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

3
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 of
different subjects into a common space so that group-level statistical inference can be
performed (see Figure 3.1). In this chapter, we propose a novel registration algorithm
to align HARDI data characterized by ODFs across subjects under the Riemannian
manifold of ODFs and the LDDMM framework introduced in Chapter 2. Our proposed
algorithm seeks an optimal diffeomorphism of large deformation between two ODF
fields in a spatial volume domain and at the same time, locally reorients an ODF in a
23
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
manner such that it remains consistent with the surrounding anatomical structure. To
this end, we first define the reorientation of an ODF when an affine transformation is
applied and subsequently, define the diffeomorphic group action to be applied on the
ODF based on this reorientation. We incorporate the Riemannian metric of ODFs for
quantifying the similarity of two HARDI images into a variational problem defined
under the LDDMM framework. We finally derive the gradient of the cost function in
both Riemannian spaces of diffeomorphisms and the ODFs, and present its numerical
implementation. Both synthetic and real brain HARDI data are used to illustrate the
performance 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
Subjects
Data
Acquisition
Figure 3.1: The role of Chapter 3 in the ODF-based analysis framework.
24
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 of ψ and the sampling directions of s. We will
now show how to derive the analytical form when a non-singular affine transformation
acts 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, where
blue 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 and

s =(r sin

θ cos ϕ, r sin

θ sin ϕ, r cos


θ)
in Cartesian coordinates. We first
assume that the change of the diffusion sampling directions due to affine transformation
A is

s = As . (3.1)
Similar to [
56
], we assume that the volume fraction of fibers with orientation near
direction
s
equals
p(s)dΩ
, where
dΩ
is the small patch. Just as in [
56
], we assume that
25
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
the volume fraction of fibers oriented towards the small patch
dΩ
remains the same
after the patch is transformed. That is,
p(s)dΩ=

p



s


s

d

Ω, (3.2)
and in polar coordinates,
p(θ, ϕ)sinθdθdφ =

p(

θ, ϕ)sin

θd

θdϕ. (3.3)
When

s = As in Cartesian coordinates, for the small volume fraction, we have
dxdydz =det(A)dxdydz,
and in polar coordinates,
r
2
sin

θd


θdϕdr =det(A)r
2
sin θdθdϕdr.
Since r =
As
s
r, we obtain
As
3
s
3
r
2
sin

θd

θdϕdr =det(A)r
2
sin θdθdϕdr,
⇒ sin

θd

θdϕ =
s
3
As
3

det(A)sinθdθdϕ. (3.4)
From Eqns. (3.3) and (3.4), we get
p(θ, ϕ)sinθdθdφ =

p(

θ, ϕ)sin

θd

θdϕ
=

p(

θ, ϕ)
s
3
As
3
det(A)sinθdθdϕ. (3.5)
26
3.1 Affine Transformation on Square-Root ODFs
Removing sin θdθdϕ from both sides yields
p

s
s

=

det(A)s
3
As
3

p

As
As

Finally, we make a change of variable from s to s → A
−1
s, giving the following

p

s
s

=
det A
−1
s
3
A
−1
s
3
p


A
−1
s
A
−1
s

.
For an ODF, s ∈ S
2
, s =1, and therefore, we have

p (s)=
det A
−1
A
−1
s
3
p

A
−1
s
A
−1
s

⇒ Aψ(s)=


det A
−1
A
−1
s
3
ψ

A
−1
s
A
−1
s

. (3.6)
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

s =
A
−1
s
A
−1
s
, (3.7)
where the transformed sampling directions


s
are normalized back into the unit sphere
S
2
. This is analogous to a pullback deformation. Notice that for
s ∈ S
2
, Eq.
(3.7)
defines an invertible function of
s
and therefore, we can find the ODF
Aψ(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
27
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
Jacobian in this case is
det A
−1
A
−1
s
3

