Registration, atlas generation, and statistical analysis of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions 3

Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions Due to the inter-subject anatomical variatio

Diffeomorphic Metric Mapping of

High Angular Resolution Diffusion

Imaging based on Riemannian

Structure of Orientation Distribution Functions

Due to the inter-subject anatomical variation, it is necessary to align ODF images ofdifferent subjects into a common space so that group-level statistical inference can beperformed (see Figure 3.1) In this chapter, we propose a novel registration algorithm

to align HARDI data characterized by ODFs across subjects under the Riemannianmanifold of ODFs and the LDDMM framework introduced in Chapter 2 Our proposedalgorithm seeks an optimal diffeomorphism of large deformation between two ODFfields in a spatial volume domain and at the same time, locally reorients an ODF in a

manner such that it remains consistent with the surrounding anatomical structure Tothis end, we first define the reorientation of an ODF when an affine transformation isapplied and subsequently, define the diffeomorphic group action to be applied on theODF based on this reorientation We incorporate the Riemannian metric of ODFs forquantifying the similarity of two HARDI images into a variational problem definedunder the LDDMM framework We finally derive the gradient of the cost function inboth Riemannian spaces of diffeomorphisms and the ODFs, and present its numericalimplementation Both synthetic and real brain HARDI data are used to illustrate theperformance of our registration algorithm.

HARDI

data

ODF images

ODF images

in common space

ODF atlas

ODF Reconstruction Registration

Atlas Generation

Biomarkers/ Inference

Statistical Analysis

serve as common space in registration

3.1 Affine Transformation on Square-Root ODFs

3.1 Affine Transformation on Square-Root ODFs

In this section, we discuss the reorientation of the√

ODF,ψ(s), when a non-singular

affine transformation A is applied As illustrated in Figure 3.2, we denote the

trans-formed √

ODF as ψ(s) = Aψ(s), reflecting the fact that an affine transformation

induces changes in both the magnitude ofψ and the sampling directions of s We will

now show how to derive the analytical form when a non-singular affine transformationacts on an ODF

Figure 3.2: Illustration of affine transformation on square-root ODFs (Similar to the shape

of ODF, the colors of ODF also indices the relative values of ODF in each direction, whereblue stands for low ODF value and red for high value.)

First of all, we denote s = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) in Cartesian

coor-dinates ands = (rsin θcos  ϕ, rsin θsin  ϕ, rcos θ) in Cartesian coordinates We first

assume that the change of the diffusion sampling directions due to affine transformation

A is

Similar to [56], we assume that the volume fraction of fibers with orientation near

direction s equalsp(s)dΩ, where dΩ is the small patch Just as in [56], we assume that

the volume fraction of fibers oriented towards the small patch dΩ remains the same

after the patch is transformed That is,

d xdydz = det (A)dxdydz,

and in polar coordinates,

r2sin θd θd ϕd r = det(A)r2

3.1 Affine Transformation on Square-Root ODFs

Removing sin θdθdϕ from both sides yields

An alternative way of obtaining the property in Eqn (3.6) is to assume that the

change of the diffusion directions due to affine transformation A is

where the transformed sampling directionss are normalized back into the unit sphere

S2 This is analogous to a pullback deformation Notice that for s ∈ S2, Eq (3.7)

defines an invertible function of s and therefore, we can find the ODF A ψ(s) using the

change-of-variable technique of PDF Recall the fundamental theorem for PDF: let X

be a continuous random variable having probability density function f X (x) Suppose

g(x) is one-to-one and differentiable function of x Then the random variable Y defined

by Y = g(X) has a probability density function given by f Y (y) = f X (g −1 (y)) |J(y)|

where J is the Jacobian of g −1 (y) Since A is a 3 × 3 matrix, the determinant of the

Jacobian in this case is detA −1

A −1 s3 In either case, the following theorem is obtained

Theorem 3.1.1 Reorientation of ψ based on affine transformation A Let Aψ(s)

be the result of an affine transformation A acting on a √

ODF ψ(s) The following

analytical equation holds true

where · is the norm of a vector.

