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# Morphology

Language of set theory

Let denote the set of all integers, and denote the set of integer pairs . The coordinates of a pixel in an image can therefore be considered as a component of .

Let and be two white regions in a black and white binary image. Then they can be considered as two sets in space, and their components and are just a pixel inside each of the regions.

In a binary image, an object of a certain shape can be represented by a set of either 4- or 8-connected white pixels in white background. To carry out certain operations on the shape of the object, the object and its back ground are mathematically represented as sets.

In general, a set is defined as a group of components all having certain properties or satisfying certain conditions:

• Object:

• Background:

where means not belong to''.

Basic definitions:

• Translation: The translation of by is

where .
• Reflection: The reflection of is

Here the reflection is with respect to a specific origin, such as a point center in the shape, e.g., the center of the shape.
• Complement: The complement of is

• Difference: The difference between and is

Below, a structuring element is a binary object used in varioius morphology operators. All elements are measured with respect to the origin located at the center of the object. For example, if is a square, then for any , .

Dilation

where represents empty set. Alternatively,

• Find reflection of set by flipping it about its origin. If is central symmetric, this step makes no difference;
• Translate (shift, slide) by displacement over ;
• is the set of all displacements 's that keep the intersect (overlap) of and non-empty (keep them in touch'').
• Dilation of a binary image shape by expands the shape by half of the size of .

As typically the structuring element is symmetric (either central symmetric or symmetric with respect to its vertical or horizontal axis), the reflection part of the dilation definition will be dropped in the following.

For these simplest structuring elements, dilation can be carried out by setting all background pixels (with value "0") 4- or 8-connected to each object pixel (with value "1") to the value "1".

Erosion

Alternatively,

• Translate (or slide) by over ;

• is the set of all translations 's that will keep the translated version of to be entirely contained in .
• Erosion of a binary image shape by shrinks the shape by half of the size of .

If the same simple structuring elements are used, erosion can be carried out by setting each object pixel (with value "1") 4- or 8-connected to a background pixel (with value "0") to the value "0".

Properties of dilation and erosion

• Commutative:

• Non-commutative:

• Associative:

• The complement of the erosion of by is the same as the dilation of complement of by :

• The complement of the dilation of by is the same as the erosion of complement of by .

Proof:

Since set is contained in , i.e., , it has no overlap with the complement of , i.e., , and the above equation can be written as:

But the complement of this set of all satisfying the condition is the set of all not satisfying the condition, i.e.,

Taking complement on both sides above, and replacing by , we get the second relation.

Opening

The opening of and is the dilation of the erosion of by .
• Dilation and erosion are not a pair of opposite operations in the sense that their effects do not cancel each other.
• The erosion carried out first eliminates small shapes (assumed to be noise) as well as shrinking the object shape, while the following dilation grows the object back (but not the noise).

Closing

The closing of and is the erosion of the dilation of by .
• The effect of dilation and erosion do not cancel each other.
• The dilation carried out first eliminates small holes inside the object shape (assumed to be noise) as well as expanding the object shape, while the following erosion shrinks the object back (but not the noise).

Grayscale dilation and erosion

Morphological operations can be generalized to grayscale images. Here neither the image nor the structuring element is binary any more, and the Boolean operations (AND and OR, union and intersect) used for binary images are replaced by addition, subtraction, maximum and minimum operations.

• Dilation:

where the maximum value is taken in the neighborhood of pixel defined by the structuring element . Dilation tends to grow the white regions of an image. If the structuring element has positive values, the resulting image tends to be brighter.
• Erosion:

where the minimum value is taken in the neighborhood of pixel defined by the structuring element . Erosion tends to shrink the white regions of an image. If the structuring element has positive values, the resulting image tends to be darker.

Applications:

• Extracting boundaries : The boundary of , denoted by , can be obtained as the difference of and its erosion or dilation :

• Filling holes:

• Thinning:

Example:

Example:

Binary morpohology (top): original image, histogram, binary image (threshold 160), erosion, dilation;

Gray-scale morphology (bottom): original image, erosion, dilation, histogram, binary image (threshold 130).

Next: A Thinning Algorithm Up: morphology Previous: morphology
Ruye Wang 2011-11-09