. 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
Aψ(s)=

det A
−1
A
−1
s
3
ψ

A
−1
s
A
−1
s

, (3.8)
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
B(Aψ)(s)=(AB)ψ(s), (3.9)
where (AB) stands for matrix muliplication between A and B.
Proof. Base on the equation (3.8), it yields
B(Aψ)(s)=B


det A
−1
A
−1
s
3
ψ

A
−1
s
A
−1
s



=

det A
−1
det B
−1
A
−1
s
3
B
−1
s
3
ψ

A
−1
B
−1
s
A
−1
B
−1
s

=


det (AB)
−1
(AB)
−1
s
3
ψ

(AB)
−1
s
(AB)
−1
s

=(AB)ψ(s)
The ODF reorientation used in this work ensures that the transformed ODF remains
28
3.2 Diffeomorphic Group Action on Square-Root ODFs
consistent with the surrounding anatomical structure and at the same time, not solely
dependent on the rotation. Rather, by constructing the change-of-variable technique
as discussed above, the reorientation takes into account the effects of the affine trans-
formation and ensures the volume fraction of fibers oriented toward a small patch
must remain the same after the patch is transformed. While [
56
] computes the ODF
reorientation numerically by computing the corresponding Jacobian at each sampling
direction 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
Theorem 3.1.1. Figure 3.3 illustrates how

Aψ(s)
varies when
A
is a rotation, shearing,
or scaling and
ψ(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 scaling
introduces anisotropy under the reorientation scheme used here. The phenomena is in
line with what is observed in [
92
]. By construction,
Aψ(s)
fulfills the definition of the
√
ODF
, i.e.,
Aψ(s)
is positive and the integration of
(Aψ(s))
2
is equal to
1
. Hence,
the similarity of
Aψ(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 ⊂ R
3
through an affine transformation. In this section, we define
an action of diffeomorphisms
φ : Ω → Ω
on
ψ
, which takes into consideration the
reorientation of
ψ
as well as the transformation of the spatial volume in
Ω ⊂ R
3
,as
illustrated in Figure 3.4. Denote
ψ(s,x)
as the
√
ODF
with the orientation direction
s ∈ S
2
located at

x ∈ Ω
. We define the action of diffeomorphisms on
ψ(s,x)
in the
29
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
Figure 3.3:
Examples of local affine transformations on an isotropic ODF in the first
row, an ODF with a single orientation fiber in the middle row, and an ODF with crossing
fibers 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 angle
θ
z
, where
A =[cosθ
z
−
sin θ
z
0; sin θ
z
cos θ
z
0; 001]
; in panel (b), a vertical shearing with factor
ρ

y
, where
A =[100; −ρ
y
10; 001]
; and in panel (c), a vertical scaling with factor
ς
y
where
A = [1 0 0; 0 ς
y
0; 001].
form of
φ · ψ(s,x)=A
φ
−1
(x)
ψ(s,φ
−1
(x)),
where the local affine transformation
A
x
at spatial coordinates
x
is defined as the
Jacobian matrix of
φ
evaluated at
x

, i.e.,
A
x
= D
x
φ
. According to Eq.
(3.8)
, the action
of diffeomorphisms on ψ(s,x) can be computed as
φ · ψ(s,x)=





det

D
φ
−1
φ

−1




D
φ

−1
φ

−1
s



3
ψ

(D
φ
−1
φ

−1
s
(D
φ
−1
φ

−1
s
,φ
−1
(x)

. (3.10)

Property 2. (The law of composition for diffeomorphic group action on ODF)
As-
sume
φ
and
ϕ
to be two diffeomorphisms and
ψ(s,x)
as the
√
ODF
with the orientation
30
3.2 Diffeomorphic Group Action on Square-Root ODFs
direction s ∈ S
2
located at x ∈ Ω. The following property holds true
ϕ · (φ · ψ)(s,x)=(ϕ ◦φ) · ψ(s,x) , (3.11)
where ◦ stands for composition between φ and ϕ.
Proof. Based on the equation (3.10), it yields
ϕ · (φ · ψ)(s,x)=ϕ ·

(A
x
ψ(s,x)) ◦ φ
−1
(x)

=


B
x
(A
x
ψ(s,x)) ◦ φ
−1

◦ ϕ
−1
(x)
=

B
φ(x)
(A
x
ψ(s,x))

