Wavelets and functional magnetic resonance imaging of the human brain
Introduction
A wavelet is a little wave or a brief wave. Unlike sine or cosine waves, which extend infinitely with a particular frequency and phase, wavelets are finitely extended or compactly supported; their oscillations decay more or less rapidly to zero (Fig. 1). Over the last 15 years or so, wavelets have emerged as powerful new mathematical tools for the analysis of complex data sets.1 Intuitively, wavelet analysis can be understood as a way of decomposing or atomizing the total energy or variance of a spatial process or time series by an orthonormal basis of wavelets, each of which is weighted by a coefficient representing the amount of energy in the data at a particular scale and location. If we think of the total energy in the data as a frequency-time or scale-space plane, then the discrete wavelet transform can be visualized as a tiling or tessellation of the plane in which each tile has the same area but tiles representing atoms of energy at fine scales have superior resolution in time or space compared to tiles representing atoms of energy at coarse scales (see Fig. 1).
This is evidently a multiresolution analysis in that the energy of the data has been partitioned among a hierarchically organized set of scales. Low-frequency components of the energy will be represented by wavelet coefficients at coarse scales of the decomposition and higher frequency components will be represented by coefficients at finer scales. In this respect, wavelet analysis is conceptually similar to Fourier analysis, which partitions the total energy of the data among an orthonormal basis set of sinusoidal functions at different frequencies. However, wavelet analysis differs importantly from Fourier analysis by virtue of its natural adaptivity to local or nonstationary features of the data within scales of the decomposition. For example, a transient “spike” in a time series will be represented with difficulty by a set of stationary sinusoidal functions, but it will be captured quite deftly in terms of a few fine scale wavelet coefficients located around the corresponding point in time. To quote Mallat (1998): “If we are interested in transient phenomena—a word pronounced at a particular time, an apple located in the left corner of an image—the Fourier transform becomes a cumbersome tool”.
These two aspects of wavelet analysis—its multiresolution nature and its adaptivity to nonstationary or local features in data—are sufficient to indicate that it will be of interest in the analysis of functional magnetic resonance imaging (fMRI) data, which we expect will include possibly nonstationary features of interest at several scales. However, there are arguably at least three additional aspects of wavelet analysis that are advantageous for fMRI data analysis.
First, the wavelet transform is often a whitening or decorrelating transform of autocorrelated data, and this may prove to be statistically convenient in various ways. For example, as we show below, whitening of an autocorrelated time series by taking its wavelet transform can facilitate resampling or efficient linear model parameter estimation. Second, the wavelet transform has proven to be a useful basis for nonparametric regression, denoising or compression of large imaging data sets in many other applications. The signal-to-noise ratio in fMRI is often not much greater than one or two, so any techniques for enhancing representation of signal components are potentially valuable. Third, the discrete wavelet transform implemented by Mallat's pyramid algorithm is remarkably quick to compute: the algorithm has O(N) complexity compared to O(N log(N)) complexity of the fast Fourier transform. Computational speed is clearly of operational value in dealing with the large volumes of data (typically in the order of gigabytes) generated by a single fMRI study.
In addition to these general technical advantages of statistical analysis in the wavelet domain, there is a related, more substantive argument favoring the use of wavelet methods specifically in analysis of brain imaging data that is based on the expectation that the brain may often demonstrate broadly fractal properties. The word fractal was originally coined by Mandelbrot (1977) to define a class of objects with the characteristic property of self-similarity (or self-affinity),2 meaning that the statistics describing the structure in time or space of a fractal process remain the same as the process is measured over a range of different scales. In other words, the structure of the process is approximately scale invariant or scale free.
An informal, familiar example of self-similarity is provided by the complex branching structure of a tree, which is approximately preserved on examination of a single branch or twig, that is, under examination at progressively finer scales of resolution. The complexity of self-similar structures can be quantified in terms of their (usually noninteger) fractal dimensions: for example, a fractal surface will have a fractal dimension (D) in the range 2 < D < 3, with more complex or space-occupying surfaces approaching the limit D = 3 and simpler, more nearly Euclidean planar surfaces having D closer to 2. Fractal time series which have 1 < D < 2, like the human electrocardiogram (ECG) or a “raw” fMRI time series (Fig. 2), typically have long-range autocorrelations (long memory) in time and spectral density S(f) related to a power law function of frequency:
Fractal or scaling time series often have γ ∼ −1 and are therefore sometimes also referred to as 1/f or 1/f-like processes.3
The fractal or scaling properties of the brain have a bearing on the suitability of wavelets for brain mapping because it has been recognized that wavelets are particularly apt for analysis and synthesis of fractal processes (Vidakovic, 1999, Wornell, 1993, Wornell, 1996). The key feature of wavelets that makes them a natural basis for analysis of self-similar or scale-invariant data is that each level of the (discrete) wavelet decomposition is a scaled (by factor 2) version of the next smallest scale: a family of wavelets is a fractal.
