Shih_Image_Processing_and_Mathematical_Morpholo.pdf

12 Image Processing and Mathematical Morphology
image; that is, an N-dimensional composition in black and white, where the
point p is black in the binary image if and only if p � A; otherwise, p is white.
A binary image in E2 is a silhouette, a set representing foreground regions
(or black pixels). A binary image in E3 is a solid, a set representing the surface
and interior of objects. The notion of a binary image correlates the notion of
black and white pixels to a Cartesian coordinate system for the binary
image.
Let A denote a set (or a binary image) in E2 . If a set contains no elements, it
is called an empty set or a null set, denoted f. Let a � A denote an element
a = (a1, a2) in A. The complement (or inverse) of the image A is the binary
image that exchanges black and white, as given by
__
A = (A) c = {a | a œ A}. (2.1)
The refl ection of an image A is the refl ected image of A across the origin
(i.e., the version of A rotated 180° on the plane), as given by
 = {w | w = - a for a Œ A}, (2.2)
in which the elements with (a 1, a 2) are negated. An example of refl ection is
shown in Figure 2.1, where the origin of coordinates is located at the center
pixel.
The union of two images A and B is a binary image in which the pixels are
black if the corresponding input pixels are black in A or black in B, as given by
A » B = {p | p Œ A or p Œ B}. (2.3)
The intersection of two images A and B is a binary image where the pixels
are black if the corresponding input pixels are black in both A and B, as
FIGURE 2.1 Refl ection.
A « B = {p | p Œ A and p Œ B}. (2.4)