◦ φ
−1
◦ ϕ
−1
(x) ,
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
ϕ · (φ · ψ)(s,x)=

B
φ(x)
A
x

ψ(s,x)

◦ φ
−1
◦ ϕ
−1
(x)
=[D
x
(ϕ ◦ φ) ψ(s,x)] ◦ (ϕ ◦φ)
−1
(x)
=(ϕ ◦ φ) · ψ(s,x) .
For the sake of simplicity, we denote φ · ψ(s,x) as
φ · ψ(s,x)=Aψ ◦ φ
−1

(x) , (3.12)
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
31
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
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
√
ODF
s, which we
employ 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 its
diffeomorphic action. Now, we state a variational problem for mapping ODFs from one
volume 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 by
assuming 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
φ
0
= 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
ψ
targ
and a template volume
ψ
temp
as the minimal length of curves
φ
t
·ψ

temp
,t∈ [0, 1],
in a shape space such that, at time
t =1
,
φ
1
·ψ
temp
= ψ
targ
. Lengths of such curves are
computed as the integrated norm
v
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
32
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
V
v
t
,u
2
, (3.13)
where we let
m, u
2
denote the

L
2
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
−1
V
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-minimizing
curves coincide with constant-speed length-minimizing curves, one can obtain the
metric distance between the template and target
√
ODF

volumes,
ρ(ψ
temp
, ψ
targ
)
,by
minimizing

1
0
m
t
,k
V
m
t

2
dt such that φ
1
· ψ
temp
= ψ
targ
at time t =1.
We associate this with the variational problem in the form of
J(m
t
) = inf

m
t
:
˙
φ
t
=k
V
m
t
(φ
t
),
φ
0
=Id
ρ(ψ
temp
, ψ
targ
)
2
+ λ

x∈Ω
E
x
(φ
1
· ψ

temp
(s,x), ψ
targ
(s,x))dx
(3.14)
with
E
x
as the metric distance between the deformed
√
ODF
template,
φ
1
· ψ
temp
(s,x)
,
and the target,
ψ
targ
(s,x)
. We use the Riemannian metric given in
§
2.1 and rewrite Eq.
33
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
(3.14) as

J(m
t
) = inf
m
t
:
˙
φ
t
=k
V
m
t
(φ
t
),
φ
0
=Id

1
0
m
t
,k
V
m
t

2

dt
+ λ

x∈Ω
log
Aψ
temp
◦φ
−1
1
(x)
(ψ
targ
(x))
2
Aψ
temp
◦φ
−1
1
(x)
dx, (3.15)
where
A = Dφ
1
, the Jacobian of
φ
1
. For the sake of simplicity, we denote
ψ

targ
(s,x)
as
ψ
targ
(x)
. Note that since we are dealing with vector fields in
R
3
, the kernel of
V
is
a matrix kernel operator in order to get a proper definition. We define this kernel as
k
V
Id
3×3
, where
Id
3×3
is an identity matrix, such that
k
V
can be a scalar kernel. In the
rest of the chapter, we shall refer to this LDDMM mapping problem as LDDMM-ODF.
3.3.1 Gradient of J with respect to m
t
The gradient of
J
with respect to

m
t
can be computed via studying a variation
m

t
=
m
t
+  m
t
on
J
such that the derivative of
J
with respect to

is expressed as a function
of
m
t
. 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
∇J(m

t
)=2m
t
+ λη
t
, (3.16)
where
η
t
= ∇
φ
1
E +

1
t

∂
φ
s
(k
V
m
s
)


(η
s
+ m

s
)ds , (3.17)
where
∂
φ
s
(k
V
m
s
)
is the partial derivative of
k
V
m
s
with respect to
φ
s
.
η
t
in Eq.
(3.17)
can be solved backward given
η
1
= ∇
φ
1

E
, where
E =

x∈Ω
E
x
dx
, which will be
discussed in the following.
34
3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs
Gradient of E with respect to φ
1
:
The computation of
∇
φ
1
E
is not straightforward
and the Riemannian structure of ODFs has to be incorporated. Let’s first compute
∇
φ
1
E
x
at a fixed location,
x
. We consider a variation

