1,628 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
ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
We introduce ChebLieNet, a group-equivariant method on (anisotropic)
manifolds. Surfing on the success of graph- and group-based neural networks, we
take advantage of the recent developments in the geometric deep learning field
to derive a new approach to exploit any anisotropies in data. Via discrete
approximations of Lie groups, we develop a graph neural network made of
anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and
unpooling layers, and global pooling layers. Group equivariance is achieved via
equivariant and invariant operators on graphs with anisotropic left-invariant
Riemannian distance-based affinities encoded on the edges. Thanks to its simple
form, the Riemannian metric can model any anisotropies, both in the spatial and
orientation domains. This control on anisotropies of the Riemannian metrics
allows to balance equivariance (anisotropic metric) against invariance
(isotropic metric) of the graph convolution layers. Hence we open the doors to
a better understanding of anisotropic properties. Furthermore, we empirically
prove the existence of (data-dependent) sweet spots for anisotropic parameters
on CIFAR10. This crucial result is evidence of the benefice we could get by
exploiting anisotropic properties in data. We also evaluate the scalability of
this approach on STL10 (image data) and ClimateNet (spherical data), showing
its remarkable adaptability to diverse tasks.Comment: submitted to NeurIPS'21, https://openreview.net/forum?id=WsfXFxqZXR
Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging
Locally adaptive differential frames (gauge frames) are a well-known
effective tool in image analysis, used in differential invariants and
PDE-flows. However, at complex structures such as crossings or junctions, these
frames are not well-defined. Therefore, we generalize the notion of gauge
frames on images to gauge frames on data representations defined on the extended space of positions and
orientations, which we relate to data on the roto-translation group ,
. This allows to define multiple frames per position, one per
orientation. We compute these frames via exponential curve fits in the extended
data representations in . These curve fits minimize first or second
order variational problems which are solved by spectral decomposition of,
respectively, a structure tensor or Hessian of data on . We include
these gauge frames in differential invariants and crossing preserving PDE-flows
acting on extended data representation and we show their advantage compared
to the standard left-invariant frame on . Applications include
crossing-preserving filtering and improved segmentations of the vascular tree
in retinal images, and new 3D extensions of coherence-enhancing diffusion via
invertible orientation scores
Toward a mathematical theory of perception
A new technique for the modelling of perceptual
systems called formal modelling is developed. This
technique begins with qualitative observations about the
perceptual system, the so-called perceptual symmetries, to
obtain through mathematical analysis certain model
structures which may then be calibrated by experiment.
The analysis proceeds in two different ways depending upon
the choice of linear or nonlinear models. For the linear
case, the analysis proceeds through the methods of unitary
representation theory. It begins with a unitary group
representation on the image space and produces what we
have called the fundamental structure theorem. For the
nonlinear case, the analysis makes essential use of
infinite-dimensional manifold theory. It begins with a
Lie group action on an image manifold and produces the
fundamental structure formula.
These techniques will be used to study the brightness
perception mechanism of the human visual system. Several
visual groups are defined and their corresponding
structures for visual system models are obtained. A new
transform called the Mandala transform will be deduced
from a certain visual group and its implications for image processing will be discussed. Several new phenomena of
brightness perception will be presented. New facts about
the Mach band illusion along with new adaptation phenomena
will be presented. Also a new visual illusion will be
presented. A visual model based on the above techniques
will be presented. It will also be shown how use of
statistical estimation theory can be made in the study of
contrast adaptation. Furthermore, a mathematical
interpretation of unconscious inference and a simple
explanation of the Tolhurst effect without mutual channel
inhibition will be given. Finally, image processing
algorithms suggested by the model will be used to process
a real-world image for enhancement and for "form" and
texture extraction
Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores
We provide the explicit solutions of linear, left-invariant,
(convection)-diffusion equations and the corresponding resolvent equations on
the 2D-Euclidean motion group SE(2). These diffusion equations are forward
Kolmogorov equations for stochastic processes for contour enhancement and
completion. The solutions are group-convolutions with the corresponding Green's
function, which we derive in explicit form. We mainly focus on the Kolmogorov
equations for contour enhancement processes which, in contrast to the
Kolmogorov equations for contour completion, do not include convection. The
Green's functions of these left-invariant partial differential equations
coincide with the heat-kernels on SE(2), which we explicitly derive. Then we
compute completion distributions on SE(2) which are the product of a forward
and a backward resolvent evolved from resp. source and sink distribution on
SE(2). On the one hand, the modes of Mumford's direction process for contour
completion coincide with elastica curves minimizing , related to zero-crossings of 2 left-invariant derivatives of the
completion distribution. On the other hand, the completion measure for the
contour enhancement concentrates on geodesics minimizing . This motivates a comparison between geodesics and elastica,
which are quite similar. However, we derive more practical analytic solutions
for the geodesics. The theory is motivated by medical image analysis
applications where enhancement of elongated structures in noisy images is
required. We use left-invariant (non)-linear evolution processes for automated
contour enhancement on invertible orientation scores, obtained from an image by
means of a special type of unitary wavelet transform
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