Wavelets, Fourier Transform, and Fractals - WSEAS · Wavelets, Fourier Transform, and Fractals RADU...

Wavelets, Fourier Transform, and Fractals

RADU MUTIHACUniversity of Bucharest

Department of Electricity and Biophysics405 Atomistilor St., 077125 Bucharest

[email protected] http://www.fizica.unibuc.ro/mutihac

Abstract: Wavelets and Fourier analysis in digital signal processing are comparatively discussed. In data pro-cessing, the fundamental idea behind wavelets is to analyze according to scale, with the advantage over Fouriermethods in terms of optimally processing signals that contain discontinuities and sharp spikes. Apart from the twomajor characteristics of wavelet analysis - multiresolution and adaptivity to nonstationarity or local features in data- there is a related application in functional neuroimaging on the basis of the fractal or scale invariant propertiesdemonstrated by the brain imaging data. Wavelets provide an orthonormal basis for multiscale analysis and decor-relation of nonstionary time series and spatial processes. The discrete wavelet transform (DWT) has applications tostatistical analysis in functional magnetic resonance imaging (fMRI) like time series resampling by wavestrapping,linear model estimation, and methods for multiple hypothesis testing in the wavelet domain.

Key–Words: Wavelets, Fourier transform, fractals, multiresolution analysis, functional magnetic resonance imag-ing.

1 Introduction

In practice, data are acquired in a raw form and,consequently, of little immediate usefulness. It is onlywhen the information is extracted via processing thatdata become meaningful. In data analysis, it is oftendesirable to reduce the dimension of the feature spacebecause there may be irrelevant or redundant featuresthat complicate subsequent inferences and model de-sign, increase the computational demand, and renderthe analysis suboptimal. A common task is to find anadequate somewhat reduced representation of a mul-tivariate data set for discovery, analysis, and recog-nition of patterns. Transform domain processing is aversatile approach widely used in data analysis. Indata compression, the energy compaction propertiesof some classes of linear transforms are employed[25]. In data processing, noise reduction is performedby generalized filtering based on the fact that the un-derlying signal tends to get concentrated into few co-efficients while the noise is spread out more evenly.The overall effect of working in the transform domainis an increase in the signal-to-noise ratio (SNR).

The WT of an image produces a multiresolutionrepresentation where each wavelet coefficient repre-sents the information content of the image at a certainresolution in a certain position. Preprocessing in im-age analysis (like noise reduction, contrast enhance-ment, ...) can be carried out by making the opera-

tions frequency dependent (i.e., split signal/image intofrequency subbands and apply different operations oneach subband).

Wavelet methods approach the analysis of statis-tical fields by estimating the signal at any resolutionamong the random fluctuations. The appearance ofexplicit orthonormal bases entailed significant impli-cations on fMRI data analysis. Unlike the traditionalFourier bases, wavelet bases offer a degree of localiza-tion in space as well as frequency. This enables the de-velopment of simple function estimates that respondeffectively to discontinuities and spatially varying de-grees of oscillations in a signal, even when the obser-vations are corrupted by noise. Wavelet applicationto statistical fields is similar to wavelet applicationsto images since statistical maps are just images withnoise with variance equal to unity. Therefore, statisti-cal maps are transformed using the DWT, the resultingcoefficients are thresholded, and then denoised statis-tical maps are reconstructed [24]. [20]. Since the con-trol of false positives is stringent in this application,the Bonferroni approach to wavelet thresholding wasinitially suggested.

The computation of the variance wavelet trans-form (WT) is more straightforward for statistical mapsthan it is for images: (i) the noise variance of the im-age is 1; and (ii) the correlation function is computedbecause pure noise images (i.e., residual images) canbe obtained by subtracting from the original scans the

Proceedings of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Bucharest, Romania, October 16-18, 2006 49

effects estimated through statistical analysis. Thenthe noise power of the field can be computed throughFourier techniques [28]. Variances of wavelet levelsare computed by the product of the power function ofthe field with the power function of the wavelet fil-ters [23].

