Jump to content

User:Renatokeshet/mm

From Wikipedia, the free encyclopedia

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:
.

Bibliography

[edit]
  • An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)