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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.

Erosion

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

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.

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

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

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:

Thinning

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

Extension to Gray-Scale Images

- Dilation Max
- Erosion Min

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