Title: Morphological Image Processing
1Morphological Image Processing
- The word morphology refers to the scientific
branch that deals the forms and structures of
animals/plants. - Morphology in image processing is a tool for
extracting image components that are useful in
the representation and description of region
shape, such as boundaries and skeletons. - Furthermore, the morphological operations can be
used for filtering, thinning and pruning. - The language of the Morphology comes from the set
theory, where image objects can be represented by
sets. For example an image object containing
black pixels can be considered a set of black
pixels in 2D space of Z2.
2Morphological Image Processing
Set Theory Fundamentals
- Given that A is a set in Z2and a(a1,a2), then
- a is an element in A
- a is not an element in A
- given sets A and B, A is said to be the subset
of B - The union of A and B is denoted by
- The intersection of A and B is denoted by
- Two sets are disjoint/mutually exclusive if
- The complement of set A is the set of elements
not contained in A, - The difference of two sets
3Morphological Image Processing
Set Theory Fundamentals
Given 2 sets A and B
4Morphological Image Processing
Set Theory Fundamentals
- The reflection of set B is defined by
- The translation of set A by point z(z1,z2) is
defined by
Translation of A by z.
Reflection of B
5Morphological Image Processing
Logic operation involving Binary Images
- Given 1-bit binary images, A and B, the basic
logical operations are illustrated
- Note that the black indicates binary 1 and
white indicates binary 0 here.
6Morphological Image Processing
Dilation and Erosion
- Dilation and erosion are the two fundamental
operations used in morphological image
processing. Almost all morphological algorithms
depend on these two operations
- Dilation Given A and B sets in Z2, the
dilation of A by B, is defined by
- The dilation of A and B is a set of all
displacements, z , such that B and A overlap by
at least one element. The definition can also be
written as
- Set B is referred to as the structuring element
and used in dilation as well as in other
morphological operations. Dilation
expands/dilutes a given image.
7Morphological Image Processing
Dilation and Erosion
- Dilation Given the structuring element B and
set A.
Shaded area is the dilation of A by B
origin
- The structuring element B enlarges the size of A
at its boundaries. Dilation simply expands a
given image.
- The structuring element B enlarges the size of A
at its boundaries, in relation to the distance
from the origin of the structuring element .
8Morphological Image Processing
Dilation and Erosion
- Dilation Given the following distorted text
image where the maximum length of the broken
characters are 2 pixels. The image can be
enhanced by bridging the gaps by using the
structuring element given below
3x3 structuring element
- Note that the broken characters are joined.
9Morphological Image Processing
Dilation and Erosion
- Erosion Given A and B sets in Z2, the erosion
of A by structuring element B, is defined by
- The erosion of A by structuring element B is the
set of all points z, such that B, translated by
z, is contained in A.
Shaded area is the erosion of A by B
structuring element
- Note that in erosion the structuring element B
erodes the input image A at its boundaries.
Erosion shrinks a given image.
10Morphological Image Processing
Dilation and Erosion
- Erosion Given the structuring element B and
set A.
Shaded line is what is left from the erosion of A
by B
structuring element
11Morphological Image Processing
Dilation and Erosion
- Erosion Given the following binary image with
squares on size 1,3,5,7,9 and 15. You can get rid
of all the squares less than size of 15 by
erosion followed by dilation of a structuring
element of 13x13.
13x13 structuring element
Erosion of A by B
12Morphological Image Processing
Dilation and Erosion
- Dilation Cont. from the previous slide. Note
that erosion followed by dilation helps to
perform filtering.
13x13 structuring element
dilation by B
13Morphological Image Processing
Opening and closing
- Opening The process of erosion followed by
dilation is called opening. It has the effect of
eliminating small and thin objects, breaking the
objects at thin points and smoothing the
boundaries/contours of the objects.
- Given set A and the structuring element B.
Opening of A by structuring element B is defined
by
- Closing The process of dilation followed by
erosion is called closing. It has the effect of
filling small and thin holes, connecting nearby
objects and smoothing the boundaries/contours
of the objects. - Given set A and the structuring element B.
