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My thesis research concerns the exploration of high dimensional
scattered data, especially in the context of applications to neuroimaging.
The underlying question of this work is how to reveal interesting
structures in such data sets in an unsupervised manner. I address this
with the development of a new manifold learning algorithm based on a
statistical representation of the data. With an adaption to shape spaces
I demonstrate the effectiveness of this approach to the analysis of
brain image populations.
List of Publications
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Principal Surfaces and Manifold Learning
Manifold learning is a specific approach to nonlinear dimensionality
reduction based on the assumption that data points are sampled from a low
dimensional manifold embedded in a high dimensional ambient space. The
aim is to uncover the low dimensional manifold structure from the
samples in the high ambient space. Many methods for manifold learning
have been proposed in the machine learning literature. Much of the
recent work focused around what can be called global or spectral methods.
These methods have a closed form solution based on the spectral
decomposition of a matrix that is compiled based on local properties of
the input data.
In many use cases of manifold learning it is necessary to map from
ambient space to manifold coordinates or construct data points given
manifold coordinates, e.g. build a generative model. Manifold learning
methods thus far are mostly concerned with finding a low dimensional
parametrization, the manifold coordinates, of the data, but often do not
provide the tools to project or construct new data points, as for example
PCA does in the linear case.
We propose an approach, kernel map manifolds, that provides the
tools to project data points onto the manifold and reconstruct data
points on the manifold. The approach is firmly rooted in the concept of
principal surfaces, a conceptual extension of principal component
analysis to nonlinear data. Informally principal surfaces pass through
the middle of a distribution. A variational formulation of principal
surface leads to generative manifold models that formally converge to
principal surfaces as the number of samples increases. Furthermore, does
this approach lead to a quantitative evaluation of the geometric manifold
fit, previously missing in most manifold learning techniques, in terms of
projection distance onto the manifold.
Related publications
Samuel Gerber, Tolga Tasdizen, Ross Whitaker
"Dimensionality Reduction and Principal Surfaces via Kernel Map Manifolds",
In Proceedings of the 2009 International Conference on Computer Vison
(ICCV 2009). [pdf]
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Brain Population Analysis with Manifold Models
In many neuroimage applications a summary or representation of a
population of brain images is needed. A common approach is to build a
template, or atlas, that represents a population. Recent work introduced
clustering based approaches, which in a data driven fashion, compute
multiple templates Each template represents a part of the population.
In a different direction, researcher proposed kernel-based regression of
brain images with respect to an underlying parameter. This yields a
continuous curve in the space of brain images that estimates the
conditional expectation of a brain image given the parameter. A natural
question that arises based on these investigations is can the space
spanned by a set of brain images be approximated by a low-dimensional
manifold? In other words, how effectively can a low-dimensional,
nonlinear model represent the variability in brain anatomy.
We adapt the kernel map manifold approach to work on a shape space
based on geometric coordinate transformations. This allows to measure
shape changes in a low dimensional Euclidean space.We apply this method
to the OASIS and ADNI brain database. We show that the learned manifold
provides a good fit in terms of projection distance and as a proxy for
statistical analysis. We perform linear regression of the learned
manifold coordinates with several clinical parameters. This provides
strong evidence that the proposed manifold representation of brain image
data sets captures important clinical trends.
Related Publications
Samuel Gerber, Tolga Tasdizen, Sarang Joshi, Ross Whitaker,
"On the Manifold Structure of the Space of Brain Images",
In Proceedings of the 2009 International Conference on Medical Image
Computing and Computer Assisted Intervention (MICCAI 2009) [pdf]
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