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Events on April 21, 2017

Ph.D. Thesis Defense

Prasanna Muralidharan Presents:

Bayesian regression and longitudinal modeling of manifold data: Applications to time-varying shape analysis

April 21, 2017 at 11:30am for 1hr
Evans Conference Room, WEB 3780
Warnock Engineering Building, 3rd floor.

Abstract:

The statistical study of anatomical shape is of crucial importance in many medical image analysis applications, specifically in the context of understanding healthy brain developmental processes and those with neurological disorders such as Alzheimer's and Huntington's disease. Shape, defined as the geometry of an object invariant to position, size and orientation, is non-linear. Therefore traditional Euclidean statistics for shape analysis are not appropriate.

However, shape is better represented on non-linear Riemannian manifolds. This dissertation develops novel statistical methodologies to analyze data that find a natural parameterization on Riemannian manifolds. These techniques are generalizations of methods developed for the Euclidean setting. The methods are then applied to study anatomical shape variability.

First, a new statistical technique called geodesic mixed-effects models is developed to study longitudinal data variability parameterized on a non-linear manifold. Recent emergence of large-scale longitudinal imaging studies necessitates development of such methods to study dynamic anatomical processes such as motion, growth, and degeneration. Geodesic mixed-effects models are natural generalizations of linear mixed-effects models developed for Euclidean longitudinal data, to the manifold setting. This model was evaluated on example anatomical shape data to study age-related anatomical variability in healthy individuals and those that have Alzheimer's disease.

Just as with geodesic mixed-effects models, existing longitudinal shape models have focused on modeling only age-related anatomical variability, but have not included the ability to handle multiple covariates, such as sex, disease diagnosis, IQ, etc. Unfortunately, this is not straightforward to setup when anatomical shape is represented on a manifold. Instead, as a first step, this dissertation proposes a Bayesian mixed-effects model, for shape represented in a linear space, that incorporates simultaneous relationships between longitudinal shape data and multiple predictors or covariates to the model. The framework also automatically selects which covariates are most relevant to the shape evolution based on observed data.

Finally, this dissertation proposes a Bayesian interpretation of the polynomial regression problem for data represented on a manifold. The Bayesian model prevents overfitting inherent to least-squares methods. By design, the model also automatically selects the most relevant regression coefficients.

Posted by: Nathan Galli

SCI Distinguished Lecture Series
Gunther Uhlmann

Gunther Uhlmann, Walker Family Endowed Professor of Mathematics, University of Washington Presents:

Travel Time Tomography

April 21, 2017 at 2:00pm for 1hr
Evans Conference Room, WEB 3780
Warnock Engineering Building, 3rd floor.

Abstract:

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, joint with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.

Posted by: Nathan Galli