Closing of A by structuring element B is defined
by
14Morphological Image Processing
Opening and closing
- Opening The opening of A by the structuring
element B is obtained by taking the union of all
translates of B that fit into A. - The opening operation can also be expressed by
the following formula
Outer boundary of A
Origin of B Circular structuring element
Shaded area complete opening
Possible translations of B in A
15Morphological Image Processing
Opening and closing
- Closing The closing has a similar geometric
interpretation except that we roll B on the
outside of the boundary. - The opening operation can also be expressed by
the following formula
Outer boundary of A
Outer boundaries of closing
Shaded area complete closing
Possible translations of B on the outer
boundaries of A
16Morphological Image Processing
Opening and closing
B circular structuring element
result of erosion of A by B
result of opening of A by B
result of dilation of A by B
result of closing of A by B
17Morphological Image Processing
Opening and closing
- Noise Filtering The morphological operations
can be used to remove the noise as in the
following example
3x3 square structuring element
result of opening followed by closing Note that
impulsive noise within the background and the
fingerprints is removed.
after opening
18Morphological Image Processing
Hit-or-Miss Transform (Template Matching)
- Hit-or-miss transform can be used for shape
detection/ Template matching. - Given the shape as the structuring element B1 the
Hit-or-miss transform is defined by
- Where B2 W-X and B1X. W is the window
enclosing B1. Windowing is used to isolate the
structuring element/object.
B2W-X, used as the second structuring element.
Complement of B1
Shape that we are searching for Used as the
structuring element (B1X)
19Morphological Image Processing
Hit-or-Miss Transform
B2
B1
Complement of A
Erosion of A by B1
Erosion of comp of AC by B2
The location of the matched object/shape,
20Morphological Image Processing
Basic Morphological Algorithms
- Boundary Extraction The boundaries/edges of a
region/shape can be extracted by first applying
erosion on A by B and subtracting the eroded A
from A.
Ex 1 3x3 Square structuring element is used for
boundary extraction.
Ex 2 The same structuring element in Ex1 is
used. Note that thicker boundaries can be
obtained by increasing the size of structuring
element.
21Morphological Image Processing
Basic Morphological Algorithms
- Region Filling Region filling can be performed
by using the following definition. Given a
symmetric structuring element B, one of the
non-boundary pixels (Xk) is consecutively diluted
and its intersection with the complement of A is
taken as follows
- Following consecutive dilations and their
intersection with the complement of A, finally
resulting set is the filled inner boundary region
and its union with A gives the filled region F(A).
22Morphological Image Processing
Basic Morphological Algorithms
This region is filled first.
Filling of all the other regions
A non-boundary pixel
Ex 1 X01 (Assume that the shaded boundary
points are 1 and the white pixels are 0)
23Morphological Image Processing
Basic Morphological Algorithms
- Connected Component Extraction The following
iterative expression can be used to determine all
the pixels in component Y which is in A.
- X01 corresponds to one of the pixels on the
component Y. Note that one of the pixel locations
on the component must be known. - Consecutive dilations and their intersection
with A, yields all elements of component Y.
Result of first iteration
Known pixel, p
Result of second iteration
Result of last iteration
24Morphological Image Processing
Basic Morphological Algorithms
- Connected Component Extraction
Input image (Chicken fillet)
Thresholded image
15 connected components with different number of
pixels
After erosion by 5x5 square structuring element
of 1s
25Morphological Image Processing
Basic Morphological Algorithms
- Thinning Thinning of A by the structuring
element B is defined by
hit-or-miss transform/template matching
- Note that we are only interested in pattern
matching of B in A, so no background operation is
required of the hit-miss-transform. - The structuring element B consists of a sequence
of structuring elements, where Bi is the rotated
version of Bi-1. Each structuring elements helps
thinning in one direction. If there are 4
structuring elements thinning is performed from 4
directions separated by 90o. If 8 structuring
elements are used the thinning is performed in 8
directions separated by 45o. - The process is to thin A by one pass with B1,
then the result with one pass of B2, and continue
until A is thinned with one pass of Bn.
26Morphological Image Processing
Basic Morphological Algorithms
- Thinning The following set of structuring
elements are used for thinning operation.
...
If there is no change any more. Declared to be
the thinned object