Some Basic Morphological Algorithm

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Chapter 9

• Some Basic Morphological Algorithm

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Some Basic Morphological Algorithm

• Boundary Extraction
• Region Filling
• Extraction of Connected Components
• Convex Hull
• Thinning
• Thickening
• Skeletons
• Pruning

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Boundary Extraction

• The boundary of a set A denoted by
• Where B is a suitable structuring element
• Its algorithm is following these
• Eroding A by B
• Performing the set difference between A and its
  erosion

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Boundary Extraction
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Boundary Extraction
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Region Filling

• Beginning with a point p inside the boundary, the
  objective is to fill the entire region with 1s

Point p
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Region Filling

• If all nonboundary (background) points are
  labeled 0, then we assign a value of 1 to p to
  begin. The following procedure then fill the
  region with 1s

Where X0 p and B is the symmetric structuring
element.
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Region Filling Algorithm

• Pick a point inside p given it value 1
• Set X0 p
• Start k 1
• Repeat getting Xk by
• Terminate process if Xk Xk-1
• The set union of Xk and A is answer

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Region Filling Algorithm
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Region Filling Algorithm
p
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Region Filling
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Connected Components Extraction

• To establish if two pixels are connected, it must
  be determined if they are neighbors and if their
  gray levels satisfy a specified criterion of
  similary.
• In practice, extraction of connected components
  in a binary image is central to many automated
  image analysis applications.

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Connected Components Extraction

• Let Y represent a connected component contained
  in a set A and assume that a point p of Y is
  known
• Following expression

Where X0 p and B is the symmetric structuring
element.
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Connected Components Extraction Algorithm

• Pick a point of Y set p
• Set X0 p
• Start k 1
• Repeat getting Xk by
• Terminate process if Xk Xk-1
• The answer set Y is Xk

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Convex Hull

• A set A is said to be convex if the straight line
  segment joining any two points in A lies entirely
  with in A
• The convex hull H of an arbitrary set S is the
  smallest convex set containing
• The set difference H-S is called the convex
  deficiency of S.

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Convex Hull

• Let Bi, i1,2,3,4 represent 4 structure
• The procedure consists of implementing the
  equation
• Let , where the subscript conv
  (convergence) in the sense that
• Then the convex hull of A is

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Convex Hull Algorithm

• Set
• Do with B1
• Repeat to apply hit-or-miss transformation to A
  with B1 until no further change occur Xn.
• Union Xn with A, called D1
• Do same as B1 with B2, B3, and B4 hence we will
  get D1, D2, D3 and D4
• Union all of D will be the answer of convex hull

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Convex Hull Algorithm
B1
B2
B3
B4
Dont care
x
Background
Foreground
A
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HIT-or-MISS A with B1
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HIT-or-MISS A with B2
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HIT-or-MISS A with B3
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HIT-or-MISS A with B4
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Convex Hull
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Thinning

• The thinning of a set A by a structuring element
  B, can be defined
• Where Bi is a rotated version of Bi-1
• Using this concept, we now define thinning as

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Thickening

• The thickening of a set A by a structuring
  element B, can be defined
• Where Bi is a rotated version of Bi-1
• Using this concept, we now define thinning as

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