| Operators |
This chapter describes operators of region morphology.
Region morphology provides a set of morphological operators that allow to modify or describe the shape of a region. The morphological operators can be used, for example, to connect or disconnect adjacent regions or to smooth the boundary of a region. In the following, we will take a closer look at the morphological operators.
To dilate or erode an input region, a structuring element is applied to the input region. This structuring element is scanned over the image line-by-line. During dilation the reference point of the structuring element is added to the resulting region whenever the structuring element and the input region have at least one pixel in common. This results in an enlarged region, as shown in the image below. Erosion reduces the area of the input region because the reference point is only added to the resulting region if the structuring element lies completely within the input region. As a result, erosion can alternatively be used to find objects.
| (1) | (2) | (3) | (4) |
These operators can be used to dilate or erode a region:
| Morphological Operators | Structuring Element | Reference Point | |
|---|---|---|---|
| dilation1 | erosion1 | arbitrary | origin |
| minkowski_add1 | minkowski_sub1 | arbitrary, transposed | origin |
| dilation2 | erosion2 | arbitrary | arbitrary |
| minkowski_add2 | minkowski_sub2 | arbitrary, transposed | arbitrary |
| dilation_circle | erosion_circle | circular | origin |
| dilation_rectangle1 | erosion_rectangle1 | rectangular | origin |
Note that Minkowski addition and dilation are identical if the structuring element is symmetric. The same applies to Minkowski subtraction and erosion. Erosion can be used to separate objects that are attached to each other. In the following, the steps that are required to separate objects are described briefly.
First, the objects of the image must be segmented, for example by using the operator threshold. Next, the operator connection is used to get multiple regions instead of a single region. As you can see in the image (3) below, the result of the connection is unsatisfactory because several objects are merged.
| (1) | (2) | (3) |
This problem can be solved using erosion. As mentioned above, erosion reduces the area of the input region. Thus, if erosion is applied prior to the operator connection, the regions are separated as desired. Lastly, dilation is applied on the separated regions to approximately get the original shape back.
| (1) | (2) | (3) |
Both operators generate the resulting region by combining dilation and erosion. Opening is an erosion followed by a dilation. It is useful to eliminate small unwanted structures. Closing is the opposite of opening, i.e., a dilation followed by an erosion. The closing operator is able to close small gaps, as shown below.
| (1) | (2) | (3) | (4) |
These operators can be used to open or close a region:
| Morphological Operators | Structuring Element | Reference Point | |
|---|---|---|---|
| opening | closing | arbitrary | origin |
| opening_circle | closing_circle | circular | origin |
| opening_rectangle1 | closing_rectangle1 | rectangular | origin |
In addition to the operators mentioned above, you can use top_hat to get the difference between the input region and the result of the opening, or bottom_hat to get the difference between the result of the closing and the input region. Furthermore, you can calculate the boundary of a region with the operator boundary.
| (1) | (2) | (3) |
The operator hit_or_miss can be used to find objects, taking the foreground and the background of the image into account. To remove unwanted branches from a skeleton, pruning is a suitable operator.
In the following list, the most important terms that are used in the context of Morphology are described.
Region which is modified by morphological operators.
Region which is used to scan the input region.
| Operators |