In general, the most attractive mathematical prop-erties of wavelets in applications are: compact support(i.e., they vanish outside a finite interval) and contin-uous differentiability. By truncating the wavelet co-efficients below a certain threshold, the data becomesparsely represented, which makes wavelets appropri-ate for data compression and quite similar to the topo-graphic independent component analysis (ICA) [10][19]. The interest in wavelets is also due to compu-tationally efficient implementation of multiresolutionanalysis (MRA). Just as Fast Fourier Transform (FFT)algorithms made the Fourier Transform (FT) a practi-cal tool for spectral analysis, the MRA has made theDWT a viable tool for computational time-scale anal-ysis.

The continuous wavelet transform (CWT) iscommonly applied in Physics, whereas the discretewavelet transform (DWT) is more common in nu-merical analysis, signal and image processing. Themain three directions of wavelet applications in imag-ing are: (i) image compression, (ii) image denoising,and (iii) image/contrast enhancement. The followingstands for that.

2 Continuous Function SpacesThe space of all functions f(t), t ∈ R that are

square integrable

∫ +∞

−∞|f(t)|2 dt < +∞ (1)

is denoted by L2(R) or, simply L2. The space L2(R)may be endowed with some properties:

1. Scalar (inner) product:

〈f(t), g(t)〉 =∫ +∞

−∞f(t)g(t) dt (2)

where the integral is taken in the Lebesgue’ssense. A space where a scalar product is definedis called a unitary space. A unitary space withfinite dimension is called an Euclidian space.

2. Norm:

‖f(t)‖ =∫ +∞

−∞|f(t)|2 dt < +∞ (3)

A norm always exists in a unitary space due tothe scalar product defined therein. However, anorm may be defined in a space which is not en-dowed with a scalar product. The energy of afunction f(t) is given by its squared L2-norm:

‖f‖2L2

= 〈f, f〉L2 (4)

thus, the notation f ∈ L2 is equivalent to thestatement that ‖f‖2

L2is finite.

3. Distance:

d(f(t), g(t)) = ‖f(t)− g(t)‖ (5)

Spaces endowed with a distance are called met-ric spaces. A unitary space is always a metricspace since it is equipped with a norm However,a distance may be defined without a norm.

4. Completness: A metric space is called completeif all Cauchy sequences converge to an elementin the space. That is, if M 6= ∅ is a metric spaceand xk ∈ M,k ∈ N and d(x, y) is the distancefunction, and if the following holds:

∀ε > 0, ∃N ∈ N : n,m > N ⇒ d(xn, xm) < ε(6)

then∃x ∈ M : lim

k→∞xk = x (7)

A Hilbert space is a complete unitary space thatcorresponds to the inner product:

〈f, g〉L2 =∫ +∞

−∞f(t)g(t) dt (8)

where the integral is taken in Lebesgue’s sense.A Banach space is a complete space endowed

with a norm of finite or infinite dimension.More generally, one defines the Lp spaces for 1 ≤

p ≤ +∞ (where L stands for Lebesgue) as the set offunctions whose Lp-norm:

‖f‖Lp =(∫ +∞

−∞|f(t)|p dt

)1/p

(9)

is finite. These are Banach spaces; there is no corre-sponding inner product whatsoever (except for p = 2).

2.1 Fourier TransformThe Fourier transform (FT) of f(t) is denoted

hereafter by f(ω); if:

f ∈ L1 ⇒ F(f(t)) = f(ω) =∫ +∞

−∞f(t) e−jωt dt

(10)


The definition can be extended for functions f ∈ Lp,as well as for generalized functions µ ∈ S′, where S′stands for Schwarz’ space of temporal distributions onR [13].

2.2 Wavelet TransformWavelets appeared as the consequence of inter-

est shifting from frequency analysis to scale analysis,which entailed the design of mathematical structuresthat vary in scale. In approximation using superpo-sition of functions, Fourier basis functions (sines andcosines) are non-local and stretch out to infinity, ren-dering them unsuited to analyze choppy signals [4].In contrast, wavelets were developed to process sig-nals that contain discontinuities and sharp spikes bycutting them up into different frequency componentsand subsequently analyzing each component with aresolution matched to its scale.

Wavelet analysis procedure is to adopt some twocontinuously-defined functions:

1. The scaling function (or father function) φ(x),which is the solution of a two-scale equation:

φ(x) =√

2∑

k∈Zh(k)φ(2x− k) (11)

where the sequence {h(k)}k∈Z is the refinementfilter.

