## Research

### Topics

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 Models of vision and 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.

### Papers

### Wp 1

→ Y. Liang, D. Yang, R. Jiang,
** Weak Musielak–Orlicz Hardy spaces and applications** on

*Mathematische Nachrichten*Vol 289 2015 pp. 634-677 — doi

→ A. Clop, R. Jiang, J. Mateu, J. Orobitg,
** A note on transport equation in quasiconformally invariant spaces** on

*Advances in Calculus of Variations*Vol 11 2016 pp. 193-202 — open access version — doi

→ 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 2016 pp. 21-55 — open access version — doi

→ 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 2016 pp. 1237-1263 — open access version — doi

→ 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 — doi

→ Z. Balogh, A. Kristály, K. Sipos,
** Geodesic interpolation inequalities on Heisenberg groupsInégalités d'interpolation géodésique sur les groupes de Heisenberg** on

*Comptes Rendus Mathematique*Vol 354 2016 pp. 916-919 — doi

→ Z. Balogh, A. Kristály, K. Sipos,
** Geometric inequalities on Heisenberg groups** on

*Calculus of Variations and Partial Differential Equations*Vol 57, Issue 2 2016 pp. 1-41 (article n° 61) — open access version — doi

→ Z. Balogh, A. Kristály, K. Sipos,
** Jacobian determinant inequality on Corank 1 Carnot groups with applications** 2017 preprint

### Wp 2

→ M. Galli,
** On the classification of complete area-stationary and stable surfaces in the sub-Riemannian Sol manifold** on

*Pacific J. Math*Vol 271 2014 pp. 143-157. — open access version — doi

→ 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 2015 pp. 737-765 — open access version — doi

→ 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 2015 pp. 2503-2516 — open access version — doi

→ 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 — doi

→ 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 — doi

→ Z. Balogh, J. T. Tyson, E. Vecchi,
** Intrinsic curvature of curves and surfaces and a Gauss-Bonnet Theorem in the Heisenberg group** on

*Mathematische Zeitschrift*Vol 287 2016 pp. 1-38 — doi

→ H. M. Nguyen, A. Pinamonti, M. Squassina, E. Vecchi,
** New characterizations of magnetic Sobolev spaces** on

*Advances in Nonlinear Analysis*2017 pp. — open access version — doi

→ A. Fiscella, A. Pinamonti, E. Vecchi,
** Multiplicity results for magnetic fractional problems** on

*Journal of Differential Equations*Vol 263 2017 pp. 4617-4633 — open access version — doi

→ A. Pinamonti, M. Squassina, E. Vecchi,
** Magnetic BV functions and the Bourgain-Brezis-Mironescu formula** on

*Advances in Calculus of Variations*2017 pp. 1-30 — open access version — doi

→ A. Pinamonti, M. Squassina, E. Vecchi,
** The Maz'ya-Shaposhnikova limit in the magnetic setting** on

*Journal of Mathematical Analysis and Applications*Vol 449 2017 pp. 1152-1159 — open access version — doi

→ S. Mukherjee, X. Zhong,
** C^1,α -Regularity for variational problems in the Heisenberg group** 2017 preprint

→ F. Ferrari, E. Vecchi,
** Hoelder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group** 2017 preprint

→ F. Colasuonno, E. Vecchi,
** Symmetry in the composite plate problem** 2017 preprint
— doi

### Wp 3

→ E. Le Donne, S. Nicolussi Golo, A. Sambusetti,
** Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary** on

*Annali di Matematica Pura ed Applicata*Vol 196 2016 pp. 1-22 — open access version — doi

→ E. Le Donne, S. Nicolussi Golo,
** Regularity properties of spheres in homogeneous groups** on

*Transactions of the American Mathematical Society*Vol 370 2017 pp. 2057-2084 — open access version — doi

→ S. Nicolussi Golo,
** Some remarks on contact variations in the first Heisenberg group** on Vol 3 2018 pp. 311-335 — open access version
— doi

→ C.Y. Guo, S. Nicolussi Golo, M. Williams,
** Quasiregular mappings between subRiemannian manifolds** 2015 preprint

### Wp 4

→ 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 — doi

→ 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 — doi

→ S. Abbasi-Sureshjani, J. Zhang, R. Duits, B.ter Haar Romeny,
** Retrieving challenging vessel connections in retinal images by line co-occurrence statistics** on

*Biological Cybernetics*Vol 111 2016 pp. 237-247 — open access version — doi

→ 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 — doi

→ 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, H. 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 2016 pp. 1117-1135 — doi

→ 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

*13th International Symposium on Biomedical Imaging (ISBI)*2016 pp. — doi

→ J. Zhang, E. Bekkers, S. Abbasi-Sureshjani, B. Dashtbozorg, B. ter Haar Romeny,
** Bridging Disconnected Curvilinear Structures via Numerical Evolutions of Completion Process in Ophthalmologic Images** on

