Biomedical Engineering Reference
In-Depth Information
coefficients by 2 when expanding and 0
.
5 when reducing. Thanks to our choice
of the expansion and reduction operators, the origin of the image does not
change.
9.4.8.7
Fast Spline Calculations
It is essential to take full advantage of the properties of splines. First, specialized
routines are used to calculate the values of a B-spline of a specific order using
a minimum number of operations. Second, as we are using tensor products of
B-splines as our basis functions, many operations can be performed in a sepa-
rable fashion, reducing the complexity of operations from
O
(
k
N
), where
N
is
the number of dimensions and
k
the size of the data, to
O
(
kN
). This is the case
for the prefiltering step required to find the B-spline coefficients, and also for
the interpolation of values of a function given by its B-spline coefficients. Third,
the compact support of B-splines simplifies many of the infinite sums in the
expressions given earlier, reducing them to sums over just a small number of
elements.
9.4.8.8
Stopping Criterion
To get a fast optimization algorithm, particular attention has to be paid to the
stopping criterion. This holds for both GD and ML algorithms. Classically, the
relative and absolute improvement of the criterion value is compared with a fixed
threshold [59]. For our class of problems, it is often advantageous to base the
stopping criterion on the changes
c
of parameter values. We stop when the
step size falls below an a priori given threshold
ε
. The size of a step that fails
gives an indication of the accuracy of the result and is therefore easy to set.
Typically, we would use the threshold of
ε
=
10
−
1
∼
10
−
3
pixels for the finest
level and slightly more for coarser levels, as there is usually not enough details
and coherence between levels.
9.4.8.9
Masking
A substantial gain in speed comes from considering only important pixels when
calculating the data criterion (9.7) and its derivatives. It is possible to deter-
mine an a priori mask of significant pixels, for example, 10
∼
50 % of the total
