Research | Samuel Gerber

personal - classes - software - research - home

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.

Dissertation

List of Publications

High Dimensional Function Visualization

An important goal of scientific data analysis is to understand the behavior of a system or process based on a sample of the system. In many instances it is possible to observe both input parameters and system outputs, and characterize the system as a high-dimensional function. Such data sets arise, for instance, in large numerical simulations, as energy landscapes in optimization problems, or in the analysis of image data relating to biological or medical parameters. This paper proposes an approach to analyze and visualizing such data sets. The proposed method combines topological and geometric techniques to provide interactive visualizations of discretely sampled high-dimensional scalar fields. The method relies on a segmentation of the parameter space using an approximate Morse-Smale complex on the cloud of point samples. For each crystal of the Morse-Smale complex, a regression of the system parameters with respect to the output yields a curve in the parameter space. The result is a simplified geometric representation of the Morse-Smale complex in the high dimensional input domain. Finally, the geometric representation is embedded in 2D, using dimension reduction, to provide a visualization platform. The geometric properties of the regression curves enable the visualization of additional information about each crystal such as local and global shape, width, length, and sampling densities. This provides succinct visual summary of the salient features of high dimensional scalar functions.

Movie with method description and several examples here

Software is available here

Related publications

Samuel Gerber, Peer-Timo Bremer, Valerio Pascucci, Ross Whitaker, "Visual Exploration of High Dimensional Scalar Functions", IEEE Transactions on Visualization and Computer Graphics to appear, Proceedings of VIS 2010 [pdf]

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]

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]

Samuel Gerber, Tolga Tasdizen, Thomas P Fletcher, Ross Whitaker, "Manifold Modeling for Brain Population Analysis", Medical Image Analysis, Volume 14, Issue 5, Special Issue on the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2009, October 2010, Pages 643-653, [pdf] Best Paper Award

april 2012