21 research outputs found
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
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
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
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
Sub-Riemannian geometry and its applications to Image Processing
Master's Thesis in MathematicsMAT399MAMN-MA
Local and global gestalt laws: A neurally based spectral approach
A mathematical model of figure-ground articulation is presented, taking into
account both local and global gestalt laws. The model is compatible with the
functional architecture of the primary visual cortex (V1). Particularly the
local gestalt law of good continuity is described by means of suitable
connectivity kernels, that are derived from Lie group theory and are neurally
implemented in long range connectivity in V1. Different kernels are compatible
with the geometric structure of cortical connectivity and they are derived as
the fundamental solutions of the Fokker Planck, the Sub-Riemannian Laplacian
and the isotropic Laplacian equations. The kernels are used to construct
matrices of connectivity among the features present in a visual stimulus.
Global gestalt constraints are then introduced in terms of spectral analysis of
the connectivity matrix, showing that this processing can be cortically
implemented in V1 by mean field neural equations. This analysis performs
grouping of local features and individuates perceptual units with the highest
saliency. Numerical simulations are performed and results are obtained applying
the technique to a number of stimuli.Comment: submitted to Neural Computatio
A bisector line field approach to interpolation of orientation fields
We propose an approach to the problem of global reconstruction of an orientation field. The method is based on a geometric model called "bisector line fields", which maps a pair of vector fields to an orientation field, effectively generalizing the notion of doubling phase vector fields. Endowed with a well chosen energy minimization problem, we provide a polynomial interpolation of a target orientation field while bypassing the doubling phase step. The procedure is then illustrated with examples from fingerprint analysis
A survey of mathematical structures for extending 2D neurogeometry to 3D image processing
International audienceIn the era of big data, one may apply generic learning algorithms for medical computer vision. But such algorithms are often "black-boxes" and as such, hard to interpret. We still need new constructive models, which could eventually feed the big data framework. Where can one find inspiration for new models in medical computer vision? The emerging field of Neurogeometry provides innovative ideas.Neurogeometry models the visual cortex through modern Differential Geometry: the neuronal architecture is represented as a sub-Riemannianmanifold R2 x S1. On the one hand, Neurogeometry explains visual phenomena like human perceptual completion. On the other hand, it provides efficient algorithms for computer vision. Examples of applications are image completion (in-painting) and crossing-preserving smoothing. In medical image computer vision, Neurogeometry is less known although some algorithms exist. One reason is that one often deals with 3D images, whereas Neurogeometry is essentially 2D (our retina is 2D). Moreover, the generalization of (2D)-Neurogeometry to 3D is not straight-forward from the mathematical point of view. This article presents the theoretical framework of a 3D-Neurogeometry inspired by the 2D case. We survey the mathematical structures and a standard frame for algorithms in 3D- Neurogeometry. The aim of the paper is to provide a "theoretical toolbox" and inspiration for new algorithms in 3D medical computer vision