  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.