2. Its associated wavelet function ψ(x) (prototypeor mother wavelet function):

ψ(x) =√

2∑

k∈Zg(k)φ(2x− k) (12)

where {g(k)}k∈Z is a suitable weighting se-quence.

Wavelets are ”small” waves: they oscillate and theircurves yield zero net area:

∫ +∞

−∞ψ(x) dx = 0 (13)

The term ”small” refers to the fact that they are lo-calized in time, in contrast to Fourier basis consist-ing of sines and cosines that are perfectly localizedin frequency space but do not decay as a function oftime (i.e., nonlocal support). Wavelets decay to zeroas x → ±∞ and exhibit good localization propertiesin frequency space.

Any function f ∈ L2 can be uniquely representedby the expansion:

f(x) =∑

j∈Z

∑

k∈Zdj(k)ψj,k(x) (14)

The wavelet coefficients {dj(k)}j,k∈Z are obtained byforming the (double infinite) sequence of inner prod-ucts:

dj(k) = 〈f, ψj,k〉L2 , j, k ∈ Z (15)

where {ψj,k}j,k∈Z is the biorthogonal basis of{ψj,k}j,k∈Z such that:

〈ψj,k, ψi,l〉 = δj−i · δk−l, i, j, k, l ∈ Z (16)

The biorthogonal basis is generated by a single tem-plate (biorthogonal wavelet) ψ(t) [3].

The wavelet coefficients are efficiently calculatedusing Mallat’s algorithm [15], which is based on a hi-erarchical application of the filterbank decomposition(Fig. 1). The only constraints on the choice of the fil-ters are the perfect reconstruction (PR) conditions; inthis case:

H(z−1)H(z) + G(z−1)G(z) = 1

H(z−1)H(−z) + G(z−1)G(−z) = 0(17)

The underlying decomposition algorithm is best de-scribed by two complementary filters h and g. Weconsider hereafter non-redundant dyadic orthogonalwavelet transforms only. The algorithm consists ofan iterated orthogonal filterbank with an analysis anda synthesis part. In a one-level WT, in the decompo-sition (analysis) step, the digital signal ci(k) is splitin two half-size sequences: an approximation partci+1(k) and a detail part di+1(k) by filtering it witha conjugate pair of low-pass (H(z−1)) and high-pass(G(z−1)) filters, respectively (Fig. 1). The resultsare subsequently down-sampled. During the recon-struction (synthesis) step that follows, the signal isreconstructed by up-sampling, filtering, and summa-tion of the components. The analysis and synthesisprocedures are flow-graph transposes of each other.This kind of 2-channel processing is one-to-one andreversible if the z-transforms of the filters satisfy thePR conditions [26]. Different transforms correspondto different sets of PR filters. For a signal vectorof length N0, the operations required by the WT are

G z( )-1~

H z( )-1~

d ki+1( )

2

2

2

2

G z( )-1

H z( )-1

c ki+1( )

c ki( ) c k

i( )

Figure 1: Decomposition and reconstruction quadra-ture mirror filters. Analysis part (left) and synthesispart (right) of the WT filterbank.


G z( )-1~

H z( )-1~

d ki+1( )

d ki+3( )

G z( )-1~

H z( )-1~

d ki+2( )

G z( )-1~

H z( )-1~

c ki+1( )

c ki+2( )

d ki+4( )

G z( )-1~

H z( )-1~

c ki+3( )

c ki( )

c ki+4( )

2

2 2

2 2

2

2

2

Figure 2: Blocks in recursive filterbank implementation of the wavelet transform.

O(N0), as compared with the standard FFT complex-ity ofO(N0 logN0). In multilevel WT, rather than us-ing a huge multichannel filterbank to encompass thefull spectrum, the WT employs recursive 2-channelfilterbanks; each subsequent ci is split into an approx-imation ci+1 and detail di+1 pair of coefficients. Theinverse WT reconstructs each ci from ci+1 and di+1

(Fig. 2).Spline bases posses the best approximation prop-

erties like the smallest L2-error [25]. Due to theirsmoothness, splines are well localized in both timeand frequency domains. Studies on wavelet applica-tion in neuroimaging data analysis emphasized the im-portance of symmetric wavelets and scaling functionsthat are free from phase distortions [20]. Orthogonalbases are mainly recommended because of the follow-ing: (i) signal features not known beforehand can bedetected and extracted in a multiresolution approachover many scales; (ii) transform of white noise intowhite noise [11]. As such, in wavelet analysis of fMRItime series, the preprocessed data are subject to spatialnon-redundant DWT, rather than spatially convolvedwith a Gaussian kernel [17].

