Abstract
We propose a new method for statistical analysis of functional magnetic resonance imaging (fMRI) data. The discrete wavelet transformation is employed as a tool for efficient and robust signal representation. We use structural magnetic resonance imaging (MRI) and fMRI to empirically estimate the distribution of the wavelet coefficients of the data both across individuals and spatial locations. An anatomical subvolume probabilistic atlas is used to tessellate the structural and functional signals into smaller regions each of which is processed separately. A frequency-adaptive wavelet shrinkage scheme is employed to obtain essentially optimal estimations of the signals in the wavelet space. The empirical distributions of the signals on all the regions are computed in a compressed wavelet space. These are modeled by heavy-tail distributions because their histograms exhibit slower tail decay than the Gaussian. We discovered that the Cauchy, Bessel K Forms, and Pareto distributions provide the most accurate asymptotic models for the distribution of the wavelet coefficients of the data. Finally, we propose a new model for statistical analysis of functional MRI data using this atlas-based wavelet space representation. In the second part of our investigation, we will apply this technique to analyze a large fMRI dataset involving repeated presentation of sensory-motor response stimuli in young, elderly, and demented subjects.
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Aguirre, G., Zarahn, E., and D’Esposito, M. (1998) A critique of the use of the Kolmogorov-Smirnov (KS) statistic for the analysis of BOLD fMRI data. Magn. Reson. Med. 39, 500–505.
Bandettini, P., Jesmanowicz, A., Wong, E., and Hyde, J. (1993) Processing strategies for time-course data sets in functional MRI of the human brain. Magn. Reson. Med. 30, 161–173.
Bayoumi, M. and Weeks, M. (2003) Discrete wavelet transform: architectures, design and performance issues. J. VLSI Signal Process. Syst. 35 (2), 155–178.
Bandettini, P. and Wong, E. (1997) A hypercapnia-based normalization method for improved spatial localization of human brain activation with fMRI. NMR Biomed. 10 (4–5), 197–203.
Bhattacharyya, A. (1943) On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35, 99–109.
Buckner, R., Snyder, A., Sanders, A., Raichle, M., and Morris, J. (2000) Functional brain imaging of young, nondemented, and demented older adults. J. Cogn. Neurosci. 12 (suppl 2), 24–34.
Bullmore, E., Brammer, M., Williams, S., et al. (1996) Statistical methods of estimation and inference for functional MR image analysis. Magn. Reson. Med. 35, 261–277.
Bullmore, E., Fadili, J., Breakspear, M., Salvador, R., Suckling, J., and Brammer, M. (2003) Wavelets and statistical analysis of functional magnetic resonance images of the human brain. Stat. Methods Med. Res. 12 (5), 375–399.
Bullmore, E., Fadili, J., Maxim, V., et al. (2004) Wavelets and functional magnetic resonance imaging of the human brain. NeuroImage 23 (suppl 1), S234-S249.
Cao, J. and Worsley, K. (1999) The detection of local shape changes via the geometry of Hotelling’s T2 fields. Ann. Stat. 27, 925–942.
Chang, C. Y. and Chung, P. C. (2000) Two-layer competitive based Hopfield neural network for medical image edge detection. Opt. Eng. 39 (3), 695–703.
Crabtree, E. C., Mega M. S., Linshield, C., et al. (2000) Alzheimer grey matter loss across time: unbiased assessment using a probabilistic Alzheimer brainatlas. Soc. Neurosci. Abstr. 26, 294.
Daubechies, I. (1988) Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996.
Daubechies, I. (1991) The wavelet transform: a method for time-frequency localization, Advances in spectrum analysis and array processing, Haykin, S., ed., Prentice-Hall, Englewood Cliffs, NJ, pp. 366–417.
Daubechies, I. (1993) Orthonormal bases of compactly supported wavelets, II. Variations on a theme. SIAM J. Math. Anal. 24, 499–519.
Desco, M., Hernandez, J., Santos, A., and Brammer, M. (2001) Multiresolution analysis in fMRI: sensitivity and specificity in the detection of brain activation. Hum. Brain Mapp. 14 (1), 16–27.
Dinov, I. D., Mega, M. S., Thompson, P. M., et al. (2000) Analyzing functional brain images in a probabilistic atlas: a validation of subvolume thresholding. J. Comput. Assist. Tomogr. 24 (1), 128–138.
Dinov, I. D., Mega, M. S., Thompson, P. M., et al. (2001) Construction of the first rest-state functional subvolume probabilistic atlas of normal variability in the elderly and demented brain. Neurology 56 (8), A248-A248.
Dinov, I. D., Mega, M. S., Thompson, P. M., et al. (2002) Quantitative comparison and analysis of brain image registration using frequency-adaptive wavelet shrinkage. IEEE Trans. Inf. Technol. Biomed. 6 (1), 73–85.
Dinov, I. D. and Sumners, D. W. (2001) Applications of frequency dependent wavelet shrinkage to analyzing quality of image registration. SIAM J. Appl. Math. 62 (2), 367–384.
Donoho, D. L. and Johnstone, I. M. (1995) Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90 (432), 1200–1224.
Donoho, D. L. and Johnstone, I. M. (1998) Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (3), 879–921.
Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Pickard, D. (1996) Density estimation by wavelet thresholding. Ann. Stat. 24 (2), 508–539.
Evans, M., Hastings, N., and Peacock, B. (2000) Statistical distributions, Wiley, New York.
Fadili, M. and Bullmore, E. (2002) Wavelet-generalized least squares: a new BLU estimator of linear regression models with 1/f errors. Neurolmage 15, 217–232.
Feilner, M., Blu, T., and Unser, M. (2000) Optimizing wavelets for the analysis of fMRI data. Proceedings of the SPIE-The International Society for Optical Engineering, SPIE-Int. Soc. Opt. Eng. 4119 (1–2), 626–637.
Friston, K., Holmes, A. P., Worsley, K. J., Poline, J. B., Frith, C. D., and Frackowiak, R. S. J. (1995) Statistical parametric maps in functional imaging: a general linear approach. Hum. Brain Mapp. 2, 189–210.
Golay, X., Kollias, S., Stoll, G., Meier, D., Valvanis, A., and Boesiger, P. (1998) A new correlation-based fuzzy logic clustering algorithm for fMRI. Magn. Reson. Med. 40, 249–260.
Goswami, J. C., and Chan, A. K. (1999) Fundamentals of wavelets: theory, algorithms, and applications, Wiley-Interscience, New York.
Gourieroux, C. and Monfort, A. (1992) Qualitative threshold ARCH models. (Autoregressive Conditional Heteroscedasticity) (ARCH Models in Finance). J. Economet. 52 (1–2), 159–200.
Grenander, U. aand Srivastava, A. (2001) Probability models for clutter in natural images. IEEE Trans. Pattern Anal. Mach. Intell. 23 (4), 424–429.
Hajnal, J. V., Myers, R., Oatridge, A., Schwieso, J. E., Young, I. R., and Byder, G. M. (1994) Artifacts due to stimulus correlated motion in functional imaging of the brain. Magn. Reson. Med. 31, 283–291.
Hess, A., Stiller, D., Kaulisch, T., Heil, P., and Schliech, H. (2000) New insights into the hemodynamic blood oxygenation level-dependent response through combination of functional magnetic resonance imaging and optical recordings in gerbil barrel cortex. J. Neurosci. 20, 3328–3338.
Hilton, M., Odgen, T., Hattery, D., Eden, G., and Jawerth, J. (1996) Wavelet denoising of functional MRI data. Wavelets in medicine and biology, Aldroubi, A. and Unser, M., eds., CRC Press, Boca Raton, pp. 93–114.
Kailath, T. (1967) The divergence and Bhattacharyya distance measures in signal selection. IEEE Trans. Commun. Tech. COM-15 (1), 52–60.
Kershaw, J., Ardekani, B. A., and Kanno, I. (1999) Application of Bayesian inference to fMRI data analysis. IEEE Trans. Med. Imaging 18, 1138–1153.
Krakow, K. Woemann, F. G., Symms, M. R., et al. (1999) EEG-triggered functional MRI of interictal epileptiform activity in patients with partial seizures. Brain 122, 1679–1688.
Krakow, K. et al. (2000) EEG recording during fMRI experiments: image quality. Hum. Brain Mapp. 10, 10–15.
Kullback, S. (1959) Information theory and statistics, Wiley, New York.
