Registration, atlas generation, and statistical analysis of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions 2

The space ofp forms a Riemannian manifold, also known as the statistical manifold, which is well-known from the field of information geometry [82].. Rao [83] introduced the notion of the

From existing literature [41, 45, 48], we know that at a specific spatial location,x ∈ Ω,

HARDI measurements can be used to reconstruct the ODF, the diffusion angular profile

of water molecules The ODF is actually a diffusion probability density function (PDF) defined on a unit sphereS2 and its space is defined as

P = {p : S2 → R+|∀s ∈ S2

, p(s) ≥ 0;



s∈S2p(s)ds = 1}

The space ofp forms a Riemannian manifold, also known as the statistical manifold,

which is well-known from the field of information geometry [82] Rao [83] introduced

the notion of the statistical manifold whose elements are probability density functions

and composed the Riemannian structure with the Fisher-Rao metric Cencov [84] showed that the Fisher-Rao metric is the unique intrinsic metric on the statistical

manifold P and therefore invariant to re-parameterizations of the functions There

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are many different parameterizations of PDFs that are equivalent but with different forms of the Fisher-Rao metric, leading to the Riemannian operations with different computational complexity In this work, we choose the square-root representation, which was used recently in ODF processing [1, 80, 85] The square-root representation

is one of the most efficient representations found to date as the various Riemannian operations, such as geodesics, exponential maps, and logarithm maps, are available in closed form, as illustrated in Figure 2.1

Figure 2.1: Illustration of the manifold of square-root ODFs (picture token and modified from [1]))

The square-root ODF ( √

ODF) is defined asψ(s) =p(s), where ψ(s) is assumed

to be non-negative to ensure uniqueness The space of such functions is defined as

Ψ = {ψ : S2 → R+|∀s ∈ S2

, ψ(s) ≥ 0;



s∈S2ψ2(s)ds = 1}. (2.1)

We see that from Eq (2.1), the functionsψ lie on the positive orthant of a unit Hilbert

sphere, a well-studied Riemannian manifold It can be shown [86] that the Fisher-Rao metric is simply theL2 metric, given as

ξ j , ξ k  ψi =



s∈S2ξ j(s)ξ k s)ds, (2.2) whereξ j , ξ k ∈ T ψi Ψ are tangent vectors at ψ i The geodesic distance between any two functionsψ i , ψ j ∈ Ψ on a unit Hilbert sphere is the angle

dist(ψ i , ψ j) = log ψi(ψ j) ψi = cos−1 ψ i , ψ j  = cos −1

s∈S2ψ i(s)ψ j(s)ds



,

(2.3)

where·, · is the normal dot product between points in the sphere under the L2 metric

For the sphere, the exponential map has the closed-form formula

exp(ψ i , ξ) = exp ψi(ξ) = cos( ξ ψi)ψ i+ sin( ξ ψi) ξ

ξ ψi , (2.4) where ξ ∈ Tψi Ψ is a tangent vector at ψ i and ξ ψi = 

ξ, ξ ψi By restricting

ξ ψi ∈ [0, π

2], we ensure that the exponential map is bijective The logarithm map

fromψ itoψ j has the closed-form formula

−−−→

ψ i ψ j = logψi(ψ j) = ψj − ψ i , ψ j ψ i

1− ψ i , ψ j 2 cos−1 ψ i , ψ j . (2.5)

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2.2.1 Diffeomorphic Metric

In the setting of large deform diffeomorphic metric mapping (LDDMM), the set of anatomical shapes are placed into a metric shape space This is modeled by assuming that the shapes are generated from one to the other via the group of diffeomorphisms,

