spherical wavelet transform for odf...

Medical Image Analysis 14 (2010) 332–342

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Medical Image Analysis

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Spherical wavelet transform for ODF sharpening

I. Kezele *, M. Descoteaux, C. Poupon, F. Poupon, J.-F. ManginNeuroSpin, CEA Saclay, IFR49, CEA, 91400 Gif-sur-Yvette, France

a r t i c l e i n f o

Article history:Received 9 February 2009Received in revised form 18 December 2009Accepted 7 January 2010Available online 1 February 2010

Keywords:Diffusion MRIq-Ball imagingSpherical deconvolutionOrientation distribution function (ODF)Spherical waveletsMultiresolution analysis‘‘À trous” algorithm

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* Corresponding author. Tel.: +33 1 69 28 90 77; faE-mail address: [email protected] (I. Kezele).

a b s t r a c t

The choice of local HARDI reconstruction technique is crucial for discerning multiple fiber orientations,which is itself of substantial importance for tractography, and reliable and accurate assessment of whitematter fiber geometry. Due to the complexity of the diffusion process and its milieu, distinct diffusioncompartments can have different frequency signatures, making the HARDI signal spread over multiplefrequency bands. Therefore, we put forth the idea of multiscale analysis with localized basis functions,ensuring that different frequency ranges are probed. With the aim of truthful recovery of fiber orienta-tions, we reconstruct the orientation distribution function (ODF), by incorporating a spherical wavelettransform (SWT) into the Funk–Radon transform. First, we apply and validate our proposed SWT methodon real physical phantoms emulating fiber bundle crossings. Then, we apply the SWT method to a realbrain data set. The analysis of the real data set suggests that different angular frequencies may capturedifferent information, thus stressing the importance of multiscale analysis. For both phantom and realdata, we compare the SWT reconstruction with state-of-the-art q-ball imaging and spherical deconvolu-tion reconstruction methods. We demonstrate the algorithm efficiency in diffusion ODF denoising andsharpening that is of particular importance for applications to fiber tracking (especially for probabilisticapproaches), and brain connectome mapping. Also, the algorithm results in considerable data compres-sion that could prove beneficial in applications to fiber bundle segmentation, and for HARDI based whitematter morphometry methods.

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1. Intoduction

In diffusion MRI, the measured signal attenuation is propor-tional to residual spin phase incoherence that results directly fromthe component of spin motion in the direction of applied diffusiongradients. Since the spin motion depends on the geometry of thesubstrate, the signal inherently encodes this dependence. A num-ber of methods for local diffusion modeling that estimate the fullthree-dimensional diffusion probability distribution (i.e., diffusionpropagator) have been reported in the literature. The state-of-the-art technique, diffusion spectrum imaging (DSI) (Wedeenet al., 2005) uses the direct relation between the average diffusionpropagator and diffusion signal via Fourier transform (Callaghan,1991), and samples densely the Cartesian grid in q-space. A DSIderivative, hybrid diffusion imaging (HYDI) (Wu and Alexander,2007), uses a lighter q-space sampling scheme on multiple q-shellsto interpolate and regrid the whole Cartesian volume.

Recently, a number of methods appeared that propose to modelthe diffusion signal in a general manner, that would not depend onthe geometry of the substrate (Ozarslan et al., 2009; Assemlal et al.,2009; Descoteaux et al., 2009b). These methods rely on diffusion

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x: +33 1 69 08 79 80.

signal decomposition using orthogonal bases and special functions.The two latter methods also propose an analytic, linear and com-pact reconstruction of diffusion propagator.

Apart from the direct q-space imaging, the mathematical mod-els that explain the diffusion signal (Stanisz et al., 1997; Assaf andBasser, 2005; Vestergaard-Poulsen et al., 2007) have also beenintroduced. While the former poses some constraints on diffusiongradient pulse duration and regularity of the imaged structure, sig-nal models can be derived with more relaxed assumptions, and inparticular can accommodate finite pulse widths. For certain simplegeometries (e.g., parallel planes, spheres or cylinders), the solutionto diffusion equation is analytic (Neuman, 1974; Murday and Cotts,1968). This can be readily exploited in modeling of signals in tis-sues approximated by combinations of finite numbers of simplegeometric elements. In addition, mathematical models can incor-porate a variety of parameters related to diverse microstructurefeatures, such as: cell density and radii, cell-membrane permeabil-ity, intrinsic diffusivities, and transverse relaxation constants (As-saf and Basser, 2005; Stanisz et al., 1997; Neuman, 1974; Stanisz,2003). However, to our knowledge, none of these have been thusfar exploited for fiber tracking. While the full three-dimensionalpropagator is of great interest to studies of microstructuralarchitecture, its angular part, orientation distribution function(ODF), obtained through angular q-space sampling methods on

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Fig. 1. Symmetrized scaling functions (a, b, c) and associated wavelets (d, e, f), asdefined in Eqs. (1) and (4). The maximum SH order is lc ¼ 16. (a) j = 0, lc ¼ 8; (b)j = 1, lc ¼ 4; (c) j = 2, lc ¼ 2; (d) j = 0, lc ¼ 16; (e) j = 1, lc ¼ 8; (f) j = 2, lc ¼ 4.

I. Kezele et al. / Medical Image Analysis 14 (2010) 332–342 333

single q-shells, like, high-angular resolution diffusion imaging(HARDI), is what fiber tracking methods require.

