Morphological Image Processing
The field of mathematical morphology contributes a wide range
of operators to image processing, all based around a few simple
mathematical concepts from
set theory
. The operators are
particularly useful for the analysis of
binary images
and
common usages include edge detection, noise removal, image
enhancement and image segmentation.
Morphological techniques typically probe an image with a small
shape or template known as a
structuring element.
The
structuring element is positioned at all possible locations in the
image and it is compared with the corresponding neighborhood
of pixels.Morphological operations differ in how they carry out
this comparison.
Some Basic Concepts from Set Theory
If a=(a1,a2) is an element of A, then we write:
a
A
If a is not an element of A , we write:
a
A
The set with no elements is called the null or empty set and is denoted by
the symbol
.
If every element of a set A is also an element of another set B, then a is
said to be a subset of B, denoted as
A
B
The
union
of two sets A and B, denoted by
C=A
B
Is the set of all elements belonging to either A,B or both
The
intersection
of two sets A and B, denoted by
D=A
B;
Is the set of all elements belonging to both A and B
Two sets A and B are said to be
disjoint or mutually exclusive
if they
have no common elements.
A
B =
Logical Operation Involving Binary Images
p
q
p AND q
p OR q
NOT p
0
0
0
0
1
0
1
0
1
1
1
0
0
1
0
1
1
1
1
0
The logical operations just descried have a one

to one
correspondence with the set operations is set theory ( union ,
intersection ) with the limitation that logical operations are
restricted to binary variables.
Fundamental Definitions
We defined an image as an (amplitude) function of two, real
(coordinate) variables
a
(
x
,
y
) or two, discrete variables
a
[
m
,
n
]. An
alternative definition of an image can be based on the notion that
an image consists of a set (or collection) of either continuous or
discrete coordinates. In a sense the set corresponds to the points
or pixels that belong to the objects in the image. This is illustrated
in Figure which contains two objects or sets
A
and
B
. Note that
the coordinate system is required. For the moment we will
consider the pixel values to be binary.
Figure
: A binary image containing two object sets
A
and
B
.
Fundamental Definitions
The object
A
consists of those pixels a that share some
common property:
Object

The background of
A
is given by
A
c
(the
complement
of
A
)
which is defined as those elements that are not in
A
:
Background

Fundamental Definitions
The fundamental operations associated
with an object are the standard set
operations
union
,
intersection
, and
complement
plus
translation
:
Fundamental Definitions
The
intersection
of any two binary images A and B , written A
B,
is the binary image which is 1 at all pixels
p
which are 1 in both A and
B
The intersection is computed by applying the rule:
}
{
B
p
and
A
p
p
B
A
The
union
of A and B, written A
B, is the binary image which is 1 at
all pixels p which are 1 in A or 1 in B ( or in both)
}
{
B
p
or
A
p
p
B
A
The
complement
of A is the binary image which interchange the 1s and
0s in A . Thus,
}
A
p
and
p
{p
A
Fundamental Definitions
* Translation

Given a vector
x
and a set
A
, the
translation
,
A
+
x
, is defined as:
Note that, since we are dealing with a digital image composed of
pixels at integer coordinate positions (
Z
2
), this implies restrictions
on the allowable translation vectors
x
.
The basic
Minkowski
set
operations

addition and subtraction

can
now be defined. First we note that the individual elements that
comprise
B
are not only pixels but also
vectors
as they have a
clear coordinate position with respect to [0,0]. Given two sets
A
and
B
:
Minkowski
addition

Minkowski
subtraction

Fundamental Definitions
–
Erosion and Dilation
From these two Minkowski operations we define the
fundamental mathematical morphology operations
dilation
and
erosion
:
Erosion

