Elsevier

NeuroImage

Volume 23, Supplement 1, 2004, Pages S234-S249
NeuroImage

Wavelets and functional magnetic resonance imaging of the human brain

https://doi.org/10.1016/j.neuroimage.2004.07.012Get rights and content

The discrete wavelet transform (DWT) is widely used for multiresolution analysis and decorrelation or “whitening” of nonstationary time series and spatial processes. Wavelets are naturally appropriate for analysis of biological data, such as functional magnetic resonance images of the human brain, which often demonstrate scale invariant or fractal properties. We provide a brief formal introduction to key properties of the DWT and review the growing literature on its application to fMRI. We focus on three applications in particular: (i) wavelet coefficient resampling or “wavestrapping” of 1-D time series, 2- to 3-D spatial maps and 4-D spatiotemporal processes; (ii) wavelet-based estimators for signal and noise parameters of time series regression models assuming the errors are fractional Gaussian noise (fGn); and (iii) wavelet shrinkage in frequentist and Bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps. We conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fMRI data.

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:S(|f|)|f|γorlogS(|f|)=c+γlog|f|.

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:Ψj,k(t)=12jΨ(t2jk2j);and by dilating and translating a “father” wavelet or scaling function ϕ with unit integral over time ∫ϕ(t)dt = 1:ϕj,k(t)=12jϕ(t2jk2j);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.

References (118)

  • K.J. Friston et al.

    To smooth or not to smooth? Bias and efficiency in fMRI time series analysis

    NeuroImage

    (2000)
  • S. Guan et al.

    A wavelet method for the characterization of spatiotemporal patterns

    Physica D

    (2002)
  • S. Hayasaka et al.

    Validating cluster size inference: random field and permutation methods

    NeuroImage

    (2003)
  • B. Horwitz

    The elusive concept of brain connectivity

    NeuroImage

    (2003)
  • V.G. Kiselev et al.

    Is the brain cortex a fractal?

    NeuroImage

    (2003)
  • J. Marchini et al.

    On bias in the estimation of autocorrelations for fMRI voxel time series analysis

    NeuroImage

    (2003)
  • S.C. Ngan et al.

    Temporal filtering of event-related fMRI data using cross-validation

    NeuroImage

    (2000)
  • C.K. Peng et al.

    Statistical properties of DNA sequences

    Physica A

    (1995)
  • J. Raz et al.

    A wavelet packet model of evoked potentials

    Brain Lang.

    (1999)
  • Y. Shimizu et al.

    Wavelet-based multifractal analysis of fMRI time series

    NeuroImage

    (2004)
  • J. Theiler et al.

    Testing for nonlinearity: the method of surrogate data

    Physica D

    (1992)
  • F. Abramovich et al.

    Thresholding of wavelet coefficients as multiple hypotheses testing procedure

  • F. Abramovich et al.

    Bayesian approach to wavelet decomposition and shrinkage

  • P. Abry et al.

    Multiscale nature of network traffic

    IEEE Signal Process. Mag.

    (2002)
  • A.N. Abu-Rezq et al.

    Best parameters selection for wavelet packet-based compression of magnetic resonance images

    Comput. Biomed. Res.

    (1999)
  • A. Aldroubi et al.

    Wavelets in Biology and Medicine

    (1996)
  • R.G. Baraniuk et al.

    Measuring time-frequency information content using the Renyi entropies

    IEEE Trans. Inf. Theory

    (2002)
  • V. Barra et al.

    Tissue segmentation on MR images of the brain by possibilistic clustering on a 3D wavelet representation

    J. Magn. Reson. Imaging

    (2000)
  • J. Beran

    Statistics for Long-Memory Processes

    (1994)
  • M. Brammer

    Multidimensional wavelet analysis of functional magnetic resonance images

    Hum. Brain Mapp.

    (1998)
  • M. Breakspear et al.

