A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.Comment: 123 pages, revised versio

Highly corrupted image inpainting through hypoelliptic diffusion

We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing), for which we obtain reconstructions comparable with the state-of-the-art.Comment: 15 pages, 10 figure

Cortical-Inspired Wilson–Cowan-Type Equations for Orientation-Dependent Contrast Perception Modelling

We consider the evolution model proposed in Bertalmío (Front Comput Neurosci 8:71, 2014), Bertalmío et al. (IEEE Trans Image Process 16(4):1058–1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson–Cowan equations (Wilson and Cowan in BioPhys J 12(1):1–24, 1972), mainly in terms of efficient representation properties. Then, in order to explicitly encode local directional information, we exploit the model of the primary visual cortex (V1) proposed in Citti and Sarti (J Math Imaging Vis 24(3):307–326, 2006) and largely used over the last years for several image processing problems (Duits and Franken in Q Appl Math 68(2):255–292, 2010; Prandi and Gauthier in A semidiscrete version of the Petitot model as a plausible model for anthropomorphic image reconstruction and pattern recognition. SpringerBriefs in Mathematics, Springer, Cham, 2017; Franceschiello et al. in J Math Imaging Vis 60(1):94–108, 2018). The resulting model is thus defined in the space of positions and orientation, and it is capable of describing assimilation and contrast visual bias at the same time. We report several numerical tests showing the ability of the model to reproduce, in particular, orientation-dependent phenomena such as grating induction and a modified version of the Poggendorff illusion. For this latter example, we empirically show the existence of a set of threshold parameters differentiating from inpainting to perception-type reconstructions and describing long-range connectivity between different hypercolumns in V1

Geometry of the Visual Cortex with Applications to Image Inpainting and Enhancement

Equipping the rototranslation group $SE(2)$ with a sub-Riemannian structure inspired by the visual cortex V1, we propose algorithms for image inpainting and enhancement based on hypoelliptic diffusion. We innovate on previous implementations of the methods by Citti, Sarti and Boscain et al., by proposing an alternative that prevents fading and capable of producing sharper results in a procedure that we call WaxOn-WaxOff. We also exploit the sub-Riemannian structure to define a completely new unsharp using $SE(2)$, analogous of the classical unsharp filter for 2D image processing, with applications to image enhancement. We demonstrate our method on blood vessels enhancement in retinal scans.Comment: Associated python package available at https://github.com/ballerin/v1diffusio

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Cortical-Inspired Wilson-Cowan-Type Equations for Orientation-Dependent Contrast Perception Modelling

19 pags., 17 figs.We consider the evolution model proposed in Bertalmío (Front Comput Neurosci 8:71, 2014), Bertalmío et al. (IEEE Trans Image Process 16(4):1058–1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson–Cowan equations (Wilson and Cowan in BioPhys J 12(1):1–24, 1972), mainly in terms of efficient representation properties. Then, in order to explicitly encode local directional information, we exploit the model of the primary visual cortex (V1) proposed in Citti and Sarti (J Math Imaging Vis 24(3):307–326, 2006) and largely used over the last years for several image processing problems (Duits and Franken in Q Appl Math 68(2):255–292, 2010; Prandi and Gauthier in A semidiscrete version of the Petitot model as a plausible model for anthropomorphic image reconstruction and pattern recognition. SpringerBriefs in Mathematics, Springer, Cham, 2017; Franceschiello et al. in J Math Imaging Vis 60(1):94–108, 2018). The resulting model is thus defined in the space of positions and orientation, and it is capable of describing assimilation and contrast visual bias at the same time. We report several numerical tests showing the ability of the model to reproduce, in particular, orientation-dependent phenomena such as grating induction and a modified version of the Poggendorff illusion. For this latter example, we empirically show the existence of a set of threshold parameters differentiating from inpainting to perception-type reconstructions and describing long-range connectivity between different hypercolumns in V1.M. B. acknowledges the support of the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 761544 (Project HDR4EU) and under Grant Agreement No. 780470 (Project SAUCE), and of the Spanish government and FEDER Fund, Grant Ref. PGC2018-099651-B-I00 (MCIU/AEI/FEDER, UE). L. C., V. F. and D. P. acknowledge the support of a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d’avenir program, through the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02 and of the research project LiftME funded by INS2I, CNRS. V. F. acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant No. 794592 and from the INdAM project Problemi isoperimetrici in spazi Euclidei e non. V. F. and D. P. also acknowledge the support of ANR-15-CE40- 0018 project SRGI - Sub-Riemannian Geometry and Interactions. B. F. acknowledges the support of the Fondation Asile des Aveugles

Cortical-inspired Wilson–Cowan-type equations for orientation-dependent contrast perception modelling

We consider the evolution model proposed in Bertalmío (Front Comput Neurosci 8:71, 2014), Bertalmío et al. (IEEE Trans Image Process 16(4):1058–1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson–Cowan equations (Wilson and Cowan in BioPhys J 12(1):1–24, 1972), mainly in terms of efficient representation properties. Then, in order to explicitly encode local directional information, we exploit the model of the primary visual cortex (V1) proposed in Citti and Sarti (J Math Imaging Vis 24(3):307–326, 2006) and largely used over the last years for several image processing problems (Duits and Franken in Q Appl Math 68(2):255–292, 2010; Prandi and Gauthier in A semidiscrete version of the Petitot model as a plausible model for anthropomorphic image reconstruction and pattern recognition. SpringerBriefs in Mathematics, Springer, Cham, 2017; Franceschiello et al. in J Math Imaging Vis 60(1):94–108, 2018). The resulting model is thus defined in the space of positions and orientation, and it is capable of describing assimilation and contrast visual bias at the same time. We report several numerical tests showing the ability of the model to reproduce, in particular, orientation-dependent phenomena such as grating induction and a modified version of the Poggendorff illusion. For this latter example, we empirically show the existence of a set of threshold parameters differentiating from inpainting to perception-type reconstructions and describing long-range connectivity between different hypercolumns in V1.The authors acknowledge the anonymous referees for their suggestions which improved significantly the quality of their manuscript. M. B. acknowledges the support of the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 761544 (Project HDR4EU) and under Grant Agreement No. 780470 (Project SAUCE), and of the Spanish government and FEDER Fund, Grant Ref. PGC2018-099651-B-I00 (MCIU/AEI/FEDER, UE). L. C., V. F. and D. P. acknowledge the support of a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d’avenir program, through the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02 and of the research project LiftME funded by INS2I, CNRS. V. F. acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant No. 794592 and from the INdAM project Problemi isoperimetrici in spazi Euclidei e non. V. F. and D. P. also acknowledge the support of ANR-15-CE40-0018 project SRGI - Sub-Riemannian Geometry and Interactions. B. F. acknowledges the support of the Fondation Asile des Aveugles