Closing (morphology)

Closing ( in German even close) is a morphological basis operation in digital image processing. Applies the operator when filtering of images; by closing can be localized dark interference suppress in an image or filter out small dark structures targeted.

Formal definition

Given a complete lattice. An operator on a ( algebraic ) Close when applies to all:

  • ; ie the operator is extensive (the result is " bigger" than the original)
  • ; ie, the order structure of the association is maintained by the operation.
  • ; ie the operator is idempotent ( a repeated applying leads to no further change in the result ).

Close in Binärbildmorphologie

In the case of Binärbildmorphologie the association is given by the power set lattice of all pixels. A binary image is thus seen as point set. The first two of the above properties can then be formulated as follows:

  • By closing no pixels are deleted, but added at most points.
  • When an image contains a picture as a subset, so is that after closing also contains the result of the result of. Note that it does not have to involve proper subsets. Consequence of this is that two different images can be displayed by closing on the same image. A closing is so i.a. not reversible ( so it will be completely deleted information ).

Close means be patterned element

A special case is the closing means be patterned element. It is defined as follows:

So it is the one after performing a dilation and an erosion each with the same structuring element. The dilatation all holes are closed, in which the structuring element is not completely fit. The subsequent erosion reduces the image back so far that it comes as close as possible to the original. The fully closed by the dilation holes no longer arise here; only partially closed holes are widened again.

Close in grayscale morphology

In the case of gray-scale morphology, the dressing is the set of all functions. Formally, the required values ​​for the definition ( a complete association to obtain ) - ∞ ∞ and . In practice, however, is of importance only in the case of discrete, finite definition and range of values.

The general characteristics of the opening are then prepared as follows:

  • ; (no pixel is given a value that is smaller than the original, ie the image is dark at any point )
  • ; ( when an image is not bright at any point as a second image, the image is bright and closed at any point )

Duality

The dual operation is to close the opening. Accordingly, the statements about the opening to the closing can be transferred. We interprete this, the image background as the foreground and vice versa. For gray scale images, this means that it takes the counter number of the brightness values ​​. Then, it executes the corresponding dual opening by (eg with the dual structuring element ) and forms from the result obtained the dual picture.

194537
de