It is now accepted that between one third and two thirds ofwhite matter (WM) voxels in common HARDI datasets contain evi-dence for multiple fiber configurations (Pierpaoli and Basser, 1996;Behrens et al., 2003; Descoteaux, 2008). It is thus important toincorporate more informative models of local diffusion into trac-tography algorithms, to capture this complex tissue architecture.Several HARDI reconstruction models can resolve non-trivial fiberconfigurations such as crossings, splits, fannings. However, thepower of these methods greatly depends on the ODF, or fiber orien-tation distribution (FOD) reconstruction algorithm. The choice ofHARDI reconstruction technique is therefore of crucial importance.To render the tractography algorithms truthful and reliable, thereconstruction method is expected to provide sharp and accuratedescriptions of the ODF/FOD. It is particularly important to getsharp ODF/FODs when using probabilistic tractography techniquesthat sample the directions around peaks. Sharper ODF/FODs yieldbetter direction samplings (i.e. the samples are more concentratedaround the peaks).

Since the directions of maximal diffusion (that are implied byaverage principal fiber orientations inside a voxel) are unknowna priori, HARDI explores the space in a uniform manner, and thusdespite its obvious gain over DTI, it still suffers from a certainamount of information redundancy. A number of approaches for‘‘simplification” and regularization of Apparent Diffusion Coeffi-cient profiles (ADCp) resulting from HARDI by projection onto pre-defined bases have been demonstrated (Frank, 2002; Alexanderet al., 2002; Ozarslan and Mareci, 2003; Descoteaux et al., 2006).The preliminary work was oriented towards spherical harmonics(SHs) (Frank, 2002; Alexander et al., 2002), that offered a suitablespherical basis for high ADCp compression as well as some fastalgorithms for ODF reconstruction (Hess et al., 2006; Descoteauxet al., 2007). In addition, some preliminary work on ODF sharpen-ing in SH basis was also presented (Descoteaux et al., 2005). How-ever, the SH decomposition is not localized and is therefore notfully adapted to the problem of extracting ODF maxima corre-sponding to white matter fiber orientations. Nonetheless, SHs havebeen successfully used in conjunction with some more local bases,e.g., in spherical deconvolution (SD) methods for FOD estimation(Tournier et al., 2004, 2007), that probe the signal locally, thanksto the finite (i.e., local) spatial support of the convolution kernel,which allows for sparse representations. A number of other sig-nal-adapted bases were described in the literature (Jian and Vemu-ri, 2007; Kaden et al., 2007; Dell’Acqua et al., 2008).

The problem with SD is the low-pass nature of the kernel thatrenders the method extremely susceptible to noise. Incorporatingadditional constraints, such as positivity and regularity (Tournieret al., 2007; Jian and Vemuri, 2007), is thus imperative for reliableFOD reconstruction. A good side of these deconvolution techniquesis that they can yield accurate FOD estimates even at relatively lowb-values, as well as they do not necessitate multiple b-value acqui-sition schemes. This is also true for other recent HARDI reconstruc-tion methods such as PAS-MRI (Jansons and Alexander, 2003) andDOT (Ozarslan et al., 2006).

We opt for a wavelet-based, localized multiscale analysis. Con-trary to other techniques, this technique superimposes scale-re-lated components of the data and improves considerably filteringof noise-peaks (inherent to, for example, SD methods), by remov-ing the noise in all scale-related components per se (Starck andMurtagh, 2006). We use the spherical wavelet transform (SWT)of (Starck et al., 2006) directly on the diffusion ODF (Kezele et al.,2008), and we incorporate the approximate relation between theHARDI signal and the ODF into the wavelet analysis, via the FRT(Tuch, 2004). In our approach, we explicitly search for the relevantand localized diffusion information on multiple scales. Our

algorithm is an extension of the well known ‘‘à trous” algorithmfrom 2D images to functions defined on the sphere (Starck et al.,2006). It is an undecimated (hence, redundant) wavelet transform,with cubic B-spline scaling function, defined on the sphere. Thetransformation redundancy actually helps us to avoid Gibbs alias-ing inherent to orthogonal or bi-orthogonal basis (Starck and Mur-tagh, 2006). Also, although the explicit data reconstruction maynot be mandatory for a number of applications (e.g., the ODF esti-mation, as it will be shown in the text that follows), the positiveside of the algorithm is that it is fully invertible (unlike some otherspherical wavelet algorithms (Starck et al., 2006)). Further, it al-lows inclusion of different data conditioned constraints for thereconstruction.

In parallel, and independently of our work (Kezele et al., 2008),in Michailovich and Rathi (2008), the authors build a ridgelet frameto decompose HARDI signal and upon this decomposition theytransform the basis functions with the Funk–Radon transform(FRT) to calculate the diffusion ODF.

We apply our method to both physical diffusion phantoms (Po-upon et al., 2008) with 90� and 45� fiber-crossings and real in vivohuman brain data. Throughout the paper we also make a specialeffort to compare our SWT results with state-of-the-art analyticalq-ball imaging (Descoteaux et al., 2007) and spherical deconvolu-tion (Tournier et al., 2007) reconstructions, and to demonstratethe added value of SWT-based reconstruction that, with its sensi-tivity even to low angular separation, and its inherent sparsenessmay prove quite beneficial for applications to fiber tracking, fiberbundle segmentation, or morphometry methods based on HARDIderived features.

The paper is organized as follows. In Section 2, we review theSWT of Starck et al. (2006) and develop the extension for HARDIand ODF sharpening. We also present the real datasets used tocompare and validate our wavelet ODF decomposition. Then, inSection 3, we compare the ODF profiles reconstructed from ourSWT, analytical q-ball and spherical deconvolution on the physicalphantoms and in vivo data. We discuss our results and point tolimitations and perspectives of this work in Section 4.

