Biomedical Engineering Reference
In-Depth Information
Using a Taylor expansion of the right side and ignoring higher-order terms,
the optical flow equation (13.13) can be rewritten as
------
dy
dt
------
dx
dt
I
I
------
dz
dt
I
I
------
------
------
-----
0
(13.14)
x
y
z
t
Alternatively this can be written as
I
I
u
0
(13.15)
where
I
is the spatial gra-
dient of the image, and
u
describes the motion between the two images. In
general, additional smoothness constraints are imposed on the motion field
u
in order to obtain a reliable estimate of the optical flow. These smoothness
constraints are discussed in more detail in the following section.
I
is the temporal difference between the images,
13.2.7
Registration as an Optimization Problem
Like many other problems in computer vision and image analysis, registra-
tion can be formulated as an optimization problem whose goal is to minimize
an associated energy or cost function. The most general form of such a cost
function is
C
C
similarity
C
deformation
(13.16)
where the first term characterizes the similarity between the source and target
image and the second term characterizes the cost associated with particular
deformations. Most of the nonrigid registration techniques discussed so far
can be formulated in this framework. From a probabilistic point of view, the
cost function in Equation (13.16) can be explained in a Bayesian framework.
40
In this context, the similarity measure can be viewed as a likelihood term
which expresses the probability of a match between source and target image.
The second term can be interpreted as a prior which represents
a
priori
knowl-
edge about the expected deformations.
The first term is the driving force behind the registration process, and aims
to maximize the similarity between both images. The different similarity mea-
sures can be divided into two main categories: point based and voxel based
similarity measures (a detailed discussion of those can be found in Chapter 3).
Point-based similarity measures minimize the distance between features such
as points, curves, or surfaces of corresponding anatomical structures and
requires prior feature extraction. In recent years, voxel-based similarity measures
such as sums of squared differences, cross correlation, or mutual information
described in Chapter 3 have become increasingly popular. These voxel-based
