Registration, atlas generation, and statistical analysis of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions 5

Geodesic Regression of Orientation Distribution Functions with its Application to Aging Study After the HARDI dataset across a population has been warped into the common atlas,the non-li

Geodesic Regression of Orientation

Distribution Functions with its

Application to Aging Study

After the HARDI dataset across a population has been warped into the common atlas,the non-linear nature of HARDI data presents a challenge to HARDI-based statisticalanalysis (see Figure 5.1) Regression analysis is a fundamental statistical tool todetermine how a measured variable is related to one or more independent variables.The most widely used regression model is linear regression, because of its simplicity,ease of interpretation, and ability to model many phenomena However, if the responsevariable takes values in a nonlinear manifold, a linear model is not applicable Suchmanifold-valued measurements arise in many applications, including those that involvedirectional data, transformations, tensors, and shapes Several works have studied theregression problem on manifolds e.g., [69, 70] In this chapter, we adapt the framework

of geodesic regression, proposed in [70], to the HARDI data and apply it to the aging

study First we derive the algorithm for the geodesic regression on Riemannian manifold

of ODFs and conduct the simulation experiment to evaluate its performance Finally,

we examine the effects of aging via geodesic regression of ODFs in a large group ofhealthy men and women, spanning the adult age range

HARDI

data

ODF images

ODF images

in common space

ODF atlas

ODF Reconstruction Registration

Atlas Generation

Biomarkers/ Inference

Statistical Analysis

serve as common space in registration

Subjects

Data

Acquisition

Figure 5.1: The role of Chapter 5 in the ODF-based analysis framework

In statistics, the simple linear regression is an approach to modeling the relationship

between a scalar dependent variableY and a non-random scalar variable denoted as X.

5.1 Geodesic Regression on the ODF manifold

A linear regression model of this relationship can be given as

whereβ0 is an unknown intercept parameter,β1 is an unknown slope parameter, and

is an unknown random variable representing the error drawn from distributions withzero mean and finite variance Givenn observations, i.e., (x i , y i ), for i = 1, 2, · · · , n,

the least square estimates, ˆβ0and ˆβ1, for the intercept and slope can be computed byminimizing the square errors

framework of geodesic regression in [70, 81]

We now formulate the geodesic regression that models the relationship of aΨ-valued

random variable Ψ and a non-random variableX ∈ R Such a geodesic relationship

cannot be simply written as the summation of the interceptβ0 and the scalar variable

X weighted by β1, whereβ0andβ1are scalar measures as shown in Eq (5.1) Rather,

in this case (see Figure 5.2), it needs to be adopted to the manifold setting in whichthe exponential map in Eq (2.4) will be used to replace the addition in the Euclideanspace Hence, we introduce the parametersψ and ξ, where ψ is an element on Ψ and

ξ ∈ T ψ Ψ is a tangent vector at ψ on Ψ This pair of parameters (ψ, ξ) provides an

interceptψ and a slope ξ, analogous to the β0 andβ1parameters in the simple linearregression in Eq (5.1) The geodesic regression model can thus be written as

Ψ = exp

where is a random variable taking values in the tangent space at exp(ψ, Xξ) Notice

that in a Euclidean space, the exponential map is simply addition, i.e., exp(ψ, Xξ) = ψ+Xξ Thus, when the manifold is a Euclidean space, the geodesic model is equivalent

to the simple linear regression in Eq (5.1) Hence, the geodesic regression is thegeneralization of the simple linear regression to the manifold setting

Figure 5.2: Geodesic regression on manifoldΨ.

5.1 Geodesic Regression on the ODF manifold

5.1.1 Least-Squares Estimation and Algorithm

Given n observations, (x i , ψ i ) ∈ R × Ψ, i = 1, 2, · · · , n, we estimate (ψ, ξ) via

solving a least-squares problem by computing the least-squares estimates ˆψ and ˆξ,

tion problem cannot be analytically solved Therefore, we seek the estimates of (ψ, ξ)

using a gradient descent algorithm In order to do so, we have to derive the gradient of

