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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 8connected 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 nonempty (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 8connected to
each object pixel (with value "1") to the value "1".
Erosion
Alternatively,
If the same simple structuring elements are used, erosion can be carried
out by setting each object pixel (with value "1") 4 or 8connected to a
background pixel (with value "0") to the value "0".
Properties of dilation and erosion
 Commutative:
 Noncommutative:
 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;
Grayscale morphology (bottom): original image, erosion, dilation, histogram, binary image (threshold 130).
Next: A Thinning Algorithm
Up: morphology
Previous: morphology
Ruye Wang
20111109