Digital Image Processing

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Digital Image Processing

• Chapter 9

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Chapter 9Morphological Image Processing

• Mathematical morphology
• A useful tool for extracting image components in
  the representation of region shape.
• Boundaries, skeletons, and convex hull.
• Set theory is usually used to describe
  mathematical morphology.
• Sets represent objects in a binary image.
• Black representing object, denoted by 1.
• White representing background , denoted by 0.

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9.1 Preliminaries

• Our interest in this chapter is sets in Z2, where
  each element denotes the coordinates of an object
  pixel.
• If a(a1, a2), we write if a is an
  element in A.
• if a is not an element in A.
• The null or empty set is denoted by .
• We use braces, , to specify the content of a
  set. For example, Cww-d, for .

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Operations of Sets
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Additional Definitions

• Translation
• Reflection

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9.1.2 Logic Operations Involving Binary Images

• The logic operations discussed in this section
  involve binary images.
• Black pixel 1.
• White pixel 0.

Note logic operations are restricted to binary
variables, which is not the case in general for
set operations.
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Logic Operations Involving Binary Images
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9.2 Dilation and Erosion

• These two operations are fundamental to
  morphological processing.
• Dilation to enlarge an object along its
  boundary.
• Erosion to shrink an object into a smaller size.

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9.2.1 Dilation

• With A and B are sets in Z2, the dilation of A by
  B, denoted A B, is defined as
• A B
• Other interpretation A B
• B is commonly referred to as the structuring
  element.
• The dilation of A by B is the set of all
  displacements, z, such that the reflection of B
  and A overlap by at least one element.

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The Illustration of Dilation
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The Implementation of Dilation

• Given a binary image f and the structuring
  element s, construct a duplicate of f, denoted by
  g.
• For each pixel p f(x, y), do the following
• If p is black
• If p is at the boundary (any of the 4-adjacent
  neighbors is white) of the object, center the
  origin of s at (x, y) in g, and fill the pixels
  black on which s covers.
• Return g.

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Application of Dilation

• One of the simplest applications of dilation is
  for bridging gaps.

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9.2.2 Erosion

• With A and B are sets in Z2, the dilation of A by
  B, denoted A B, is defined as
• A B
• The erosion of A by B is the set of all points z
  such that B, translated by z, is contained in A.

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The Implementation of Dilation

• Given a binary image f and the structuring
  element s, construct a duplicate of f, denoted by
  g.
• For each pixel p f(x, y), do the following
• If p is white
• If p is adjacent to the boundary of the object,
  center the origin of s at (x, y) in g, and fill
  the pixels white on which s covers.
• Return g.

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Application of Dilation

• One of the simplest uses of erosion is for
  eliminating irrelevant detail (in terms of size)
  from a binary image.

Note that objects are represented by white
pixels, rather than by black pixels.
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9.3 Opening and Closing

• Opening to break narrow isthmuses and to
  eliminate thin protrusions.
• Closing to fuse narrow breaks and long thin
  gulfs, to eliminate small holes, and to fill gaps
  in the contour.

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Illustration of Opening and Closing
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Example 9.4 Application of Opening and Closing
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9.4 The Hit-or-Miss Transformation

• The morphological hit-or-miss transform is a
  basic tool for shape detection or pattern
  matching.
• Let B denote the set composed of X and its
  background.
• B (B1, B2), where B1X, B2W-X.
• The match of B in A, denoted by A B, is

To find objects that may contain X
To find objects that may be contained in X
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The Hit-or-Miss Transformation

• Other interpretation
• If B is 3x3, the matching can be done directly
  rather than computing the background image.

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9.5 Some Basic Morphological Algorithms

• Boundary extraction
• Region filling
• Extraction of connected components
• Convex Hull
• Thinning
• Skeletons
• Pruning

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9.5.1 Boundary Extraction

• The boundary of a set A, denoted by ß(A), can be
  obtained by first eroding A by B and then
  performing the set difference between A and its
  erosion.

