Closing

Closing (in English also Close ) is a morphological basic operation in the digital image processing . The operator is used when filtering images; By closing, locally limited dark disturbances in an image can be suppressed or small dark structures can be specifically filtered out.

Formal definition

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

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

Closing 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: ${\ displaystyle V}$

Closing does not delete any pixels, but only adds points.
If an image contains an image as a subset, the following applies: after closing, the result of also contains the result of . Note that they do not have to be real subsets. It follows from this, among other things, that two different images can be mapped onto the same image by closing. Closing is therefore generally not reversible (information is therefore completely deleted). ${\ displaystyle y}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle x}$

Closing by means of a structuring element

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

{\ displaystyle A \ bullet X = (A \ oplus X) \ ominus X}

Closing a binary image with a circle element

It is therefore a matter of performing a dilation and an erosion one after the other, each with the same structuring element. The dilation closes all holes into which the structuring element does not fully fit. The subsequent erosion reduces the image again so that it comes as close as possible to the original. The holes that are completely closed by the dilation no longer arise; only partially closed holes are widened again.

Closing 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 range of definitions and values is important. ${\ displaystyle V}$ ${\ displaystyle D \ mapsto \ mathbb {R}}$

The general properties of opening are then shown as follows:

${\ displaystyle \ varphi (f (x)) \ geq f (x)}$ ; (no image point receives a value that is smaller than the original, i.e. the image does not become darker at any point) ${\ displaystyle \ forall x \ in D}$
${\ displaystyle f (x) \ leq g (x) \ Rightarrow \ varphi (f (x)) \ leq \ varphi (g (x))}$ ; (if an image is not lighter than a second image at any point , the closed image is also not lighter than at any point ) ${\ displaystyle \ forall x \ in D}$ ${\ displaystyle f}$ ${\ displaystyle g}$ ${\ displaystyle \ varphi (f (x))}$ ${\ displaystyle \ varphi (g (x))}$

duality

The dual operation of closing is opening . Correspondingly, the statements about opening can be transferred to closing. One interprets the background as the foreground and vice versa. For gray value images this means that the opposite number is used for the brightness values . Then the corresponding dual opening is carried out (e.g. with the dual structuring element) and the dual image is formed from the result obtained.

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