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. The method is illustrated on several synthetic examples of two dimensional functions. Two use cases, using data sets from the UCI machine learning repository, demonstrate the utility of the proposed approach on real data. Finally, in collaboration with domain experts the proposed method is applied to two scientific challenges. The analysis of parameters of climate simulations and their relationship to predicted global energy flux and the concentrations of chemical species in a combustion simulation and their integration with temperature.
This paper describes a method for building efficient representations of large sets of brain images. Our hypothesis is that the space spanned by a set of brain images can be captured, to a close approximation, by a low-dimensional, nonlinear manifold. This paper presents a method to learn such a low-dimensional manifold from a given data set. The manifold model is generative-brain images can be constructed from a relatively small set of parameters, and new brain images can be projected onto the manifold. This allows to quantify the geometric accuracy of the manifold approximation in terms of projection distance. The manifold coordinates induce a Euclidean coordinate system on the population data that can be used to perform statistical analysis of the population. We evaluate the proposed method on the OASIS and ADNI brain databases of head MR images in two ways. First, the geometric fit of the method is qualitatively and quantitatively evaluated. Second, the ability of the brain manifold model to explain clinical measures is analyzed by linear regression in the manifold coordinate space. The regression models show that the manifold model is a statistically significant descriptor of clinical parameters.
This paper presents a method for correcting the geometric and greyscale distortions in diffusionweighted MRI that result from in-homogeneities in the static magnetic field. These inhomogeneities may due to imperfections in the magnet or to spatial variations in the magnetic susceptibility of the object being imaged-so called susceptibility artifacts. Echo-planar imaging (EPI), used in virtually all diffusion weighted acquisition protocols, assumes a homogeneous static field, which generally does not hold for head MRI. The resulting distortions are significant, sometimes more than ten millimeters. These artifacts impede accurate alignment of diffusion images with structural MRI, and are generally considered an obstacle to the joint analysis of connectivity and structure in head MRI. In principle, susceptibility artifacts can be corrected by acquiring (and applying) a field map. However, as shown in the literature and demonstrated in this paper, field map corrections of susceptibility artifacts are not entirely accurate and reliable, and thus field maps do not produce reliable alignment of EPIs with corresponding structural images. This paper presents a new, image-based method for correcting susceptibility artifacts. The method relies on a variational formulation of the match between an EPI baseline image and a corresponding T2-weighted structural image but also specifically accounts for the physics of susceptibility artifacts. We derive a set of partial differential equations associated with the optimization, describe the numerical methods for solving these equations, and present results that demonstrate the effectiveness of the proposed method compared with field-map correction.
Non-linear dimensionality reduction of noisy data is a challenging problem encountered in a variety of data analysis applications. Recent results in the literature show that spectral decomposition, as used for example by the Laplacian Eigenmaps algorithm, provides a powerful tool for non-linear dimensionality reduction and manifold learning. In this paper, we discuss a significant shortcoming of these approaches, which we refer to as the repeated eigendirections problem. We propose a novel approach that combines successive 1-dimensional spectral embeddings with a data advection scheme that allows us to address this problem. The proposed method does not depend on a non-linear optimization scheme; hence, it is not prone to local minima. Experiments with artificial and real data illustrate the advantages of the proposed method over existing approaches. We also demonstrate that the approach is capable of correctly learning manifolds corrupted by significant amounts of noise.
This paper introduces a novel partition-based regression approach that incorporates topological information. Partition-based regression typically introduce a quality-of-fit-driven decomposition of the domain. The emphasis in this work is on a topologically meaningful segmentation. Thus, the proposed regression approach is based on a segmentation induced by a discrete approximation of the Morse-Smale complex. This yields a segmentation with partitions corresponding to regions of the function with a single minimum and maximum that are often well approximated by a linear model. This approach yields regression models that are amenable to interpretation and have good predictive capacity. Typically, regression estimates are quantified by their geometrical accuracy. For the proposed regression, an important aspect is the quality of the segmentation itself. Thus, this paper introduces a new criterion that measures the topological accuracy of the estimate. The topological accuracy provides a complementary measure to the classical geometrical error measures and is very sensitive to over-fitting. The Morse-Smale regression is compared to state-of-the-art approaches in terms of geometry and topology and yields comparable or improved fits in many cases. Finally, a detailed study on climate-simulation data demonstrates the application of the Morse-Smale regression. Supplementary materials are available online and contain an implementation of the proposed approach in the R package msr, an analysis and simulations on the stability of the Morse-Smale complex approximation and additional tables for the climate-simulation study.
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