Kernel-Based Morphometry of Diffusion Tensor Images

Abstract

Voxel-based group-wise statistical analysis of diffusion tensor imaging (DTI) is an integral component in most population-based neuroimaging studies such as those studying brain development during infancy or aging, or those investigating structural differences between healthy and diseased populations. The majority of studies using DTI limit themselves by testing only certain properties of the tensor that mainly include anisotropy and diffusivity. However, the pathology under study may affect other aspects like the orientation information provided by the tensors. Therefore, for detecting subtle pathological changes it is important to perform group-wise testing on the whole tensor, which encompasses the changes in anisotropy, diffusivity and orientation. This is rendered challenging by the fact that conventional linear statistics cannot be applied to tensors. Moreover, the pathology over the population is unknown and could be non-linear, further complicating the group-based statistical analysis. This chapter gives a perspective on performing voxel-wise morphometry of tensor data using kernel-based approach. The method is referred as Kernel-based morphometry (KBM) as it models the tensor distribution using kernel principal component analysis (kPCA), which linearizes the data in high dimensional space. Subsequently a Hotelling T ² test is performed on the high dimensional kernelized data to determine statistical group differences. We apply this method on simulated and real datasets and show that KBM can effectively identify the underlying tensorial distribution. Thus it can potentially elucidate pathology-induced population differences, thereby establishing a kernelized full tensor framework for population studies.

Appendix

Proof of the equivalence between Hotelling’s T ² test and FDA [24].

Let x _1i , i = 1, ⋯ , N ₁ be the p-dimensional data vectors belonging to class 1 and let x _2i , i = 1, ⋯ , N ₂ be the p-dimensional data vectors belonging to class 2. Let $\bar{\mathbf{x}}_{1},\bar{\mathbf{x}}_{2}$ denote the means for classes 1 and 2. Let $S_{x} = ( \frac{1} {N_{1}+N_{2}-2})\Big(\sum _{i=1}^{N_{1}}(\mathbf{x}_{ 1i} -\bar{\mathbf{x}}_{1})(\mathbf{x}_{1i} -\bar{\mathbf{x}}_{1})^{T} +\sum _{ i=1}^{N_{2}}(\mathbf{x}_{ 2i} -\bar{\mathbf{x}}_{2})(\mathbf{x}_{2i} -\bar{\mathbf{x}}_{2})^{T}\Big)$.

Hotelling’s T ² test: Then the T ² test statistic is:

$$\displaystyle{ T_{x}^{2} = \frac{N_{1}N_{2}} {N_{1} + N_{2}}(\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})^{T}S_{ x}^{-1}(\bar{\mathbf{x}}_{ 1} -\bar{\mathbf{x}}_{2}) }$$

(3)

Assuming normal distributions for x _1i , i = 1, ⋯ , N ₁ and x _2i , i = 1, ⋯ , N ₂ implies that $F_{x} = \frac{N_{1}+N_{2}-p-1} {(N_{1}+N_{2}-2)p}T_{x}^{2}$ has the cdf $F(p,N_{1} + N_{2} - p - 1)$. Therefore, the parametric form of Hotelling’s T ² test is sometimes refered to as the F-test.

FDA: The FDA optimal linear discriminant direction is $w = S_{x}^{-1}(\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})$ and the corresponding scalar mapping is $y = (\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})^{T}S_{x}^{-1}x$.

Therefore, $\bar{y}_{1} -\bar{ y}_{2} = (\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})^{T}S_{x}^{-1}(\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})$,

$$\displaystyle\begin{array}{rcl} S_{y}& =& ( \frac{1} {N_{1} + N_{2} - 2})\Big(\sum _{i=1}^{N_{1} }(y_{1i} -\bar{ y}_{1})(y_{1i} -\bar{ y}_{1})^{T} +\sum _{ i=1}^{N_{2} }(y_{2i} -\bar{ y}_{2})(y_{2i} -\bar{ y}_{2})^{T}\Big) \\ & =& ( \frac{1} {N_{1} + N_{2} - 2})(\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})^{T}S_{ x}^{-1}\Big(\sum _{ i=1}^{N_{1} }(\mathbf{x}_{1i} -\bar{\mathbf{x}}_{1})(\mathbf{x}_{1i} -\bar{\mathbf{x}}_{1})^{T} {}\end{array}$$

(4)

$$\displaystyle\begin{array}{rcl} & & +\sum _{i=1}^{N_{2} }(\mathbf{x}_{2i} -\bar{\mathbf{x}}_{2})(\mathbf{x}_{2i} -\bar{\mathbf{x}}_{2})^{T}\Big)S_{ x}^{-1}(\bar{\mathbf{x}}_{ 1} -\bar{\mathbf{x}}_{2}) \\ & =& (\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})^{T}S_{ x}^{-1}(\bar{\mathbf{x}}_{ 1} -\bar{\mathbf{x}}_{2}) {}\end{array}$$

(5)

and

$$\displaystyle\begin{array}{rcl} T_{y}^{2}& =& \frac{N_{1}N_{2}} {N_{1} + N_{2}}(\bar{y}_{1} -\bar{ y}_{2})^{T}S_{ y}^{-1}(\bar{y}_{ 1} -\bar{ y}_{2}) \\ & =& \frac{N_{1}N_{2}} {N_{1} + N_{2}}(\bar{\mathbf{x}}_{1} -\bar{\mathbf{x}}_{2})^{T}S_{ x}^{-1}(\bar{\mathbf{x}}_{ 1} -\bar{\mathbf{x}}_{2}) \\ & =& T_{x}^{2} {}\end{array}$$

(6)

Thus, the T ² statistic computed on vectorial input samples, i.e. T _x ², is mathematically equivalent to a one-dimensional T ² statistic T _y ² computed on the scalar samples obtained by performing Fisher discriminant analysis on the input vectorial samples.