Diffeomorphic Mapping for Cortical Surfaces

, such that $f\in \mathbf {V}^{(r)}$ is a function at resolution r, $r\in [0,R]$ , with the level of detail increasing as r increases.

Let $T=(\{x_i\},\{\varSigma _{ijk}\})$ be a triangular surface mesh, where $\{x_i\},i=1,\dots ,N$ is a set of vertices and $\{\varSigma _{ijk}\}$ a set of simplices, with each simplex $\varSigma _{ijk}$ as a three tuple of points

. Given a mesh $T^{(r)}$ at level r, with coordinates $X^{(r)} = [x_1,\dots , x_i, \dots ,x_{N^{(r)}}]$ , where $x_i\in \mathbb {R}^3$ and $N^{(r)}$ is the number of vertices on $T^{(r)}$ . The new vertices on $T^{(r+1)}$ , denoted as $\hat{X}^{(r+1)}$ , can be given as

$\begin{aligned} \hat{X}^{(r+1)} = X^{(r)}A_{N^{(r)}\times M}, \end{aligned}$

(1)

where A is an $N^{(r)}\times M$ matrix for a simple subdivision scheme (where all elements of

except for $A_{ij}=A_{kj}=0.5$ ) and M is the number of the new vertices on $T^{(r+1)}$ , $X^{(r+1)} = [X^{(r)},\hat{X}^{(r+1)}]$ . Given an “averaging matrix” $B_{N^{(r+1)}\times N^{(r+1)}}$ , a general subdivision scheme can be rewritten as

$\begin{aligned} X^{(r+1)} = X^{(r)}[I_{N^{(r)}\times N^{(r)}} \quad A]B = \tilde{X}^{(r+1)}B = X^{(r)}P^{(r)}, \end{aligned}$

(2)

where $\tilde{X}^{(r+1)} = X^{(r)}[I_{N^{(r)}\times N^{(r)}} \quad A]$ , $P^{(r)} = [I_{N^{(r)}\times N^{(r)}} \quad A]B$ .

As explained in [12], surfaces can be parametrized with a function S(y), where y is defined on a base (coarsest) mesh $T^{(0)}$ , i.e. y is a point on one of the simplices in $\varSigma _{ijk}^{(0)}$ and can be tracked through a predefined subdivision process to a limit surface. We begin by first defining $S^{(0)}(y):=y$ . Let $S^{(r-1)}(y)$ be found in a simplex $\varSigma _{abc}^{(r)}$ with vertices $(\tilde{x}_a,\tilde{x}_b,\tilde{x}_c)$ , $\tilde{x}$ found in $\tilde{X}^{(r)}$ . Using the barycentric coordinates $(\lambda _a,\lambda _b,\lambda _c)$ such that $S^{(r-1)}(y) = \lambda _a\tilde{x}_a + \lambda _b\tilde{x}_b + \lambda _c\tilde{x}_c$ , we can induce a bijective map $S^{(r-1)}(y)\rightarrow S^{(r)}(y)$ where

$\begin{aligned} S^{(r)}(y) = \lambda _ax_a + \lambda _bx_b + \lambda _cx_c, \qquad x\in X^{(r)} \end{aligned}$

(3)

and $(x_{a},x_{b},x_{c})$ corresponds to $(\tilde{x}_a,\tilde{x}_b,\tilde{x}_c)$ of the simplex $\varSigma _{abc}^{(r)}$ . Then, $S(y)\!\!:=\lim _{r\rightarrow \infty }S^{(r)}(y)$ . In matrix form,

$\begin{aligned} S^{(r)}(y) = \varvec{\lambda }^{(r)}(y)(X^{(r)})^T. \end{aligned}$

(4)

It follows that

$\begin{aligned} S^{(r)}(y)&= \varvec{\lambda }^{(r)}(y) (X^{(r-1)}P^{(r-1)})^T \end{aligned}$

(5)

$\begin{aligned}&= \varvec{\lambda }^{(r)}(y) (P^{(r-1)})^T\cdots (P^{(0)})^T(X^{(0)})^T. \end{aligned}$

(6)

In other words, surfaces when parameterized into meshes, can be henceforth understood as functions from a small collection of triangles into $\mathbb {R}^3$ . The subdivision of triangles allows us to move from one resolution to another, providing a family of surfaces for registration. We show three resolutions of the brain cortex in Fig. 1.

Fig. 1.

Increasing levels of $X^{(r)}$ from left to right. Subcaptions indicates the corresponding levels, number of vertices, and number of faces – “Level-r (no. of vertices, no. of faces)”.

2.2 Multiresolution Large Deformation Diffeomorphic Metric Mapping for Surfaces

Now, we state a variational problem for mapping two surfaces under the framework of LDDMM. LDDMM assumes that transformation can be generated from one to another via flows of diffeomorphisms $\varphi _t$ , which are solutions of ordinary differential equations $\dot{\varphi }_t = v_t (\varphi _t), t \in [0,1],$ starting from the identity map $\varphi _0={{\mathtt {Id}}}$ . They are therefore characterized by time-dependent velocity vector fields $v_t, t \in [0,1]$ . We define a metric distance between a target surface $S_{targ}$ and an atlas surface $S_{atlas}$ as the minimal length of curves $\varphi _t \cdot S_{atlas}, t \in [0,1],$ in a shape space such that, at time

