Biomedical Engineering Reference
In-Depth Information
this resolution is used as the starting estimate for registration at a higher res-
olution, and so on.
48
Multiresolution approaches do not entirely solve the problem of multiple
optima in the parameter space. It might be thought that the optimization
problem involves finding the globally optimal solution within the parameter
space, and that a solution to the problem of multiple optima is to start the
optimization algorithm with multiple starting estimates, resulting in multi-
ple solutions, and choose the solution which has the best value of the similar-
ity measure. This sort of approach, called “multistart” optimization, can be
effective for surface-matching algorithms. For voxel similarity measures,
however, the problem is more complicated. The desired optimum when reg-
istering images using voxel similarity measures is frequently
not
the global
optimum, but is one of the local optima. The following example serves to
illustrate this point. When registering images using joint entropy, an
extremely good value of the similarity measure can be found by transforming
the images such that only air in the images overlaps. This will give a few pix-
els in the joint histogram with very high probabilities, surrounded by pixels
with zero probability. This is a very low entropy situation and will tend to
have lower entropy than the correct alignment. The global optimum in
parameter space will, therefore, tend to correspond to an obviously incorrect
transformation. The solution to this problem is to start the algorithm within
the ''capture range” of the correct optimum; that is, within the portion of the
parameter space in which the algorithm is more likely to converge to the cor-
rect optimum than the incorrect global one. In practical terms, this requires
that the starting estimate of the registration transformation is reasonably
close to the correct solution. The size of the capture range depends on the fea-
tures in the images and cannot be known
a priori
, so it is difficult to know in
advance whether the starting estimate is sufficiently good. This is not, how-
ever, a very serious problem, as visual inspection of the registered images can
easily show convergence outside the capture range. In this case, the solution
is clearly and obviously wrong (e.g., relevant features in the image do not
overlap at all). If this sort of failure of the algorithm is detected, the registra-
tion can be restarted with a better starting estimate obtained, for example, by
interactively transforming one image until it is approximately aligned with
the other.
3.5
Image Transformation
Image registration involves determining the transformation
T
that relates the
domain of image
A
to image
B
. This transformation can then be used to trans-
form one image into the coordinates of the second within the region of over-
lap of the two domains As discussed in section 3.2, this process
involves interpolation and needs to take into account the difference in sample
spacing in image
A
and
B
.
T
A
,
B
.