φ

1
= φ
1
+ h
of
φ
1
and denote
the corresponding variation in
A
as
A

, where
A = D
x
φ
1
and
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.
∂

E
x
|
=0
(3.18)
=2

log
Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),

∂ log
A

ψ
temp
◦(φ

1
)
−1
(x)
ψ
targ
(x)
∂
|
=0

Aψ
temp
◦φ
−1
1
(x)
= − 2

log
Aψ
temp
◦φ

−1
1
(x)
ψ
targ
(x),
∂ log
Aψ
temp
◦φ
−1
1
(x)
A

ψ
temp
◦ (φ

1
)
−1
(x)
∂
|
=0

Aψ
temp
◦φ

−1
1
(x)
= −2

log
Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),
∂ log
Aψ
temp
◦φ
−1
1
(x)
Aψ
temp
◦ (φ

1
)
−1
(x)

∂
|
=0

Aψ
temp
◦φ
−1
1
(x)
  
term (A)
−2

log
Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),
∂ log
Aψ
temp
◦φ
−1
1

(x)
A

ψ
temp
◦ (φ
1
)
−1
(x)
∂
|
=0

Aψ
temp
◦φ
−1
1
(x)
  
term (B)
=2

log
Aψ
temp
◦φ
−1
1

(x)
ψ
targ
(x),


(D
x
φ
1
)
−
∇
x
(Aψ
temp
),h


◦ φ
−1
1
(x)

Aψ
temp
◦φ
−1
1
(x)

  
term (A)
+2

⎡
⎢
⎢
⎢
⎢
⎣
div

log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
1
x

Aψ
temp
(x)

div


log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
2
x

Aψ
temp
(x)

div

log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
3
x


Aψ
temp
(x)

⎤
⎥
⎥
⎥
⎥
⎦
,h

◦ φ
−1
1
(x)

 
term (B)

,
35
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
where

denotes the matrix transpose and
e
i

is a
3 × 1
vector with the
i
th element as
one and the rest as zero.
·, ·
Aψ
temp
(x)
is the Fisher-Rao metric defined in Eq.
(2.2)
.
∇
x
(Aψ
temp
)
in term (A) is the first derivative of the
√
ODF
,
Aψ
temp
, with respect to
x
. Since
Aψ
temp
also lies in the Riemannian manifold of

√
ODF
s,
∇
x
(Aψ
temp
)
is a
vector with each element being a logarithm map of Aψ
temp
and is defined as
∇
x
[Aψ
temp
(x)] =
⎡
⎢
⎢
⎢
⎢
⎣
1
|e
1
|
log
Aψ
temp

(x)
Aψ
temp
(x + e
1
)
1
|e
2
|
log
Aψ
temp
(x)
Aψ
temp
(x + e
2
)
1
|e
3
|
log
Aψ
temp
(x)
Aψ
temp
(x + e

3
)
⎤
⎥
⎥
⎥
⎥
⎦
,
where
e
1
, e
2
and
e
3
indicate small variations in three orthonormal directions of
R
3
, respectively.
In term (B) of Eq.
(3.18)
, we define
L
x
as a
3 ×3
matrix of logarithm maps with its
ith column written as

L
i
x
=(D
x
φ
1
)
−1
su
i
−
1
2
Aψ
temp
(s,x)w
i
,
where
w
i
is the
i
th column of
(D
x
φ
1
)

−1
. Denote
s =

D
x
φ
1

−1
s
.
u
i
is the
i
th element
of vector
u = −

det

D
x
φ
1

−1
(D
x

φ
1
)
−
∇
s

ψ

s
s
,x


s
3

.
In sum,
∇
φ
1
E
can be computed by integrating
∇
φ
1
E
x
over the image space. With a