    Spatio-temporal wavelet resampling for functional neuroimaging data

    Hum. Brain Mapp.

    (2004)
  • J.H. Brown et al.

    The fractal nature of nature: power laws, ecological complexity and biodiversity

    Philos. Trans. R. Soc. B

    (2002)
  • A. Bruce et al.

    Applied Wavelet Analysis with S-PLUS

    (1996)
  • E.T. Bullmore et al.

    Fractal analysis of the boundary between white matter and cerebral cortex in magnetic resonance images: a controlled study of schizophrenic and manic-depressive patients

    Psychol. Med.

    (1994)
  • E. Bullmore et al.

    Statistical methods of estimation and inference for functional MR image analysis

    Magn. Reson. Med.

    (1996)
  • E. Bullmore et al.

    Global, voxel and cluster tests, by theory and permutation, for a difference between two groups of structural MR images of the brain

    IEEE Trans. Med. Imaging

    (1999)
  • E. Bullmore et al.

    Methods for diagnosis and treatment of stimulus-correlated motion in generic brain activation studies using fMRI

    Hum. Brain Mapp.

    (1999)
  • E. Bullmore et al.

    Colored noise and computational inference in neurophysiological fMRI time series analysis: resampling methods in time and wavelet domains

    Hum. Brain Mapp.

    (2001)
  • E. Bullmore et al.

    In praise of tedious permutation

  • E.T. Bullmore et al.

    Wavelets and statistical analysis of functional magnetic resonance images of the human brain

    Stat. Methods Med. Res.

    (2003)
  • E. Carlstein et al.

    Matched block bootstrap for dependent data

    Bernouilli

    (1998)
  • I. Daubechies

    Ten Lectures on Wavelets

    (1992)
  • A.C. Davison et al.

    Bootstrap Methods and Their Application

    (1997)
  • M. Desco et al.

    Multiresolution analysis in fMRI: sensitivity and specificity in the detection of brain activation

    Hum. Brain Mapp.

    (2001)
  • R.W. Dijkerman et al.

    On the correlation structure of the wavelet coefficients of fractional Brownian motion

    IEEE Trans. Inf. Theory

    (1994)
  • I.D. Dinov et al.

    Quantitative comparison and analysis of brain image registration using frequency-adaptive wavelet shrinkage

    IEEE Trans. Inf. Technol. Biomed.

    (2002)
  • D.L. Donoho et al.

    Adapting to unknown smoothness via wavelet shrinkage

    J. Am. Stat. Assoc.

    (1995)
  • M.J. Fadili et al.

    A comparative evaluation of wavelet-based methods for multiple hypothesis testing of brain activation maps

    NeuroImage

    (2004)
  • M.J. Fadili et al.

    Wavelet methods for characterising mono-and poly-fractal noise structures in shortish time series: an application to functional MRI

    Proc. IEEE Int. Conf. Image Process.

    (2001)
  • Y. Fan

    On the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes

    IEEE Trans. Inf. Theory

    (2003)
  • Cited by (211)

    • Functional connectivity of EEG is subject-specific, associated with phenotype, and different from fMRI

      2020, NeuroImage
      Citation Excerpt :

      Recently, more sophisticated methods for the estimation of FC in fMRI have been developed to extend the traditional approach on FC (Bullmore and Sporns, 2009; Bullmore and Bassett, 2011; Smith et al., 2011). These attempt to capture different time scales (Bullmore et al., 2004), reduce common sources of variance (Salvador et al., 2005), capture delayed correlations (Kitzbichler et al., 2009), or capture causation (Reid et al., 2019). Some of them have also been linked to phenotypic variables such as age (Meunier et al., 2009; Mowinckel et al., 2012), fluid intelligence (Ezaki et al., 2019), and schizophrenia (Fornito et al., 2012), and different methods for correlating with phenotype have been benchmarked (Dadi et al., 2019).

    View all citing articles on Scopus
    View full text