The project will introduce a new interdisciplinary and multidisciplinary endeavor where it will possible to study metric analysis, starting from measure theory and PDE. These deep mathematical instruments will be the main building blocks for modelling the visual cortex, or more abstract problems of minimal surfaces and soap films. The concrete application to technological problems span from artificial vision to robotics, eye tracking and urban traffic.
The main problems to be studied are:
Wp1 Geometric measure theory: The project will provide a complete characterisation of rectifiability in terms of mass transportation and tangent measures. The relation between rectifiability and Hausdorff dimension distortion will be studied, together with the dimension distortion problem for Sobolev Maps.
Wp2 Sub-Riemannian PDE: Nowadays the main open questions arise in the study of non linear PDE: p-laplacian and curvature flow in groups. In case of curvature the problem of characteristic points has to be faced- In addition, we will study problems related to k-forms or systems in Lie groups.
Wp3 Soap films and Minimal surfaces: One of the most fascinating topics, at the frontier between geometric measure theory and PDE are minimal surfaces. Even in the Euclidean setting, the theory is far from being complete, and a better description of minimal is needed.
Wp4 Models of vision: It is well known that the visual cortex can be modelled as a contact structure with a sub-Riemannian metric are It would be extremely challenging to extend these results in order to understand how to use Lie groups to model the modular structure of the visual cortex, and perform medical image processing.
Wp5 Mass transportation for traffic simulation: We will introduce new models of traffic simulation, with instruments of mass transportation and GMT.
Wp6 Image analysis: In close cooperation with advanced clinical partners, we aim to demonstrate that substantial progress can be made in detection and analysis of blood vessel structure in optical images of the retina.
Wp7 Eye path tracking: Here we apply the new models of the brain functionality together with analysis of symbolic sequences in order to characterize deep properties of eye movements.
Wp8 Lie groups in robotics: Robotics can be naturally described in the Lie group of rigid motions. Models of robotics are strictly connected with models of area V6 of the visual cortex.
→ Y. Liang, D. Yang, R.Jiang Weak Musielak–Orlicz Hardy spaces and applications on Mathematische Nachrichten Vol 289, issue 5-6 2015 pp. 634-677
→ A. Clop, R. Jiang, J. Mateu, J. Orobitg, Linear transport equations for vector fields with subexponentially integrable divergence on Calculus of Variations and Partial Differential Equations Vol 55, issue 1 2016 pp. 21-55 open access version
→ A. Clop, R. Jiang, J. Mateu, J. Orobitg Flows for non-smooth vector fields with subexponentially integrable divergence on Journal of Differential Equations Vol 261, issue 2 2016 pp. 1237-1263 open access version
→ R. Jiang, H. C. Zhangg Hamilton’s gradient estimates and a monotonicity formula for heat flows on metric measure spaces on Nonlinear Analysis, Theory, Methods and Applications Vol 131 2016 pp. 32-47 open access version
→ A. Clop, R. Jiang, J. Mateu, J. Orobitg A Note on Transport Equation in Quasiconformally Invariant Spaces preprint
→ A. Clop, H. Jylhä, J. Mateu, J. Orobitg Well-posedness for the continuity equation for vector fields with suitable modulus of continuity preprint
→ M. Galli, M. Ritoré Area-stationary and stable surfaces of class C1 in the sub-Riemannian Heisenberg group H1 on Advances in Mathematics Vol 18, issue 74 2015 pp. 737-765 open access version
→ M. Galli, M. Ritoré Regularity of C1 surfaces with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds on Calculus of Variations and Partial Differential Equations Vol 54, issue 3 2015 pp. 2503-2516 open access version
→ M. Galli On the classification of complete area-stationary and stable surfaces in the sub-Riemannian Sol manifold on Pacific J. Math Vol 271, issue 1 2014 pp. 143-157. open access version
→ M. Galli The regularity of Euclidean Lipschitz boundaries with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds on Nonlinear Analysis, Theory, Methods and Applications Vol 136 2016 pp. 40-50 open access version
→ B. Franceschiello, A. Sarti, G. Citti A neuro-mathematical model for geometrical optical illusions preprint
