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

2 Background 2.1 Riemannian Manifold of ODFs 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 sphere S2 and its space is defined as P = {p : S2 → R+ |∀s ∈ S2 , p(s) ≥ 0; s∈S2 p(s)ds = 1} . The space of p 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 15 2. BACKGROUND 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 as ψ(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; 16 s∈S2 ψ (s)ds = 1}. (2.1) 2.1 Riemannian Manifold of ODFs We see that from Eq. (2.1), the functions ψ 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 the L2 metric, given as ξj , ξ k ψi = s∈S2 ξj (s)ξk (s)ds, (2.2) where ξj , ξk ∈ Tψi Ψ are tangent vectors at ψi . The geodesic distance between any two functions ψi , ψj ∈ Ψ on a unit Hilbert sphere is the angle dist(ψi , ψj ) = logψi (ψj ) ψi = cos−1 ψi , ψj = cos−1 s∈S2 ψi (s)ψj (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(ψi , ξ) = expψi (ξ) = cos( ξ ψi )ψi where ξ ∈ Tψi Ψ is a tangent vector at ψi and ξ ξ ψi + sin( ξ ψi ) ξ , ξ ψi = ξ, ξ ψi . ψi (2.4) By restricting ∈ [0, π2 ], we ensure that the exponential map is bijective. The logarithm map from ψi to ψj has the closed-form formula −−−→ ψj − ψi , ψj ψi cos−1 ψi , ψj . ψi ψj = logψi (ψj ) = − ψi , ψj 17 (2.5) 2. BACKGROUND 2.2 Large Deformation Diffeomorphic Metric Mapping 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 φt ∈ G as a flow generated via ordinary differential equations (ODEs), where φt , t ∈ [0, 1] obeys the following equation (see Figure 2.2), φ˙t = vt (φt ), φ0 = Id, t ∈ [0, 1], (2.6) where Id denotes the identity map and vt are the associated velocity vector fields. The vector fields vt are 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 vt to lie in a smooth Hilbert space (V , · V) with s-derivatives having finite integral square and zero boundary [88, 89]. In our case, we model V as a reproducing kernel Hilbert space with a linear operator L associated with the norm square u V = Lu, u , where ·, · denotes the L2 inner product. The group of diffeomorphisms G(V ) are the solutions of Eq. (2.6) with the vector fields satisfying vt V dt < ∞. We define a metric distance between a target shape Itarg and a template shape Itemp as the minimal length of curves φt · Itemp , t ∈ [0, 1], in a shape space such that, at time 18 2.2 Large Deformation Diffeomorphic Metric Mapping Figure 2.2: Flow equation. t = 1, φ1 · Itemp is as similar as possible to Itarg . The latter notation represents the group action of φ1 on Itemp . For instance, in the image case (see Figure 2.3), for which −1 Itemp = Itemp (x), x ∈ Ω ⊂ R3 , it is φ1 · Itemp = Itemp ◦ φ−1 = Itemp (φ1 (x)). Lengths of such curves are computed as the integrated norm vt V of the vector field generating the transformation. Figure 2.3: Mapping one shape to another via the group action of diffeomorphic transformation. On the right, the diffeomorphisim φ is shown on the square grid. Using the duality isometry in Hilbert spaces, one can equivalently express the lengths in terms of mt , interpreted as momentum such that for each u ∈ V , mt , u ◦ φt = kV−1 vt , u , where kV is the reproducing kernel of V . We let m, u 19 (2.7) denote the L2 inner product 2. BACKGROUND between m 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 mt = kV−1 vt , where φ∗t is referred to as the pullback operation on a vector measure, mt . Using the identity vt V = kV−1 vt , vt = mt , kV mt and 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 mt < mt , vt ◦ φt >2 dt (2.8) The minimum being computed over all mt such that : φ˙ t = vt (φt ), vt = kV (φ∗t mt ), φ0 = id and φ1 · Itemp = Itarg . Note that since in this thesis we are dealing with vector fields in R3 , kV (x, y) is a matrix kernel operator in order to get a proper definition. 2.2.2 Conservation Law of Momentum One can prove that mt satisfies the following property at all times [90, 91]. Conservation Law of Momentum. For all u ∈ V , mt , u = m0 , (Dφt )−1 u(φt ) . (2.9) Eq. (2.9) uniquely specifies mt as a linear form on V , given the initial momentum m0 and the evolving diffeomorphism φt . We see that by making a change of variables and obtain the following expression relating mt to the initial momentum m0 and the geodesic φt connecting Itemp and Itarg , −1 −1 mt = |Dφ−1 t |(Dφt ) m0 ◦ φt . 20 (2.10) 2.2 Large Deformation Diffeomorphic Metric Mapping where stands for the transpose of matrix. As a direct consequence of this property, given the initial momentum m0 and the initial diffeomorphism φ0 , one can generate a unique time-dependent diffeomorphic transformation and consquently the evolving shape with time, φt · 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 momentum m0 , 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 momentum m0 , provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied. 21 2. BACKGROUND 22 . assumed to be non-negative to ensure uniqueness. The space of such functions is defined as Ψ = {ψ : S 2 → R + |∀s ∈ S 2 , ψ(s) ≥ 0;  s∈S 2 ψ 2 (s)ds =1}. (2. 1) 16 2. 1 Riemannian Manifold of ODFs We. ODF, the diffusion angular profile of water molecules. The ODF is actually a diffusion probability density function (PDF) defined on a unit sphere S 2 and its space is defined as P = {p : S 2 → R + |∀s. intrinsic metric on the statistical manifold P and therefore invariant to re-parameterizations of the functions. There 15 2. BACKGROUND are many different parameterizations of PDFs that are equivalent