E with respect to ψ as well as with respect to ξ We refer readers to §5.1.1.1 for the

detailed derivation of these two gradient terms

Briefly, the gradient ofE with respect to ψ (see Eq 5.7), denoted as ∇ ψ E, can be

i ψ i⊥



, (5.6)

i ψ i

⊥are given as

logψˆ

i ψ i



=logψˆ

i ψ i , ξ ξ i

i  ψˆi

ˆ

and (·)  and (·) ⊥denote components of (·) parallel and orthogonal to ξ irespectively

We employ the gradient descent algorithm to find the minimizer to the least-squaresproblem found in Eq (5.4) At the beginning of the optimization, we initializeξ = 0

and ψ to be the Karcher mean of the observed ODF, ψ i , i = 1, 2, · · · , n, where the

Karcher mean is computed using the Karcher mean algorithm given in [1] During eachiteration of the optimization, the estimates of (ψ, ξ) are respectively updated using

Eq (5.5) and Eq (5.6) The above computation is repeated until the change ofE is

sufficiently small

5.1.1.1 Derivation of the Least-Squares Estimation

We now show how to compute the gradient of E in Eq (5.4) with respect to the

interceptψ and the slope ξ via the calculus of variation method The reader can skip

this subsection without losing the flow of the exposition by assuming that the gradient

5.1 Geodesic Regression on the ODF manifold

ofE in Eq (5.4) holds ture For the simplicity, we denote ˆ ψ i = exp(ψ, x i ξ) in the

following derivation

We first compute the gradient of E with respect to ψ, denoted as ∇ ψ E Let

ψ ε = exp(ψ, εμ) where ε is a scalar and μ ∈ T ψ Ψ is a tangent vector at ψ We now

take the derivative ofE at ε = 0 We have

i ψ i , ∂ε ∂ logexp(ψ ε ,x i ξ) ψ i

ε=0

ˆ

tangent space of ˆψ ito that ofψ Therefore,

Next, we compute the derivative of E with respect to ξ denoted as ∇ ξ E Let

ξ ε = ξ + εμ, where ε is a scalar and μ ∈ T ψ Ψ is a tangent vector at ψ We now take

the derivative ofE at ε = 0 We have

i ψ i , ∂ε ∂ logexp(ψ,xi ξ ε)ψ i

ε=0

ˆ

ψ i ,

where (a) is obtained from the first order approximation of logexp(ψ,xi ξ ε)ψ i based

on Taylor expansion of the logarithm map According to the analytical form of theexponential map given in Eq (2.4), we have

5.1 Geodesic Regression on the ODF manifold

onξ at ψ by parallel transport Therefore, it yields

logψˆ

i ψ i

⊥are given as

logψˆ

i ψ i



=logψˆ

i ψ i , ξ ξ i

i  ψˆi

ˆ

ψ as the least-squares solution to Eq (5.9).

Now, to measure the amount of explained variance of the ODF by the variableX,

we use a generalization of the coefficient of determination, i.e.,R2 statistic,

Finally, we empirically compute the distribution of theR2statistic via a permutationtest by calculating all possible values of theR2 statistic under the permutation of thelabels on the observed data points In each randomized trial,X is randomly assigned

5.2 Experiments

to individual subjects and theR2statistic is computed based on Eq (5.10) We repeatthis forN times The p-value can be calculated as the percentage of the R2 greater thanthat from the original data without permutation If thep-value is less than a significance

level of 0.05, we conclude that the independent variable X is significantly related to the

ODF Otherwise, the relationship between the ODF andX is insignificant.

For voxel-based analysis on the ODF image, we can apply the above procedure

to every voxel However, in each randomized trial, X is only randomly assigned to

individual subjects once and theR2 statistic is computed for all voxels in the ODFimage The correction of multiple comparisons throughout the ODF image can beachieved using false discovery rate (FDR) as described in [106]

In this section, we demonstrate the performance of the geodesic regression on the mannian manifold of ODFs using synthetic and real brain data Synthetic experiments

Rie-on ODFs, generated by the multi-tensor method [107], are shown in§5.2.1 In §5.2.2,

we examine aging effects on the brain white matter via geodesic regression of ODFs innormal adults aged 21 years old and above

5.2.1 Experiments on Synthetic ODF Data

We now evaluate the performance of the geodesic regression using synthetic ODF data

As illustrated in Figure 5.3, two ODFs are generated were first generated using themulti-tensor method proposed in [45] The first represents a single fiberψ0, while thesecond represents a crossing fiberψ1 We consider ψ = ψ0 and the logarithm maprelating the two ODFsξ = log ψ0ψ1 as the ground truth of the geodesc regression in

this experiment.

