Opening (morphology)

Opening ( in German also open or opening) is a morphological basis operation in digital image processing. The opening serves, inter alia, the suppression of local interference from bright pixels or the filtering of small structures. The opening dual operation is closing.

  • 3.1 Opening means be patterned element
  • 3.2 Opening by reconstruction filter

Formal definition

Given a complete lattice. An operator on a ( algebraic ) open if for all:

  • ; ie the operator is anti -extensive (the result is " smaller" 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 ).

Open 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 opening any additional pixels are set, but at most points away.
  • When an image contains a picture as a subset, so is that after opening 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 imaged by an opening on the same image. An opening is so i.a. not reversible ( so it will be completely deleted information ).

This definition is very broad; in practice, various methods have been established, which are outlined briefly below.

Open means be patterned element

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

Thus, it is the sequentially performing erosion and dilation respectively with the same structuring element. By erosion of all of the structures will be erased, which are smaller than the structuring element. The subsequent dilation makes the erosion of the remaining undo.

Illustrative, the Definition when it circumscribes to

Which represents around moved item. Opening an image with a structuring element is thus the union of all shifted versions of which are completely contained in.

Open by size

When opening by size all related structures are deleted, the less pixels contain than a certain threshold. This operator also satisfies the formal definition of the opening.

Opening by reconstruction filter

The conditional dilation of having the condition is defined as

So you dilated with and "cuts " then from all points that do not lie in. (Ie the neighboring pixels of a point ) is chosen as the structuring element of the unit environment, this is called the geodesic dilation

The nth geodetic dilation is defined as

So you have taken one by one all neighboring pixels which will be in the immediate vicinity of the image and checks whether they are still in. If you repeat this process as often as you eventually reach the point at which changes nothing. This is known as the reconstruction of the marker from

If the marker is obtained from the image by means of an opening be patterned element, this is referred to as an opening by reconstruction

Example

The following figure shows the results of different methods. The original image is shown in (a ), which has been opened in ( b ) by size. In an opening with a circle as be patterned element ( shown in yellow ) one obtains the result ( c). The left lower structure is completely deleted because the circle does not " fit ". The image ( d) Finally, the reconstruction of (a) from (c), ie a reconstruction open to the circular element.

Open 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, of importance is only 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 higher than the original, ie the image is brighter at any point )
  • ; ( when an image is not bright at any point as a second image, as the active screen is bright, as well as at any point ).

Analogous to Binärbildmorphologie there are various established methods here.

Open means be patterned element

The definition is analogous to Binärbildmorphologie.

The intuition is ( almost) analogous to that in the case of Binärbildmorphologie. Again, the structures are preserved in the fully fits the structuring element. However, the image is defined herein as the mountains on one plane (the gray values ​​to determine the amount, the image coordinates of the point in the plane ) is interpreted. The structuring element scans the mountain from below.

As a structuring element are usually "flat" structuring elements used, ie the value of the element is 0 in the displayed structure and otherwise - ∞.

Opening by reconstruction filter

The opening with reconstruction filter is defined analogously to Binärbildmorphologie. Here is the flat unit environment as a structuring element for the realization of the unit area ia.

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