Hit-or-miss transform

In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.
Mathematical definition.
In binary morphology, an image is viewed as a subset of a Euclidean space formula_1 or the integer grid formula_2, for some dimension "d". Let us denote this space or grid by "E".
A structuring element is a simple, pre-defined shape, represented as a binary image, used to probe another binary image, in morphological operations such as erosion, dilation, opening, and closing.
Let formula_3 and formula_4 be two structuring elements satisfying formula_5. The pair ("C","D") is sometimes called a "composite structuring element". The hit-or-miss transform of a given image "A" by "B"=("C","D") is given by:
where formula_7 is the set complement of "A".
That is, a point "x" in "E" belongs to the hit-or-miss transform output if "C" translated to "x" fits in "A", and "D" translated to "x" misses "A" (fits the background of "A").
Some applications.
Thinning.
Let formula_8, and consider the eight composite structuring elements, composed of:
and the three rotations of each by 90°, 180°, and 270°. The corresponding composite structuring elements are denoted formula_13.
For any "i" between 1 and 8, and any binary image "X", define
where formula_15 denotes the set-theoretical difference.
The thinning of an image "A" is obtained by cyclically iterating until convergence: