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**|**HHS Author Manuscripts**|**PMC2861508

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- Abstract
- 1 Introduction
- 2 Riemannian Metric for Positive-Valued Real Functions
- 3 Groupwise Registration of 4th-Order Tensor Fields
- 4 Experimental Results
- 5 Conclusions
- References

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Med Image Comput Comput Assist Interv. Author manuscript; available in PMC 2010 April 29.

Published in final edited form as:

Med Image Comput Comput Assist Interv. 2009 October 1; 5761: 640–647.

doi: 10.1007/978-3-642-04268-3_79PMCID: PMC2861508

NIHMSID: NIHMS144136

CISE Department, University of Florida, Gainesville, FL 32611, USA

Angelos Barmpoutis: ude.lfu.esic@uopmraba; Baba C. Vemuri: ude.lfu.esic@irumev

See other articles in PMC that cite the published article.

Registration of Diffusion-Weighted MR Images (DW-MRI) can be achieved by registering the corresponding 2nd-order Diffusion Tensor Images (DTI). However, it has been shown that higher-order diffusion tensors (e.g. order-4) outperform the traditional DTI in approximating complex fiber structures such as fiber crossings. In this paper we present a novel method for unbiased group-wise non-rigid registration and atlas construction of 4th-order diffusion tensor fields. To the best of our knowledge there is no other existing method to achieve this task. First we define a metric on the space of positive-valued functions based on the Riemannian metric of real positive numbers (denoted by ^{+}). Then, we use this metric in a novel functional minimization method for non-rigid 4th-order tensor field registration. We define a cost function that accounts for the 4th-order tensor re-orientation during the registration process and has analytic derivatives with respect to the transformation parameters. Finally, the tensor field atlas is computed as the minimizer of the variance defined using the Riemannian metric. We quantitatively compare the proposed method with other techniques that register scalar-valued or diffusion tensor (rank-2) representations of the DWMRI.

Group-wise image registration is a challenging task in medical imaging which is related to the problem of computing an atlas, i.e. the image of the average subject from a set of co-registered subjects. There are two prevalent approaches for atlas construction. The first one is based on group-wise alignment of 3D shapes [1,2], while the second one is uses alignment of 3D image intensity maps.

In this paper we focus on the second category, and therefore we review only techniques that are based on intensity map registration. Joshi et al. [3] proposed a method for group-wise image registration and simultaneous atlas construction. In this method the atlas is formed by minimizing the distance between the displacement fields that warp the images and therefore it is not biased toward a specific subject data. The estimated atlas does not belong to the set of registered subjects unlike the method presented in [4], which perform pair-wise registration of all the subjects and select the least biased target as the atlas.

The aforementioned methods perform scalar-valued image registration. It has been shown, however, that registration of diffusion tensor-valued images (DTI) produces more accurate alignments of fibrous tissues [5]. In this approach the tensors should be re-oriented appropriately after the warping of the DTI images in order to preserve the micro-structural geometry in the subjects. One way to avoid the tensor re-orientation is to register rotation invariant quantities or other highly structured features extracted from DTI [6]. A DTI similarity measure that uses the full information in the tensors and performs their re-orientation using locally affine transformations was employed in [7]. Furthermore, two methods for diffeomorphic non-rigid DTI registration were proposed in [8] and [9] both of which use analytic derivatives of the reorientation term in the corresponding energy functions.

All the above techniques perform pair-wise DTI registration. Multi-subject registration for DTI atlas construction was proposed in [10] by extending the scalar-image framework in [3]. Another group-wise DTI registration technique which unfolds the manifold described by the Geodesic-Loxodromes metric on diffusion tensors and produces vector-valued images that are being warped in order to estimate the DTI atlas was recently proposed in [11].

Although the methods for DTI registration and atlas construction yield richer representations than the corresponding scalar-image based techniques, they fail in regions of fiber crossings and other complex tissue geometries since 2* ^{nd}*-order tensors cannot account for multiple peaks in the diffusivity function. This problem can be resolved by using 4

In this paper we present a novel method for unbiased 4* ^{th}*-order tensor field atlas construction. Our method (significantly) generalizes the unbiased diffeomorphic scalar image atlas construction framework in [3] to the case of symmetric positive definite higher-order tensors. The atlas is computed simultaneously with the non-rigid deformation fields using a functional minimization procedure. We define a novel cost function using the Riemannian metric on positive valued functions which is a generalization of the Riemannian metric on

The key contributions of this work are: To the best of our knowledge, this is the first report in literature for higher-order tensor field atlas construction. Our method outperforms the existing methods that register derived scalar images or 2* ^{nd}*-order tensor fields from DWMRI, both of which fail to accurately warp datasets with complex local tissue structures such as fiber crossings. Furthermore, we employ a novel metric based on the Riemannian geometry of positive-valued spherical functions and we show that it produces more accurate results compared to the standard Euclidean metric. Finally our cost function has analytic derivatives with respect to the unknown transformation parameters that lead to an efficient and easily scalable implementation of our framework.

