Polynomial whitening for high-dimensional data

The inverse square root of a covariance matrix is often desirable for performing data whitening in the process of applying many common multivariate data analysis methods. Direct calculation of the inverse square root is not available when the covariance matrix is either singular or nearly singular, as often occurs in high dimensions. We develop new methods, which we broadly call polynomial whitening, to construct a low-degree polynomial in the empirical covariance matrix which has similar properties to the true inverse square root of the covariance matrix (should it exist). Our method does not suffer in singular or near-singular settings, and is computationally tractable in high dimensions. We demonstrate that our construction of low-degree polynomials provides a good substitute for high-dimensional inverse square root covariance matrices, in both

A morphological visualization analysis of porous structures in phase inversion processes

Porous membranes are an important technology that continually become more established as steps in research lead to its better understanding. Pore formation within the porous membranes is an emerging field, and current research is focused on understanding their development in phase inversion processes. This thesis attempts to aid in the analyis of these types of simulations by providing a visualization tool to extract information about the material and its pores. A morphological analysis was implemented on the structure of three simulated data sets. The analysis consisted of extracting the pore network from the data and creating sets of visualizations to interpret the structure and properties of the material. A special focus with regard to shape of the pores was applied when creating these visual tools. The results showed a good agreement with an intuitive visual analysis of the input data set and the resulting output, and they resulted in new visual tools to better understand the structure of porous membrane development and point to promising work for the future