2. Methods

2.1. Spherical wavelet transform – ‘‘à trous” algorithm

For our application, we employed an isotropic transform, withcubic B-spline scaling function. This function is shown to be veryclose to Gaussian, converges to zero rapidly, and in addition, it ful-fills the dilation equation (Starck et al., 2006). The algorithm is de-rived directly from the Fast Fourier Transform (FFT)–based wavelettransform (Starck and Murtagh, 2006), and being defined on thesphere, it relies on the SH transform. The scaling function (seeFig. 1, panels a–c), Ulc ðh;/Þ, where lc is the cutoff frequency, andwhere ðh;/Þ follow the physics convention (i.e., h is longitudeand / is azimuth) exhibits azimuthal symmetry, and thus, its SHtransform does not depend on the phase m:

334 I. Kezele et al. / Medical Image Analysis 14 (2010) 332–342

Ulc ðh;/Þ ¼ Ulc ðhÞ ¼Xlc

l¼0

Ulc ðl;0ÞYl;0ðh;/Þ; ð1Þ

where Yl;m is the SH of order l and phase m, and bUlc is the SH trans-form of Ulc .

The SH representation greatly simplifies the convolution withthe spherical function hðh;/Þ to be analyzed, which reduces to:

c0ðl;mÞ ¼ dðUlc � hÞðl;mÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi

4p2lþ 1

rUlc ðl; 0Þhðl;mÞ; ð2Þ

where operator � stands for the convolution, c0 are the SH coeffi-cients of the resulting convolution, and hðl;mÞ are the SH coeffi-cients of spherical function h.

The multiresolution decomposition of h is performed on a dya-dic scale, by convolving h with the rescaled versions of the scalingfunction Ulc (dyadically dividing the cutoff frequency lc by powersof 2): cj ¼ U2�j lc

� h, where j (j ¼ 0; . . . ; J ) is the scale, and U2�j lcis a

rescaled Ulc with 2j times lower cutoff frequency. The decomposi-tion can be done recursively as cjþ1 ¼ cj � fj (settingc0ðh;/Þ ¼ hðh;/Þ), where fj represents a low-pass filter associatedto each scale j. This fj is defined in terms of the scaling functionUlc and its SH transform Fj is defined as follows:

F jðl;mÞ ¼bU

2�ðjþ1Þ lcðl;mÞbU

2�j lcðl;mÞ

if l < 2�ðjþ1Þlc and m ¼ 0

0 else

8><>: ð3Þ

Following the approach of the ‘‘à trous” algorithm, the waveletcoefficients are then defined as the difference between two consec-utive low-pass filtered versions of h:(cf. Fig. 1, panels d–f):

wjþ1 ¼ cj � cjþ1: ð4Þ

This relation implies that the high-pass filter g to obtain thewavelet coefficients directly, is given in SH basis by:Gjðl;mÞ ¼ 1� Fjðl;mÞ, where fj is defined in SH, as in Eq. (3) above,at each scale j. The scaling function itself is defined in SH space:

Ulc ¼32

B32llc

� �;

where B3ðxÞ ¼1

12ðjx� 2j3 � 4jx� 1j3 þ 6jxj3 � 4jxþ 1j3

þ jxþ 2j3Þ:

ð5Þ

In direct space, the scaling function U is proportional to thesinc4 px

2xc

� �(where xc ¼ 1=lc) (Starck and Murtagh, 2006), and is as

such closely approximated by a Gaussian, and converges to zerorapidly.

Finally, Eq. (4) above implies a straightforward reconstructionscheme of the original signal c0:

c0ðh;/Þ ¼ cJðh;/Þ þXJ

j¼1

wjðh;/Þ: ð6Þ

2.1.1. Wavelet coefficients filteringTo meet the request on data sparsity, which directly leads to its

compression, we introduce an additional step that follows after thesignal decomposition of Eq. (6). This step concerns wavelet coeffi-cients shrinkage (or filtering), which is shown to be extremelyeffective in data denoising and contrast sharpening (Mallat,1999). Wavelet filtering is, in general, a non-linear transformationof wavelet coefficients, at each analyzed scale. It is well known thatwith the aim of sparse signal representation, the filtering should bedone by minimizing the L0 norm of these coefficients. It is alsoknown that the L0 minimization leads to a NP-hard problem, andthat the L1 norm minimization results in the sparsest solution,closest to the one obtained by L0 minimization. It is worthwhile

noting that if the wavelet basis were orthogonal, filtering basedon hard thresholding would provide us with the exact solution tothe L0 minimization (Starck and Murtagh, 2006).

Since the noise of diffusion MRI in each direction follows Riciandistribution, it is difficult to estimate the exact distribution of noiseon wavelet coefficients that is, at each scale, given as a convolutionof noise over different directions with the wavelet band-pass filter.Hence, to define a ð1� aÞ% (e.g. 95%) threshold for wavelet coeffi-cient shrinkage (where a is the significancy level), one possibilitywould be to perform permutations of signal coefficients along dif-ferent directions, calculate the wavelet coefficients of this synthe-sized signals, order them increasingly by their magnitude, andlocate the value at ð1� aÞ%. The average of this value from all per-mutations defines the ð1� aÞ% threshold. At each wavelet scaleonly those coefficients whose magnitude is larger than the thresh-old are preserved. We tested this approach for 1-voxel experimentson phantom data. However, permutation tests are highly costly incomputation time (e.g., no less than 105 permutations are to begenerated at each voxel), and thus are not practically applicableto larger data sets. For these reasons, we define the threshold withrespect to the percentage error in wavelet synthesis. In the waveletsynthesis relation given in Eq. (4), we keep the subset of waveletcoefficients with highest magnitude so that the overall synthesizedODF percentage error with respect to ODF synthesized using all thewavelet coefficients does not exceed 1.5% for phantoms, and 3% (onthe average) for real data. This thresholding scheme gives besttrade-off between compression and denoising. No more than 10wavelet coefficients per scale are needed. This way, the signal isclosely preserved while denoised and compressed. The number ofcoefficients for its final representation is highly reduced (e.g., fora maximum SH order of 8 or 16, instead of 45 or 153 SH coeffi-cients, respectively, we have only 20 wavelet coefficients (10 perscale) to faithfully represent the spherical signal).

