Opening (image processing)

Opening (in German also opening or opening) is a morphological basic operation in digital image processing . The opening serves u. a. the suppression of local disturbances through bright pixels or the filtering out of small structures. The dual operation to opening is closing .

Formal definition

A complete association is given . An operator on is an (algebraic) opening if the following applies to all : ${\displaystyle V}$ ${\displaystyle \gamma }$ ${\displaystyle V}$ ${\displaystyle x,y\in V}$

• ${\displaystyle \gamma (x)\leq x}$; ie the operator is anti-extensive (the result is "smaller" than the original)
• ${\displaystyle x\leq y\Rightarrow \gamma (x)\leq \gamma (y)}$; ie the order structure of the association is retained by the operation.
• ${\displaystyle \gamma \left(\gamma (x)\right)=\gamma (x)}$; ie the operator is idempotent (repeated use does not change the result any more).

Open in binary image morphology

In the case of binary image morphology, the association is given by the power set association of all image points. A binary image is thus understood as a point set. The first two of the above properties can then be formulated as follows: ${\displaystyle V}$

• Opening no additional pixels is set, but at most points are removed.
• If an image contains an image as a subset, the following applies: after opening, the result of also contains the result of . Note that they do not have to be real subsets. This means, among other things, that two different images can be mapped onto the same image by opening them. Opening is therefore generally irreversible (information is completely deleted). ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$

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

Opening by means of a structuring element

Opening a binary image with a circle as a structuring element.

A special case is opening using a structuring element. It is defined as follows:

${\displaystyle A\circ X=(A\ominus X)\oplus X}$

It is therefore a matter of executing an erosion and a dilatation one after the other, each with the same structuring element. The erosion erases all structures that are smaller than the structuring element. The subsequent dilation reverses the erosion for the remainder.

The definition becomes clearer if it is described as

${\displaystyle A\circ X=\bigcup _{\{y|X_{y}\subseteq A\}}X_{y}}$

where represents the element moved by . So opening an image with a structuring element is the union of all shifted versions of that are completely contained in. ${\displaystyle X_{y}}$ ${\displaystyle y}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle A}$

Open by size

When opening with size, all connected structures that contain fewer pixels than a certain threshold value are deleted. This operator also satisfies the formal definition of opening.

Open using the reconstruction filter

The conditional dilation of with under the condition is defined to ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle B}$

${\displaystyle A\oplus _{B}X=(A\oplus X)\cap B}$.

So you dilate with and then "cut" off all points that are not in . If one chooses the unit environment (i.e. the neighboring pixels of a point) as the structuring element , one speaks of geodetic dilation ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle B}$ ${\displaystyle X}$

${\displaystyle \delta _{B}^{(1)}(A)}$.

The nth geodetic dilation is defined as

${\displaystyle \delta _{B}^{(n)}(A)=\delta _{B}^{(1)}(\delta _{B}^{(n-1)}(A))}$.

So you gradually add all the neighboring pixels that are in the immediate vicinity of the image and check whether they are also in . If you repeat this process as often as you like, you will eventually reach the point where nothing changes anymore. This is known as reconstructing from the marker ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A}$

${\displaystyle \rho _{A}(B)=\bigcup _{n\geq 1}\delta _{B}^{(n)}(A)}$.

If the marker was obtained from the image by opening it by means of a structuring element , this is referred to as opening by reconstruction ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X}$

${\displaystyle \mu _{X}(B)=\rho _{A}(B);A=B\circ X}$.

example

The following figure shows the results of the different procedures. The original image is shown in (a), which was opened for size in (b). If you open it with a circle as a structuring element (shown in yellow), you get the result (c). The lower left structure is completely deleted because the circle does not "fit in". Finally, the image (d) is the reconstruction of (a) from (c), i.e. a reconstruction opening with the circle element.

Open in the gray value morphology

In the case of gray value morphology, the association is the set of all functions . Formally, you need the values ​​-∞ and + ∞ for the definition (to get a complete lattice). In practice, however, only the case of a discrete, finite domain of definition and value is important. ${\displaystyle L}$ ${\displaystyle D\mapsto \mathbb {R} }$

The general properties of opening are then shown as follows:

• ${\displaystyle \gamma (f(x))\leq f(x)}$; (no image point receives a value that is higher than the original, i.e. the image does not become brighter at any point) ${\displaystyle \forall x\in D}$
• ${\displaystyle f(x)\leq g(x)\Rightarrow \gamma \left(f(x)\right)\leq \gamma \left(g(x)\right)}$; (If an image is not lighter than a second image at any point , the opened image is also not lighter than at any point ). ${\displaystyle \forall x\in D}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle \gamma \left(f(x)\right)}$ ${\displaystyle \gamma \left(g(x)\right)}$

Analogous to binary image morphology, there are also various established methods here.

Opening by means of a structuring element

The definition is analogous to the binary image morphology.

${\displaystyle A\circ X=(A\ominus X)\oplus X}$

The view is (almost) analogous to that in the case of binary image morphology. The structures into which the structuring element fits completely are also retained here. However, the image is interpreted here as a mountain range above a plane (the gray values ​​determine the height, the image coordinates the point in the plane). The structuring element scans the mountains from below.

Mostly “flat” structuring elements are used as structuring elements, ie the value of the element is 0 in the area of ​​the structure to be displayed and otherwise -∞.

Open using the reconstruction filter

Opening with a reconstruction filter is defined analogously to the binary image morphology. The flat unit environment is generally used as a structuring element for the representation of the unit environment.

See also

• Closing
• erosion
• Dilation

literature

• Image Processing and Mathematical Morphology . Jean Serra. Academic Press, London, 1982
• Image Processing and Mathematical Morphology, Part II: Theoretical Advances . Jean Serra. Academic Press, London, 1988
• Methods of digital image signal processing . Piero Zamperoni, Vieweg Verlag, 1989
• Granulometries in gray value morphology . Martin Pfeiffer. Shaker Verlag Aachen, 1999. ISBN 3-8265-4784-5