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Digital Image Processing (2000)

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6.2 Set Description of Binary Images

The language of mathematical morphology is that of set theory. Sets in mathematical morphology represent the shapes that are visible in binary images. The set of all the white (or black) pixels in a binary image constitutes a complete description of the image. Each element of the set is a 2-tuple of integer values identifying the Cartesian coordinates of each white (or black) pixel with respect to some assumed coordinate center. It is typical to assume that the coordinate center coincides with a corner element of the array.

Set-theoretic functions.

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Here is an example binary image.
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The set description of image A is obtained by locating all the ones in the image. We assume the default origin location at position {1, 1} in array A and return all positions as offsets from the origin. Different choices for the location of the origin result in different set descriptions . The set descriptions differ by a translation.
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To obtain the set complement (or background image) of some set , it is necessary to define the global set , a set of all coordinate pairs of an N1 × N2 image.
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Here is the set complement.
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Here we view the complement image.
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The set description of a translation of a set is the vector sum of the positions of the elements of the set with a translation vector. Consider a translation of set by the vector {1,1}.
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The set description of a reflection of a set with respect to element 0 is the vector difference of 0 and the position of each element of the set. Typically, 0 is the origin. However, in the context of a matrix implementation of reflection, it is more convenient to set 0={N1-1,N2-1}, where N1, N2 are the row and column dimensions of the matrix.
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In matrix notation, reflection of an N1 × N2 array A denoted , is defined
    
where is the rth and cth element of array . The last result is the same as that obtained using the function Reflect.
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The set theoretic concepts of translation and reflection will be used in the next section to define the fundamental operators of binary morphology