9 research outputs found
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
of rototranslations of the plane. Here, we propose a semi-discrete
version of this theory, leading to a left-invariant structure over the group
, 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 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 WilsonâCowan-Type Equations for Orientation-Dependent Contrast Perception Modelling
We consider the evolution model proposed in BertalmĂo (Front Comput Neurosci 8:71, 2014), BertalmĂo et al. (IEEE Trans Image Process 16(4):1058â1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used WilsonâCowan equations (Wilson and Cowan in BioPhys J 12(1):1â24, 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):307â326, 2006) and largely used over the last years for several image processing problems (Duits and Franken in Q Appl Math 68(2):255â292, 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):94â108, 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 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 , 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 BertalmĂo (Front Comput Neurosci 8:71, 2014), BertalmĂo et al. (IEEE Trans
Image Process 16(4):1058â1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used WilsonâCowan equations (Wilson and Cowan in
BioPhys J 12(1):1â24, 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):307â326, 2006) and largely used over the last years for several image processing problems (Duits and Franken
in Q Appl Math 68(2):255â292, 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):94â108, 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 Unionâs
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 dâavenir 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 Unionâs Horizon 2020 research and innovation programme under the Marie SkĆodowska-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 WilsonâCowan-type equations for orientation-dependent contrast perception modelling
We consider the evolution model proposed in BertalmĂo (Front Comput Neurosci 8:71, 2014), BertalmĂo et al. (IEEE Trans Image Process 16(4):1058â1072, 2007) to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used WilsonâCowan equations (Wilson and Cowan in BioPhys J 12(1):1â24, 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):307â326, 2006) and largely used over the last years for several image processing problems (Duits and Franken in Q Appl Math 68(2):255â292, 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):94â108, 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 Unionâs 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 dâavenir 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 Unionâs Horizon 2020 research and innovation programme under the Marie SkĆodowska-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