Orthogonal wavelet basis functions can be foundby appropriate choice of the sequences {h(k)}k∈Zand {g(k)}k∈Z or, equivalently, φ and ψ, such that{ψj,k}j,k∈Z constitutes an orthonormal basis of L2.Hence

∀f ∈ L2, f(x) =∑

j∈Z

∑

k∈Zdj(k)ψj,k+

∑

j∈Z

∑

k∈Zcj(k)φj,k

(18)where the wavelet coefficients {dj(k)}j,k∈Z and theapproximation coefficients {cj(k)}j,k∈Z, due to or-thogonality, are obtained by inner products with thecorresponding basis functions:

dj(k) = 〈f, ψj,k〉, cj(k) = 〈f, φj,k〉, (19)

The decomposition of any f ∈ L2 is practically car-ried out on a finite number of scales only, say J , so

that:

f(x) =J∑

j=1

∑

k∈Zdj(k)ψj,k +

∑

k∈ZcJ(k)φJ,k (20)

Orthogonality imposes:

H(z) = H(z−1) and G(z) = G(z−1) (21)

where H(z) is the the synthesis scaling filter, thatis, the transfer function (z-transform) of the low-passrefinement filter h, and H(z) is the associated anal-ysis scale filter. Likewise, G(z) is the the synthe-sis wavelet filter, that is, the transfer function (z-transform) of the high-pass filter g, and G(z) is theassociated analysis wavelet filter. The high-pass filterg is the modulated version of h given by:

G(z) = z ·H(−z−1) (22)

The filters must obey the quadrature mirror filter(QMF) conditions for PR (17), which, in terms of thelow-pass filter h only, equate:

H(z)H(z−1) + H(−z)H(−z−1) = 2

H(1) =√

2 ⇔ H(−1) = 0(23)

The orthogonality property means that the innerproduct of the mother wavelet with itself is unity, andthe inner products between the mother wavelet andthe shifts and dilates of the mother wavelet are zero.The collection of shifted and dilated wavelet func-tions is called a wavelet basis. The grid in shift-scalespace on which the wavelet basis functions are definedis called the dyadic grid. Any continuous function(e.g., signals or images) is uniquely projected ontothe wavelet basis functions and expressed as a linearcombination of the basis functions. The set of coef-ficients which weight the wavelet basis functions inrepresenting any continuous function are referred toas the wavelet transform (WT) of the given function.


Remarkably, the WT yields an efficient representationfor functions which have similar character to the func-tions in the wavelet basis. There are wavelet familiesof orthornormal basis functions like the biorthogonalwavelets or wavelet basis functions not orthogonal inany sense. The large number of known wavelet fam-ilies and functions provides a rich space in which tosearch for a wavelet which optimally represent anyfunction of interest.

The continuous-time interpretation of the waveletdecomposition is based on the fundamental conceptof scaling (or scale-varying) function. Scale-varyingbasis functions render signal processing less sensitiveto noise because it measures the average fluctuationsof the signal at different scale. Since the original datacan be represented in terms of wavelet expansion (i.e.,a linear combination of the wavelet functions), anyoperations on data can be carried out using the cor-responding wavelet coefficients only. Decompositionof functions in terms of orthonormal basis functionsis performed by a Fourier expansion as well. Thewavelet basis functions have compact support, in con-trast with the Fourier basis functions. That is, thewavelet basis functions are non-zero only on a finiteinterval, whereas the sinusoidal basis functions of theFourier expansion are infinite in extent. The compactsupport allows the WT to adequately represent func-tions or signals which have localized features. Thisrepresentation is suitable in data compression, signaldetection and denoising. The point is that the struc-tured component of a signal is well represented bya relatively few wavelet coefficients, whereas the un-structured component on the signal like noise projectsquasi-equally onto all of the basis functions. Subse-quently, the structured and unstructured parts of thesignal are separated in the WT domain.