Property 1 Assume A and B to be two matrices of affine transformations and ψ is a square-root ODF The following property holds true

where (A B) stands for matrix muliplication between A and B.

Proof Base on the equation (3.8), it yields

3.2 Diffeomorphic Group Action on Square-Root ODFs

consistent with the surrounding anatomical structure and at the same time, not solelydependent on the rotation Rather, by constructing the change-of-variable technique

as discussed above, the reorientation takes into account the effects of the affine formation and ensures the volume fraction of fibers oriented toward a small patchmust remain the same after the patch is transformed While [56] computes the ODFreorientation numerically by computing the corresponding Jacobian at each samplingdirection via a series of transformations and applying it to transform the orientation,there is in fact an analytical closed form formula for the reorientation as provided by

trans-Theorem 3.1.1 Figure 3.3 illustrates how A ψ(s) varies when A is a rotation, shearing,

or scaling andψ(s) is an isotropic ODF, an ODF with a single fiber, or an ODF with

crossing fibers From Figure 3.3, one immediately observes that a shearing or scalingintroduces anisotropy under the reorientation scheme used here The phenomena is in

line with what is observed in [92] By construction, A ψ(s) fulfills the definition of the

√

ODF, i.e., A ψ(s) is positive and the integration of (Aψ(s))2 is equal to 1 Hence,

the similarity of A ψ(s) to the square-root ODFs can be quantified in the Riemannian

structure given in§2.1 for the HARDI registration.

3.2 Diffeomorphic Group Action on Square-Root ODFs

We have shown in§3.1 how to reorient ψ located at a fixed spatial position x in the

image volume Ω ⊂ R3 through an affine transformation In this section, we define

an action of diffeomorphisms φ : Ω → Ω on ψ, which takes into consideration the

reorientation ofψ as well as the transformation of the spatial volume in Ω ⊂ R3, asillustrated in Figure 3.4 Denoteψ(s, x) as the √ODF with the orientation direction

s ∈ S2 located at x ∈ Ω We define the action of diffeomorphisms on ψ(s, x) in the

Figure 3.3: Examples of local affine transformations on an isotropic ODF in the firstrow, an ODF with a single orientation fiber in the middle row, and an ODF with crossingfibers in the bottom row From left to right, three types of affine transformations,A, on

the ODFs are demonstrated: in panel (a), a rotation with angleθ z, whereA = [cos θ z −

sinθ z 0; sinθ z cosθ z0; 0 0 1]; in panel (b), a vertical shearing with factor ρ y, where

A = [1 0 0; −ρ y 1 0; 0 0 1]; and in panel (c), a vertical scaling with factorς y where

As-3.2 Diffeomorphic Group Action on Square-Root ODFs

direction s ∈ S2 located at x ∈ Ω The following property holds true

where ◦ stands for composition between φ and ϕ.

Proof Based on the equation (3.10), it yields

where we denote A x = D x φ, B x = D x ϕ, and B φ(x) = D φ(x) ϕ.

Using the property from the equation (3.9), we have

where it will be used in the rest of the chapter

Since φ · ψ(s, x) is in the space of √ODF, the Riemannian distance given in§2.1

Figure 3.4: Illustration of diffeomorphic group action on square-root ODFs

can be directly used to quantify the similarity of φ · ψ(s, x) to other √ODFs, which weemploy in the HARDI registration described in the following section

3.3 Large Deformation Diffeomorphic Metric Mapping

for ODFs

The previous sections equip us with an appropriate representation of the ODF and itsdiffeomorphic action Now, we state a variational problem for mapping ODFs from onevolume to another We define this problem in the “large deformation” setting of Grenan-der’s group action approach for modeling shapes, that is, ODF volumes are modeled byassuming that they can be generated from one to another via flows of diffeomorphisms

φ t, which are solutions of ordinary differential equations ˙φ t = v t (φ t ), t ∈ [0, 1], starting

from the identity map φ0 = Id They are therefore characterized by time-dependent

velocity vector fields v t , t ∈ [0, 1] We define a metric distance between a target volume

