Cross-Entropy, Reversed Cross-Entropy, and Symmetric Divergence Similarity Measures for 3D Image Registration: A Comparative Study - Biomedical Image Analysis - page 398

For the reversed cross-entropy maximization, if the marginal probability
is zero, the contribution to the reversed cross-entropy is zero due to the fact
that 0 log 0 = 0. If the joint probability is zero and the marginal probability is
not zero, then the contribution to the reversed cross-entropy is infinite. Since
the reversed cross-entropy is being maximized and we assume the optimized
reversed cross-entropy has a finite value, this instability due to the zero joint
pdf terms must be addressed. As shown later in this chapter, if the zero joint
probability terms are ignored, the profile of the reversed cross-entropy will
have a zig-zag shape, which adversely affects the optimization process and ren-
ders the reversed cross-entropy an inappropriate measure for registration. We
elected to assign a small probability value to the zero joint probabilities, which
is

1

binsize × binsize × al pha , where α was selected empirically to be 100. We will discuss
the impact of the α selection in a later section.
For the reversed cross-entropy minimization, if the priori probability is zero,
then the contribution to the reversed cross-entropy is zero due to the fact that
0 log 0 = 0. If the joint probability is zero and the priori probability is not zero,
the contribution to the reversed cross-entropy is infinite. In the later case, since
we assume the minimized reversed cross-entropy has a lower bound, their con-
tributions are ignored.
The symmetric divergence method utilizes the cross-entropy and reversed
cross-entropy components and its numerical stability is determined by those
two terms.

10.4 Registration Experimental Setup

10.4.1 Dataset Description

MR, nuclear transmission (Tx) and nuclear emission (Em) scans were used as
test volumes. They were primary brain images, with some inclusion of the neck
and chest. The image data consists of slices. The x -axis is directed horizontally
from right to left, the y -axis horizontally from front to back, and the z -axis
vertically from bottom to top.
This study involved seven patients. All the pertinent image file information is
tabulated in Table 10.1. The MR images were acquired with an OutLook MR scan-
ner (Philips Medical Systems, formerly Marconi Medical Systems, Cleveland,