2.2. Spherical wavelet transform of the ODF

Our goal is to come up with a sharp and sparse representation ofthe diffusion ODF, with the capacity for resolving multiple fiberbundle configurations, even with low separation angles, and evenfor low SNR. Such a depiction of diffusion process is expected toboost the power of fiber tracking applications that rely stronglyon the quality of local models, and their ability to delineate distinctdiffusion compartments along close directions. In addition, wesearch for sparse signal representations, to: diminish the uncer-tainty in recovered fiber track directions, and to obtain compactrepresentations for lighter computational loads. Implicitly, com-pact signal models result also in a small number of interesting fea-tures that may prove useful for applications like fiber bundlesegmentation or for morphometry studies based on HARDI deriveddescriptors.

If we assume that the ODF is given as a finite sum of probabilitydistribution functions on the sphere, where each of these functionsdescribes the angular probability of finding a WM fiber bundlealong a predefined set of directions (centered at a prescribed direc-tion), then a natural decomposition of the ODF would be onto a ba-sis of such functions. Typically, for voxels with only a single fiberbundle oriented along a certain direction d, its contribution tothe ODF in the observed voxel can be obtained analytically (Desco-teaux et al., 2009a; Descoteaux, 2008) based on the Gaussian diffu-sion assumption. However, due to complex WM architecture,including different types of fiber-crossings and similar non-trivialfiber configurations, as well as due to limited capacity of imagingtools to resolve such configurations, the bandwidth of the elemen-tary ODF-building function cannot be assumed uniform across theimaged WM. For that reason, we favor multiscale approach for ODFanalysis.

1 Note that we do not use the recursive scheme presented in Section 2.1 in ordernot to cancel out the 2pPln ð0Þ term from UB .


We thus apply the spherical wavelet transform (SWT) to the dif-fusion ODF. However, we would like to work directly on HARDI sig-nal. This is possible by incorporating the aforementionedapproximate relation between the signal and its ODF via theFunk–Radon transform. Starting with the spherical scaling functionU, and the associated wavelet function W, we derive in this section,step-by-step, and through FRT, the multiscale spherical filters UB

and WB, that enable us to perform the SWT of ODF by convolvingthe HARDI signal with the derived filters directly. We thus avoidany explicit ODF calculations.

Since both the HARDI signal and its ODF are symmetric and realfunctions on the sphere, they can be represented by a modified SHbasis, Yn, that is symmetric and real, taking into account only evenorder l, as defined in Descoteaux et al. (2006, 2007). If we unfoldthe recursive multiscale scheme described in Section 2.2, thenthe scaling functions Uj convolve the ODF, successively, at eachscale j, j ¼ 0; . . . ; J, to produce the low-pass components of theODF. Referring to Eq. (4), the associated wavelet functions, Wj,are now simply given by:

Wj ¼ Uj�1 �Uj; ð7Þ

and are also convolved with the ODF at each scale j, to provide theODF band-pass details.

The signal symmetry also allows to symmetrize the cubic B-spline scaling function U, and its associated wavelet function W.As the two functions are real, they can be likewise representedby the same symmetric and real SH basis, Yn.

Letting ~u represent a unit direction on the sphere and N themaximal order of modified SH basis corresponding to the chosencutoff frequency lc , we first express the convolution on the spherebetween the ODF, W, and the scaling function U as

ðW �UÞð~uÞ ¼XN

n¼1

bWnbUnYnð~uÞ: ð8Þ

Then, FRT of the signal, expressed as great circle integrals overperpendicular directions to ~u on the unit sphere X, is solved ana-lytically (Descoteaux et al., 2007) yielding:

Wð~uÞ ¼Z~v2X

dð~u;~vÞhð~vÞd~v|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}FRT

! bWn ¼ 2pPln ð0ÞbHn; ð9Þ

where d is spherical Dirac delta function, h the HARDI signal definedon the sphere, Pln the Legendre polynomial of order ln correspondingto order n of the modified SH basis Yn, and, bWn; bUn, and bHn stand forthe nth component of the spherical harmonic transform of W; U,and h respectively. Eqs. (8) and (9) imply the following identity:


n¼1

2pPln ð0ÞbHnbUnYnð~uÞ: ð10Þ

By grouping the SH coefficients of scaling function U withLegendre polynomials, Pln ð0Þ (scaled by 2p), of the correspondingorder:


n¼1

2pPlnð0ÞbUn|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}bUBn

bHnYnð~uÞ; ð11Þ

we obtain the following identity:


n¼1

UBnbHnYnð~uÞ: ð12Þ

bUBn represents the SH coefficients of a derived filter UB, thatrepresents (refer to the relation given by Eq. (9)), the Funk–Radontransform of the scaling function U. This relation implies that wecan calculate the convolution of the ODF W and the scaling function

U by calculating the convolution of the HARDI signal h and thederived filter UB.1

The derived low-pass function UB, that we employ to transferthe SWT analysis from diffusion ODF to HARDI signal directly, isillustrated in Fig. 2 (panels a–c). Shown are three different scalesthat correspond to modified SH cutoff frequencies of 8, 4, and 2respectively. It is interesting to note that the Gaussian profiles ofUB scaling functions match naturally HARDI response functionsfor single fiber bundle orientation inside the voxel (Tournieret al., 2004, 2007; Michailovich and Rathi, 2008). Different fibercharacteristics give rise to different frequency content of theseresponse functions. Fig. 2 shows the UB that correspond to HARDIresponse functions for diffusion within the bundles oriented alongthe z-axis. Similarly, simple rotations of UB kernels accommodatesingle orientation fiber bundle response functions along otherdirections.

