Image Processing Algorithms Implementation Using Quantum Cellular Automata - Cellular Automata in Image Processing and Geometry - page 65

Fig. 4.1 The erosion of an object by a structuring element

structuring elements. It will be assumed that in binary images, white pixels repre-
sent background regions, while black pixels denote foreground, although in some
implementations this convention is reversed.
The implementation details for all operations can be generalized, by considering
that the output image is produced by applying a set operator (intersection, union,
inclusion, complement) between the input image and the structuring element.
To compute a mathematical operation on a binary image using a structuring el-
ement, each of the pixels in the input image (usually called input pixel) should be
considered in turn. For each pixel, we superimpose the structuring element over the
input image, so that the origin of the structuring element (usually the central pixel)
coincides with the input pixel position. The set operator applied on the image's and
the structuring element's pixels values gradually produce the output image.
The most basic morphological operation is that of erosion. As it is shown in
Fig. 4.1, suppose A is a binary object in the input image and S is a structuring
element (only the black pixels). If S is placed with its origin pixel at every pixel of
the input image

. Then the erosion of A by S is defined to
be the set of all pixel locations for which S placed at that pixel is contained within
A . This is denoted A

( i , j ) we denote it by S ( i , j )

S and is written as:

= ( A A S i , j ) : S ( i , j ) ↂ (4.1)

Fig. 4.1 presents the input image with the binary object A , the structuring element
S which is a 3 S where the grey pixels are the

eroded pixels that are subtracted from the object (in fact they are white).
Dilation can be defined as a complementary to the erosion operation, but an al-
ternative definition that helps to understand how it is applied is the following. Let as
assume that S is the reflection of S . Obviously, S =

× 3 cross and the eroded image A

S for symmetrical structuring

S contains all the pixels lying in any S ( elements. Then A ↕ i , j ) for which ( i , j ) ∈ A .