Research Interests
Broadly speaking,
my interests lie in the fields of
Pattern Theory,
Computer Vision,
Probability, Information Theory, Machine Learning,
Control Theory,
Statistical Filtering and Signal Processing.
More specifically, I am working on the theory
of Shape Spaces and its applications
(see brief description below) with my PhD advisor, Professor
David
Mumford
of the
Division of Applied Mathematics here
at
Brown University.
In a past life I have also worked on
Random Sampling of Continuous-Time Stochastic Dynamical Systems
and State Estimation in Stochastic Hybrid Systems at UC Berkeley
and the University of Padova, Italy.
Current research area: the Geometry of Shape Spaces.
For a more detailed description, see my Research Statement.
I am currently working on my PhD thesis. The topic is connected to the theory of shape deformation, which is very central in computer vision. Direct applications of such theory include object recognition, target detection and tracking, classification of biometric data, and automated medical diagnostics, to name a few.
One of the main ideas in this area has been to use fluid flow ideas, which lead to a Riemannian metric on many deformation related spaces such as the space of closed plane curves, the space of n-tuples of landmark points, the spaces of images (scalar multivariate functions), and others. However, the geometry of these Riemannian manifolds has remained a mystery until very recently, when researchers started addressing certain fundamental questions: for example, the curvature of such manifolds is completely unknown in most cases. I am working on the curvature in one of the simplest ones, which is that of landmark points.
Knowledge of curvature on a Riemannian manifold is essential in that it allows to infer about the uniqueness of geodesics connecting two shapes, the convergence or divergence of geodesics (that depart from a common shape but with different initial velocities), the well-posedness of the problem of computing the implicit mean and higher statistical moments of samples on the shape manifold.
The latter issue is of fundamental importance since it allows to build templates, i.e. shape classes that represent typical situations in certain applications. For example, templates can used for the identification of structures in medical imagery, such as x-rays of hands or Magnetic Resonance Images (MRI) of brains. A template can represent the prototypical structure of a healthy person's brain, or the sturcure of the brain of someone developing Alzheimer's disease: such templates are matched to the MRI scan of an individual patient, and the geodesic distances between the data and the templates can then be used to formulate a diagnosis on the patient's health.
I am now embarking on the crucial phase of my PhD research, which involves the analytical calculation of sectional curvature of the space of n-tuples of landmark points and the numerical computation of geodesics on such Riemannian manifold. These computations will also be used to analyze medical data deriving from Magnetic Resonance Imaging of the human brain.
For more details, see my
publications page (preprints on the Geometry
of Shape Spaces).
My advisor, Professor
David
Mumford,
taught a graduate course on the above topic:
AM282-01,
The Mathematics of Shape,
with Applications to Computer Vision in the Spring of 2006.
I was a guest lecturer and teaching assistant for the course.
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Past research:
Random Sampling of Continuous-Time Stochastic Dynamical Systems.
We consider a dynamical system where the state equation is given by a
linear SDE and noisy measurements occur at discrete times, in correspondence
of the arrivals of a Poisson process. Such system models a network of
a large number of sensors that are not synchronized with one another,
so that the waiting time between two measurements is suitably modelled
by an exponential random variable. We formulate a Kalman Filter-based
state estimation algorithm. The sequence of estimation error covariance
matrices (which measure the effectiveness of the estimation algorithm)
is not deterministic as for the ordinary Kalman Filter, but is
a stochastic process itself: in fact we show that it is a homogeneous
Markov process. In the one-dimensional case we compute a complete statistical
description of this process: such description depends on the Poisson sampling
rate (which is proportional to the number of sensors on a network) and on
the dynamics of the continuous-time system represented by the state equation.
In particular we find that unstable dynamical systems are harder to track
than stable ones (which is in accordance with physical intuition) especially
when the sampling rate is too low. As far as stable systems are concerned,
when prior knowledge on state is poor it is convenient to wait until state
is sufficiently close to the origin before starting state estimation: this
explains the apparently paradoxical fact that in some situations increasing
the sampling rate does not lower the estimation error variance. Finally,
in the situation where it is possible to choose the sampling rate (e.g.
by increasing the number of sensors in a network) we have found a lower
bound on such rate that allows to limit the estimation error variance
below an arbitrary threshold with an arbitrary probability.
For more details, see my
publications page (Master's Thesis, MTNS 2002 conference paper). The research work was done in collaboration with Professor
Michael I. Jordan of
the EECS Department at the University of California,
Berkeley.
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State Estimation in Stochastic Hybrid Systems.
The approach to theory of Hybrid Systems that has been developed in the past few years
is mainly deterministic (with only a few isolated exceptions). In my opinion
such point of view is somewhat limited since the modeling of physical systems and the
consequent analysis are often made more efficient (and interesting) by adding random
quantities to the model. Since there is no universally-accepted notion of
Stochastic Hybrid System, we formulated a definition roughly as follows:
a continuous variable x(t) evolves in time according to a stochastic
ordinary differential equation whose parameters depend on q(t), a
discrete state (that takes values in a finite or at most countable
set); state q(t) may evolve in time as a Markov process, although in principle
its evolution may also depend on the continuous variable x(t).
My research has mainly explored the problem of estimating the discrete state
from noisy measurements of the continuous state, with applications to
fault detection (e.g., think of a device
that works in two discrete states: good and faulty;
such states cause different continuous state dynamics, and one has the interest
of estimating whether the device is working properly by observing
the evolvution of the continuous state). Many ideas originate from the Statistics
literature: in particular, we have been inspired by
Conditional Dynamical Linear Models.
For more details, see my
publications page (UC Berkeley Report, MTNS 2004
and CDC 2004 conference papers, IEEE TAC Journal paper).
The research work was done in collaboration with Professor
Giorgio
Picci
and Dr.
Eugenio
Cinquemani of the
Department of Information Engineering at the University
of Padova, Italy.
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