This paper studies a variational formulation of the image matching problem. We consider a scenario in which a canonical representative image
T
T
is to be carried via a smooth change of variable into an image that is intended to provide a good fit to the observed data. The images are all defined on an open bounded set
G
⊂
R
3
G \subset {R^3}
. The changes of variable are determined as solutions of the nonlinear Eulerian transport equation
\[
d
η
(
s
;
x
)
d
s
=
v
(
η
(
s
;
x
)
,
s
)
,
η
(
τ
;
x
)
=
x
,
(
0.1
)
\frac {{d\eta \left ( s; x \right )}}{{ds}} = v\left ( \eta \left ( s; x \right ),s \right ), \qquad \eta \left ( \tau ; x \right ) = x, \qquad \left ( 0.1 \right )
\]
with the location
η
(
0
;
x
)
\eta \left ( 0; x \right )
in the canonical image carried to the location
x
x
in the deformed image. The variational problem then takes the form
\[
arg
min
v
[
‖
v
‖
2
+
∫
G
|
T
o
η
(
0
;
x
)
−
D
(
x
)
|
2
d
x
]
,
(
0.2
)
\arg \min \limits _v {\kern -0.1pt} \left [ {{{\left \| v \right \|}^2} + \int _G {{{\left | {T o \eta \left ( {0; x} \right ) - D\left ( x \right )} \right |}^2}dx} } \right ], \qquad \left ( {0.2} \right )
\]
where
‖
v
‖
\left \| v \right \|
is an appropriate norm on the velocity field
v
(
⋅
,
⋅
)
v( \cdot , \cdot )
, and the second term attempts to enforce fidelity to the data.