In parallel to our work (Kezele et al., 2008), in Michailovich andRathi (2008), the authors derive a ridgelet framework, startingwith the spherical wavelets, derived from Gauss–Weirestrass ker-nels of a specified bandwidth following the work of Candes andDonoho (1999). The authors demonstrate that the ridgelet generat-ing function is the Funk–Radon transform of the spherical waveletscaling function. Hence, our derived filter UB is very similar to theridgelet generating function of Michailovich and Rathi (2008).

Similarly to the UB definition, we define the wavelet-derivedfilters WB as FRT of the spherical wavelet functions from Eq. (7):

dWB n ¼ 2pPln ð0ÞcW n ð13Þ

Then, from the linearity of Funk–Radon transform, and Eq. (7) itdirectly follows:

WBjþ1 ¼ UBj �UBjþ1; ð14Þ

which together with Eq. (12) results in:

ðW �WÞð~uÞ ¼XN

n¼1

WBnbHnYnð~uÞ: ð15Þ

The wavelet-derived function WB is illustrated in Fig. 2, panelsd–f. Shown are three different scales that correspond to SH cutofffrequencies of 16, 8, and 4, respectively. These wavelet-derivedfunctions are related to scaling function-derived filters UB (Fig. 2,panels a–c) through Eq. (14).

Similar to the relation in Eq. (6), we have, combining Eqs. (7)–(15):

Wð~uÞ ¼ ðW �UJÞð~uÞ þXJ

j¼1

ðW �WjÞð~uÞ

¼ ðh �UBJÞð~uÞ þXJ

j¼1

ðh �WBjÞð~uÞ ¼ Wð~uÞL þWð~uÞH; ð16Þ

where WL and WH denote the low and high frequency componentsof W. The final step in this ODF decomposition is wavelet coefficientfiltering, as described in Section 2.1.1. This way, we obtain a sparseand denoised representation of the ODF. Using the convolution the-orem it is straightforward to show that the spherical wavelet coef-ficients of the ODF can be obtained by convolving the HARDI signalwith the same wavelet basis and integrating the result along thegreat circles perpendicular to the given ODF directions.

Despite a high level of ODF wavelet compression, we see thatthe low frequency part of the ODF is still present in the ODFreconstruction of Eq. (16). In practice, we typically limit this low

Fig. 2. Funk–Radon transform of the scaling functions (a, b, c) and associated wavelets (d, e, f) (see Fig. 1), as defined in Eqs. (11), (13) and (14). These functions are used totransfer the ODF wavelet analysis directly onto the HARDI signal (refer to Section 2.2). The maximum SH order is lc ¼ 16. In (a) j = 0, lc ¼ 8; (b) j = 1, lc ¼ 4; (c) j = 2, lc ¼ 2; (d)j = 0, lc ¼ 16; (e) j = 1, lc ¼ 8; (f) j = 2, lc ¼ 4.

Fig. 3. ODF and FOD reconstruction for the 90� and 45� crossing phantoms in the centermost voxel. Shown are in rows 1 and 2: the full analytical ODF (sh-q-ball), maximumSH order lc ¼ 16; (a, d) wavelet coefficients of the first resolution scale (highest frequencies of WH from Eq. (16), j ¼ 2, maximum SH order lc ¼ 16 for the 90�-, and lc ¼ 14 forthe 45�-crossing phantom) after thresholding; (b, e) wavelet coefficients of the second resolution scale (lower frequencies of WH from Eq. (16), j ¼ 1, maximum SH order lc ¼ 8for the 90�-, and lc ¼ 7 for the 45�-crossing phantom) after thresholding; (c, f) the sharp ODF (WH from Eq. (16)) reconstructed with wavelet coefficients at both scales afterthresholding; the last row illustrates the spherical deconvolution (SD) profiles, from filtered SD (fSD) and constrained regularized SD (CSD) with SH order lc ¼ 16 for the 90�-,and lc ¼ 14 for the 45�-crossing phantom.


frequency component to lc ¼ 2 (the SH order corresponding to the2nd-rank tensor) that, effectively, carries no relevant informationon fiber bundle angular distribution. Recall that our main goal isto extract a small number of interesting ODF components thatwould facilitate WM fiber tracking and similar applications. Thus,instead of representing our sparse angular distribution functionby both low and high frequency component, we rather focus onthe ‘‘high” frequency part only, effectively neglecting the lowest

frequency ODF contribution. In other words, we focus on the sharpODF, WH , as opposed to the ODF W in its entirety. In this respect,our sharp ODF is expected to resemble a FOD that is in essence asort of a high-pass filtering adapted to data. This link betweenthe two methods can be better appreciated if we redefine thehigh-frequency part of the ODF ðWð~uÞHÞ from Eq. (16) above,changing the band-pass filters WBj with high-pass filters WHj asthe following:


Wð~uÞH ¼XJ

j¼1

ðh �WHjÞð~uÞ ¼XJ

j¼1

ðh � R�1j Þð~uÞ ð17Þ

where WHj ¼ R�1j stands for the high-pass inverse of the low-pass

filter Rj at scale j, with Rj defined similar to the convolution kernelof Tournier (Tournier et al., 2004, 2007) (scale j determines the fil-ters’ cutoff frequencies). For this resemblance between the meth-ods, we compare our results with both the analytical SH q-ballmethod (Descoteaux et al., 2007), and the FOD reconstructions fromspherical deconvolution (SD) reconstructions (Tournier et al., 2004,2007).

