65 research outputs found
H\"older equivalence of the value function for control-affine systems
We prove the continuity and the H\"older equivalence w.r.t.\ an Euclidean
distance of the value function associated with the cost of the
control-affine system \dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q), satisfying
the strong H\"ormander condition. This is done by proving a result in the same
spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The
techniques used are based on a reduction of the control-affine system to a
linear but time-dependent one, for which we are able to define a generalization
of the nilpotent approximation and through which we derive estimates for the
shape of the reachable sets. Finally, we also prove the continuity of the value
function associated with the cost of time-dependent systems of the form
.Comment: 25 pages, some correction
Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces
We study the evolution of the heat and of a free quantum particle (described
by the Schr\"odinger equation) on two-dimensional manifolds endowed with the
degenerate Riemannian metric , where , and the parameter . For
this metric describes cone-like manifolds (for it is
a flat cone). For it is a cylinder. For it is a
Grushin-like metric. We show that the Laplace-Beltrami operator is
essentially self-adjoint if and only if . In this case the
only self-adjoint extension is the Friedrichs extension , that does
not allow communication through the singular set both for the heat
and for a quantum particle. For we show that for the
Schr\"odinger equation only the average on of the wave function can
cross the singular set, while the solutions of the only Markovian extension of
the heat equation (which indeed is ) cannot. For we
prove that there exists a canonical self-adjoint extension , called
bridging extension, which is Markovian and allows the complete communication
through the singularity (both of the heat and of a quantum particle). Also, we
study the stochastic completeness (i.e., conservation of the norm for the
heat equation) of the Markovian extensions and , proving
that is stochastically complete at the singularity if and only if
, while is always stochastically complete at the
singularity.Comment: 29 pages, 2 figures, accepted versio
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 and analysis of control-affine systems: motion planning, heat and Schr\uf6dinger evolution
This thesis is dedicated to two problems arising from geometric control theory, regarding control-affine systems , where is called the drift. In the first part we extend the concept of complexity of non-admissible trajectories, well understood for sub-Riemannian systems, to this more general case, and find asymptotic estimates. In order to do this, we also prove a result in the same spirit as the Ball-Box theorem for sub-Riemannian systems, in the context of control-affine systems equipped with the L1 cost. Then, in the second part of the thesis, we consider a family of 2-dimensional driftless control systems. For these, we study how the set where the control vector fields become collinear affects the diffusion dynamics. More precisely, we study whether solutions to the heat and Schr\uf6dinger equations associated with this Laplace-Beltrami operator are able to cross this singularity, and how its the presence affects the spectral properties of the operator, in particular under a magnetic Aharonov\u2013Bohm-type perturbation
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
On the mathematical replication of the MacKay effect from redundant stimulation
In this study, we investigate the intricate connection between visual
perception and the mathematical modelling of neural activity in the primary
visual cortex (V1), focusing on replicating the MacKay effect [Mackay, Nature
1957]. While bifurcation theory has been a prominent mathematical approach for
addressing issues in neuroscience, especially in describing spontaneous pattern
formations in V1 due to parameter changes, it faces challenges in scenarios
with localised sensory inputs. This is evident, for instance, in Mackay's
psychophysical experiments, where the redundancy of visual stimuli information
results in irregular shapes, making bifurcation theory and multi-scale analysis
less effective. To address this, we follow a mathematical viewpoint based on
the input-output controllability of an Amari-type neural fields model. This
framework views the sensory input as a control function, cortical
representation via the retino-cortical map of the visual stimulus that captures
the distinct features of the stimulus, specifically the central redundancy in
MacKay's funnel pattern ``MacKay rays''. From a control theory point of view,
the exact controllability property of the Amari-type equation is discussed both
for linear and nonlinear response functions. Then, applied to the MacKay effect
replication, we adjust the parameter representing intra-neuron connectivity to
ensure that, in the absence of sensory input, cortical activity exponentially
stabilises to the stationary state that we perform quantitative and qualitative
studies to show that it captures all the essential features of the induced
after-image reported by MacKa
Cortical origins of MacKay-type visual illusions. A case for the non-linearity
To study the interaction between retinal stimulation by redundant geometrical
patterns and the cortical response in the primary visual cortex (V1), we focus
on the MacKay effect (Nature, 1957) and Billock and Tsou's experiments (PNAS,
2007). Starting from a classical biological model of neuronal fields equations
with a non-linear response function, we use a controllability approach to
describe these phenomena. The external input containing a localised control
function is interpreted as a cortical representation of the static visual
stimuli used in these experiments. We prove that while the MacKay effect is
essentially a linear phenomenon (i.e., the nonlinear nature of the activation
does not play any role in its reproduction), the phenomena reported by Billock
and Tsou are wholly nonlinear and depend strongly on the shape of the
nonlinearity used to model the response function
Beyond sparse coding in V1
Growing evidence indicates that only a sparse subset from a pool of sensory
neurons is active for the encoding of visual stimuli at any instant in time.
Traditionally, to replicate such biological sparsity, generative models have
been using the norm as a penalty due to its convexity, which makes it
amenable to fast and simple algorithmic solvers. In this work, we use
biological vision as a test-bed and show that the soft thresholding operation
associated to the use of the norm is highly suboptimal compared to
other functions suited to approximating with
(including recently proposed Continuous Exact relaxations), both in terms of
performance and in the production of features that are akin to signatures of
the primary visual cortex. We show that sparsity produces a denser
code or employs a pool with more neurons, i.e. has a higher degree of
overcompleteness, in order to maintain the same reconstruction error as the
other methods considered. For all the penalty functions tested, a subset of the
neurons develop orientation selectivity similarly to V1 neurons. When their
code is sparse enough, the methods also develop receptive fields with varying
functionalities, another signature of V1. Compared to other methods, soft
thresholding achieves this level of sparsity at the expense of much degraded
reconstruction performance, that more likely than not is not acceptable in
biological vision. Our results indicate that V1 uses a sparsity inducing
regularization that is closer to the pseudo-norm rather than to the
norm
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