Dilation

wher
e
Fundamental Definitions
–
Erosion and Dilation
While either set A or B can be thought of as an
"image", A is usually considered as the image and B is
called a structuring element. The structuring element
is to mathematical morphology what the convolution
kernel is to linear filter theory.
Dilation
, in general, causes objects to dilate or grow in size;
erosion
causes objects to shrink. The amount and the way
that they grow or shrink depend upon the choice of the
structuring element. Dilating or eroding without specifying
the structural element makes no more sense than trying to
lowpass filter an image without specifying the filter.
Structuring Elements
The structuring element is sometimes called the
kernel
, but we
reserve that term for the similar objects used in convolutions.
The structuring element consists of a pattern specified as the
coordinates of a number of discrete points relative to some origin.
Normally Cartesian coordinates are used and so a convenient way
of representing the element is as a small image on a rectangular
grid. Figure 1 shows a number of different structuring elements of
various sizes. In each case the origin is marked by a ring around that
point. The origin does not have to be in the center of the structuring
element, but often it is. As suggested by the figure, structuring
elements that fit into a 3
×
3 grid with its origin at the center are the
most commonly seen type.
Structuring Elements
Figure 1
Some example structuring elements.
Sstructuring Element
Note that each point in the structuring element may have a value. In the
simplest structuring elements used with binary images for operations
such as erosion, the elements only have one value, conveniently
represented as a one. More complicated elements, such as those used
with
thinning
or
grayscale morphological operations
, may have other
pixel values.
The
structuring element
is already just a set of point coordinates
(although it is often represented as a binary image). It differs from
the input image coordinate set in that it is normally much smaller,
and its coordinate origin is often not in a corner, so that some
coordinate elements will have negative values. Note that in many
implementations of morphological operators, the structuring
element is assumed to be a particular shape (
e.g.
a 3
×
3 square) and
so is hardwired into the algorithm.
Sstructuring Element
The two most common structuring elements
(given a Cartesian grid) are the 4

connected
and 8

connected sets,
N
4
and
N
8
. They are
illustrated in Figure .
Figure: The standard structuring elements
N
4
and
N
8
.
Binary Images
For a binary image,
white pixels are normally taken to represent
foreground regions
, while black pixels denote background. (Note
that in some implementations this convention is reversed, and so it
is very important to set up input images with the correct polarity for
the implementation being used). Then the set of coordinates
corresponding to that image is simply the set of two

dimensional
Euclidean coordinates of all the foreground pixels in the image,
with an origin normally taken in one of the corners so that all
coordinates have positive elements.
Gray Scale Images
For a grayscale image, the
intensity value
is taken to represent height
above a base plane, so that the grayscale image represents a surface in
three

dimensional Euclidean space. Figure 2 shows such a surface. Then
the set of coordinates associated with this image surface is simply the set
of three

dimensional Euclidean coordinates of all the points within this
surface
and also all points below the surface, down to the base plane
.
Note that even when we are only considering points with integer
coordinates, this is a lot of points, so usually algorithms are employed
that do not need to consider all the points.
Fundamental Morphological Operations
Erosion
and
dilation
work (at least conceptually) by translating the
structuring element to various points in the input image, and
examining the intersection between the translated kernel coordinates
and the input image coordinates.
For instance, in the case of erosion, the output coordinate set
consists of just those points to which the origin of the structuring
element can be translated, while the element still remains entirely
`within' the input image.
Virtually all other mathematical morphology operators can be defined
in terms of combinations of erosion and dilation along with set
operators such as intersection and union. Some of the more important
are
opening
,
closing
and
skeletonization
.
Fitting and Hitting
When we place a structuring element in a binary image, each of
its pixels is associated with the corresponding pixel of the
neighborhood under the structuring element. In this sense, a
morphological operation resembles a “binary correlation”. The
operation is logical rather than arithmetic in nature.
The structuring element is said to
fit
the image if,
for each of its
pixels that is set to 1
, the corresponding image pixel is also 1.
The structuring element is said to
hit
, an image if
for any of its
pixels that is set to 1
, the corresponding image pixel is also 1.
Erosion
The basic effect of the operator on a binary image is to
erode
away the boundaries of regions of
foreground pixels
(
i.e.
white pixels, typically). Thus areas of foreground pixels shrink
in size, and holes within those areas become larger
Figure 2
Effect of erosion using a 3
×
3 square structuring element
Strip away a layer of pixels from an object, shrinking it in the
process.
Erosion

How It Works
The erosion operator takes two pieces of data as inputs. The first is the
image which is to be eroded. The second is a (usually small) set of
coordinate points known as a structuring element (also known as a
kernel
). It is this structuring element that determines the precise effect
of the erosion on the input image.
The mathematical definition of erosion for
binary
images is as
follows:
* Suppose that
X
is the set of Euclidean coordinates corresponding
to the input binary image, and that
K
is the set of coordinates for
the structuring element.
* Let
Kx
denote the translation of
K
so that its origin is at
x
.
* Then the erosion of
X
by
K
is simply the set of all points
x
such
that
Kx
is a subset of
X
.
Erosion

