Research Interests

My research interests lie in the fields of Differential Geometry, Pattern Theory, Computer Vision, Probability, Information Theory, and Control Theory. More specifically, these days I am working almost full-time on two research areas:

(i) on the theory of Shape Spaces and its applications to Pattern Theory, with David Mumford of Brown University and Peter Michor of the University of Vienna;

(ii) and on the recovery of images a ffected by ground-level turbulence, with Yifei Lou, Stefano Soatto and Andrea Bertozzi of UCLA and Angelo Cenedese of the University of Padova, Italy.

Summaries are reported below. See also my current Research Statement.

Here at the Department of Mathematics of UCLA I am part of the Computational and Applied Mathematics (CAM) group and my mentors are Andrea Bertozzi and Tony Chan. I am also co-organizing the Image Processing Seminars with Luminita Vese. My research is supported by ONR Grants #N000140910256 and #N000141010808.

Below you will also find a brief description of my past (engineering-related) research projects, namely on Stochastic Hybrid Systems and Random Sampling of Continuous-Time Stochastic Dynamical Systems, which were my research areas at UC Berkeley and the University of Padova, Italy. In a previous life I also did some work on a probabilistic approach to Motion Field Recovery, with applications to Vision-based Autonomous Navigation: cool stuff, but that really belongs to the past.

Main research area #1: the Differential Geometry of Shape Spaces.

The theory of shape deformation 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 concepts, 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. Technically, a Riemannian structure is induced by the action of a Lie group of diffeomorphisms (with a given metric) on the shape manifold. 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 of the Riemannian Manifolds of Landmark points (or "feature points"), which is one of the simplest since it is finite-dimensional.

Knowledge of curvature on a Riemannian manifold is essential in that it allows one 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.

The dimension of the manifold of Landmarks is n=ND, where N is the number of landmarks and D is the dimension of the ambient space in which they live. When we endow the manifold with the above Riemannian structure (i.e. the one induced by the action of groups of diffeomorphisms) the metric tensor may be written, in any set of coordinates, as a finite dimensional matrix. It turns out that the inverse of the metric, i.e. the cometric, has a relatively simple structure in that each of its elements only depends on 2D of the ND coordinates. So we have developed a formula (which is valid for any Riemannian manifold) that conveniently expresses sectional curvature in terms of the first and second partial derivatives of the cometric matrix, since (for Landmarks) the structure of the first and second derivatives of the cometric is very sparse. We have then applied such formula to the computation of sectional curvature for the manifold of Landmarks, and analyzed the effects of curvature on the qualitative dynamics of the cogeodesic flow.

For more details see the paper Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks and also my PhD Thesis; one more paper (with an infinite-dimensional version of the formula for sectional curvature and some applications to Differential Geometry) is in preparation. Some reference material can be downloaded from the home page of AM282-01, The Mathematics of Shape, with Applications to Computer Vision, which David Mumford taught at Brown University.

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Main research area #2: Recovery of Images affected by ground-level Turbulence

The problem of reconstructing an image that is altered by ground-level atmospheric turbulence is still largely unsolved, due to the anisotropic nature of the distortion phenomenon. On the other hand, there is a large literature on optical astronomy, which is concerned with the problem of reconstructing images that are mostly obtained by telescopes that operate in quasi-isoplanatic atmospheric conditions. These methods are however inadequate under the conditions that are typical of ground-level atmospheric turbulence, i.e. with distortion, blurring and noise that are markedly not space- nor time-invariant.

The typical situation is the one shown in Figure (a) below, that shows a frame within a sequence of images taken at the Naval Air Weapons Station at China Lake, in Southern California. The image shows a silhouette and a patterned board observed at a distance of 1 km (0.621 miles); the images are taken at a rate of 30 frames per second. The distortion due to turbulence is not stationary in space since it depends on the distance between the imaging system and the observed objects: for example, the faraway background is a ected by larger turbulence. Several image sequences, corresponding to di erent lighting and temperature conditions, are currently being used in our experiments.

We approach the problem with an algorithm that combines Dynamic Texture, Shape-from-defocus, and Total Variation Blind Deconvolution. A paper is in preparation with A. Bertozzi, S. Soatto and Y. Lou of UCLA, and with A. Cenedese of the University of Padova, Italy.

(a) Original frame. (b) Detail from original frame. (c) Reconstructed detail.

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Past research:

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. Such point of view is somewhat limited in that 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 (MTNS 2004 and CDC 2004 conference papers, a technical report and an IEEE TAC journal paper). The research work was done in collaboration with Dr. Eugenio Cinquemani of the Department of Information Technology and Electrical Engineering at ETH, Switzerland, and with Professor Giorgio Picci of the Department of Information Engineering at the University of Padova, Italy.

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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 and the Department of Statistics at the University of California, Berkeley.

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