G The diffeomorphisms are introduced as transformations of the coordinates on the background space Ω ⊂ R3, i.e., G : Ω → Ω One approach, proposed by [87] and

adopted in this thesis, is to construct diffeomorphisms φ t ∈ G as a flow generated

via ordinary differential equations (ODEs), where φ t , t ∈ [0, 1] obeys the following

equation (see Figure 2.2),

˙

φ t = v t (φ t ), φ0 = Id, t ∈ [0, 1], (2.6)

where Id denotes the identity map and v tare the associated velocity vector fields The vector fieldsv tare constrained to be sufficiently smooth, so that Eq (2.6) is integrable and generates diffeomorphic transformations over finite time The smoothness is ensured by forcing v t to lie in a smooth Hilbert space (V , · V) withs-derivatives

having finite integral square and zero boundary [88, 89] In our case, we modelV as

a reproducing kernel Hilbert space with a linear operatorL associated with the norm

square u 2

V = Lu, u2, where·, ·2 denotes the L2 inner product The group of diffeomorphismsG(V ) are the solutions of Eq (2.6) with the vector fields satisfying

1

0 v t V dt < ∞.

We define a metric distance between a target shapeItargand a template shapeItemp

as the minimal length of curvesφ t · Itemp, t ∈ [0, 1], in a shape space such that, at time

Figure 2.2: Flow equation.

t = 1, φ1 · Itemp is as similar as possible toItarg The latter notation represents the group action ofφ1 onItemp For instance, in the image case (see Figure 2.3), for which

I temp = I temp (x), x ∈ Ω ⊂ R3, it isφ1· I temp = I temp ◦ φ −1

1 = I temp (φ −11 (x)) Lengths

of such curves are computed as the integrated norm v t V of the vector field generating the transformation

Figure 2.3: Mapping one shape to another via the group action of diffeomorphic

Using the duality isometry in Hilbert spaces, one can equivalently express the lengths in terms ofm t, interpreted as momentum such that for eachu ∈ V ,

m t , u ◦ φ t 2 =k −1

wherek V is the reproducing kernel ofV We let m, u2 denote theL2 inner product

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betweenm and u, but also, with a slight abuse, the result of the natural pairing between

m and v in cases where m is a pointwise density This identity is classically written as

φ ∗ t m t = k V −1 v t, whereφ ∗ t is referred to as the pullback operation on a vector measure,

m t Using the identity v t 2

V =k −1

V v t , v t 2 =m t , k V m t 2and the standard fact that energy-minimizing curves coincide with constant-speed length-minimizing curves, one can obtain the metric distance between the template and target shapes,ρ(Itemp, Itarg), as

ρ(Itemp, Itarg)2 = inf

m t

 1

0 < m t , v t ◦ φ t >2 dt (2.8) The minimum being computed over allm t such that : ˙φ t = v t (φ t ), v t = k V (φ ∗ t m t),

φ0 = id and φ1· Itemp = Itarg Note that since in this thesis we are dealing with vector fields inR3,k V (x, y) is a matrix kernel operator in order to get a proper definition.

2.2.2 Conservation Law of Momentum

One can prove thatm tsatisfies the following property at all times [90, 91]

Conservation Law of Momentum For all u ∈ V ,

m t , u2 =m0, (Dφ t)−1 u(φ t)2. (2.9)

Eq (2.9) uniquely specifiesm t as a linear form on V , given the initial momentum

m0and the evolving diffeomorphismφ t We see that by making a change of variables and obtain the following expression relatingm tto the initial momentumm0 and the geodesicφ tconnectingItemp andItarg,

m t=|Dφ −1

t |(Dφ −1

t ) m0◦ φ −1

wherestands for the transpose of matrix.

As a direct consequence of this property, given the initial momentumm0and the initial diffeomorphismφ0, one can generate a unique time-dependent diffeomorphic transformation and consquently the evolving shape with time,φ t · Itemp, as shown in Figure 2.4

Figure 2.4: Geodesic specified by initial momentum

In the study of anatomical shapes and variation, pioneered by [87], we consider the shapes as a manifold embedded in a high-dimensional space, where each shape is considered as a point on the manifold As a direct consequence of conservation law of momentum, given the initial momentumm0, one can generate a unique time-dependent diffeomorphic transformation which connects one shape to another Given a fixed shape, referred to as atlas,Iatlas, the space of the initial momentumm0, provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied

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