Morphological Image Processing

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Morphological Image Processing

• by
• Rajesh Narra

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Introduction
Morphology deals with form and Structure of
animals and plants
Mathematical Morphology deals with set theory
Sets in Mathematical Morphology represents
objects in an Image
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How to represent images as sets ?
For Binary Images it is an 2-D integer space
where sets represents Black pixels (White pixels).
Y axis
X axis
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Contd....
Gray Scale images are represented as 3-D integer
space where Z axis refers to gray level value.
Y axis
X axis
Z axis
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Basic set operators
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Basic Morphology Operators

• Dilation
• Erosion
• Opening
• Closing
• Hit or Miss Transformation

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Basic Components in Morphology

• Every Operation has two elements are present .
• 1) Input Image (almost Binary)
• 2) Structuring element
• Mainly operations results depends upon the
  structure element that is chosen by us.

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Dilation

• Dilation of A by B is denoted and defined as
• A ? B z ( B )z ? A ?? ?

Theoretical way of generation Obtaining
reflection of B about its origin and shifting
this reflection by z the dilation of A by B then
is set of all displacements, z such that B and A
overlap by at least one element
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How to implement this practically
Structuring element
Resultant image
Input Image
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Gray Scale Images
Structuring element
Input Image
Resultant Image
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Erosion
Erosion of A by B is denoted and defined as A? O
B z ( B )z ? A
Theoretical way of generation Erosion of A by B
is the set of all points such that B, translated
by z is contained in A.
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Binary Images
Structuring element
Output image
Input Image
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Gray scale Images
Structuring element
Input Image
Output image
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• What happens when we do dilation after erosion
  and vice versa

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Answer

• Opening
• Erosion followed by dilation
• Closing
• Dilation followed by Erosion

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Opening

• Opening of A by B is denoted by
• A o B ( A
  ? O B ) ? B.
• It is less destructive than the Erosion.

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Application
Structuring element is an 11pixel array with a
circle shape
Input Image
Resultant Image
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Closing

• Closing of A by B is denoted and defined as
• A ? B ( A ? B ) O B
• Closing is less destructive of the original
  boundary shape than dilation

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Application
Structure element is in between the smaller
circle and large circle
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Hit or Miss Transform

• Basic operation where all other operators
• can be derived
• It Hit or miss transform of A by B is denoted by
• A B
• The difference is in structure element from
  others

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Example
Structuring elements
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Thinning

• Thinning is used to remove selected pixel from
  foreground pixels
• Thinning of A by B is denoted and defined as
• A? B A- H/M(A,B)
• Thinning is applied repeatedly until no change is
  observed in Image
• A? B ((.. (A? B1) ? B2) ? B3).. ) ? Bn)

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Thinning Illustration
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Example
Input Image
Output result Image
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Thickening

• Thickening is used to grow selected regions of
  foreground pixels in binary images
• Thinning of A by B is denoted and defined as

?
A
B A U H/M(A,B)
Similarly Thickening is also repeatedly applied
to image until no change is observed.
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Thickening Illustration
45 degrees oriented Structuring elements
Output result Image
Input Image
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Duals
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Simple Morphology Applications

• Boundary Extraction
• Region Filling
• Connected Components

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Boundary Extraction

• Boundary of a set A is denoted by ?(A) and is
  defined as
• ß(A) A (A O B )
• Where B is a suitable Structuring element
• This is obtained by first eroding A by B and then
  performing the set difference between A and its
  erosion.

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Example
Input image
Output Image
Structure element used is a 3X3 all 1s matrix
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• What is difference between the edge detection
  filters and the current approach

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Region Filling

• Based on set dilations , complementation and
  intersections.
• Procedure
• Xk (Xk-1 ?? B) ? Ac k 1,2,3..
• Where X0 p and B is symmetric structure element
• Terminates if Xk Xk-1

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Example
Input Image
Output result Image
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Extraction of Connected Components

• Procedure
• Assume that a point p of Y ( a connected
  component in A) is known.
• Following iterative expression yields all the
  elements of Y.
• Xk (Xk-1 ?? B) ? A k 1,2,3..
• Where X0 p and B is symmetric structure element
• Terminates if Xk Xk-1

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References

• R. C. Gonzalez and R. E. Woods Digital Image
  Processing, Addison-Wesley, New York, 199, Chap
  9.
• E. Davies Machine Vision Theory, Algorithms and
  Practicalities, Academic Press, 1990, pp 149 -
  161.
• R. Haralick and L. Shapiro Computer and Robot
  Vision, Vol. 1, Addison-Wesley Publishing
  Company, 1992, Chap. 5.
• A. Jain Fundamentals of Digital Image Processing,
  Prentice-Hall, 1989, Chap. 9.
• http//www.cee.hw.ac.uk/hipr/html/morops.html
  University of Edinburgh UK
• http//www.mathworks.com/ MATLAB website