Thesis (PDF) - Signal & Image Processing Lab

32 CHAPTER 2. THEORETICAL BACKGROUND
Keshet also defines in [16] the notion of boundary topographic distance (BTD)
of a pixel x of a bounded image f, as the topographic distance between x and the
boundary of f. That is, the BTD of x is the least topographic distance between x
and any point on the boundary of f. The BTD function, BTf, is the mapping from
each pixel x to its BTD, BTf(x).
2.5.2 An extension of boundary topographic distance defini-
tion
The notion of BTD, defined by Keshet, can be easily extended, allowing the boundary
of the image f to be any connected flat zone of f. The notion of flat zone is discussed
in section 2.2. In some cases, a boundary of an image isn’t the best starting point
for topographic distance calculation, because it is arbitrary set by the properties and
position of the camera used to take the picture. When choosing another flat zone,
as a reference for the topographic distance, the choice can be more meaningful and
suited for the problem to be solved.
For instance, in some applications it can be useful to define the boundary as the
largest flat zone of the image. Another example is shown in Fig. 2.9. In this example
a root flat zone is chosen to a be a symmetric center of the image - the circle center.
2.5.3 Boundary Topographic Variation (BTV) Transform
Let πf(x) be a connected path linking the boundary of an image f to a pixel x and
ˆπf(x) be one such path, with least topographic distance between x and the boundary.
The topographic distance (total variation) on ˆπf(x) is exactly BTf(x) and ˆπf(x) is
called a minimal-BTD path for x. There may be more than one minimal-BTD path
for a given point x.
Assume that a minimal-BTD path is associated with an alternating sequence (AS).
The AS describes the “ups and downs” that occur on that path. These “ups and
downs” are called path variation. For instance, consider a grayscale image f, and
suppose that a minimal-BTD path ˆπf(x) from the boundary to a pixel x has the
following function values:
f(ˆπf(x)) = (0, 2, 6, 9, 8, 8, 5, 1, 3, 7, 7, 4, 5). (2.19)