36
3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs
change of variable from x to φ
−1
1
(x) in the integration, ∇
φ
1
E can be written as
∇
φ
1
E =2

x∈Ω
det(φ
1
(x))
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
(D

x
φ
1
)
−
(3.19)
⎡
⎢
⎢
⎢
⎢
⎣
 log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)) ,
1
|e
1
|
log
Aψ
temp
(x)
Aψ

temp
(x + e
1
) 
Aψ
temp
(x)
 log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)) ,
1
|e
2
|
log
Aψ
temp
(x)
Aψ
temp
(x + e
2
) 
Aψ

temp
(x)
 log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)) ,
1
|e
3
|
log
Aψ
temp
(x)
Aψ
temp
(x + e
3
) 
Aψ
temp
(x)
⎤
⎥
⎥

⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎣
div

log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
1
x

Aψ
temp
(x)

div


log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
2
x

Aψ
temp
(x)

div

log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
3
x


Aψ
temp
(x)

⎤
⎥
⎥
⎥
⎥
⎦
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
dx .
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

1
and thus, do not consider the variation in
A
, i.e.,
A

is ignored. Therefore, in [
2
], the gradient of
E
with respect to
φ
1
only incorporates
term (A) of Eq.
(3.18)
. This term is similar to the scalar image matching case and only
takes into account image shape difference in the volume space. We illustrate this in
Figure 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 at
each voxel inside the circle are oriented horizontally. Figure 3.5 (b) shows the first target
image, where its overall image shape is an ellipsoid and the ODFs inside its voxels are
oriented horizontally. Figure 3.5 (c) shows the second target image, where its overall
image shape is circular as the template image but the ODFs at each voxel inside the
circle are oriented at
45
◦
. The results obtained using only term (A) as proposed in [

2
]
37
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
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 in
the 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 such
deformations as the image shapes are the same, resulting in the deformation field being
zero. Figures 3.5 (d, e) show the results using both terms (A) and (B) as proposed in
this current chapter. From Figure 3.5 (d), we see that the proposed algorithm gives a
deformation 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 φ
1
We now elaborate on the derivation of terms (A) and (B) in Eq.

(3.18)
. The reader
can skip this subsection without any discontinuation by assuming that the derivation of
terms (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
and
rewrite
E
A
= −2

log
Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),
∂ log
Aψ
temp
◦φ

−1
1
(x)
Aψ
temp
◦ (φ

1
)
−1
(x)
∂
|
=0

Aψ
temp
◦φ
−1
1
(x)
.
Given
∂(φ
1
+h)
−1
(x)
∂
|

=0
= −[(Dφ
1
)
−1
h] ◦ φ
−1
1
(x),wehave
E
A
=2

log
Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),


(D
x
φ
1
)

−
∇
x
(Aψ
temp
),h


◦ φ
−1
1
(x)

Aψ
temp
◦φ
−1
1
(x)
.
38
3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs
Term (B): We denote term (B) of Eq. (3.18) as E
B
and rewrite
E
B
= −2

log

Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),
∂ log
Aψ
temp
◦φ
−1
1
(x)
A