Figure 5.3: Illustration of synthetic ODFs for single (a) and crossing fibers (b)

We constructed a series of simulation data by randomly generating the error term,

, of the geodesic regression in Eq (5.3) We first drawed x i , i = 1, , n from a

uniform distribution on [0, 1] The error term  i was then generated from an isotropicGaussian distribution in the tangent space of exp(ψ, x i ξ), with the standard deviation

σ = Mξ ψ, where M is a constant determining the level of noise The resulting

data (x i , ψ i) was considered as observations in the geodesic regression to estimatethe parameters ( ˆψ, ˆξ) according to the algorithm described in §5.1 An illustration of

synthetic data at different levels of noiseM is shown in Figure 5.4 This experiment

was repeated for 1 000 times for each sample size (n = 2 k , k = 3, , 8) and each level

of noise (M = 0.1, 0.5, 1.0, 2.0), respectively We calculate the mean squared error

(MSE) between the estimated parameters ( ˆψ, ˆξ) and the ground truth (ψ, ξ) as

5.2 Experiments

Figure 5.4: Illustration for synthetic√

ODF data, regression result and ground truth underfour levels of noise (M = 0.1, 0.5, 1.0, 2.0): In each panel, each column shows the ODFs at

x i = 0, 0.2, 0.4, 0.6, 0.8, 1.The first five rows illustrate the synthetic ODFs, while the next

row shows the regression result The bottom row shows the ground truth for the geodesicregression

for each experiment with a certain sample size and a noise level T = 1000 is the

number of repeated trials, and ( ˆψ t , ˆξ t ) is the estimate from the t-th trial It is important

to note that ˜ξ t ∈ T ψ Ψ is the transformed ˆξ tfrom the tangent space of ˆψ tto the tangentspace ofψ through parallel transportation Figure 5.5 shows the plots of the resulting

MSE of ( ˆψ, ˆξ) As expected, the MSE approaches zero as the sample size increases for

all noise levels When the noise level is high, a larger sample size is required to achieve

an accurate estimation

To check for consistency of the results with metrics other than the geodesic distanceused in the regression framework, we compared the MSE of the geodesic distance withthe MSE of theL2 norm of spherical harmonics coefficients, and the MSE of symmetricKullback-Leibler divergence between ODFs [51], in Figure 5.6 From Figure 5.6, wesee that the different metrics exhibit the same behavior

Figure 5.5: Evaluation of the geodesic regression accuracy using synthetic√

ODF Panels(a) and (b) show the plots of the mean square error of ˆψ and ˆξ for estimated geodesic

regression at four noise levels (M = 0.1, 0.5, 1, 2) against the number of observations n

respectively

We calculated mean variance of the synthetic data MSE(ψ i , ¯ ψ), mean squared

residualsMSE(ψ i , ˆ ψ i ), and R2 for the estimated geodesic regression at all noise levelsunder three distance measures of ODFs including the geodesic distance, theL2 norm

of spherical harmonics coefficients, and the symmetric Kullback-Leibler divergencebetween ODFs, in Figure 5.7 As designed in this experiment, the higher level of noise,the larger the mean variance of the synthetic data (see the first column) From the results,

5.2 Experiments

Figure 5.6: Consistency of results under different metrics including the geodesic distance,the L2 norm of spherical harmonics coefficients, and the symmetric Kullback-Leibler

divergence between ODFs,

we see that theR2value decreases dramatically as the noise increases Low values ofR2statistically mean that the regression geodesic explains a small portion of the variance ofthe synthetic data This finding is expected due to the noise being large (forM ≥ 0.5)

and high-dimensionality of the underlying space However, it is important to note herethat the lowR2 value does not imply that the parameters estimated by regression can

be found by a chance Even when the noise level is high, the regression still provides

an accurate estimation, as demonstrated in Figure 5.5 In addition, when comparingthe mean variance, mean squared residuals, andR2 across each row, it demostrates theconsistency of the regression results under different distance measures

To investigate the effects of the order of spherical harmonics on geodesic regressionresults, we generated a simulated √

