Biomedical Engineering Reference
In-Depth Information
The optimum parameters of the non-rigid registration transformation
T
are determined by a line search algorithm similar to the steepest descent
method [44]. The target function of the optimization is the NMI similarity of
the reference and the transformed floating image. We start by computing a dis-
crete approximation of the gradient of the target function with respect to the
parameters of the transformation
T
. This is achieved by a simple finite difference
scheme. Despite the high-dimensional parameter space, gradient approximation
can be performed very efficiently; due to the compact support of the B-spline
functions, each parameter of
T
influences only a small volume in image space
(i.e., the local 4
×
4
×
4 control point neighborhood). When moving any single
control point, all voxels of the floating image outside this area remain in the
same location. Their contribution to the similarity measure can therefore be
precomputed and reused [70].
In order to capture large deformations as well as small ones, the algo-
rithm incorporates a multiresolution deformation strategy based on multilevel
B-splines [31]. After finishing optimization at one control point resolution, the
spacing between the control points is reduced by a factor of 2 before registration
continues. The positions of the control points in the refined grid are determined
in a way that exactly preserves the current deformation [18, 58].
Using adaptive grid refinement [55, 67] and a parallel multiprocessor im-
plementation [56], we are able to keep computation times within reasonable
bounds. For example, we can complete a non-rigid registration of an image to
an atlas, each about the size as described earlier in this chapter, within about
10 minutes on a modern PC (Intel Pentium 4, 3.0 GHz, hyperthreading enabled).
11.3.4
Regularization of the Non-Rigid Transformation
Confocal microscopy imaging is a substantially less controlled image formation
process than typical medical imaging modalities. Varying concentrations of the
chromophor within one structure, laser power fluctuations, tiling artifacts, and
absorption of emitted light from deep structures lead to substantial imaging arti-
facts. As illustrated in Fig. 11.5, these artifacts can cause severe problems for the
non-rigid registration, leading to grossly incorrect coordinate transformations.
These can, to some extent, be prevented by regularizing the image similarity cost
function with an additional constraint term that controls the geometric proper-
ties of the coordinate mapping. The total optimization function thus becomes