Assume *a, b* ^{+}, i.e. are elements of the space of positive real numbers. The Logarithmic map at location *a* is given by *Log _{a}*(

$$\mathit{\text{dist}}(a,b)=\left|\text{log}\frac{a}{b}\right|$$

(1)

which satisfies scale invariance, i.e. *dist*(*sa, sb*) = *dist*(*a, b*) ∀, *b, s* ^{+}, additionally to the properties of distance measures.

The Riemannian metric in ^{+} can also be used to define distances between positive-valued functions *f _{a}*(

$${\mathit{\text{dist}}}^{2}({\mathbf{\text{D}}}_{1},{\mathbf{\text{D}}}_{2})={{\int}_{{S}^{2}}\left|log\frac{d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{1})}{d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{2})}\right|}^{2}dg.$$

(2)

Note that the integral in Eq. 2 is over *S*^{2}, i.e. the space of unit vectors **g**. This distance function is invariant with respect to 3D rotations and scale, i.e. *dist*(*s***R** **D**_{1}, *s***R** **D**_{2}) = *dist*(**D**_{1}, **D**_{2}) ∀*s* _{+} and **R** *SO*_{3}.

Similarly, the distance between ordered n-tuples whose elements are positive real numbers can be defined using the Riemannian metric in ^{+}. In this case the distance between *A* = {*a*_{1}, *a*_{2}, … , *a _{n}*} and

In the next section we will employ the above distance measure in order to achieve simultaneous group-wise registration and atlas construction of fields of spherical functions modeled using Cartesian tensor bases of order 4.

Cartesian tensor bases of various orders have been used for approximating physical quantities computed from DW-MRI datasets. 4* ^{th}* -order tensors

In the case of 4* ^{th}*-order generalized diffusion tensors, the diffusivity is a positive-valued function and can be computed using the parametrization in [15]. This produces fields of positive-valued spherical functions whose processing can be achieved using the Riemannian metric presented in Sec. 2.

The problem of group-wise registration of *N* tensor-fields and simultaneous atlas estimation can be formulated as a functional minimization problem. By using Eq. 2 the energy function to be minimized is given by

$$E({\phi}_{n},{\mathbf{\text{D}}}_{\mu})=\sum _{n=1}^{N}{\int}_{\Omega}{{\int}_{{S}_{2}}\left(log\frac{d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{n}\circ {\phi}_{n})}{d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{\mu})}\right)}^{2}d\mathbf{\text{g}}\mathit{dx}+\sum _{n=1}^{N}{\int}_{\Omega}\mathit{\text{cost}}({\phi}_{n})\mathit{\text{dx}}$$

(3)

where **D*** _{μ}* is the 4

Note that the tensor coefficients are dependent on the local rotation of the coordinate system [12]. Hence, given a deformation * _{n}* the transformed spherical function field at location

$$d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{n}\circ {\phi}_{n})=\sum _{i,j,k,l}{D}_{n}^{i,j,k,l}(x\circ {\phi}_{n}){({\mathbf{\text{R}}}_{x}\mathbf{\text{g}})}_{i}{({\mathbf{\text{R}}}_{x}\mathbf{\text{g}})}_{j}{({\mathbf{\text{R}}}_{x}\mathbf{\text{g}})}_{k}{({\mathbf{\text{R}}}_{x}\mathbf{\text{g}})}_{l}$$

(4)

where **R*** _{x}* is the rotation of deformation

The deformation can be parametrized as a time varying vector field such that * _{n}*(

We will minimize the energy function (Eq. 3) by evolving the deformation fields * _{n}* using a greedy iterative scheme which approximates the solution to the above minimization problem, similar to the technique in [3]. For this purpose we will construct a field of forces by computing the first order variation of the first term in Eq. 3 with respect to the transformation parameters as follows

$${F}_{n}=-2{\int}_{{S}^{2}}log\left(\frac{d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{n}\circ {\phi}_{n})}{d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{\mu})}\right)[{\nabla}_{\mathit{\text{trans}}}+{\nabla}_{\mathit{\text{rot}}}]log(d(\mathbf{\text{g}};{\mathbf{\text{D}}}_{n}\circ {\phi}_{n}))dg$$

(5)

where the variation * _{trans}* is related with the local translation (i.e. variation of
${D}_{n}^{i,j,k,l}(x\circ {\phi}_{n})$ in Eq. 4) and

After the estimation of the fields of forces *F _{n}, n* = 1 …

Finally, the tensor coefficients of the atlas can be updated by also minimizing the first term in Eq. 3 with respect to the parameters of a positive definite 4* ^{th}*-order tensor using the parametrization in [15].