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Example 9.5 Boundary Extraction

• Binary 1s are shown in white and 0s in black.
• Using 5x5 structuring element would result in a
  boundary between 2 and 3 pixels thick.

The structuring element in this example is 3x3
therefore, the boundary is one pixel thick.
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9.5.2 Region Filling

• Goal given a point p inside the boundary (Fig.
  (a)), fill the entire region with 1s.
• Let X0 p. The filled set Xk can be obtained by

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The Procedure of Region Filling

• The algorithm terminates at iteration step k if
  XkXk-1.
• The result is obtained from the union of Xk and
  the boundary in A.

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9.5.3 Extraction of Connected Components

• Goal given a point p, find the component that
  connects to p.
• Let X0 p. The set Xk can be obtained by
• The algorithm terminates at iteration step k if
  XkXk-1.
• The result Y is obtained from Xk.

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The Procedure of Finding Connected Components
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Example 9.7
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9.5.4 Convex Hull

• A set A is said to be convex.
• If the straight line joining any two points in A
  lies entirely within A.
• The convex hull H of a set S is the smallest
  convex set containing S.
• The set H-S is called the convex difference,
  which is useful for object description.
• The procedure is to implement the equation
• With Xi0A. Let DiXiconv, where conv indicates
  that XikXik-1. The convex hull of A is

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Convex Hull
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Limiting Growth of Convex Hull
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9.5.5 Thinning

• The thinning of a set A by a structuring element
  B, denoted A B, is defined by
• Each B is usually a sequence of structuring
  elements
• B1, B2,are different rotated versions of B.
• The result of thinning A by one pass is the union
  of the results obtained by thinning by Bi by one
  pass.

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Thinning Procedure
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9.5.6 Thickening

• The thickening of a set A by a structuring
  element B, denoted A B, is defined by
• A more efficient scheme is to obtain the
  complement of A, say Ac, and then to compute Cc,
  where C is the thinned result of Ac and Cc is its
  complement.

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9.5.7 Skeletons

• The dot line the skeleton of A, S(A).

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The Procedure of Skeletonization
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9.5.8 Pruning
spur
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The Procedure of Pruning

• Thinning an input set A to eliminate the short
  line segment by
• To restore the character to its original form
• Find the set containing all the end points by
• Dilate the end points and find the intersection
  with A
• The union of X3 and X1 yields the desired result
  

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9.6 Extensions to Gray-Scale Images

• Dilation
• Let Df and Db be the domains of f and b, where b
  is the structuring element.
• The dilated image tends to be brighter.
• The dark details either reduce or eliminated,
  depending on their values and shapes relate to
  the structuring element.
• Erosion
• The eroded image tends to be darker.
• The bright details either reduce or eliminated.

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Example 9.9
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9.6.3 Opening and Closing
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Example 9.10

• In (a), the decreased sizes of the small, bright
  details, with no appreciable effect on the darker
  gray levels.
• In (b), the decreased sizes of the small, dark
  details, with relatively little effect on the
  bright features.

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9.6.4 Applications of Gray-Scale Morphology

• Morphological smoothing
• i.e. performing opening followed by a closing.
• Morphological gradient
• Let g denote the operation, and then
• Depending less on edge directionality.

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9.6.4 Applications of Gray-Scale Morphology

• Top-hat transformation

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9.6.4 Applications of Gray-Scale Morphology

• Textural segmentation
• Use closing operation to eliminate the left half.
• Apply opening to restore and join the right half.
• Threshold the result to draw the boundary.

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9.6.4 Applications of Gray-Scale Morphology

• Granulometry (????)
• Apply opening with different sizes of structuring
  elements.
• Calculate image difference.
• Draw the histogram to evaluate the difference
  with respect to various sizes of structuring
  elements.
• For some x, particles with similar size of x have
  higher responses in the histogram.