2.3. Data acquisition

We will first demonstrate our SWT method on two real physicalphantoms emulating two fiber bundles crossing at 90� and 45�,respectively (Poupon et al., 2008). The phantom data were ac-quired on a 1.5T Signa MR system (GE Healthcare, Milwaukee),TE/TR = 130 ms/4.5 s, 12.0 s (45� and 90� phantom, respectively),BW = 200 KHz. To enhance the SNR (keeping SNRmin > 4), large

Fig. 4. Estimates of the peak angular separation for SWT-based ODF sharpening methocrossing phantoms. SH q-balls (max SH order = 16, k ¼ 0:006) are shown in the second rowin the fourth row. Maximum SH order was lc ¼ 16 for the 90�-, and lc ¼ 14 for the 45�-detected.

voxel dimensions were used (FOV = 32 cm, matrix size of32 � 32). We analyze the data acquired at a b-value ofb ¼ 2000 s mm�2, along 4000 uniformly distributed orientations(see Fig. 3). For comparisons with real brain data sets, the direc-tions were subsampled to 200, uniformly distributed (using a geo-desic interpolation algorithm (Tuch, 2004)).

The SWT is also applied to a real in vivo data sets of a healthyvolunteer. The data set was acquired on a 1.5T Signa MR system(GE Healthcare, Milwaukee), TE/TR = 100.2 ms/19 s, BW = 200 KHz,FOV = 24 cm on a 128 � 128 matrix, TH = 2 mm, 60 axial slices,b ¼ 0=3000 s mm�2 (SNR � 2), 200 uniformly distributed orienta-tions. This dataset is a part of the publicly available HARDI data-base (Poupon et al., 2006).

3. Results

3.1. Spherical wavelet transform of the ODF on the physical phantoms

We applied our method for spherical wavelet decomposition ofODF to the 90�- and 45�-crossing real physical phantoms. To enable

d. Shown are the results from the 3 � 3 voxels ROI (first row) of both 90� and 45�. Sharp ODFs by SWT method with peak angular separation at each voxel are shown

crossing phantom; j = 2 wavelet scales. ”X” represents the voxels with no crossing

Fig. 5. Estimates of the peak angular separation for SD-based methods. Shown arethe results from the 3 � 3 voxels ROI (first row) of both 90� and 45� crossingphantoms (Fig. 4). The first row illustrates the spherical deconvolution (SD) profilesfrom filtered SD (fSD) and the second row the SD profiles from constrainedregularized SD (CSD) with SH. The maximum SH order for both methods was lc ¼ 16for the 90�-, and lc ¼ 14 for the 45�-crossing phantom.


the subsequent comparison of results with the results of similaranalysis performed on the real data of two healthy volunteers,we subsampled the directions to the same set of 200 directionsas our real dataset (Section 2.3). Also, to adapt our analysis fre-quency bands to the frequency content of the data, the original cu-bic B-spline kernel was twice dilated in the frequency space. Thecutoff frequency was set to lc ¼ 16 for the 90�-crossing phantom.Due to noisy decomposition onto the SH basis (seen in both SWTand SD-based methods), the cutoff frequency was set to lc ¼ 14for the 45�-crossing phantom. The datasets were decomposed onto2 resolution scales; scale 1 corresponding to maximum SH ordersof lc ¼ 16, and lc ¼ 14, for the 90�- , and 45�-crossing phantom,respectively; and similarly, scale 2 corresponding to maximumSH order lc ¼ 8=7, respectively. We analyzed first the central10 � 10 � 14 mm voxel in both phantoms.

For the 90�-crossing phantom, since the configuration is quitesimple, only the four most significant wavelet coefficients at eachscale were kept. For the 45�-crossing phantom, since the separa-tion angle is quite low, the 10 most significant wavelet coefficientsat each scale were kept. For both scales, this thresholding resultedin less than 1.5% error when the overall sum of the lowest fre-quency ODF component and two thresholded wavelet scales wascompared to the estimated full SH analytical q-ball. To regularizeSH coefficients estimates for q-ball analysis, we employed La-place–Beltrami smoothing of k ¼ 0:006 as explained in (Desco-teaux et al., 2007). Fig. 3 depicts the results for the central voxelof 90� and 45� crossing phantoms. For comparison purposes, wealso reconstructed analytical q-ball as well as two types of FODs(following the algorithms of Tournier et al. (2004, 2007)) with fil-tered spherical deconvolution (FSD) and with regularized con-strained spherical deconvolution (CSD), at different order l(depicted in Fig. 3). The low-pass filter for FSD was [1 1 1 0.5 0.10.02 0.002 0.0005 0.0001] applied to SH order l = [0 2 4 6 8 12 1416]. For CSD, we used a regularization parameter k ¼ 1, and thethreshold for positivity constraint s ¼ 0:1. The maximum SH orderwas set lc ¼ 16 for the 90�, and lc ¼ 14 for the 45� phantom. Thedeconvolution kernel for fSD and CSD was estimated from a man-ually defined ROI in the single fiber part of the ex vivo phantom.

With the purpose of quantitative evaluation, we compared thepeak angular separation for SWT-based sharpening method tothe peak angular separation of the two SD techniques, for the3 � 3 voxels central ROI of the 90�-crossing and 45�-crossing phan-toms. For the 90�-crossing phantom, five most significant coeffi-cients per scale are kept; for the 45�-crossing phantom, thisnumber was set at 10. All other parameters for SWT, and the twoSD methods were as above. The calculated angles are shown in tab-ular form in Figs. 4 and 5, overlayed on the grid of the ROIs’ sharp-ODFs calculated by SWT at each voxel for both phantoms (Fig. 4),and on the grid of ROIs’ FODs for the two phantoms, in the caseof the two SD methods (Fig. 5).

