development) Co-direction with G. Citti
II) University Courses: 2014-2015 and 2015-2016. Teaching assistant in "Wavelets and signal processing" (Chair: E. Hernández). Master in Mathematics, Universidad Autónoma de Madrid.

The strong developments obtained on the characterization of reproducing systems and invariant spaces associated to noncommutative group actions have opened the way towards the production of a large class of new results concerning geometric properties of sampling and approximation in far more general settings than those previously considered in the literature. Some of these results are currently under investigation, while others are part of a new research line that will involve the collaboration with colleagues in Spain, the US, and South America. On the other hand, other new results planned by the project, concerning noncommutative groups and the reproducing properties of their orbits in a common but degenerate setting, have already been announced in contributed and invited conference and constitute the subject of two works in preparation.
The harmonic analysis approach to the modeling of the visual cortex has reached a higher level of maturity and the framework prepared by this project is now open to include new more general phenomena, as already shown by the discussion in one of the last papers of mechanisms of auditory perception. The perspectives of this research line will rely on new collaboration opened in approximation theory and machine learning for the development of biologically inspired new mechanisms of anisotropic feature extraction and symmetry learning.
Applications of the developed research will be considered also with respect to industrial innovation, due to the contact with two highly technological private companies, one located in Madrid (GMV) and one located in Bologna (Marposs) that has been promoted during the final stages of the project. The perspectives may consider also the supervision of joint applied mathematics Master thesis projects, as the one developed during the first year of HAViX with a master student in Madrid and the spanish cardiovascular research center CNIC, or the one currently developing with a master student in Bologna and the robotic prosthetic research division of the italian institute for work accidents INAIL.

HAViX project lies within in the frameworks of theoretical and applied harmonic analysis and of mathematical models of brain’s low level cognition. The interdisciplinarity of the project has promoted transfer of knowledge between such areas. Applications have technological and industrial impact at the level of data analysis and artificial vision. The topics of frame theory, group representations and nonlinear approximation addressed by HAViX play a central role in harmonic analysis. Applications of these results allow a better understanding of the neural mechanisms of feature extraction in primary visual cortex.

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