ψtargand a template volumeψtempas the minimal length of curves φ t ·ψtemp, t ∈ [0, 1],

in a shape space such that, at time t = 1, φ1· ψtemp =ψtarg Lengths of such curves arecomputed as the integrated normv t  V of the vector field generating the transformation,

where v t ∈ V , where V is a reproducing kernel Hilbert space with kernel k V and norm

3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs

 ·  V

To ensure solutions are diffeomorphisms, V must be a space of smooth vector fields

[88] Using the duality isometry in Hilbert spaces, one can equivalently express the

lengths in terms of m t , interpreted as momentum such that for each u ∈ V ,

m t , u ◦ φ t 2 = k −1

where we let 2 denote the L2 inner product 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 singular (e.g., a measure) This identity is classically written as φ ∗ t m t = k V −1 v t, where

φ ∗ 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 2 and the standard fact that energy-minimizingcurves coincide with constant-speed length-minimizing curves, one can obtain themetric distance between the template and target√

ODF volumes, ρ( ψtemp, ψtarg), byminimizing1

0 m t , k V m t 2dt such that φ1· ψtemp =ψtargat time t = 1.

We associate this with the variational problem in the form of

with E xas the metric distance between the deformed√

ODF template, φ1· ψtemp(s, x),

and the target,ψtarg(s, x) We use the Riemannian metric given in §2.1 and rewrite Eq.

where A = Dφ1, the Jacobian of φ1 For the sake of simplicity, we denoteψtarg(s, x)

asψtarg(x) Note that since we are dealing with vector fields inR3, the kernel of V is

a matrix kernel operator in order to get a proper definition We define this kernel as

k VId3×3, where Id3×3 is an identity matrix, such that k V can be a scalar kernel In therest of the chapter, we shall refer to this LDDMM mapping problem as LDDMM-ODF

3.3.1 Gradient of J with respect to mt

The gradient of J with respect to m t can be computed via studying a variation m  t =

m t +  mt on J such that the derivative of J with respect to  is expressed as a function

ofmt According to the general LDDMM framework derived in [93, 94], we directly

give the expression of the gradient of J with respect to m t as

where ∂ φ s (k V m s ) is the partial derivative of k V m s with respect to φ s η tin Eq (3.17)

can be solved backward given η1 = ∇ φ1E, where E = 

x∈Ω E x dx, which will be

discussed in the following

3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs

Gradient of E with respect to φ1: The computation of∇ φ1E is not straightforward

and the Riemannian structure of ODFs has to be incorporated Let’s first compute

∇ φ1E x at a fixed location, x We consider a variation φ 1 = φ1+ h of φ1and denote

the corresponding variation in A as A  , where A = D x φ1and A  = D x φ 1 Here, we

directly give the expression of ∂  E x | =0 and the reader is referred to§3.3.1.1 for the full

derivation of terms (A) and (B) in the following equation

=2



logAψtemp◦φ −1

1 (x) ψtarg(x), ∂ log A  ψtemp◦(φ 1)−1(x) ψtarg(x)



Aψtemp◦φ −11 (x)

=− 2logAψtemp◦φ −1

1 (x) ψtarg(x), ∂ log Aψtemp◦φ −11 (x) A

=−2logAψtemp◦φ −1

1 (x) ψtarg(x), ∂ log Aψtemp◦φ −11 (x) A ψtemp◦ (φ 

−2logAψtemp◦φ −1

1 (x) ψtarg(x), ∂ log Aψtemp◦φ −11 (x) A

logAψtemp(x) ψtarg(φ1(x)), L2x

Aψtemp (x)

div

logAψtemp(x) ψtarg(φ1(x)), L3

wheredenotes the matrix transpose and ei is a 3× 1 vector with the ith element as

one and the rest as zero Aψtemp(x) is the Fisher-Rao metric defined in Eq (2.2)

