ALGORITHMS FOR PROCESSING CONNECTED COMPONENTS WITH GRAY VALUES - Fundamentals of Three-Dimensional Digital Image Processing - page 203

[STEP II] Move the current voxel by two voxels in the trace direction. If
the continuity condition is satisfied at the current voxel after the above
movement, then write value 1 into the corresponding voxel of the work
image F W and go to [STEP I] . Otherwise move the current voxel back

to the previous position and go to [STEP III] .
[STEP III] Change the tracing direction into the direction connecting the
current voxel and adjacent voxels and select such a direction that the
amount of angle change is less than 90 degrees. Then move the current
voxel by one voxel along the newly selected direction. If the continuity
condition is satisfied at the current voxel after the movement, then write
value 1 into the corresponding voxel of an image

F W and go to [STEP
I] . Otherwise move the current voxel back to the starting point of the
trace and terminate the tracing.

All directions traced in the past by the tracing procedure at all visited voxels
should be recorded, and the tracing procedure is terminated if the direction
of tracing in each step of the procedure is found to be the same as the one
that was traced in the past. This is required in order to avoid the same voxel
being traced more than once in the same direction.

Remark 6.4. The concrete content of continuity condition is not fixed here.
As was stated in Chapter 3, however, only 20 cases occur in the size relation
among three principal curvatures. The continuity condition will be fixed by
giving which of those 20 cases are adopted as a ridge point. In the experiments
presented in the text, for example, the following two cases:
(Set 1) Pattern 5, 6, 9, and 10 in Table 3.4
and
(Set 2) Two or more principal curvatures are positive, and only | k 3 | among

| k 1 | , | k 2 | ,and | k 3 | is not a local maximum

were tested.
The first set (Set 1) above was seen frequently on the centerline of an
artificial image used here (Fig. 6.7). The centerline is really located along a
line with the highest density value in this artificial figure, and is considered
as a ridgeline intuitively. The second set (Set 2) above includes as the cases
No. 1

∼

11 (excluding No. 4 and 8) of Table 3.4 and the case No. 19. Since (Set
2) contains (Set 1), the set of voxels extracted by using it includes the one
extracted by (Set 1). In the mathematical analysis of a 3D curved surface,
a curved surface such that H< 0 and K< 0 is sometimes called ridge
surface, and the one such that H< 0 and K = 0 is called saddle ridge,
where H and K represent the mean curvature and the Gaussian curvature,
respectively (Table 3.4) [Haralick83]. By the analogy with this, the above (Set
2) will contain relatively more of patterns like the ridge surface and parts of
saddle ridge of a 4D curved surface.