How It Works
As an example of binary erosion, suppose that the structuring
element is a 3
×
3 square, with the origin at its center as shown in
Figure 2. Note that in this and subsequent diagrams, foreground
pixels are represented by 1's and background pixels by 0's.
The erosion of a binary image
A
by a binary image
B
is 1 at a
pixel
p
if and only if every 1 pixel in the translation of B to p is
also 1 in A.
Figure 2
Erosion

How It Works
To compute the erosion of a binary input image by this structuring
element, we consider each of the
foreground
pixels in the input image
in turn. For each foreground pixel (which we will call the
input pixel
)
we superimpose the structuring element on top of the input image so
that the origin of the structuring element coincides with the input pixel
coordinates.
If for
every
pixel in the structuring element, the
corresponding pixel in the image underneath is a foreground pixel,
then the input pixel is left as it is. If any of the corresponding
pixels in the image are background, however, the input pixel is
also set to background value.
For our example 3
×
3 structuring element, the effect of this operation is
to remove any foreground pixel that is not completely surrounded by
other white pixels (assuming
8

connectedness
). Such pixels must lie
at the edges of white regions, and so the practical upshot is that
foreground regions shrink (and holes inside a region grow).
Erosion

Guidelines for Use
Most implementations of this operator will expect the input image to be
binary, usually with foreground pixels at intensity value 255, and
background pixels at intensity value 0. Such an image can often be
produced from a grayscale image using
thresholding
. It is important to
check that the
polarity
of the input image is set up correctly for the
erosion implementation being used.
The structuring element may have to be supplied as a small binary
image, or in a special matrix format, or it may simply be hardwired into
the implementation, and not require specifying at all. In this latter case, a
3
×
3 square structuring element is normally assumed which gives the
shrinking effect described above. The effect of an erosion using this
structuring element on a binary image is shown in Figure 3
Erosion

Guidelines for Use
The 3
×
3 square is probably the most common structuring element used
in erosion operations, but others can be used. A larger structuring
element produces a more extreme erosion effect, although usually very
similar effects can be achieved by repeated erosions using a smaller
similarly shaped structuring element.
Erosions can be made directional by using less symmetrical
structuring elements. For example, a structuring element that is 10
pixels wide and 1 pixel high will erode in a horizontal direction only.
Similarly, a 3
×
3 square structuring element with the origin in the
middle of the top row rather than the center, will erode the bottom of
a region more severely than the top.
Erosion

Guidelines for Use
This image is the result of eroding four times
with a disk shaped structuring element 11 pixels
in diameter.
It shows that the hole in the
middle of the image increases in size as the
border shrinks
. Note that the shape of the
region has been quite well preserved due to the
use of a disk shaped structuring element. In
general, erosion using a disk shaped structuring
element will tend to round concave boundaries,
but will preserve the shape of convex
boundaries.
Erosion

Guidelines for Use
There are many specialist uses for erosion. One of the more
common is to separate touching objects in a binary image so that
they can be counted using a
labeling algorithm
. The image shows
a number of dark disks (coins in fact) silhouetted against a light
background
The result of
thresholding
the image at pixel value 90 yields :
It is required to count the coins.
However, this is not going to be
easy since the touching coins
form a single fused region of
white, and a counting algorithm
would have to first segment this
region into separate coins
before counting, a non

trivial
task
Erosion

Guidelines for Use
.The situation can be much simplified by eroding the image
The image shows the result of eroding
twice using a disk shaped structuring
element 11 pixels in diameter. All the
coins have been separated neatly and the
original shape of the coins has been
largely preserved. At this stage a
labeling
algorithm
can be used to count the coins.
The relative sizes of the coins can be used
to distinguish the various types by, for
example, measuring the area of each
distinct region.
Erosion

Guidelines for Use
The image is derived from the
same input picture, but a 9
×
9
square structuring element is used
instead of a disk (the two
structuring elements have
approximately the same area). The
coins have been clearly separated
as before, but the square
structuring element has led to
distortion of the shapes, which is
some situations could cause
problems in identifying the regions
after erosion.
Erosion