Denoising and sharpening obtained very good results, for bothphantoms, while reducing the number of necessary coefficientsfor ODF presentation from 153 to: 8 for 90�-crossing phantom,and to 20 for 45�-crossing phantom. Directions of maximal diffu-sion are distinguishable for all but the two edge ROI voxels forthe 45�-crossing phantom, with the error in the recovered separa-tion angle within the sampling angular resolution limit for both90� and 45�-crossing phantom. The inability to discern the twolobes at those two edge voxels is likely the consequence of the jointeffects of unequal contribution of the two diffusion compartmentsand Bessel blurring (Tuch, 2004), due to Funk–Radon basedapproximation of the radial projection of the full diffusion proba-bility density function. The approaches to minimize the letter willbe the part of the future work (see Section 4).

Overall we reach a valuable result, demonstrating the SWTmethod’s capacity to resolve the fiber bundle crossings at angles

as low as 45�, at a single q-shell, that standard q-ball transformcannot achieve. For 90�-crossing phantom, q-ball transform man-ages to extract the principle diffusion/fiber directions, however,the result is more blurred compared to our sharp ODF. On 45�-crossing phantom, the directions of maximal diffusion are no long-er discernable.

The CSD results in sharper FODs than the ODFs of our SWTmethod. The peak angular separation is also closer to the groundtruth for the former (Figs. 4 and 5). The CSD deconvolution kernelwas fully adapted to the phantom data. Also, the CSD is a non-lin-ear regularized method. Although, the SWT produces less sharpODFs, the good side is that is linear. Further, no regularizationneeds to be imposed. Although the two methods, SWT and SDare differently founded, the former being based on signal projec-tions and convolution, and the latter on reverse process of signaldeconvolution, there are some similarities, especially betweenthe SWT and CSD. The high-pass filtering inherent to SD with theimposed constraint on the number of non-zero directions is verysimilar in logic to SWT-based sharpening with wavelet filteringto account for signal sparsity. This is the reason why both CSDand SWT outperform the fSD that does not take into account anyforms of sparsity constraints. The FODs appear blurred, withnumerous negative values of important magnitude, due to whichthe principal fiber directions are visibly less prominent. Obviously,fSD lacks an additional condition to cope with the noise at higher


SH orders (that are needed to discriminate signals along directionswith ‘‘low” separation angles).

3.2. Spherical wavelet transform of the ODF on real in vivo data

Similar analysis was performed on a ROI of a real data set, onlyfor these experiments, the cutoff frequency was set to lc ¼ 8 toaccommodate the SH coefficients’ estimate to the number of avail-

Fig. 6. q-Ball ODF and sharp ODF reconstructions on real data. From left to right, andanatomical MRI image; full ODF calculated analytically (Descoteaux et al., 2007) in ththresholded WH (Eq. (16)) (the maximum number of the significant coefficients was 10 atof WH from Eq. (16), j ¼ 2, maximum lc ¼ 8) after thresholding; (c) wavelet coefficients(16), j ¼ 1, maximum lc ¼ 4); (d) the sharp ODF (WH from Eq. (16)) reconstructed with wahow different frequency scales capture complementary directional information (see tex

able measuring directions, and to diminish the high-frequencynoise as much as possible (given the rather poor SNR in real datasets). We analyzed the data at two scales, and kept 10 most signif-icant wavelet coefficients at each scale (with the average recon-struction error of 2.6%). To regularize SH coefficients estimateswe employed Beltrami–Laplace smoothing as explained in Desco-teaux et al. (2007). For real data, the negative values are hard thres-holded to zero. For the first data set, along 200 directions, we also

from top to bottom: analyzed ROI; zoom at ROI on the fused diffusion RGB ande SH space with l ¼ 8; (a) full ODF W reconstructed as a sum of the WL and theboth scales); (b) wavelet coefficients of the first resolution scale (highest frequenciesof the second resolution scale after thresholding (lower frequencies of WH from Eq.velet coefficients at both scales after thresholding; (e) example WM voxels showingt for details).

Fig. 7. FOD reconstructions on real data (the same dataset, ROI and frequency scales as in Fig. 6). The first row shows filtered and constrained spherical deconvolution (fSD,CSD) for maximum SH order lc ¼ 4, and the second row shows fSD and CSD for maximum SH order lc ¼ 8.


show some typical WM voxels for which the multiscale analysisproves to be particularly important.

The results of the analysis on the real data set are shown in Figs.6 and 7. The overall sum of the thresholded wavelet coefficients atboth scales and the lowest signal component (Fig. 6 a) differs fromthe estimated SH analytical q-ball by 2.6% on the average. Denois-ing is already visible, even with no wavelet shrinkage. Our sharpODF (Fig. 6 d) extracts highly relevant information on fiber com-partments. With only 10 coefficients per scale we manage to cap-ture the most significant diffusion/fiber directions at each voxel.Panel e of Fig. 6 suggests the importance of the multiscale ap-proach: different scales may assess complementary informationon the underlying diffusion milieu, i.e., distinct diffusion compart-ments may give rise to different frequency content of the signal(Fig. 6 e). (Note that the shown ODFs are min–max scaled.) The firstrow of panel f, corresponds to the ROI voxel lying exactly on theborder between WM and cortical mantle (position (7, 1) (row, col-umn) in panels a–d). The two principal diffusion directions appearto be related to fibers of corona radiata: passing tangentially to thecortical mantle, and protruding into the cortical mantle. The sec-ond and third row are associated with the two more central WMROI positions: (7,8) (likely the cortico-spinal tract and corona rad-iata crossing), and (7,9) (the crossing of calosal fibers, and radiationfibers from corpus calosum) (in panels a–e), respectively. Sphericaldeconvolution results on the same two resolution scales are shownin Fig. 7. The results of SWT and CSD look quite similar. (For faircomparisons of the results of SWT-based reconstruction method,and the SD method, we keep the maximum SH order of the CSDtechnique at the same value as for SWT.) The ground true beingcurrently unavailable, it is not possible to compare the accuracyof the two methods, however, the visual inspection suggests thatthe SWT outperforms both SD methods in real data. Future workwill include more extensive analysis on phantom data emulatinglarger number of crossing bundles and with different crossing an-gles. As for fSD, very low SNR in this real data set appears to bequite challenging, and the technique fails to capture the local tissuearchitecture.