ODF data at the noise level M = 0.5 with the

number of observationsn = 64 with the coefficients of spherical harmonics from the

2nd-order and up to the 8th-order The results of geodesic regression under differentspherical harmonics orders are shown in Figure 5.8 We observed that the results ofgeodesic regression on ODFs of 2nd-order of spherical harmonics produce “smoother”ODFs Despite a slight loss of information when due to the truncation resulting fromlower order spherical harmonics, the results of geodesic regression on ODFs of 4th-order

Figure 5.7: Results of geodesic regression for simulated√

ODF data at four noise levels(M = 0.1, 0.5, 1.0, 2.0) against the number of observations n Three types of metric be-

tween ODFs: geodesic distance; L2 norm of spherical harmonics coefficients and symmetricKullback-Leibler divergence are calculated, one for each row Under one type of ODFmetrics of that row, the mean variance of synthetic data,MSE(ψ i , ¯ ψ); the mean squared

residuals of the geodesic regression,MSE(ψ i , ˆ ψ i ); and R2of the geodesic regression are

shown in each column respectively

and above are relatively stable to the order choice

5.2 Experiments

Figure 5.8: Results of geodesic regression under different spherical harmonics orders

5.2.2 Experiments on Real Human Brain Data: Aging Study

In this section, we examined the effects of aging on brain white matter via geodesicregression of ODFs in a large group of normal Chinese subjects, spanning the entireadult age range

5.2.2.1 Image Acquisition and Preprocessing

We first briefly describe the demographic information of human subjects and HARDIdata processing used in this study The dataset was acquired from 185 Chinese partici-pants (79 males and 106 females) ranging from 22 to 79 years old (mean± standard

deviation (SD): 47.7 ± 15.9 years) All participants had minimental state examination

(MMSE) greater than 26 and have no history of major illnesses and mental disorders.Each participant underwent HARDI using a 3T Siemens Magnetom Trio Tim scannerwith a 32-channel head coil at Clinical Imaging Research Center of the National Uni-

versity of Singapore The image protocols were as follows: (i) isotropic high angularresolution diffusion imaging (single-shot echo-planar sequence; 48 slices of 3 mmthickness; with no inter-slice gaps; matrix: 96× 96; field of view: 256 × 256 mm;

repetition time: 6800 ms; echo time: 85 ms; flip angle: 90◦; 91 diffusion weightedimages (DWIs) withb = 1150 s/mm2, 11 baseline images without diffusion weighting);(ii) isotropic T2-weighted imaging protocol (spin echo sequence; 48 slices with 3 mmslice thickness; no inter-slice gaps; matrix: 96× 96; field of view: 256 × 256 mm;

repetition time: 2600 ms; echo time: 99 ms; flip angle: 150◦)

The DWIs of each subject were first corrected for motion and eddy current tions using affine transformation We followed the procedure detailed in [103] to correctgeometric distortion of the DWIs due to b0-susceptibility differences over the brain

distor-in a sdistor-ingle subject Briefly summarizdistor-ing, the T2-weighted image was considered asthe anatomical reference The deformation that relates the baseline b0 image withoutdiffusion weighting to the T2-weighted image characterized the geometric distortion

of the DWI For this, intra-subject registration was first performed using FSL’s LinearImage Registration Tool [104] to remove linear transformations (rotations and trans-lations) between the diffusion weighted images and the T2-weighted image Then,

we used the brain warping method, large deformation diffeomorphic metric mapping(LDDMM) [93], to find the optimal nonlinear transformation that deformed the baselineimage without the diffusion weighting to the T2-weighted image This diffeomorphictransformation was then applied to every DWI in order to correct the nonlinear geomet-ric distortion Finally, we estimated the ODFs represented by 4th-order real sphericalharmonics using the approach considering the solid angle constraint based on DWIimages described in [48]

The ODFs of each subject were then warped to an ODF atlas using an

repetition... regression on ODFs of 2nd-order of spherical harmonics produce “smoother”ODFs Despite a slight loss of information when due to the truncation resulting fromlower order spherical harmonics, the... behavior

Figure 5. 5: Evaluation of the geodesic regression accuracy using synthetic√

ODF Panels(a) and (b) show the plots of the mean square error of ˆψ and ˆξ for estimated