Guidelines for Use
Erosion can also be used to remove small spurious bright spots
(`salt noise‘ )
in images.
We can also use erosion for
edge detection
by taking the erosion
of an image and then
subtracting
it away from the original
image, thus highlighting just those pixels at the edges of objects
that were removed by the erosion.
Finally, erosion is also used as the basis for many other
mathematical morphology operators.
Dilation
The basic effect of the operator on a binary image is to
gradually
enlarge
the boundaries of regions of
foreground pixels
(
i.e.
white
pixels, typically). Thus areas of foreground pixels grow in size while
holes within those regions become smaller.
Dilation

How It Works
The dilation operator takes two pieces of data as inputs. The first is
the image which is to be dilated. The second is a (usually small) set of
coordinate points known as a structuring element (also known as a
kernel
). It is this structuring element that determines the precise effect
of the dilation on the input image.
The mathematical definition of dilation for
binary
images is as follows:
•
Suppose that
X
is the set of Euclidean coordinates corresponding to the
input binary image, and that
K
is the set of coordinates for the structuring
element.
•
Let
Kx
denote the translation of
K
so that its origin is at
x
.
•
Then the dilation of
X
by
K
is simply the set of all points
x
such that the
intersection of
Kx
with
X
is non

empty.
Dilation

How It Works
To compute the dilation of a binary input image by this structuring
element, we consider each of the
background
pixels in the input image in
turn. For each background pixel (which we will call the
input pixel
) we
superimpose the structuring element on top of the input image so that the
origin of the structuring element coincides with the input pixel position.
If
at least one
pixel in the structuring element coincides with a
foreground pixel in the image underneath, then the input pixel is set
to the foreground value. If all the corresponding pixels in the image
are background, however, the input pixel is left at the background
value.
For our example 3
×
3 structuring element, the effect of this operation is to
set to the foreground color any background pixels that have a
neighboring foreground pixel (assuming 8

connectedness). Such pixels
must lie at the edges of white regions, and so the practical upshot is that
foreground regions grow (and holes inside a region shrink).
Dilation is the
dual
of erosion
i.e.
dilating foreground
pixels is equivalent to eroding the background pixels
Guidelines for Use
Most implementations of this operator expect the input image to be
binary, usually with foreground pixels at pixel value 255, and
background pixels at pixel value 0. Such an image can often be produced
from a grayscale image using
thresholding
. It is important to check that
the
polarity
of the input image is set up correctly for the dilation
implementation being used.
A larger structuring element produces a more extreme dilation effect,
although usually very similar effects can be achieved by repeated
dilations using a smaller but similarly shaped structuring element. With
larger structuring elements, it is quite common to use an approximately
disk shaped structuring element, as opposed to a square one.
Guidelines for Use
This image was produced by two dilation
passes using a disk shaped structuring
element of 11 pixels radius. Note that the
corners have been rounded off. In general,
when dilating by a disk shaped structuring
element, convex boundaries will become
rounded, and concave boundaries will be
preserved as they are.
Guidelines for Use
There are many specialist uses for dilation. For instance it can be used to
fill in small spurious holes (`
pepper noise'
) in images
The image shows the result of dilating
this image with a 3
×
3 square
structuring element. Note that
although the noise has been effectively
removed, the image has been degraded
significantly.
Guidelines for Use
Dilation can also be used for
edge detection by taking the
dilation of an image and then subtracting
away the
original image, thus highlighting just those new pixels at the
edges of objects that were added by the dilation. For
example, starting with
Effect of Dilation and Erosion
•
Original Binary Image: Black
•
Structure Element: Red
•
Resultant Image: White
a)Dilation
b)Erosion
B) We must imagine that we are sliding the structuring
element around the boundary on the
inside
of the object
The simples dilation and erosion algorithms for
binary images
The processing of boundary pixels instead of object pixels
means that, computational complexity can be reduced from
O
(
N
2
) to
O
(
N
) for an
N
x
N
image. A number of "fast"
algorithms can be found in the literature that are based on this
result . The simplest dilation and erosion algorithms are
frequently described as follows.
* Dilation

