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Morphological Image Processing

Morphology

- Morphology
- The branch of biology that deals with the form

and structure of organisms without consideration

of function - Mathematical Morphology
- Mathematical tool for processing shapes in image,

including boundaries, skeletons, convex hulls,

etc. - Use of set theoretical approach

Set Theory Definitions and Notations

- SET (?)
- A collection of objects (elements)
- membership (?)
- If ? is an element (member) of a set ?, we write

? ? ? - Subset (?)
- Let A, B are two sets. If for every a ? A, we

also have a ? B, then the set A is a subset of B,

that is, A ? B - If A ? B and B ? A, then A B.
- Empty set (?)

- Complement set
- If A ? ?, then its complement set Ac ? ? ? ?,

and ??A - Union (?)
- A ?B ? ? ?A or ? ?B
- Intersection (?)
- A ? B ? ? ?A and ? ?B
- Set difference (-)
- B\A B ?Ac
- Note that B-A ?A-B
- Disjoint sets
- A and B are disjoint (mutually exclusive) if A ?

B ?

Set Relations

Translation and Reflection

- Translation (A)z c c a z, for a ?A
- Reflection

Logic Operations Between Binary Images

Dilation and Erosion

- Dilation
- B structure element
- Erosion
- A ? B z (B)z ? A
- Relations
- (A ? B)c

Example of Dilation

Example of Erosion

Opening

A ? B (A ? B) ? B

Closing

A ? B (A ? B) ? B

Example Opening Closing

Finger Print Processing using Opening and Closing

Hit-or-Miss Transformation for shape detection

(e)

(d)

Figure 9.12 (a) Set A, (b) A window W and the

local Background of X w.r.t. W, W-X. (c) Ac. (d)

A?X

Intersection of (d) and (e) shows the location of

the origin of X, as desired.

Hit-or-Miss Transform

- Denote B1 object, B2 local background of B1,

then, - or
- Reason to have a local background
- Two or more objects are distinct only if they

form disjoint (disconnected) sets. This is

guaranteed by requiring that each object have at

least a one-pixel-thick background around it.

Hit-or-Miss Transform

- Previous example does not contain dont care

entries. - In structure element
- 1 foreground
- 0 background
- X dont care
- Output is 1 if exact match of both foreground and

background pixels.

- Hitnmiss.m
- 1 foreground
- -1 background
- 0 dont care

Hitnmiss.m

Morphological Boundary Extraction

?(A) A - (A ? B) (9.5-1)

Example of Boundary Extraction

Region Filling

Fig915.m

Region Filling Example

Connected Component Extraction

- Y connected component in set A,
- p a known point in Y

Fig915.m

Thinning

Thinning is often accomplished using a sequence

of rotated structuring elements (a). Given a set

A (b), results of thinning with first element is

shown in (c), and the next 7 elements (d) (i).

There is no change between 7th and 8th elements,

and no change after first 3 elements. Then it

converges to a m-connectivity.

Fig921.m

Thickening

A?B A ?hitnmiss(A,B) A?B (((A?B1) ?B2)

?Bn) Thickening is the dual of thinning

operation. Usually, thickening a set A is

accomplished by thinning Ac, and then complement

the result. Then a post-processing prunning

process is applied to remove disconnected points

as shown to the left.

Skeleton

- A skeleton of a set A consists of points z that

is the center of a maximum disk - A maximum disk is a circle in A that can not be

enclosed by another circle that is also in A. - Figure 9.23. (a) set A, (b), (c) sets of possible

maximum disks. (d) dotted line is the skeleton.

Skeleton Equations

- Define k consecutive erosions of A as
- A?kB ( (A?B)?B)? )?B) (9.5-13)
- Sk(A) (A?kB) - (A?kB)?B

(9.5-12) - Let K maxk (A?kB) ?? (9.5-14)
- Then the skeleton can be found as

Illustration of Skeleton Computation

Figure 9.24 Implementation of eq.

(9.5-11)-(9.5-15). The original set is at the top

left and its morphological skeleton is at the

bottom of the 4th column. The reconstructed set

is at the bottom of the 6th column. Define k

consecutive erosions of A as A?kB ( (A?B)?B)?

)?B) (9.5-13) Sk(A) (A?kB) - (A?kB)?B

(9.5-12) Let K maxk (A?kB) ??

(9.5-14) Then the skeleton can be found

as

Pruning

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