Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T12:01:42.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI

Part of: Geometric probability and stochastic geometry Inference from stochastic processes Stochastic processes General convexity

Published online by Cambridge University Press:  01 July 2016

Keith J. Worsley
Keith J. Worsley*
Affiliation:
McGill University
*
∗ Postal address: Department of Mathematics and Statistics, McGill University, 805 ouest, rue Sherbrooke, Montréal, Québec, Canada H3A 2K6. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN. The test statistic was the maximum of a Gaussian random field in an N+1 dimensional ‘scale space’, N dimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution for N=2 and N=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2 fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

Keywords

Euler characteristicdifferential topologyintegral geometryimage analysiswavelet thresholdingadaptive filteringmultiscalemultiresolution

MSC classification

Primary: 60G60: Random fields 62M09: Non-Markovian processes
Secondary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 773 - 793
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the Natural Sciences and Engineering Research Council of Canada, and the Fonds pour la Formation des Chercheurs et l'Aide à la Recherche de Québec.

References

Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, New York.Google Scholar
Adler, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Prob. 10, 1–74.Google Scholar
Büchel, C. et al. (1996). Non-linear regression in parametric activation studies. NeuroImage 4, 60–66.Google Scholar
Bullmore, E. T. et al. (1996). Statistical methods of estimation and inference for functional MR image analysis. Magnetic Resonance in Medicine 35, 261–277.CrossRefGoogle ScholarPubMed
Cao, J. and Worsley, K. J. (1999a). The detection of local shape changes via the geometry of Hotelling's T2 fields. Ann. Statist. 27, 925–942.Google Scholar
Cao, J. and Worsley, K. J. (1999b). The geometry of correlation fields, with an application to functional connectivity of the brain. Ann. Appl. Prob. 9, 1021–1057.Google Scholar
Crivello, F. et al. (1995). Intersubject variability in functional neuroanatomy of silent verb generation: assessment by a new activation detection algorithm based on amplitude and size information. NeuroImage 2, 253–263.Google Scholar
Daubechies, I. (1992). Ten lectures on wavelets (CBMS-NSF Regional Conf. Ser. Appl. Math. 61). SIAM, Philadelphia.Google Scholar
Fletcher, P. C. et al. (1996). Is multivariate analysis of PET data more revealing than the univariate approach? Evidence from a study of episodic memory retrieval. NeuroImage 3, 209–215.CrossRefGoogle Scholar
Friston, K. J., Jezzard, P. and Turner, R. (1994). Analysis of functional MRI time series. Human Brain Mapping 1, 153–171.Google Scholar
Klain, G. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge University Press.Google Scholar
Kwong, K. K. et al. (1992). Dynamic magnetic resonance imaging of human brain activity during primary sensory stimulation. Proc. Nat. Acad. Sci. USA 89, 5675–5679.Google Scholar
Lange, N. and Zeger, S. L. (1997). Non-linear Fourier time series analysis for human brain mapping by functional magnetic resonance imaging (with discussion). J. R. Statist. Soc. C 46, 1–29.Google Scholar
Lindeberg, T. (1994). Scale-space Theory in Computer Vision. Kluwer, Boston.Google Scholar
Ogawa, S. et al. (1992). Intrinsic signal changes accompanying sensory stimulation: functional brain mapping with magnetic resonance imaging. Proc. Nat. Acad. Sci. USA 89, 5951–5955.Google Scholar
Ouyang, X., Pike, G. B. and Evans, A. C. (1994). fMRI of human visual cortex using temporal correlation and spatial coherence analysis. In 13th Annual Symp. Soc. Magnetic Resonance in Medicine.Google Scholar
Poline, J.-B. and Mazoyer, B. M. (1994a). Enhanced detection in activation maps using a multifiltering approach. J. Cerebral Blood Flow Metabolism 14, 690–699.Google Scholar
Poline, J.-B. and Mazoyer, B. M. (1994b). Analysis of individual brain activation maps using hierarchical description and multiscale detection. IEEE Trans. Med. Imaging 13, 702–710.Google Scholar
Rosenfeld, A. and Kac, A. C. (1982). Digital Picture Processing, Vol. 2. Academic Press, Orlando, FL.Google Scholar
Shafie, K., Sigal, B., Siegmund, D. O. and Worsley, K. J. (2001). Rotation space random fields. Submitted.Google Scholar
Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23, 608–639.Google Scholar
Solo, V., Purdon, P., Weisskoff, R. and Brown, E. (2001). A signal estimation approach to functional MRI. IEEE Trans. Med. Imaging 20, 26–35.Google Scholar
Worsley, K. J. (1986). Confidence regions and tests for a change-point in a sequence of exponential family random variables. Biometrika 73, 91–104.Google Scholar
Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of χ2, F and t fields. Adv. Appl. Prob. 26, 13–42.Google Scholar
Worsley, K. J. (1995a). Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23, 640–669.Google Scholar
Worsley, K. J. (1995b). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. Appl. Prob. 27, 943–959.Google Scholar
Worsley, K. J. (1996). The geometry of random images. Chance 9, 27–40.Google Scholar
Worsley, K. J. and Friston, K. J. (1995). Analysis of fMRI time-series revisited—again. NeuroImage 2, 173–181.Google Scholar
Worsley, K. J., Marrett, S., Neelin, P. and Evans, A. C. (1996). Searching scale space for activation in PET images. Human Brain Mapping 4, 74–90.Google Scholar