Take each binary object pixel (with value "1") and
set all background pixels (with value "0") that are
C

connected
to that object pixel to the value "1".
* Erosion

Take each binary object pixel (with value "1") that is
C

connected to a background pixel and set the object pixel
value to "0".
The simples dilation and erosion algorithms for
binary images
Comparison of these two procedures to eq. where
B
=
N
C=4
or
N
C=8
shows that they are equivalent to the
formal definitions for dilation and erosion. The
procedure is illustrated for
dilation
in next Figure .
Figure
: Illustration of
dilation
. Original object pixels are in gray;
pixels added through
dilation
are in black
(a)
B
=
N
4
(b)
B
=
N
8
Thus,
dilation
and
erosion
on binary images can be
viewed as a form of convolution over a Boolean algebra.
More examples
Finally, dilation is also used as the basis for many
other mathematical morphology operators, often in
combination with some
logical operators
. A simple
example is
region filling
which is illustrated using
the image
This image and all the following results were zoomed with a factor of
16
for a better display,
i.e.
each pixel during the processing corresponds to a
16
×
16
pixel square in the displayed images. Region filling applies
logical NOT
,
logical AND
and dilation iteratively. The process can be
described by the following formula:
where Xk is the region which after convergence fills the boundary,
J
is the structuring element and A
not
is the negative of the boundary.
This combination of the dilation operator and a logical operator is also
known as
conditional dilation
.
More examples
Imagine that we know
Xo
,
i.e.
one pixel which lies inside the region
shown in the above image,
e.g.
First, we dilate the image containing the single
pixel using a structuring element resulting in
To prevent the
growing region from
crossing the
boundary, we AND
it with
which is the
negative of the
boundary.
Dilating the
resulting
image,
More examples
Repeating these two steps until convergence, yields
and finally
ORing
this image with the initial boundary
yields the final result, as can be seen in
Opening
The basic effect of an
opening
is somewhat like
erosion
in that it tends to remove some of the
foreground (bright) pixels from the edges of regions
of foreground pixels. However it is less destructive
than erosion in general. As with other morphological
operators, the exact operation is determined by a
structuring element. The effect of the operator is to
preserve
foreground
regions that have a similar shape
to this structuring element, or that can completely
contain the structuring element, while eliminating all
other regions of foreground pixels.
Opening

How It Works
Very simply, an opening is defined as an erosion followed by a
dilation
using the same structuring element for both operations
.
The opening operator therefore requires two inputs: an image to be
opened, and a structuring element.
Opening

Opening is the
dual
of closing,
i.e.
opening the foreground pixels
with a particular structuring element is equivalent to closing the
background pixels with the same element
Closing

Opening and Closing

Properties
Duality

Translation

For the
opening
with structuring element
B
and images
A
,
A
1
,
and
A
2
, where
A
1
is a subimage of
A
2
(
A
1
A
2
):
Antiextensivity

Increasing monotonicity

Idempotence

Opening and Closing

Properties
For the
closing
with structuring element
B
and images
A
,
A
1
, and
A
2
, where
A
1
is a subimage of
A
2
(
A
1
A
2
):
Extensivity

Idempotence
Some operators have the special property that applying them
more than once to the same image produces no further change
after the first application. Such operators are said to be
idempotent
. Examples include the morphological operators
opening and closing.
Increasing monotonicity

Idempotence

Opening

Guidelines for Use
While erosion can be used to eliminate small clumps of undesirable
foreground pixels,
e.g.
`salt noise', quite effectively, it has the big
disadvantage that it will affect
all
regions of foreground pixels
indiscriminately. Opening gets around this by performing both an erosion
and a dilation on the image. The effect of opening can be quite easily
visualized. Imagine taking the structuring element and sliding it around
inside
each foreground region, without changing its orientation.
All
pixels which can be covered by the structuring element with the
structuring element being entirely within the foreground region will
be preserved.
However, all foreground pixels which cannot be reached
by the structuring element without parts of it moving out of the
foreground region will be eroded away. After the opening has been
carried out, the new boundaries of foreground regions will all be such
that the structuring element fits inside them, and so further openings with
the same element have no effect. The property is known as
idempotence
.
Opening

