sharing the characteristic of either decreasing or increasing the size of an object.
The output ranges of CL erosion and dilation satisfy Eqs. (11.18) and (11.20),
Various filter structures may be considered, depending on the size of the edge,
the location of the origin, and the type of CLOs and on whether the origin is taken
into account. Suggestively, one characteristic rhombus structuring element B is
described by
* [,] 9
where the asterisk inside the square brackets denotes the location of the origin
(i, j) and the surrounding asterisks denote the pixels in the structuring element.
The structure of a 2D CL filter corresponding to this structuring element is given
f(i,j) = g(i- 1,j) o g(i,j - 1) o g(i,j) o g(i + 1,j) o g(i,j + 1).
Using the filter structure of Eq. (11.21), the erosion of the image G using CL filters
is given by
= g(i- 1,j) CANDg(i, j - 1) CANDg(i, j) CANDg(i + 1,j) CANDg(i, j + 1).
Since the new state of each pixel depends only on the present state of that pixel
and those of its neighbors, the new state for every pixel in the filtered image can
be computed independently and simultaneously. In a cellular context, the "neigh-
borhood" of a given cell is defined as the extent of the structure of the CL filter
when the origin of the structure is centered on that cell.
1 1.4
Properties of Coordinate Logic Filters
This section discusses the most fundamental properties of the operations of CL
erosion, dilation, opening, and closing. These properties hold for both binary and
gray-level signals and images.
The duality properties for CL dilation and erosion, which allow for the inter-
change of the functionality of the image and its complementary image, are respec-
tively as follows:
G D = NOT ( NOT G)B,
G E = NOT ( NOT G)D.
The proof of Eqs. ( 11.23) and ( 11.24) results from the application of the De Morgan
laws in coordinate logic [Die71]. These properties can also be extended to derive
the duality properties for CL opening and closing, which are, respectively,
(GE) D
= NOT (( NOT
G) D)B (11.25)
(GD) E = NOT ((NOTG)E)B. (11.26)

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