Chapter 9
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Chapter 9. Morphological Image Processing. 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.

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Chapter 9

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Chapter 9

Morphological Image Processing


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.


Scope

  • Will focus on binary images.

  • Applicable to other situations. (Higher-dimensional space)


Set Theory

  • Empty set

  • Subset

  • Union

  • Intersection

  • Disjoint sets

  • Complement

  • Difference

  • Reflection of set B:

  • Translation of set A by point z=(z1,z2):


Logic Operations

  • AND

  • OR

  • NOT


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.


Illustration


Example


Erosion

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

  • Dilation and erosion are duals:


Illustration


Example: Removing image components


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


Opening

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


Closing

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


Illustration


Example


Hit-or-Miss Transform

  • Shape detection tool


Boundary Extraction

  • Definition:


Region Filling

  • Beginning with a point p inside the boundary, repeat:

    with X0=p

  • Until Xk=Xk-1

  • Conditional dilation


Example


Extraction of Connected Component

  • Beginning with a point p of the connected component, repeat:

    with X0=p

  • Until Xk=Xk-1

  • The connected component Y=Xk


Illustration


Example


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.


Algorithm

  • Four structuring elements: Bi, i=1,2,3,4

  • Repeat

    with X0i =A until Xki=Xk-1i to obtain Di

  • Theconvex hull of A is:


Illustration


Thinning

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


Illustration


Thickening


Skeleton


Skeleton: Definition


Illustration


Pruning


Extension to Gray-Scale Images

  • Dilation Max

  • Erosion Min


Illustration


Opening and Closing


Smoothing and Gradient


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