*Ophthalmic Medical Image Analysis*2016 pp. 156-157 — doi

→ B. Dashtbozorg, S. Abbasi-Sureshjani, J. Zhang, F. Huang, E. Bekkers, B. ter Haar Romeny,
** Infrastructure for Retinal Image Analysis** on 2016 pp. — doi

→ 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 56 2016 pp. 158-172 — open access version — doi

→ G. Citti, B. Franceschiello, G. Sanguinetti, A. Sarti,
** Sub-Riemannian mean curvature flow for image processing** on

*SIAM Journal on Imaging Sciences*Vol 9 2016 pp. 212-237 — open access version — doi

→ M. Favali, G. Citti, A. Sarti,
** Local and global gestalt laws: A neurally based spectral approach** on

*Neural Computation*Vol 29 2017 pp. 394-422 — open access version — doi

→ Z. Li, F. Huang, J. Zhang, B. Dashtbozorg, S. Abbasi-Sureshjani, Y. Sun, X. Long, Q. Yu, B. Ter Haar Romeny, T. Tan,
** Multi-modal and multi-vendor retina image registration** on

*Biomedical Optics Express*Vol 9 2018 pp. 410-422 — doi

→ S. Abbasi-Sureshjani, M. Favali, G. Citti, A. Sarti, B. ter Haar Romeny,
** Curvature Integration in a 5D Kernel for Extracting Vessel Connections in Retinal Images** on

*IEEE Transactions on Image Processing*Vol 27 2018 pp. 606 - 621 — open access version — doi

### Wp 5

→ H. Jylä, T. Rajala,
** L∞ estimates in optimal mass transportation** on

*Journal of Functional Analysis*Vol 270 2016 pp. 4297-4321 — open access version — doi

→ A. Clop, H. Jylhä, J. Mateu, J. Orobitg,
** Well-posedness for the continuity equation for vector fields with suitable modulus of continuity** 2016 preprint

→ A. Clop, R. Giova, F. Hatami, A. Passarelli di Napoli,
** Congested traffic dynamicsand very degenerate elliptic equationsunder supercritical Sobolev regularity** 2018 preprint

→ F. Hatami, A. Clop, A. Ruiz de Vila,
** New approaches in adjustment matrix** in preparation

→ A. Clop, R. Giova, F. Hatami, A. Passarelli di Napoli,
** Non-autonomous very degenerate elliptic equations with applications to traffic models** in preparation

### Wp 6

→ A. Mashtakov, Y. L. Sachkov.,
** Integrability of left-invariant sub-Riemannian structures on the special linear group SL2(R)** on

*Differential Equations*Vol 50 2014 pp. 1541-1547 — doi

→ E. J. Bekkers, R. Duits, A. Mashtakov, G. Sanguinetti,
** A PDE approach to data-driven sub-Riemannian geodesics in SE(2)** on

*SIAM Journal on Imaging Sciences*Vol 8 2015 pp. 2740-2770 — open access version — doi

→ A. Mashtakov, Y. L. Sachkov,
** Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups** on

*Differential Equations*Vol 51 2015 pp. 1476-1483 — open access version — doi

→ 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 — doi

→ R. Duits, A. Ghosh, T. Dela Haije, A. Mashtakov,
** On sub-Riemannian geodesics in SE(3) whose spatial projections do not have cusps** on

*Journal of Dynamical and Control Systems*Vol 22 2016 pp. 771-805 — open access version — doi

→ A. Mashtakov, A. ArdentovYuri, L. Sachkov,
** Relation between Euler’s elasticae and sub-Riemannian geodesics on SE(2)** on

*Regular and Chaotic Dynamics*Vol 21 2016 pp. 832-839 — doi

→ A. Mashtakov, R. Duits, Yu. Sachkov, E. J. Bekkers, I. Beschastnyi,
** Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in {\text {SO(3)}}** on

*Journal of Mathematical Imaging and Vision*Vol 58 2017 pp. 239-264 — doi

### Wp 7

→ B. Franceschiello, A. Mashtakov, G. Citti, A. Sarti,
** Modelling of the Poggendorff Illusion via sub-Riemannian geodesics in the roto-translation group** on

*Lecture Notes in Computer Science*Vol 10590 2017 pp. 37-47 — doi

→ B. Franceschiello, A. Sarti, G. Citti,
** A neuro-mathematical model for geometrical optical illusions** on

*Journal of Mathematical Imaging and Vision*Vol 60 2018 pp. 94-108 — open access version — doi

### Wp 8

→ E. Baspinar, G. Citti, A. Sarti,
** A geometric model of multi-scale orientation preference maps via Gabor functions** on

*Journal of Mathematical Imaging and Vision*Vol 60 2018 pp. 1–13 — open access version — doi

→ E. Baspinar, G. Citti,
** A diffusion driven curvature flow** 2016 preprint

→ E. Baspinar, G. Citti,
** Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures** 2016 preprint