Three ways in which wavelets are technically attractive for statistical analysis of fractal processes are as follows: (i) wavelets effect a multiresolution decomposition that is advantageous for analysis of scale-invariant processes that, by definition, will demonstrate self-similar structure on several scales of measurement; (ii) wavelets are theoretically optimal whitening or decorrelating filters for 1/f-like processes (Fan, 2003) and many issues in estimation and hypothesis testing are simplified by independence; and (iii) wavelets can be used to construct good estimators for fractal dimensions, the Hurst exponent and other measures of complexity (Baraniuk et al., 2002). We will return to these advantageous aspects of wavelet analysis in greater detail below; first, we briefly review some of the prior literature on fractal properties of natural and specifically neurobiological data.
Following Mandelbrot (1977), it has become increasingly clear that fractal, scale-invariant or scaling properties are shared by a wide variety of natural and social phenomena—ranging from Internet traffic (Abry et al., 2002) and econometric time series (Mandelbrot, 1997) through DNA base sequences (Peng et al., 1995) to collaborative and affiliative social networks (Strogatz, 2001) and ecosystems (Brown et al., 2002). The pathophysiology of the heart is arguably the human biological system most productively investigated to date using mathematical tools from fractal geometry and wavelet theory. It has been shown that the branching structures of the coronary arterial and His–Purkinje fiber trees are self-similar and have fractional dimensions. Moreover, the fractal geometry of cardiac anatomy has been related to the power law dynamics of the ECG and various fractal and wavelet-based measures of complexity of ECG data have been developed and shown to improve diagnosis of coronary artery disease and prognosis of otherwise sudden arrhythmias (Ivanov et al., 1996). More generally, the fractal geometry of metabolite exchange surfaces and vascular transport systems has been used persuasively to explain the widespread prevalence of non-Euclidean allometric scaling laws in biology (West et al., 1999).
There have been some comparable applications of fractal analysis to neurobiological data. Dendritic branching patterns of single neurons have been quantified in terms of fractal dimensions (Caserta et al., 1995). Fractal dimensions and 1/f spectral properties have been measured in electroencephalographic (EEG) signals (Bullmore et al., 1994a, Linkenkaer-Hansen et al., 2001, Senhadji et al., 1995). The fractal properties of anatomical surfaces and boundaries segmented in human MRI data have been measured (Blanton et al., 2001, Bullmore et al., 1994b, Free et al., 1996, Kiselev et al., 2003, Thompson et al., 1996). Fractal methods have been applied to analysis of radioligand SPET and PET images (Kuikka and Tiihonen, 1998), and imaging-orientated models for cerebral blood flow have been proposed on the basis of the probably fractal geometry of cerebrovascular architecture (Turner, 2001). There have also been some preliminary investigations of 1/f spectral properties in fMRI time series (Fadili and Bullmore, 2002, Fadili et al., 2001, Shimizu et al., 2004, Zarahn et al., 1997).
It is probably also relevant to note that wavelets are increasingly invoked in the theoretical and numerical study of complex dynamical systems. For example, wavelets have been shown parsimoniously to capture the rich dynamics of morphological phenomena such as microbial growth and nonequilibrium chemical reactions (Guan et al., 2002); to display the flow of information between scales in nonequilibrium fluid flows (Nakao et al., 2001); and to predict the behavior of spatially extended nonlinear dynamical equations (Parlitz and Meyer-Kress, 1995). These aspects of wavelets may be leading indicators of future applications to fMRI and electrophysiological data because they show how wavelets can shed light on the underlying mechanisms of pattern formation and information flow in complex systems like the brain.
In short, scale-invariant processes are abundant in nature and there has already been some successful work applying ideas from fractal geometry to analysis of several modalities of human brain mapping data. There is a case for considering wavelets as more than “just another basis,” one among many possible and equally plausible mathematical domains for statistical analysis of fMRI data.
Previous general reviews of wavelets in biomedical image processing, including some early work on fMRI, are provided by Aldroubi and Unser (1996), Laine (2000) and Bullmore et al. (2003). Statistical issues in wavelet analysis of time series are addressed comprehensively by Percival and Walden (2000). Wornell, 1993, Wornell, 1996 makes a detailed case for the general optimality of wavelets for analysis of fractal signals. Bruce and Gao (1996) describe the implementation of wavelet methods in S-PLUS.
