Composition of basic operators#
Opening#
Opening (french: ouverture) consists of an erosion followed by a dilation. The erosion removes small objects but also decreases the size of bigger objects. To avoid this, the result is dilated with the same structuring element.
Fig. 70 Example of an opening on a small image \(I\) by the structuring element \(E_c\) (with the origin \(c\) is at the centre and represented by the blue dot).#
Property#
Opening is an idempotent operation, that is to say, applying twice the same opening gives the same result as only one opening:
\[ (I \circ E) \circ E = I \circ E \]
Closing#
Contrary to opening, closing (french: fermeture) is firstly a dilation, then an erosion. Indeed, expansion closes holes but enlarges objects. To avoid the widening of the objects, an erosion can be applied with the same structuring element.
Fig. 71 Example of closing on a small image \(I\) by the structuring element \(E_c\) (with the origin \(c\) is at the centre and represented by the blue dot).#
Properties#
Similarly to opening, closing is an idempotent operation:
\[ (I \bullet E) \bullet E = I \bullet E \]
Hit-or-miss transform#
The hit-or-miss transform (french: transformée tout-ou-rien) is used to detect objects of a particular shape. It is the intersection of the two sets given by:
the erosion by a first structuring element \(E_1\): \(I \ominus E_1\),
and the erosion of the background by a second structuring element \(E_2\): \(I^\mathrm{c} \ominus E_2\)
with \(E_1 \cap E_2 = \varnothing\) (the structuring elements must be disjointed).
The hit-or-miss transform by the two structuring elements \(E_1\) and \(E_2\) is noted \(I \otimes (E_1, E_2)\):
Fig. 72 Example of a hit-or-miss transform applied on the image \(I\) by the structuring elements \(E_1\) and \(E_2\). The origins of the structuring elements are marked by the blue and green dots.#
Sometimes, the two structuring elements are combined into a single structuring element whose pixels have the following values:
\(1\): pixels that belong to the object to detect,
\(-1\): pixels that do not belong to the object to detect (i.e pixels of the background),
\(0\): unused pixels (also called “don’t care pixels”).
With this notation, the structuring element of the hit-or-miss transform in Fig. 72 writes