Volume 2, Algorithms - mipav - National Institutes of Health

M I P A V
M e d i c a l I m a g e P r o c e s s i n g, A n a l y s i s, & V i s u a l i z a t i o n
Chapter 2, Using MIPAV Algorithms: B-Spline Automatic Registration
HX ( ) = ⎛–---
1⎞ ⎝ N⎠∑n k
⋅ ( logn k
– logN)
k
where log n k can be a precomputed lookup table of logarithms of integer
values.
The range of values for this measure is [0,1].
Improving performance
The gradient descent applied to a control point at a time can be quite slow.
Performance can be improved by taking advantage of the local control
aspect of using B-splines. Since only one control point is being moved at a
time, the local control aspect of using B-splines means that only some of the
registered source image values will be different. Note that for an image of a
given size, the number of image values, under local control, affected by
moving a control point decreases as the number of control points increases.
One performance improvement can be achieved by precomputing the B-
spline basis function values once for each possible sample for each dimension.
Since the registration occurs by transforming the source image into
the space of the target image, all samples for B-spline basis function evaluation
always occur at the discrete sample coordinates for the target image.
Since the number of target images samples for a dimension is xed, the B-
spline basis function will be evaluated on the [0, 1] interval at equally
spaced intervals (we are using open, uniform B-splines). For each dimension,
a 2D array is created which stores B-splines basis function evaluations
for each sample with the weight associated with each control point. Because
of local control, many of these weights will be zero indicating that a sample
is not affected by a particular control point. In the process of creating this
2D array of precomputed basis function evaluations, the index range of
samples affected by each control point can be determined. The algorithm to
determine this range does not examine the weights as they are computed;
instead, the range of control points affecting a single sample are determined
from the computed knot index and the known B-spline degree.
Another performance improvement involves maintaining intermediate
error measure calculations. For example, in the case of the least squares
error measure, an image is maintained of the current error calculations,
that is, the target image subtracted from the current registered source
MIPAV User’s Guide, Volume 2, Algorithms 105
12/5/08