The decomposition (19) can be extended to multi-ple dimensions (e.g., 2D or 3D) by using tensor prod-uct basis functions, which amounts to successively ap-plying the 1D decomposition algorithm along each di-mension in multidimensional data. By iteration, 2q

different type of basis functions are generated in q di-mensions. The corresponding qD separable scalingfunctions with x = (x1, x2, ..., xq) are:

φj,k(x) =q∏

i=1

φj,ki(xi) (24)

where k = (k1, ..., kq) is the vector integer index.The rest of 2q − 1 types of wavelet basis functionsare obtained by replacing one or more factors in (24)with wavelet terms of the form ψj,ki(xi), j ∈ Z, i =1, 2, ..., q. Define b = (b1, ..., bq) a binary vector such

as:

bi =

{1 if φj,ki

is replaced by ψj,ki

0 otherwise, i = 1, 2, ..., q

(25)and

ϕj,ki=

{ψj,ki

if bi = 1φj,ki

otherwise, j ∈ Z, i = 1, 2, ..., q

(26)then the mixed tensor product wavelets can be rewrit-ten [20]

wmj,k(x) =

q∏

i=1

ϕj,ki(xi), m = 1, 2, ..., 2q−1 (27)

with

m =q∑

i=1

bi2i−1 (28)

Here, m indicates a preferential spatial orientationsince φ is low-pass and ψ is high-pass. As for in-stance, in the 2D case, wm

j,k(x) for m = 1, 2, 3 corre-spond to wavelets oriented along the horizontal, diag-onal, and vertical directions, respectively. The corre-sponding multidimensional coefficients:

cj(k) = 〈f, φj,k〉dm

j (k) = 〈f, wmj,k〉

(29)

are iteratively obtained by successive filtering anddownsampling by a factor of two. In the case of mul-tilevel FWT of 2D images, each approximation coeffi-cient ci is split into an approximation coefficient ci+1

and three detail coefficients d1i+1, d2

i+1, and d3i+1, for

horizontally, vertically, and diagonally oriented de-tails, respectively. The biorthogonal scaling functionand its corresponding wavelet function are also plot-ted (Fig. 3a). These wavelets are employed to run 2Dthree-level WT of an axial MR brain slice (Fig 3b).Further on, the first three coarse approximation levelsare separately shown in Fig. 4 for better visual com-parison.

The time-frequency resolution difference be-tween the FT and the WT is best revealed by lookingat the basis function coverage of the time-frequencyplane [27]. A windowed FT (WFT) is shown in Fig.5a, where the window is a square wave that truncatesthe Fourier basis function (sine or cosine) to fit a par-ticular window width. Because a single window sizeis used throughout, the resolution of the analysis isidentical at all locations in the time-frequency plane.Contrarily, the window size varies in the WT, so that


c3

d3

2

d2

2

d1

2

d3

1

d2

1

d1

1d

3

3

d2

3

d1

3

Figure 3: Approximation and detailed wavelet coefficients in a level-three 2D WT (left); 2D wavelet decompositionof a typical MR axial slice. A coarse (approximation) image at a resolution level L is represented by 2L pixelsin each direction. The detail images at a particular level L are produced by horizontal, vertical, and diagonaldifferences between successive levels. The set of coefficients consists of the lowest coarse level image and thehigher level detail images. The original image is 28 × 28 pixels.

Figure 4: Coarse images of a typical MR axial sliceat various approximation levels: (a) Approximation1 (resolution level L = 7); (b) Approximation 2(L = 6); (c) Approximation 3 (L = 5). All imagesare rescaled to the same size for better comparison.

sharp discontinuities are isolated by short high fre-quency basis functions and detailed frequency anal-ysis is allowed by long low frequency ones (Fig. 5b).The FT employs sine and cosine basis functions only,whereas the set of wavelet transforms is infinite.

2.3 Families of WaveletsThere is a large variety of wavelet functions avail-

able. Various wavelet families differ by the compro-mise between their spatial compactness and smooth-ness. Some of particular interest in neuroimaging dataanalysis are presented hereafter (Fig. 6).

