User:Renatokeshet/mm
In mathematical morphology, reconstruction is an operation that...
Mathematical definition
[edit]Let X and Y be subsets of an Euclidean space or the integer grid , for some dimension d, such that . Also, let B be a structuring element.
The reconstruction of X from Y is given by:
- ,
where
- ,
and denotes the conditional dilation of Y inside X:
- .
The symbol denotes morphological dilation.
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 and be two structuring elements satisfying . The pair (C,D) is sometimes called composite structuring element. The hit-or-miss transform of a given image A by B=(C,D) is given by:
- ,
where 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
[edit]- Pattern detection. By definition, the hit-or-miss transform indicates the positions where a certain pattern (characterized by the composite structuring element B) occurs in the input image.
- Thinning. 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.
- The thinning of an image A is obtained by cyclically iterating until convergence:
- .
- Computing the Euler number.
Bibliography
[edit]- An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)