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
, and consider the eight composite structuring elements, composed by:
:
and
,
:
and
and the three rotations of each by
,
, and
. The corresponding composite structuring elements are denoted
.
For any ''i'' between 1 and 8, and any binary image ''X'', define
::
,
where
denotes the
set-theoretical difference and
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:
:
.
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
where
denotes the
set-theoretical difference and
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
is the structural element and
is the image being operated on.
References
{{Reflist
Mathematical morphology
Digital geometry