Kullback, S. and Leibler, R. A. (1951) On information and sufficiency. Ann. Math. Stat. 22, 79–86.
Kuppusamy, K., Lin, W., and Haacke, E. (1997) Statistical assessment of cross-correlation and variance methods and the importance of electrocardiogram gating in functional magnetic resonance imaging. Magn. Reson. Imaging 15, 169–181.
Logothetis, N., Pauls, J., Augath, M., Trinath, T., and Oeltermann, A. (2001) Neurophysiological investigation of the basis of the fMRI signal. Nature 412, 150–157.
Mallat, S. (1989) A theory for multi-resolution signal decomposition: the wavelet representation. IEEE-TPAMI 11, 674–693.
Mazziotta, J., Toga, A., Evans, A., et al. (2001) A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM). Philos. Trans. R. Soc. Lond. B Biol. Sci. 356 (1412), 1293–1322.
Mazziotta, J. C., Toga, A. W., Evans, A., et al. (1995) A probabilistic atlas of the human brain: theory and rationale for its development. The International Consortium for Brain Mapping (ICBM). NeuroImage 2 (2), 89–101.
Mega, M., Dinov, I. D., Thompson, P., et al. (2005) Automated brain tissue assessment in the elderly and demented population: construction and validation of a sub-volume probabilistic brain atlas. NeuroImage 26, 1009–1018.
Mega, M. S., Cummings, J. L., O’Connor, S. M. et al. (2001) Cognitive and metabolic responses to metrifonate therapy in Alzheimer disease. Neuropsychiatry Neuropsychol. Behav. Neurol. 14 (1), 63–68.
Mega, M. S., Dinov, I. D., Lee, L., et al. (2000) Orbital and dorsolateral frontal perfusion defect associated with behavioral response to cholinesterase inhibitor therapy in Alzheimer’s disease. J. Neuropsychiatry Clin. Neurosci. 12 (2), 209–218.
Mega, M. S., Thompson, P. M., Dinov, I. D., et al. (2000) The UCLA Alzheimer’s atlas: structural and func-tional applications. Ann. Neurol. 48 (3), 427–427.
Mekle, R., Laine, A., Perera, G., and DeLaPaz, R. (2000) Activation detection in fMRI data via multi-scale singularity detection. Proceedings of the SPIE-The International Society for Optical Engineering, SPIE-Int. Soc. Opt. Eng. 4119 (1–2), 615–625.
Meyer, F. (2003) Wavelet-based estimation of a semiparametric generalized linear model of fMRI time-series. IEEE Trans. Med. Imaging 22 (3), 315–322.
Ogawa, S., Lee, T.-M., Kay, A. R., and Tank, D. W. (1990) Brain magnetic resonance imaging with contrast dependent on blood oxygenation. Proc. Natl. Acad. Sci. USA 87 (24), 9868–9872.
Ruttimann, U. E., Unser, M., Rawlings, R. et al. (1998) Statistical analysis of functional MRI data in the wavelet domain. IEEE Trans. Med. Imaging 17 (2), 142–154.
Thompson, P. M., Mega, M. S., Woods, R. P. et al. (2000) Early cortical change in Alzheimer’s disease detected with a disease-specific, population-based, probabilistic brain atlas. Neurology 54 (7 suppl 3), A475-A476.
Tsai, A., Fisher, J., Wible, C., Wells, W., Kim, J., and Willsky, A. (1999) Analysis of functional MR1 data using mutual information, Presented at Medical Imaging and Computer-Assisted Intervention-MICCAI’99, In: Taylor, C. and Colchester, A., eds., Cambridge, England, pp. 473–480.
Worsley, K. (1994) Local maxima and the expected Euler characteristic of excursion sets of X2, F and T fields. Adv. Appl. Probab. 26, 13–42.
Xu, Z. H. and Chan, A. K. (2002) Encoding with frames in MRI and analysis of the signal-to-noise ratio. IEEE Trans. Med. Imaging 21 (4), 332–342.
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Dinov, I.D., Boscardin, J.W., Mega, M.S. et al. A wavelet-based statistical analysis of fMRI data. Neuroinform 3, 319–342 (2005). https://doi.org/10.1385/NI:3:4:319
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DOI: https://doi.org/10.1385/NI:3:4:319