RKHS approach for signal detection in rotation and scale space random fields

Akbar Abravesh,K Shafie

doi:10.2991/jsta.2016.15.3.2

Abstract

Two important papers of Worsley, Siegmund and coworkers consider rotation and scale space random fields for detecting signals in fMRI (functional magnetic resonance imaging) brain images. They use the global maxima of images for detection of a signal. In the current work, we utilize a reproducing kernel Hilbert space (RKHS) approach to show for both rotation and scale space random fields the global maximum of the image is indeed the likelihood ratio test statistic.

Highlights

For the past two decades, Gaussian random fields have been used in a variety of applications in astronomy, neural imaging and genetics to model images produced by modern sensor technologies
Two papers of [1] and [2] consider rotation and scale space random fields for detecting signals in fMRI brain images. They use the global maxima of images for detection of a signal
For the scale space random fields, [3] uses techniques of Gaussian measures on Hilbert spaces to present a proof. We give another proof which works for both rotation and scale space random fields

Summary

Introduction

For the past two decades, Gaussian random fields have been used in a variety of applications in astronomy, neural imaging and genetics to model images produced by modern sensor technologies. Two papers of [1] and [2] consider rotation and scale space random fields for detecting signals in fMRI (functional magnetic resonance imaging) brain images. [4, 5] used the RKHS approach to find the likelihood ratio for detection problem in the Gaussian case These methods, for different situations, were extended by [6], [7], [8] among others. RKHS approach for signal detection in rotation and scale space random fields is an (N + 1)-dimensional zero mean Gaussian random field with the covariance function. They use two different approaches, the expected Euler characteristic of the excursion set or the volume of tubes, to find an approximate P-value

Rotation space random fields

A Gaussian rotation space random field is defined as

Reproducing kernel Hilbert spaces and random fields

Likelihood ratio test for scale and rotation space random fields

Discussion