∇ x (A ψtemp) in term (A) is the first derivative of the√

ODF, A ψtemp, with respect to

x Since A ψtemp also lies in the Riemannian manifold of√

where wi is the ith column of (D x φ1)−1 Denotes =D x φ1 −1

s u i is the ith element

3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs

change of variable from x to φ −11 (x) in the integration, ∇ φ1E can be written as

log Aψtemp (x) ψtarg(φ1(x)) , |e1

1|logAψtemp (x) A ψtemp(x + e1) Aψtemp(x)

log Aψtemp (x) ψtarg(φ1(x)) , |e1

2|logAψtemp (x) A ψtemp(x + e2) Aψtemp(x)

log Aψtemp (x) ψtarg(φ1(x)) , |e1

3|logAψtemp (x) A ψtemp(x + e3) Aψtemp(x)

logAψtemp(x) ψtarg(φ1(x)), L2

x

Aψtemp (x)

div

logAψtemp(x) ψtarg(φ1(x)), L3

We now would like to emphasize the difference of the above gradient derivation

from our previous work [2] The fundamental difference is that in [2], we assume that A does not change under the variation φ 1and thus, do not consider the variation in A, i.e.,

A  is ignored Therefore, in [2], the gradient of E with respect to φ1only incorporatesterm (A) of Eq (3.18) This term is similar to the scalar image matching case and onlytakes into account image shape difference in the volume space We illustrate this inFigure 3.5, where we have one template image and two target images Figure 3.5 (a)shows the template image, where its overall image shape is circular and the ODFs ateach voxel inside the circle are oriented horizontally Figure 3.5 (b) shows the first targetimage, where its overall image shape is an ellipsoid and the ODFs inside its voxels areoriented horizontally Figure 3.5 (c) shows the second target image, where its overallimage shape is circular as the template image but the ODFs at each voxel inside thecircle are oriented at 45◦ The results obtained using only term (A) as proposed in [2]

are shown in Figures 3.5 (f, g) In Figure 3.5 (f), we see that because of the contribution

of term (A) in Eq (3.18), the deformation field and its corresponding momentum inthe target space point to the direction that enlarges the circle to the ellipsoid However,

in Figure 3.5 (g), we see that term (A) in Eq (3.18) is unable to account for suchdeformations as the image shapes are the same, resulting in the deformation field beingzero Figures 3.5 (d, e) show the results using both terms (A) and (B) as proposed inthis current chapter From Figure 3.5 (d), we see that the proposed algorithm gives adeformation field that enlarges the circle to the ellipsoid, similar to that of Figure 3.5 (f).More importantly, as shown in Figure 3.5 (e), we see that the deformation that amounts

to rotating the ODFs is captured by term (B) of Eq (3.18), which is a property that [2]does not possess

3.3.1.1 Derivation of the gradient of E x with respect to φ1

We now elaborate on the derivation of terms (A) and (B) in Eq (3.18) The readercan skip this subsection without any discontinuation by assuming that the derivation ofterms (A) and (B) in Eq (3.18) holds true

Term (A): For the sake of simplicity, we denote term (A) of Eq (3.18) as E A andrewrite

3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs

Term (B): We denote term (B) of Eq (3.18) as E Band rewrite

1

−1

We now derive the above equation in order to express it in an explicit form of h Before

doing so, we first define a 3× 3 identity matrix as Id3×3 = [e1, e2, e3], where ei is

a 3× 1 vector with the ith element as one and the rest as zero Denote (D x φ1)−1 =

[w1, w2, w3], where wi is the ith column of (D x φ1)−1 Thus, the trace of D x h(D x φ1)−1

We introduce the following lemma [95] that leads to a simple expression of E B.

Lemma 3.3.1 For smooth vector fields, h, u, w, defined in a bounded open domain in

where u i is the ith element of u.

As a consequence, when defining L i x = (D x φ1)−1 su i − 1

x

Aψtemp (x)

div

logAψtemp(x) ψtarg(φ1(x)), L2

x

Aψtemp (x)

div

logAψtemp(x) ψtarg(φ1(x)), L3

3. 3.1.1 Derivation of the gradient of E x with respect to φ1

We now elaborate on the derivation of terms (A) and (B) in Eq (3. 18) The readercan... subsection without any discontinuation by assuming that the derivation ofterms (A) and (B) in Eq (3. 18) holds true

Term (A): For the sake of simplicity, we denote term (A) of Eq (3. 18)... (f), we see that because of the contribution

of term (A) in Eq (3. 18), the deformation field and its corresponding momentum inthe target space point to the direction that enlarges the circle