In recent years, very detailed images of the brain, produced by modern sensor technologies, have given the neuroscientist the opportunity to study the functional activation of the brain under different conditions. The main statistical problem is to locate the isolated regions of the brain where activation has occurred (the signal), and separate them from the rest of the brain where no activation can be detected (the noise). To do this the images are often spatially smoothed before analysis by convolution with a filter f (t) to enhance the signal to noise ratio, where t is a location vector in N dimensional space. The motivation for this comes from the Matched Filter Theorem of signal processing, which states that signal added to white noise is best detected by smoothing with a filter whose shape matches that of the signal. The problem is that the scale of the signal is usually unknown. It is natural to consider searching over filter scale as well as location, that is, to use a filter s-N/2ft/s with scale s varying over a predetermined interval [ s1,s2 ]. This adds an extra dimension to the search space, called scale space (see Poline and Mazoyer, 1994). Siegmund and Worsley (1995) establish the relation between searching over scale space with the problem of testing for a signal with unknown location and scale and find the approximate P-value of the maximum of the scale-space filtered image using the expected Euler characteristic of the excursion set. In this thesis we study the extension of the scale space result to rotating filters of the form | S|--1/4f (S --1/2t), where S is now an N x N positive definite symmetric matrix that rotates and scales the axes of the filter.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.35614 |
| Date | January 1998 |
| Creators | Shafie H., Khalil. |
| Contributors | Worsley, K. J. (advisor) |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Doctor of Philosophy (Department of Mathematics and Statistics.) |
| Rights | All items in eScholarship@McGill are protected by copyright with all rights reserved unless otherwise indicated. |
| Relation | alephsysno: 001641755, proquestno: NQ44578, Theses scanned by UMI/ProQuest. |
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