Guidelines for Use
Figure :
Effect of opening using a 3
×
3 square structuring element
As with erosion and dilation, it is very common to use this 3
×
3
structuring element. The effect in the above figure is rather subtle since
the structuring element is quite compact and so it fits into the foreground
boundaries quite well even before the opening operation
To increase the effect, multiple erosions are often performed with this
element followed by the same number of dilations. This effectively
performs an opening with a larger square structuring element.
Opening

Guidelines for Use
A binary image containing a mixture of
circles and lines. Suppose that we want to
separate out the circles from the lines, so that
they can be counted. Opening with a disk
shaped structuring element 11 pixels in
diameter gives
Some of the circles are slightly distorted, but in general, the lines have
been almost completely removed while the circles remain almost
completely unaffected.
Opening

Guidelines for Use
Suppose that this time we wish to separately
extract the horizontal and vertical lines.
The result of an opening
with a 3
×
9 vertically
oriented structuring
element is shown in
The image shows what happens if we use a 9
×
3
horizontally oriented structuring element instead.
Note that there are a few glitches in this last image
where the diagonal lines cross vertical lines.
Opening

Guidelines for Use
As we have seen, opening can be very useful for separating out
particularly shaped objects from the background, but it is far from
being a universal 2

D object recognizer/segmenter. For instance if
we try and use a long thin structuring element to locate, say,
pencils in our image, any one such element will only find pencils
at a particular orientation. If it is necessary to find pencils at other
orientations then differently oriented elements must be used to
look for each desired orientation. It is also necessary to be very
careful that the structuring element chosen does not eliminate too
many desirable objects, or retain too many undesirable ones, and
sometimes this can be a delicate or even impossible balance.
Opening

Guidelines for Use
The image contains
two kinds of cell:
small, black ones and
larger, gray ones.
Thresholding the
image at a value of
210
yields
We want to retain only the
large cells in the image,
while removing the small
ones. This can be done
with straightforward
opening. Using a
11
pixel
circular structuring
element yields
Closing
Closing is similar in some ways to
dilation
in that it tends to
enlarge the boundaries of foreground (bright) regions in an
image (and shrink background color holes in such regions), but
it is less destructive of the original boundary shape. As with
other morphological operators, the exact operation is
determined by a structuring element. The effect of the operator
is to preserve
background
regions that have a similar shape to
this structuring element, or that can completely contain the
structuring element, while eliminating all other regions of
background pixels.
Closing

How It Works
Closing is opening performed in reverse. It is defined simply as a
dilation followed by an erosion
using the same structuring element
for both operations
..The closing operator therefore requires two
inputs: an image to be closed and a structuring element.
Closing

Closing

Guidelines for Use
One of the uses of dilation is to fill in small background color holes in
images,
e.g.
`pepper noise'. One of the problems with doing this, however,
is that the dilation will also distort
all
regions of pixels indiscriminately. By
performing an erosion on the image after the dilation,
i.e.
a closing, we
reduce some of this effect. The effect of closing can be quite easily
visualized. Imagine taking the structuring element and sliding it around
outside
each foreground region, without changing its orientation. For any
background boundary point, if the structuring element can be made to touch
that point, without any part of the element being inside a foreground region,
then that point remains background. If this is not possible, then the pixel is
set to foreground. After the closing has been carried out the background
region will be such that the structuring element can be made to cover any
point in the background without any part of it also covering a foreground
point, and so further closings will have no effect. This property is known as
idempotence
Closing

Guidelines for Use
Figure:
Effect of closing using a 3
×
3 square structuring element
Closing

Guidelines for Use
An image containing large holes and small
holes. If it is desired to remove the small holes
while retaining the large holes, then we can
simply perform
a closing with a
disk

shaped
structuring element with a diameter larger
than the smaller holes, but smaller than the
large holes.
The image is the result of a
closing with a 22 pixel diameter
disk.
Links
Image Processing Fundamentals

Morphology

based Operations
http://www.mmorph.com/resources.html
http://cmm.ensmp.fr/~beucher/wtshed.html
http://www.cwi.nl/projects/morphology/
http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip

Morpholo.html
http://www.mathworks.com/access/helpdesk/help/toolbox/images
/images.shtml
http://www

dsv.cea.fr/thema/shfj/web/demo_extraction/english/cerveau.htm
Comments 0
Log in to post a comment