Randomized Wavelets on Arbitrary Domains and Applications to Functional MRI Analysis

TL;DR

This paper introduces a novel method for constructing randomized wavelets on complex, arbitrary brain domains, specifically the human cortex, and demonstrates their effectiveness in enhancing spatial localization in fMRI data analysis.

Contribution

It develops a lifting scheme-based approach for wavelet construction on arbitrary volumes with randomness, improving analysis power and spatial localization in fMRI studies.

Findings

Enhanced spatial localization in fMRI analysis.

Multiple realizations improve analysis robustness.

Applicable to complex brain structures like the human cortex.

Abstract

Wavelets have been shown to be effective bases for many classes of natural signals and images. Standard wavelet bases have the entire vector space $\mathbb R^n$ as their natural domain. It is fairly straightforward to adapt these to rectangular subdomains, and there also exist constructions for domains with more complex boundaries. However those methods are ineffective when we deal with domains that are very arbitrary and convoluted. A particular example of interest is the human cortex, which is the part of the human brain where all the cognitive activity takes place. In this thesis, we use the lifting scheme to design wavelets on arbitrary volumes, and in particular on volumes having the structure of the human cortex. These wavelets have an element of randomness in their construction, which allows us to repeat the analysis with many different realizations of the wavelet bases and averaging the results, a method that improves the power of the analysis. Next, we apply this type of wavelet transforms to the statistical analysis to fMRI data, and we show that it enables us to achieve greater spatial localization than other, more standard techniques.