The Daubechies family of wavelets are only oneof a number of wavelet families. Remarkably, the

wavelet function (mother wavelet) is orthogonal to allfunctions which are obtained by shifting the motherright or left by an integer amount. Furthermore, themother wavelet is orthogonal to all functions whichare obtained by dilating or stretching the mother bya factor of 2j and shifting by multiples of 2j units.Due to the orthonormality of the Daubechies wavelets,any continuous function is uniquely projected onto thewavelet basis functions and expressed as a linear com-bination of the basis functions.

Symmlets are wavelets within a minimum sizesupport for a given number of vanishing moments,but they are as symmetrical as possible, as opposedto the Daubechies filters which are highly asymmet-rical. They are indexed by the number of vanish-ing moments, which is equal to half the size of thesupport. Fractional splines of a real-valued degreewere proposed to produce wavelet bases [26], suchas symmetric and causal, orthogonal, and biorthog-onal. A reasonable trade-off seems to lead to sym-metric, orthonormal cubic spline wavelets. Thoughsymmetric, orthonormal, smooth wavelet basis func-tions cannot have compact support, they exhibit ex-ponential decay [3]. Symmetric basis functions donot introduce phase distortions, hence a better local-ization of the signal is achieved in the wavelet do-main. Orthogonal spline wavelets were selected be-cause of the following: (i) orthogonality is requiredby the subsequent statistical analysis; (ii) the result-


Fre

qu

en

cy

Fre

qu

en

cy

Time(a)

Time(b)

Figure 5: Tile coverage of the time-frequency plane: (a) by the Fourier basis functions; (b) by the S8 symmlet.

Figure 6: Various types of wavelets and their scaling functions: (a,b) Haar wavelet; (c,d) Nearly symmetric S8symmlet with 8 vanishing moments, (e,f) Daubechies 4 wavelet, and (g,h) Meyer wavelet.

ing family of transforms use symmetric basis func-tions; (iii) the use of splines reduces spectral overlapbetween resolution channels by increasing the degreeof spline n [20]. Nevertheless, small spectral overlapincreases data decorrelation [21], which raises the de-tection sensitivity. The decorrelation ability of orthog-onal spline wavelets stems from the fact that splineswith degree n yield L = n + 1 vanishing moments.The uncertainty principle limits the level of decorre-lation across scale since the correlation suppressioncomes at the expense of a loss in spatial localizationexpressed in the decay rate of the filter coefficients.Besides, selecting the degree of splines depends tosome extent on the assumed smoothness of the sig-nal to be detected. Smooth wavelet bases are asymp-

totically near-optimal for estimating signals that maycontain some points of discontinuity [8].

Donoho and Johnstone [7] proposed a methodto find the threshold that minimizes the estimate ofthe mean squared error (MSE). The approach equatesto applying a soft thresholding nonlinearity, with thethreshold selected by the Stein’s unbiased risk esti-mate (SURE) in the interval [0,

√2log(n)], where

n = 2J is data number and J is the number of scales.This was proved to posses various optimality proper-ties for the MSE estimation. The WT was performedusing nearly symmetric wavelet with 8 vanishing mo-ments (S8). The coefficients in an S8 wavelet ex-pansion decay with scale more rapidly than the corre-sponding Haar coefficients. The SURE shrinkage car-


ried out the best reconstruction of the original signalboth in terms of noise suppression and sharp struc-ture preservation in the neighborhood of the highly-variable spatial components.

When applying the ”MiniMax” thresholding ruleto the noisy signal, the reconstruction suppresses al-most entirely the noise, while preserving its sharpstructural details . The soft thresholding nonlinear-ity has the threshold set to the magic number λn. Inthe case of ”Visu” shrinkage thresholding , the re-construction looks even better. The difference is dueto the soft thresholding nonlinearity, which has thethreshold set to

√2 log(n) [6].

Cycle-spinning is proposed by Coifman to re-move artifacts from wavelet thresholding. Fullytranslation-invariant denoising, using the stationaryWT is a method of removing artifacts from waveletthresholding associated with discontinuities and sin-gularities. It entails a hard thresholding nonlinearity,with threshold also set to

√2log(n).