Several research groups have pioneered applications of wavelets to various issues in fMRI data analysis. The most popular application to date has been image compression or denoising (Abu-Rezq et al., 1999, Alexander et al., 2000, Angelidis, 1994, Iyriboz et al., 1999, Maldjian et al., 1997, Weaver et al., 1991, Wink and Roerdink, 2004, Wood and Johnson, 1999, Zaroubi and Goelman, 2000). Multiresolution hypothesis testing of spatial maps of fMRI time series statistics has been explored by Brammer (1998), Desco et al. (2001), and Ruttimann et al. (1998). Linear model estimation in the wavelet domain has been described by Fadili and Bullmore (2002), Meyer (2003) and Müller et al. (2003). Resampling of fMRI data in the wavelet domain has been developed by Breakspear et al. (2004), Bullmore et al. (2001a), Hossien-Zadeh et al. (2003) and Laird et al. (2004). Additionally, there have been applications of wavelets to the image processing problems of registering individual fMRI data sets in a standard anatomical space (Dinov et al., 2002) and correcting unidirectional geometric distortions in echoplanar imaging data (Kybic et al., 2000).
There have also been a number of interesting applications of wavelets to analysis of human brain mapping data in other modalities. Turkheimer et al., 2000, Turkheimer et al., 2003 developed methods for multiresolution analysis and linear modeling (ANOVA, etc.) of multisubject positron emission tomography (PET) studies in the wavelet domain; Cselenyi et al. (2002) explored two and three-dimensional wavelet transforms as spatial filters of radioligand binding potential maps measured using PET; Raz et al. (1999) used wavelet packet analysis to decompose auditory electrophysiological potentials into component waveforms; and Barra and Boire (2000) reported a technique for brain tissue classification or segmentation of structural MRI based on fuzzy clustering of wavelet coefficients.
In what follows, we first provide a brief formal introduction to some key properties of the discrete wavelet transform (DWT)4 and then discuss in more detail its application to three aspects of statistical analysis of fMRI data: (i) resampling of fMRI time series in time and space; (ii) time series modeling in the context of fractional Gaussian noise (fGn); and (iii) wavelet shrinkage or multiple hypothesis testing methods to control false discovery rate and to threshold Bayesian posterior probabilities.
Section snippets
Notation and definitions
Wavelets can be formally defined as families of functions that form an orthonormal basis for a large class of physically relevant (square integrable) functions. A wavelet family is obtained by dilating and translating a compactly supported “mother” wavelet ψ with zero integral over time ∫ψ(t)dt = 0:and by dilating and translating a “father” wavelet or scaling function ϕ with unit integral over time ∫ϕ(t)dt = 1:where j = 1, 2, 3,…,j indexes the scale Sj
Data resampling in the wavelet domain or “wavestrapping”
Data resampling by permutation or bootstrap offers many advantages for inference on functional neuroimaging data—in particular, it obviates the need to make unrealistic assumptions about spatial auto-covariance and other distributional aspects of the data. Perhaps for these reasons, an appropriate nonparametric test can have superior sensitivity compared to a parametric alternative (Bullmore et al., 2001b, Hayasaka and Nichols, 2003, Nichols and Holmes, 2002). Moreover, there are many
Time series modeling in the wavelet domain
The existence of serially dependent noise in fMRI time series not only complicates resampling but also impacts on the efficiency of estimation of the linear model parameter vector β. It is well known that ordinary least squares (OLS) will be the best linear unbiased (BLU) estimator of β if the residual series ɛ is white. However, if ɛ is autocorrelated, OLS will be less than optimally efficient and will severely underestimate the standard error of β. One response to this problem is to formulate
Multiresolution hypothesis testing
Some work on wavelet-based denoising has taken a more probabilistic approach to defining threshold values and rules (Hyvarinen, 1999). It is easy to see that, by this slight change of emphasis towards a more statistical perspective, algorithms for wavelet shrinkage applied to fMRI can be reformulated as methods for massively univariate hypothesis testing and control of type I error in the context of multiple comparisons.
Ruttimann et al. (1998) exploited the multiresolution and decorrelating
Conclusions
These are early days in the application of wavelets to the particular challenges of fMRI data analysis and much remains to be tried and tested. However, it already seems clear that the wavelet domain is a rich source of relatively new concepts and techniques to enhance the power of statistical analysis of scale-invariant time series and spatial processes. We have highlighted applications to the problems of resampling, time series modeling, and multiple hypothesis testing in fMRI, but there are
Acknowledgments
This neuroinformatics research was funded by a Human Brain Project grant from the National Institute of Biomedical Imaging and Bioengineering and the National Institute of Mental Health. This work was also supported by the Wellcome Trust and GlaxoSmithKline. The Wolfson Brain Imaging Centre is supported by a Medical Research Council (UK) Cooperative Group grant.
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