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Thinning is the transformation of a
digital image A digital image is an image composed of picture elements, also known as ''pixels'', each with '' finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions f ...
into a simplified, but topologically equivalent image. It is a type of
topological skeleton In shape analysis, skeleton (or topological skeleton) of a shape is a thin version of that shape that is equidistant to its boundaries. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity, ...
, but computed using
mathematical morphology Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be emp ...
operators.


Example

Let E=Z^2, and consider the eight composite structuring elements, composed by: :C_1=\ and D_1=\, :C_2=\ and D_2=\ and the three rotations of each by 90^o, 180^o, and 270^o. The corresponding composite structuring elements are denoted B_1,\ldots,B_8. For any ''i'' between 1 and 8, and any binary image ''X'', define ::X\otimes B_i=X\setminus (X\odot B_i), where \setminus denotes the set-theoretical difference and \odot denotes the
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-m ...
. The thinning of an image ''A'' is obtained by cyclically iterating until convergence: :A\otimes B_1\otimes B_2\otimes\ldots\otimes B_8\otimes B_1\otimes B_2\otimes\ldots.


Thickening

Thickening is the dual of thinning that is used to grow selected regions of foreground pixels. In most cases in image processing thickening is performed by thinning the background \text(X, B_i) = X\cup (X\odot B_i) where \cup denotes the set-theoretical difference and \odot denotes the
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-m ...
, and B_i is the structural element and X is the image being operated on.


References

{{Reflist Mathematical morphology Digital geometry