ψ
temp
◦ (φ
1
)
−1
(x)
∂
|
=0

Aψ

temp
◦φ
−1
1
(x)
= −2

log
Aψ
temp
◦φ
−1
1
(x)
ψ
targ
(x),
∂A

ψ
temp
(s,x) ◦φ
−1
1
(x)
∂
|
=0

Aψ

temp
◦φ
−1
1
(x)
.
According to Theorem 3.1.1, we have
A

ψ
temp
(s,x) ◦φ
−1
1
(x)=
⎡
⎢
⎣





det

D
x
φ

1


−1




D
x
φ

1

−1
s



3
ψ

(D
x
φ

1

−1
s
(D
x

φ

1

−1
s
,x

⎤
⎥
⎦
◦ φ
−1
1
(x) .
Denote s =

D
x
φ
1

−1
s.
Given
∂(D
x
φ

1

)
−1
(x)
∂
|
=0
=
∂(D
x
φ
1
+D
x
h)
−1
(x)
∂
|
=0
= −(D
x
φ
1
)
−1
D
x
h(D
x
φ

1
)
−1
,we
can now compute
∂A

ψ
temp
(s,x) ◦φ
−1
1
(x)
∂
|
=0
=

−

det

D
x
φ
1

−1

∇

s

ψ

s
s
,x


s
3

, (D
x
φ
1
)
−1
D
x
h(D
x
φ
1
)
−1
s

−
1

2
Aψ
temp
(s,x)trace

D
x
h(D
x
φ
1
)
−1


◦ φ
−1
1
(x)
=


D
x
h(D
x
φ
1
)
−1

s,u

−
1
2
Aψ
temp
(s,x)trace

D
x
h(D
x
φ
1
)
−1


◦ φ
−1
1
(x) ,
where
u = −

det

D
x

φ
1

−1
(D
x
φ
1
)
−
∇
s

ψ

s
s
,x


s
3

.
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

Id
3×3
=[e
1
, e
2
, e
3
]
, where
e
i
is
a
3 × 1
vector with the
i
th element as one and the rest as zero. Denote
(D
x
φ
1
)
−1
=
[w
1
, w
2
, w

3
]
, where
w
i
is the
i
th column of
(D
x
φ
1
)
−1
. Thus, the trace of
D
x
h(D
x
φ
1
)
−1
39
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
can be written as
trace


D
x
h(D
x
φ
1
)
−1

=
3

i=1

D
x
hw
i
, e
i

.
It yields
∂A

ψ
temp
(s,x) ◦ φ
−1
1

(x)
∂
|
=0
=


D
x
h(D
x
φ
1
)
−1
s,u

−
3

i=1
1
2
Aψ
temp
(s,x)

D
x
hw

i
, e
i


◦ φ
−1
1
(x) .
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
R
3
,

Dh w, u

2
= −

⎡

⎢
⎢
⎢
⎢
⎣
div(u
1
w)
div(u
2
w)
div(u
3
w)
⎤
⎥
⎥
⎥
⎥
⎦
,h

,
where u
i
is the ith element of u.
As a consequence, when defining
L
i
x

=(D
x
φ
1
)
−1
su
i
−
1
2
Aψ
temp
(s,x)w
i
,
it can
be easily shown that
E
B
=2

⎡
⎢
⎢
⎢
⎢
⎣
div


log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
1
x

Aψ
temp
(x)

div

log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
2
x


Aψ
temp
(x)

div

log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)),L
3
x

Aψ
temp
(x)

⎤
⎥
⎥
⎥
⎥
⎦
,h


◦ φ
−1
1
(x) .
40
3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs
3.3.2 Euler-Lagrange Equation for LDDMM-ODF
To summarize, LDDMM-ODF can be written in the form of the variational problem as
Eq. (3.15) with its Euler-Lagrange optimality conditions as the following theorem.
Theorem 3.3.1.
Given the energy
J
in the following form of Eq.
(3.15)
, the gradient
of
J
with respect to
m
t
given by the following equation has fixed points satisfying the
necessary minimizer of the objective function J.
∇J(m
t
)=2m
t
+ λη
t
, (3.20)
where

η
t
= ∇
φ
1
E +

1
t

∂
φ
s
(k
V
m
s
)


(η
s
+ m
s
)ds , (3.21)
where
∂
φ
s
(k

V
m
s
)
is the partial derivative of
k
V
m
s
with respect to
φ
s
.
η
t
in Eq.
(3.17)
can be solved backward given
η
1
= ∇
φ
1
E
, where
∇
φ
1
E
is the gradient given in Eq.

(3.19).
Refer to [93, 94] for the proof of this theorem.
3.3.3 Numerical Implementation
So far we derived
J
and its gradient
∇J(m
t
)
in the continuous setting. In this section,
we elaborate the numerical implementation of our algorithm under the discrete setting,
in particular, the numerical computation of ∇
φ
1
E.
In discretization of the spatial domain, we first represent the ambient space,
Ω
, using
a finite number of points on the image grid,
Ω
∼
=
{(x
i
)
N
i=1
}
. In this setting, we can
assume

m
t
to be the sum of Dirac measures such that
m
t
=

N
i=1
α
i
(t) ⊗δ
φ
t
(x
i
)
and
41
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
therefore:
ρ(ψ
temp
, ψ
targ
)
2
=


1
0
n

i=1
n

j=1
α
i
(t)


k
V

φ
t
(x
i
),φ
t
(x
j
)

α
j
(t)


,
where
α
i
(t)
is the momentum vector at
x
i
and time
t
. In discretization of the spherical
domain
S
2
, we discretize it into
N
S
equally distributed diffusion sampling directions on
the sphere. For each diffusion sampling direction
k
, it can be represented as
3
D vector
with unit length
s
k
in Cartesian coordinates and
(r
k

,θ
k
,ϕ
k
)
in spherical coordinates.
We use a conjugate gradient routine to perform the minimization of
J
with respect to
α
i
(t)
. We summarize steps required in each iteration during the minimization process
below:
1. Use the forward Euler method to compute the trajectory based on the flow equa-
tion
dφ
t
(x
i
)
dt
=
N

j=1
k
V
(φ
t

(x
i
),φ
t
(x
j
))α
j
(t) . (3.22)
2. Compute ∇
φ
1
(x
i
)
E in Eq. (3.19), which is described in details below.
3.
Solve
η
t
=[η
i
(t)]
N
i=1
in Eq.
(3.17)
using the backward Euler integration, where
i
indices x

i
.
4. Compute the gradient ∇J(α
i
(t)) = 2α
i
(t)+η
i
(t).
5.
Evaluate
J
when
α
i
(t)=α
old
i
(t) − ∇J(α
i
(t))
, where

is the adaptive step size
determined by a golden section search.
Since steps
1, 3 − 5
only involve the spatial information, we follow the numerical
computation proposed in the previous LDDMM algorithm [94].
42

3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs
We now discuss how to compute
∇
φ
1
(x
i
)
E
in Eq.
(3.19)
, which involves the
√
ODF
interpolation in the spherical coordinate for
Aψ
temp
(x
i
)
atafixed
x
i
and the
√
ODF
interpolation in the image spatial domain for
ψ
targ
(φ

1
(x))
. To do so, we rewrite
Aψ
temp
(x
i
)
as
Aψ
temp
(s
k
,x
i
)
and
ψ
targ
(φ
1
(x
i
))
as
ψ
targ
(s
k
,φ

1
(x
i
))
. For the
√
ODF
interpolation in the spherical coordinate for
Aψ
temp
(x
i
)
at a fixed
x
i
, we compute
Aψ
temp
(s
k
,x
i
)
according to Eq.
(3.8)
using angular interpolation on
S
2
based on

spherical harmonics. For the
√
ODF
interpolation in the image spatial domain for
ψ
targ
(φ
1
(x))
, we compute
ψ
targ
(s
k
,φ
1
(x
i
))
under the Riemannian framework in
§
2.1
as
ψ
targ

s
k
,φ
1

(x
i
)

=exp
ψ
targ

s
k
,φ
1
(x
i
)


j∈N
i
w
j
log
ψ
targ

s
k
,φ
1
(x

i
)

(ψ
targ
(s
k
,x
j
)

,
where
N
i
is the neighborhood of
x
i
, and
w
j
is the weight of
x
j
based on the distance
between
φ
1
(x
i

)
and
x
j
. The exponential maps and logarithm maps can be computed
via Eq. (2.4) and Eq. (2.5) respectively. Finally, the inner product in Eq. (3.19),
 log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)) ,
1
|e
i
|
log
Aψ
temp
(x)
Aψ
temp
(x + e
i
) 
Aψ
temp

