Morphological Image Processing

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Chapter 9

• Morphological Image Processing

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Preview

• Morphology denotes a branch of biology that
  deals with the form and structure of animals and
  plants.
• Mathematical morphology tool for extracting
  image components that are useful in the
  representation and description of region shapes.
• Filtering, thinning, pruning.

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Scope

• Will focus on binary images.
• Applicable to other situations.
  (Higher-dimensional space)

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Set Theory

• Empty set
• Subset
• Union
• Intersection
• Disjoint sets
• Complement
• Difference
• Reflection of set B
• Translation of set A by point z(z1,z2)

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Logic Operations

• AND
• OR
• NOT

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Dilation

• With A and B as sets in Z2, the dilation of A by
  B is defined as
• Or, equivalently,
• B is commonly known as the structuring element.

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Illustration
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Example
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Erosion

• With A and B as sets in Z2, the erosion of A by B
  is defined as
• Dilation and erosion are duals

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Illustration
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Example Removing image components
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Opening and Closing

• Opening of set A by structuring element B
• Erosion followed by dilation
• Closing of set A by structuring element B
• Dilation followed by erosion

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Opening

• Opening generally smoothes the contour of an
  object, breaks narrow isthmuses, eliminate thin
  protrusions.

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Closing

• Closing tends to smooth contours, fuse narrow
  breaks and long thin gulfs, eliminate small
  holes, fill gaps in the contour.

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Illustration
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Example
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Hit-or-Miss Transform

• Shape detection tool

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Boundary Extraction

• Definition

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Region Filling

• Beginning with a point p inside the boundary,
  repeat
• with X0p
• Until XkXk-1
• Conditional dilation

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Example
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Extraction of Connected Component

• Beginning with a point p of the connected
  component, repeat
• with X0p
• Until XkXk-1
• The connected component YXk

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Illustration
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Example
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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
  within A.
• The convex hull H of an arbitrary set S is the
  smallest convex set containing S.
• H-S is called the convex deficiency of S.
• C(A) convex hull of a set A.

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Algorithm

• Four structuring elements Bi, i1,2,3,4
• Repeat
• with X0i A until XkiXk-1i to obtain Di
• The convex hull of A is

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Illustration
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Thinning

• The thinning of a set A by a structuring element
  B is defined as

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Illustration
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Thickening
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Skeleton
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Skeleton Definition
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Illustration
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Pruning
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Extension to Gray-Scale Images

• Dilation ?Max
• Erosion ?Min

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Illustration
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Opening and Closing
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Smoothing and Gradient