1 ece533 digital image processing morphological image processing

1ECE533 Digital Image Processing

Morphological Image Processing

2ECE533 Digital Image Processing(c) 2003-2006 by Yu Hen Hu

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: BbbwwB for ,|ˆ


Logic Operations Between Binary Images


Dilation and Erosion

DilationB: structure element

ErosionA B = {z | (B)z A}

Relations(A B)c =

AABz

ABzBA

z

z

ˆ|

ˆ|

ˆcA B


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

Figure 9.12 (a) Set A, (b) A window W and the localBackground of X w.r.t. W, W-X. (c) Ac. (d) AX

Intersection of (d) and (e) shows the locationof the origin of X, as desired.

(d)

(e)

A


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 don’t care entries.

In structure element» 1 – foreground» 0 – background» X – don’t care

Output is 1 if exact match of both foreground and background pixels.

Hitnmiss.m» +1: foreground» -1: background» 0: don’t care

Hitnmiss.m

match not :

match :

111

010

111

111

111

111

*.

111

010

111

111

010

111

111

111

111

*.

111

010

111


Morphological Boundary Extraction

(A) = A − (A B) (9.5-1)


Example of Boundary Extraction


Region Filling

AXY

k

ABXX

k

ckk

,3,2,1

;1

Fig915.m


Region Filling Example


Connected Component Extraction

Y: connected component in set A,

p: a known point in Y

k

kk

kk

XY

XX

ABXX

pX

then

if 1

1

0

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.

n

n

BBBABA

BBBB

BAhitnmissABA

21

21 ,,

,

Fig921.m


Thickening

AB = A hitnmiss(A,B)A{B} =((…(AB1) 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 diskA 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

1

0

( ) ( ) (9.5-11)

( ( ) ) (9.5-15)

K

kk

K

kk

S A S A

A S A kB

Define k consecutive erosions of A as:AkB = ( …(AB)B) …)B) (9.5-13)

Sk(A) = (AkB) − (AkB)B (9.5-12)

Let K = max{k | (AkB) } (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:AkB = ( …(AB)B) …)B) (9.5-13)Sk(A) = (AkB) − (AkB)B (9.5-12)Let K = max{k | (AkB) } (9.5-14)Then the skeleton can be found as:

1

0

( ) ( ) (9.5-11)

( ( ) ) (9.5-15)

K

kk

K

kk

S A S A

A S A kB


Pruning