(x)
,
and
 log
Aψ
temp
(x)
ψ
targ
(φ
1
(x)) ,L
i
x

Aψ
temp
(x)
can be computed using Eq. (2.2), where e
i
is the voxel size.
43
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
3.4 Synthetic Data
We first illustrate that the HARDI model is useful to align crossing fibers, especially
when crossing fibers have equal orientation distributions. To do so, we construct two
synthetic datasets, template and target, where there are two identical fibers perpendicu-
larly crossing each other (Figure 3.6 (a, b)). The orientations of the two crossing fibers

differ from the template image (Figure 3.6 (a)) to the target image (Figure 3.6 (b)).
We will compare the performance of LDDMM-ODF to the LDDMM algorithm based
on DTI (LDDMM-DTI) [
96
]. We refer the reader to [
96
] for detailed mathematical
derivation for LDDMM-DTI.
In the HARDI model, such orientation differences are encoded by the ODFs, while
in the DTI model, the diffusion tensors of both the template and target data look
like disks, where the first two eigenvalues being equal and the third eigenvalue being
almost zero. Although the overall image shapes are the same in both the template and
target HARDI data, the LDDMM-ODF algorithm is able to characterize the orientation
difference of the ODFs between them by generating the deformation shown in Figure
3.6 (d) with the help of term (B) of Eq.
(3.18)
. Note that a mask is used here and only
the ODFs in the circle is transformed. LDDMM-DTI fails to find any deformation
(Figure 3.6 (e)) even though the reorientation of the tensor is taken into account in the
tensor mapping.
44
3.5 HARDI Data of Children Brains
Figure 3.5:
The first and second rows respectively illustrate the original HARDI and their
enlarged images. Compared to the image on panel (a), the image on panel (b) has the
same ODFs but a different ellipsoidal image shape, while the image on panel (c) shows
different ODFs but the same circular image shape. Panels (d) and (e) show the deformations
(grid) and the corresponding momenta (arrows), calculated using
∇
φ

1
E
in Eq.
(3.19)
, for
mapping the image on panel (a) to panels (b) and (c), respectively. Panels (f) and (g) show
the deformations and the corresponding momenta, calculated using the gradient in our
previous work [2], for mapping the image on panel (a) to panels (b) and (c), respectively.
45
3. DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR
RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN
STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS
Figure 3.6:
Comparison between the LDDMM-ODF and LDDMM-DTI algorithms. Panels
(a, b) respectively show the template and target HARDI and their enlarged images, where
the ODF or diffusion tensor at each location contains two crossing fibers with equal
orientation distribution. Panel (c) illustrates the template HARDI image transformed via
the deformation given in panel (d), the result of the LDDMM-ODF algorithm. Panel (e)
illustrates no deformation found via the LDDMM-DTI algorithm and thus the template
HARDI image remains.
46
3.5 HARDI Data of Children Brains
3.5 HARDI Data of Children Brains
3.5.1 Comparison of LDDMM-FA, LDDMM-DTI and LDDMM-
ODF
In this section, we apply our proposed algorithm to real HARDI data. We evaluate the
mapping accuracy of our LDDMM-ODF algorithm by comparing it with the LDDMM-
image mapping based on FA (LDDMM-FA) and the LDDMM-DTI mapping based
on diffusion tensors using the brain datasets of
26

young children (
6
years old). All
three algorithms are developed under the LDDMM framework as given in
§
3.3 with
the exception that the matching functional,
E
, is the least square difference between
two image intensities for the image mapping, LDDMM-FA, and the Frobenius norm
between two tensors for the DTI mapping, LDDMM-DTI. More precisely, LDDMM-FA
is based on the method developed by [
97
] and LDDMM-DTI is based on the method
developed by [96]. In our implementation however, we optimize the deformation with
respect to the momentum rather than the velocity (see [
93
]). It is important to note that
all three mapping algorithms used in the following evaluation have the same numerical
scheme, such that any potential errors due to numerical related issues are avoided and
we can make a fair comparison.
Our image data are acquired using a
3T
Siemens Magnetom Trio Tim scanner with a
32
-channel head coil at the National University of Singapore. Diffusion weighted imag-
ing protocol is a single-shot echo-planar sequence with
55
slices of
2.3mm

thickness,
with no inter-slice gaps, imaging matrix
96×96
, field of view
220×220mm
2
, repetition
time=
6800ms
, echo time=
89ms
, flip angle
90
◦
.
61
diffusion weighted images with
b=
900s/mm
2
,
7
baseline (b
0
) images without diffusion weighting are acquired. Notice
47