METODOLOGÍA DE LA INVESTIGACIÓN

One of the most commonly used measures of anisotropy of a diffusion tensor D is Fractional Anisotropy (FA) [31] F A(D) = s 3[(λ1− ¯λ)2+ (λ2− ¯λ)2+ (λ3− ¯λ)2] 2(λ2 1+ λ22+ λ23) , (3.12)

where ¯λ = M D(D). F A(D) ranges from 0 in the case of perfect isotropy to 1 in the case of linear anisotropy (λ1 > λ2 = λ3 = 0). A closely related measure of anisotropy is Procrustes Anisotropy (PA), which computes the fractional anisotropy of√λ1,

√

λ2, and √

λ3 (i.e. P A(D) = F A(√D)) [15].

3.3.1 Common Frameworks for Sym-Plus(p)

In this subsection we review some established geometric frameworks for Sym+(p), including a Eucidean framework, a Log-Euclidean framework ([2]), a Riemannian affine-invariant frame- work (citations needed), and matrix square root based approaches including a Cholesky- based framework, a root-Euclidean framework, and a framework based on the Procrustes Size-and-Shape metric of [15].

We first discuss a Euclidean framework for Sym+_{(p). Let X, Y ∈ Sym}+_{(p). Since} Sym+_{(p) ⊂ Sym(p), which is isomorphic to R}p(p+1)_{2 and can be metrized with the Frobenius} norm k.kF, one can can measure the distance between X and Y via the distance function

dE(X, Y ) = kX − Y kF.

The distance dE(X, Y ) measures the length of the line

χE(t) = (1 − t)X + tY

for t ∈ [0, 1].

We next move on to the Log-Euclidean framework for Sym+_{(p). Again, let X, Y ∈} Sym+(p). Recall that Exp : Sym(p) 7→ Sym+(p) is bijective, so any M ∈ Sym+(p) can be uniquely identified with Log(M ) ∈ Sym(p). Hence, Sym+_{(p) can be thought of as a “log-} linear” space, and one can measure the distance between X, Y ∈ Sym+(p) via the distance function

dLE(X, Y ) = kLog(X) − Log(Y )kF. Geodesics corresponding to dLE(., .) are curves of the form

χLE(t) = Exp((1 − t)Log(X) + tLog(Y ))

for t ∈ [0, 1]. We refer the reader to [2] for a more detailed account of the geometric properties of the Log-Euclidean characterization of Sym+_(p).

We finally present an affine-invariant Riemannian framework for Sym+_{(p). Let X, Y ∈} Sym+(p). Note that we can write Y as Y = MX(M_X−1Y (M_X−1)T)M_XT, where MX ∈ GL(p)

is a square root of X (i.e. MXMXT = X) and M −1 X Y (M

−1

X )T ∈ Sym+(p) since Sym+(p) is closed under conjugation by matrices in GL(p). For t ∈ [0, 1], the curve

χAI(t) = MXExp(Log(MX−1Y (M −1 X )

T_)t)MT X

traces a geodesic connecting X and Y under the usual Riemannian inner product for Sym+(p) that is invariant under the group action. This curve has length

dAI(X, Y ) = kLog(MX−1Y (M −1 X )

T )kF.

It is important to note that the curve χAI(t) and its corresponding distance function dAI(.) are invariant with respect to the choice of square root for X. For more information on this affine-invariant Riemannian framework, we refer the reader to [37].

We next present frameworks based on square root decompositions, beginning with the Cholesky decomposition. Recall that the Cholesky decomposition uniquely factors a matrix X ∈ Sym+(p) as X = QQT, where Q is a lower triangular matrix with positive diagonal entries. Given X, Y ∈ Sym+_{(p), suppose that X and Y have Cholesky decompositions} X = QXQTX and Y = QYQTY. Via Cholesky composition, one can define a smooth curve on Sym+_{(p) running from X to Y using the formula}

χC(t) = [QX + t(QY − QX)][QX + t(QY − QX)]T

for t ∈ [0, 1]. This curve traces a geodesic on Sym+_{(p) from X to Y with length}

dC(X, Y ) = kQY − QXkF.

Another unique square root decomposition of an SPD matrix is the symmetric square root decomposition. Given X ∈ Sym+(p) with eigen-decomposition X = U DUT, we define its symmetric square root as X1/2 _{= U D}1/2_UT_{, where D}1/2 _{is a diagonal matrix whose diagonal} entries are the positive square roots of the diagonal entries of D. Given X, Y ∈ Sym+(p), one can create the following curve

which traces a geodesic on Sym+_{(p) from X to Y for t ∈ [0, 1] with length} dH(t) = kY1/2− X1/2kF.

This geometric framework for Sym+(p) based on the symmetric square root is referred to as the Root-Euclidean framework in [48].

Note that the Cholesky decomposition and symmetric square root decomposition meth- ods yield two out of infinitely many possible square roots of an SPD matrix. Indeed, if X ∈ Sym+(p) and X = LLT then LR, where R ∈ O(p), is also a square root of X since (LR)(LR)T _{= X. Given X, Y ∈ Sym}+_{(p) with square roots Q}

X and QY, one can define a measure of distance between X and Y as

dS(X, Y ) = inf

R∈O(p)kQX − QYRkF,

which finds the square root of Y that is closest to the initial square root QX of X under the Frobenius norm. Given initial square roots QX and QY for X and Y , respectively, it can be shown that

ˆ

R = arg inf

R∈O(p)kQX − QYRkF = U W T

where W, U ∈ O(p) come from the singular value decomposition QXQTY = W ΛUT. A geodesic from X to Y on Sym+(p) with respect to this metric will be of the form

χS(t) = [QX + t(QYR − Qˆ X)][QX + t(QYR − Qˆ X)]T for t ∈ [0, 1]. More information on this framework can be found in [15].

Like the Scaling-Rotation framework, the Log-Euclidean and affine-invariant Riemannian frameworks for Sym+_{(p) are also intrinsic frameworks since the geodesics χ}

LE(t) and χAI(t) lie in Sym+_{(p) for all t ∈ R. While the curve χ}E(t) lies in Sym(p) for all t ∈ R, there will be some t for which χE(t) will fail to be positive-definite. The curves χC(t), χH(t), and χS(t) will be non-negative definite, symmetric matrices for all t ∈ R, but can be less than full rank for some t. Hence, the Euclidean, Cholesky, Root-Euclidean, and Procrustes frameworks are all extrinsic frameworks for Sym+(p).

Of all the frameworks discussed in this section, the Procrustes size-and-shape framework perhaps shares the most conceptual similarities with the Scaling-Rotation framework. Both

frameworks parameterize Sym+_{(p) via non-unique matrix decompositions (matrix square} root and eigen-decomposition), and the distance between elements X, Y ∈ Sym+_{(p) is de-} fined as the minimum distance between all possible decompositions of X and Y .