3 Wavelet Basis as FractalsFunctional features of the brain signals are com-

plex, largely unknown, and difficult to mathemati-cal modeling, so that optimal basis functions can-not be specified in advance. Wavelet MRA circum-vents this problem by detecting and extracting the keysignal features over many scales. Fractals [14] de-fine a class of objects with the characteristic prop-erty of self-similarity (or self-affinity), meaning thatthe statistics describing the structure in time or spaceof a fractal process remain the same as the process ismeasured over a range of different scales (or scale in-variant). Fractals are complex, patterned, statisticallyself-similar or self-affine, scaling or scale-invariantstructures with non-integer dimensions, generated bysimple iterative rules widespread in real and syntheticsystems (Fig. 7). A wavelet basis is fractal (Fig. 8)and so a natural choice of basis for analysis of frac-tal data. Wavelet methods are particularly adequatein brain imaging data analysis due to broadly fractalproperties exhibited by the brain in space and time.Hence wavelets may be more than just another basisfor analysis of fMRI data (Fig. 11) [2].

Specific wavelet applications in neuroimaging in-clude: (i) resampling (up to 4D); (ii), smoothing bywavelet shrinkage [7] (which allows locally adaptivebandwidth so that the power to detect spatial fea-tures of varying extent is not constrained by the ar-bitrary choice of a single kernel size [16]); (iii) esti-mation (robust and informative models); (iv) hypoth-esis testing (multiresolution and enhanced false de-tection rate control) [18]. Wavelet-based statistical

Figure 7: Brain images and fractal model counter-parts.

analysis of fractal processes benefits are: (i) wavelet-based methods perform a multiresolution decompo-sition suitable for scale-invariant processes analysis;(ii) wavelet analysis is optimally whitening and pro-vide Karhunen-Loeve (KL) expansions [25] for long-memory (1/f -like) processes, which is the case infMRI; and (iii) wavelets provide good estimators forthe noise process parameters.

Figure 9: The CWT of fractal fMRI time series (afterBullmore [2]).

4 ConclusionIn image denoising, the aim is to suppress noise

while preserving as much as possible of the image fea-tures. Thresholding in the wavelet domain is on theassumption of white Gaussian noise. In the orthonor-mal wavelet domain, most image information is con-tained in a few largest wavelet coefficients, while thenoise is uniformly spread out across all coefficients.Thus thresholding mostly affects the noise rather thanthe signal. This behavior is in contrast with traditionallinear methods of smoothing, which perform noisesuppression at the expense of significantly broadeningthe signal features by spatial Gaussian filtering with


Figure 8: The fractal self-similarity of Daubechies 4 mother wavelet generated by WaveLab802 [1].

a single kernel. The risk of missing to detect spa-tial features of the smoothing kernel size or lower isalleviated by wavelets, which allow locally adaptivebandwidth, so that the power to detect spatial featuresof varying extent is not constrained by the arbitrarychoice of a single kernel size [9]. Virtually for allwavelet-based denoising methods, the output SNR isa linear function of the input SNR, that is, the waveletmethods, contrarily to Gaussian smoothing, improvethe SNR of the input images that already have a highSNR. Wavelet methods perform as well as Gaussiansmoothing for low SNR’s, and better than Gaussiansmoothing for higher SNR’s. Wavelet-based denois-ing methods, by introducing less smoothing, preservethe sharpness of images and retain the original shapesof the active regions.

Ideal Fourier damping of noisy data performs im-age reconstruction by setting to zero in the Fourier do-main all coefficients smaller than the noise level. Yetno means to determine which Fourier coefficients ex-ceed the noise level exist. Nevertheless, ideal Fourierdamping is not able to suppress noise while preservingimage resolution.

Wavelet analysis is also computationally efficient:the DWT plays a similar role for computational time-scale analysis in MRA as the FFT does for spectralanalysis.

Wavelet analysis is optimal in terms of detecting

transients events in data and adapts well to conditionswhere responses change significantly in amplitudeduring experiments. Simple nonlinear wavelet meth-ods outperform the traditional linear methods such assplines, Fourier series, and kernel-based smoothers.

Acknowledgements: The author is grateful to Dr.Dimitri Van De Ville, Biomedical Imaging Group(BIG), Ecole Polytechnique Federale de Lausanne(EPFL), Switzerland for clearing up subtleties of theWSPM software.

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