→ G.Citti, E.Baspinar A diffusion driven curvature flow preprint
→ G.Citti, E.Baspinar Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures preprint
→ E. Le Donne, S. Nicolussi Golo, A. Sambusetti Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary preprint
→ C.Y. Guo, S. Nicolussi Golo, M. Williams Quasiregular mappings between subRiemannian manifolds preprint
→ E. Le Donne, S. Nicolussi Golo Regularity properties of spheres in homogeneous groups preprint
→ S. Nicolussi Golo Some remarks on contact variations in the first Heisenberg group and Bernstein's Problem preprint
→ Z. Balogh, F. Ferrari, B. Franchi, E. Vecchi, K.Wildrick Steiner's formula in the Heisenberg group on Nonlinear Analysis, Theory, Methods and Applications Vol 126 2015 pp. 201-217 open access version
→ Balogh, Z.M., Tyson, J. T., Vecchi, E. Intrinsic curvature of curves and surfaces and a Gauss-Bonnet Theorem in the Heisenberg group on Mathematische Zeitschrift 2016 pp. 1-38
→ A.Pinamonti, M.Squassina, E.Vecchi Magnetic BV functions and the Bourgain-Brezis-Mironescu formula preprint
→ A.Pinamonti, M.Squassina, E.Vecchi The Maz'ya-Shaposhnikova limit in the magnetic setting on Journal of Mathematical Analysis and Applications Vol 449, issue 2 2017 pp. 1152-1159 open access version
→ S. Abbasi-Sureshjani, I. Smit-Ockeloen, J. Zhang, B. Ter Haar Romeny Biologically-Inspired Supervised Vasculature Segmentation in SLO Retinal Fundus Images on Lecture Notes in Computer Science Vol. 9164 2015 pp. 325-334
→ J. Zhang , E. Bekkers, S. Abbas-Sureshaji, B. Dashtbozorg, B. Ter Haar Romeny Robust and Fast Vessel Segmentation via Gaussian Derivatives in Orientation Scores on Lecture Notes in Computer Science Vol 9279 2015 pp. 537-547 open access version
→ M. Favali, G. Citti, A. Sarti Local and global gestalt laws: A neurally based spectral approach preprint
→ M. Favali, S. Abbasi, A. Sarti, B. H. ter Romeny Analysis of Vessel Connectivities in Retinal Images by Cortically Inspired Spectral Clustering on Journal of Mathematical Imaging and Vision Vol 55 2016 pp. 1-15 open access version
→ G. Citti, B. Franceschiello, G. Sanguinetti, A. Sarti Sub-Riemannian mean curvature flow for image processing on SIAM Journal on Imaging Sciences Vol 9, Issue 1 2016 pp. 212-237 open access version
→ Z. M. Balogh, A. Kristály, K. Sipos Geometric inequalities on Heisenberg groups preprint
→ Z. M. Balogh, A. Kristály, K. Sipos Jacobian determinant inequality on Corank 1 Carnot groups with applications preprint
→ F. Huang, B. Dashtbozorg, J.Zhang, E. Bekkers, S. Abbasi-Sureshjani, T. TJM Berendschot, B.M ter Haar Romeny Reliability of Using Retinal Vascular Fractal Dimension as a Biomarker in the Diabetic Retinopathy Detection on British Journal of Ophthalmology Vol 2016 2016 pp. 1-13
→ B. M. ter Haar Romeny , E. J. Bekkers , J. Zhang , S. Abbasi-Sureshjani , F. Huang , R. Duits , B. Dashtbozorg , T. T. J. M. Berendschot , I. Smit-Ockeloen , K. A. J. Eppenhof , J. Feng , J. Hannink , J. Schouten , M. Tong , H. Wu , Han W. van Triest , S. Zhu , D. Chen , W. He , L.Xu , P. Han , Y.Kang Brain-inspired algorithms for retinal image analysis on Machine Vision and Applications Vol 27, Issue 8 2016 pp. 1117-1135
→ S. Abbasi-Sureshjani, I. Smit-Ockeloen, E. Bekkers, B. Dashtbozorg, B. ter Haar Romeny Automatic detection of vascular bifurcations and crossings in retinal images using orientation scores on IEEE Xplore Digital Library 2016 pp.
→ E. J. Bekkers, R. Duits, A. P. Mashtakov, G. R. Sanguinetti A PDE approach to data-driven sub-Riemannian geodesics in SE(2) on SIAM Journal on Imaging Sciences Vol 8, issue 4 2015 pp. 2740-2770 open access version
→ A. Mashtakov, Y. L. Sachkov. Integrability of left-invariant sub-Riemannian structures on the special linear group SL2(R) on Differential Equations Vol 50, issue11 2014 pp. 1541-1547
→ A. Mashtakov, Y. L. Sachkov Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups on Differential Equations Vol 51, Issue 11 2015 pp. 1476-1483 open access version
→ G. Sanguinetti, E. Bekkers, R. Duits, M. Janssen, A. Mashtakov, J. M. Mirebeau Sub-Riemannian Fast Marching in SE(2) on Lecture Notes in Computer Science Vol 9423 2015 pp. 366-374 open access version
→ R. Duits, A. Ghosh, T. Dela Haije, A. Mashtakov On sub-Riemannian geodesics in SE(3) whose spatial projections do not have cusps preprint