4. Discussion

In this study, we employed a linear spherical wavelet transformfor mutliscale decomposition of the diffusion ODF. We tested andcompared our method with respect to state-of-the-art q-ball andspherical deconvolution reconstructions on real physical phantomsand real brain data. The experiments allowed us to demonstratethe algorithm efficiency to represent the relevant features withonly a very small number of wavelet coefficients. The sphericalwavelet transform thus provides a high level of data compressionand denoising. For example, a reconstruction of the q-ball or spher-ical deconvolution in SH basis of order 16 required 153 coefficientswhereas our reconstruction used at most 20 spherical waveletcoefficients (10 per scale).

More importantly, our spherical wavelet transform of the ODFhas a desired sharpening effect when focusing on the high-fre-quency part of the decomposition. This proves to be efficient forextracting pertinent angular information, concealed by signalspread out over frequency scales that cannot be seen with classicalODF alone. The results at some voxels of the real data set suggestthat different angular frequency scales may encode complemen-tary diffusion information (i.e., distinct diffusion compartments),and consequently, complementary information on tissue architec-ture. This may prove valuable for WM fiber tracking, particularly inplaces where WM fibers cross at relatively small angles. This couldalso potentially lead to new multiscale scalar/vector measures sen-sitive to different micro-architectual properties of biological tis-sues than classical DTI and HARDI-based scalar measures (suchas FA, GFA, and others).

As aforementioned, CSD acts like a high-pass filter, with anadditional constrain on the maximum number of non-negativedirections, and in that sense approaches the philosophy of waveletanalysis with signal sparsity constraint. However, CSD is computedwith a constrained regularization at the cost of a non-linear opti-mization that is computationally more expensive than the fSD orour SWT. A possible advantage of SD techniques lies in dataadapted deconvolution kernel. Even so, and especially in real data


that are much less uniform than the manufactured phantoms, theestimated deconvolution kernel may not be equally optimal for allthe investigated regions. It is possible that it should be defined dif-ferently depending on the region of the brain. Furthermore, SDmethods assume a deconvolution kernel that is data-dependentand thus, ‘‘b-value dependent”, whereas our approach does not.Although the CSD with the deconvolution kernel fully adapted tothe data at hand outperforms the SWT, the experiments on realdata suggest that the SWT is no less capable of coping with noisethan the CSD. We find the results on real data rather encouraging.Thanks to the multiscale approach and denoising at different reso-lution levels, our ODF sharpening technique performs quite well,even at maximum SH order of only eight, and even with no explicitconstraints put on the signal reconstruction from its significantwavelet coefficients. Incorporating data adapted constraints willalso be part of our future work. Perhaps more care needs also tobe taken when defining wavelet scaling functions, and render themmore adaptive to the data, which is one of the issues that we planto address in the future work.

Some very recent work (Tristàn-Vega et al., 2009) also dealtwith the Bessel blurring, inherent to all reconstruction methodsthat use Funk–Radon transform, as is the presented SWT. Futurework will be oriented to algorithmic modifications to include theproposed techniques to correct for this blurring. Also, even thoughthere are practical advantages of SWT application to single q-shellDWI acquisitions, like the possibility to use it in typical clinical set-tings, future work will look at expanding the SWT to analyse thesignals on multiple shells in q-space (Canales-Rodríguez et al.,2009; Aganj et al., 2009).

It has been shown recently that an antipodally symmetricspherical function (ASSF) can be equivalently described in the SHbasis, in the symmetric high-order Cartesian tensor basis con-strained to the sphere, and in the homogeneous polynomial basisconstrained to the sphere (Ghosh et al., 2008; Bloy and Verma,2008). By computing the stationary points of constrained polyno-mial functions, the maxima and minima of ASSF can be deduced.The ASSF maxima correspond to fiber orientations in the underly-ing diffusion substrate. It is still not understood though whetherextracting the ODF maxima by these methods would outperformthe methods that search for the fiber orientations directly, likeour SWT-sharpening technique, or SD techniques described inTournier et al. (2004, 2007). Another point worthwhile checkingin these comparisons would be the level of technical requirementsfor each technique.

To conclude, the purpose of this report was to stress the poten-tial and importance of spherical, localized, and multiscale bases inHARDI studies. The spherical wavelet basis was shown to havepromising properties that might outperform the popular sphericalharmonics basis, as demonstrated by the experiments on phan-toms and real in vivo human brain data. Our results suggest thatthe high frequency decomposition of the diffusion ODF can revealinteresting angular structure, especially when multiple angularfrequency scales are analyzed. The next step would now be toincorporate this multiscale spherical wavelet analysis in tractogra-phy algorithms. The multiscale property of SWT may open newavenues of research of diffusion that is inherently multiscale in hu-man tissues in vivo.

Acknowledgments

The authors would like to thank Bertrand Thirion, Philippe Ciu-ciu, Daniel Alexander, Pierrick Abrial, and Jean-Luc Starck for help-ful discussions. This work was supported by a Marie Curie Actionsindividual postdoctoral fellowship, EU.

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