Morphological Image Processing

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Morphological 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.

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Morphological 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

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Morphological Image Processing
Set Theory Fundamentals
Given 2 sets A and B
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Morphological 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
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Morphological 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.

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Morphological 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.

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Morphological 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 .

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Morphological 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.

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Morphological 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.

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Morphological 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
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Morphological 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
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Morphological 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
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Morphological 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

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Morphological 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
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Morphological 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
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Morphological 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
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Morphological 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
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Morphological 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)
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Morphological 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,
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Morphological 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.
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Morphological 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).

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Morphological Image Processing
Basic Morphological Algorithms

• Region Filling

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)
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Morphological 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
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Morphological 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
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Morphological 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.

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Morphological 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