Biomedical Engineering Reference
In-Depth Information
Figure 9.11: Example of a mask selecting 10 % of the pixels with the highest
gradient values for the Lena image.
number of pixels, and to consider only those pixels in subsequent calculations.
The contributions of individual pixels to the change of the criterion is directly
proportional to the amplitude of the directional derivatives at the respective
points, see (9.12). Therefore, a reasonable strategy is to construct the mask
by thresholding the gradient of the image at each multiresolution level (see
Fig. 9.11).
9.4.9
Invertibility and Regularization
In some applications, it is useful to add an extra regularization term
E
r
to the
difference measure
E
, and to look for a minimum of the combined criterion
E
c
=
E
+
γ
E
r
(9.21)
The factor
E
r
is used to make the solution well-posed, or to privilege likely
solutions based on our a priori knowledge.
First, we consider a
penalty term
designed to enforce the invertibility of the
deformation, generalizing the concept from [95] and similar to [113]. Its moti-
vation comes from the fact that if the Jacobian det(
∇
x
g
) is positive everywhere,