In general, the integral over the sphere in Eq. 5 cannot be computed analytically when the Cartesian tensor parametrization is used for modeling the diffusivity function. On the other hand the Riemannian space of ordered n-tuples (see Sec. 2) leads to analytic calculations and therefore we used it in our implementation. We constructed an m-tuple space by using a set of unit vectors **g*** _{m} m* = 1 …

The above discretization helps us also in reducing the time complexity of atlas computation, which can now be efficiently computed by

$${d}_{\mu}({\mathbf{\text{g}}}_{m})=exp(\frac{1}{N}\sum _{n=1}^{N}log(d({\mathbf{\text{g}}}_{m};{\mathbf{\text{D}}}_{n}\circ {\phi}_{n})))$$

(6)

where *d _{μ}*(

$${F}_{n}=-2\sum _{m=1}^{M}{L}_{m,n}(x)\nabla {I}_{n,m}+\sum _{\left|y-x\right|=1}{L}_{m,n}(y)\frac{{\nabla}_{{\mathbf{\text{R}}}_{y}{\mathbf{\text{g}}}_{m}}d({\mathbf{\text{g}}}_{m};{\mathbf{\text{D}}}_{n}(y\circ {\phi}_{n}))}{d({\mathbf{\text{g}}}_{m};{\mathbf{\text{D}}}_{n}(y\circ {\phi}_{n}))}\nabla {\mathbf{\text{R}}}_{y}{\mathbf{\text{g}}}_{m}$$

(7)

where
${L}_{m,n}=log\left(\frac{{I}_{n,m}(x)}{d({\mathbf{\text{g}}}_{m};{\mathbf{\text{D}}}_{\mu}(x\circ {\phi}_{n}))}\right)$, *I _{n, m}* is simply the spatial gradient of a scalar valued image and the second term in Eq. 7 correspond to the gradient related to the tensor re-orientation. In this term the rotation

Finally, after the termination of the iterative minimization procedure, the 4* ^{th}*-order tensor coefficients can be computed by fitting the tensorial model to the estimated values

In order to compare the Riemannian metric presented in Sec. 2 with a Euclidean metric in terms of fiber orientation accuracy of the atlas estimated by each metric, we performed the following experiment. We synthesized a 2-fiber crossing DW-MRI dataset (in a single voxel) using the realistic adaptive kernel model shown in Fig.3 of [17] (81 gradient directions and *b* = 1250*s/mm*^{2}). We computed a 4* ^{th}*-order tensor (shown in Fig. 1 upper left) from the synthetic dataset using the algorithm in [15]. Then we generated 100 more datasets by applying small rotations to the simulated crossing and by adding outliers (few of them are shown in Fig. 1 left). The computed atlases (average tensors) are compared in the bar chart of Fig. 1. As expected, the Riemannian mean outperforms the Euclidean mean since the physical space of the data is that of positive-valued functions.

Comparison of the 4^{th}-order tensor atlases computed by various metrics: a) Euclidean mean, b) Riemannian mean (computed in the space presented in Sec. 2)

To motivate the use of 4* ^{th}*-order tensors in registering DW-MRI, we also simulated a fiber crossing dataset and synthesized a deformation field (Fig. 2). Then we computed the corresponding FA, DTI and 4

Finally, we computed the 4* ^{th}*-order tensor field atlas from four hippocampal datasets. Each dataset consists of 21 diffusion-weighted images collected with a 415 mT/m diffusion gradient (

In this paper we presented a novel groupwise registration and atlas construction algorithm for DWMRI data sets each of which is represented by a 4th order tensor field. To the best of our knowledge, there is no existing literature on this topic. The key contribution of this work is the definition of a novel metric for positive valued spherical functions which was then used in the unbiased group-wise registration and atlas construction. Experimental results on comparisons with scalar and DTI registration techniques are favourable to our method.

^{*}This research was in part funded by the NIH grant EB007082 to BCV.

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