, $\varphi _1 \cdot S_{atlas} = S_{targ}$ . Lengths of such curves are computed as the integrated norm $\Vert v_t \Vert _V$ of the vector field, where $v_t \in V$ and V is a reproducing kernel Hilbert space with kernel

and norm $\Vert \cdot \Vert _V$ . To ensure solutions are diffeomorphisms, V must be a space of smooth vector fields. The duality isometry in Hilbert spaces allows us to express the lengths in terms of $m_t\in V^*$ , interpreted as momentum such that $\forall u\in V$ , $\langle m_t, u \circ \varphi _t\rangle _2 = \langle k_V^{-1}v_t, u\rangle _2$ , and $\langle m, u\rangle _2$ denotes the $\mathbb {L}^2$ inner product between m and u, With a slight abuse of symbols, it is the result of the natural pairing between m and v in cases where m is singular (e.g., a measure). This identity is classically written as $\varphi _t^* m_t = k_V^{-1} v_t$ , where $\varphi _t^*$ is referred to as the pullback operation on a vector measure,

. Using the identity $\Vert v_t\Vert _V^2 = \langle k_V^{-1}v_t, v_t\rangle _2=\langle m_t,k_Vm_t\rangle _2$ and the fact that energy-minimizing curves coincide with constant-speed length-minimizing curves, we obtain the metric distance between the atlas and target, $\rho (S_{atlas},S_{targ})$ , by minimizing $\Vert v_t\Vert _V^2$ , such that $\varphi _1 \cdot S_{atlas}=S_{targ}$ at time

[5]. We associate this with the variational problem in the form of

$\begin{aligned} J(m_t) =&\inf \nolimits _{m_t: \dot{\varphi }_t = k_Vm_t(\varphi _t), \varphi _0={\mathtt {Id}}} \rho (S_{atlas},S_{targ})^2 \nonumber \\&+ \gamma E(\varphi _1 \cdot S_{atlas},S_{targ}), \end{aligned}$

(7)

where E is defined based vector-valued measure as introduced in [18]. For any two surfaces

and

is defined as

$\begin{aligned} \begin{array}{ll} E(S_1,S_2) &{}= \sum \nolimits _{f,g} N^{t}_{f} k_W(c_g,c_f) N_g - 2\sum \nolimits _{f,q} N^t_f k_W(c_f,c_q) Nq \\ &{}\qquad + \sum \nolimits _{q,p} N_q^t k_W(c_q,c_p) N_r, \end{array} \end{aligned}$

(8)

where f, g are simplices from

while q, p are simplices from

is then the normal vector pointing out of the centre,

, of simplex g.

is a Gaussian kernel with bandwidth $\sigma _W$ . The metric distance $\rho (S_{atlas},S_{targ})^2$ could be easily computed as $\int _{0}^{1} ||v_t ||_V^2 dt$ .

We now construct the multiresolution diffeomorphic mapping for surfaces under the framework of LDDMM. In the previous section, we show that a surface, S, may be sequentially subsampled into meshes of decreasing resolution $T^{(r)}\dots T^{(1)}$ . With a slight abuse of notation, let us define these meshes as the discretization of the surface, rewriting $T^{(r)}$ as $S^{(r)}$ , such that $\lim \limits _{r\rightarrow \infty }S^{(r)}\rightarrow S$ . The duality isometry of

with

allows defining the smooth vector field,

through

, where

can sparsely anchor at the vertices on $S^{(r)}$ . Therefore, it is natural to seek $m_t^{(r)}$ defined at the vertices on $S^{(r)}$ and then construct the smooth vector field, $w_t^{(r)}= k_V^{(r)}m_t^{(r)}$ , where the size of $k_V^{(r)}$ can be adapted to the sparse level of the vertices on $S^{(r)}$ . From this construction, $w_t^{(r)}, r=0, 1, \dots , R$ can be defined via momentum $m_t^{(r)}\otimes \delta _x, x\in S^{(r)}, r=0, 1, \dots , R$ and construct independent tangent spaces of diffeomorphisms, $w^{(r)}\in W^{(r)}$ . The family of vector fields forms reproducing kernel Hilbert spaces, which could be summed across multiple resolutions, i.e., $\vartheta _t(w_t)= \sum _{r=0}^R w_t^{(r)}$ , to form one single vector field for the flow equation $\dot{\varphi _{t}^{\vartheta }} = \vartheta _t(\varphi ^{\vartheta }_{t})$ . Through this family of vector fields, we redefine $\rho ^{MRA}(S_{atlas},S_{targ})$ by minimizing $\int _0^1 \sum _{r=0}^R \Vert w_t^{(r)} \Vert _{W^{(r)}}^2 dt$ such that $\varphi _1^{\vartheta } \cdot S_{atlas}=S_{targ}$ at time

, where $$$\Vert w_t^{(r)} \Vert _{W^{(r)}}^2 = \big <(k_V^{(r)})^{-1}w_t^{(r)} , w_t^{(r)} \big >_2=\big <m_t^{(r)}, k_V^{(r)}m_t^{(r)} \big >_2$$” src=”/wp-content/uploads/2016/09/A339424_1_En_24_Chapter_IEq102.gif”></SPAN>. This construction of <SPAN id=IEq103 class=InlineEquation><IMG alt=$ is in turn similar to that proposed for the large deformation diffeomorphic kernel bundle mapping (LDDKBM) for the registration of images [16].

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