Morphological Image Processing

1
Morphological Image Processing

• Preliminaries
• Applications
• -- extracting image components for region
  shape representation and description, e.g.,
  boundary, skeleton, structure, etc.
• -- image post pre-processing, e.g.,
  morphological filtering, thinning and pruning.
• Set theory the language of mathematical
  morphology

2
Preliminary

• Set theory
• -- If a(a1, a2) is an element of A (A is a set
  in 2D integer space Z2), then a ?A
• Example A a, a(x,y) coordinates of
  pixels
• If b is not an element of A, then b ?A

3
Preliminary (contd)

• Relationship of set A and set B
• (a) A ? B
• A is a set of B (any element in A is also the
  element of B)
• (b) C A ?B
• Union of two sets A and B (all elements belong
  to either A, B or both)
• (c) D A? B
• Intersection of two sets A and B (all elements
  belong to both A and B)

4
Preliminary (contd)

• Relationship of set A and set B (contd)
• (d) A? B ?
• Disjoint or mutually exclusive (no common
  elements)
• (e) Ac w w ?A
• Complement of A (set of elements not contained
  in A)
• (f) A-B ww ?A, w ?B A ? Bc
• Difference of two sets A and B (set of
  elements that belong to A, but not to B)

5
Preliminary (contd)

• Relationship of set A and set B (contd)
• (g) (B) w w -b for b ? B
• reflection of set B
• (h) (A )z c c az, for a ? A
• translation of set A by point z (z1, z2)
• (f) A-B ww ?A, w ?B A ? Bc
• Difference of two sets A and B (set of
  elements that belong to A, but not to B)

6
Preliminary (contd)

• Logic operation on Binary images
• -- principal logic operations
• AND, OR, NOT (complement), XOR (exclusive
  OR),
• NOT-AND (not(A) AND (B))
• -- Logic operations are performed on a
  pixel-by-pixel basis between corresponding pixels
  of two or more images
• -- Note
• - set operation is a pixel-coordinates
  operation geometrically
• - logic operation is a pixel-value operation.

7
Dilation on Binary Image

• Binary dilation of A by B (A and B are
  two sets in space Z2)
• -- Definition 1
• A ? B Z (B)z? A
  ??
• - B is called structuring element. (B)z is the
  operation as follow
• First, reflecting B about its origin Second,
  shifting this reflection (B) by Z
• - Dilation of A by B is the set of all
  displacements (Z), which satisfies the following
  condition (B) and A are overlapped by at least
  one element.
• - We can rewrite the definition as A ? B Z
  (B)z? A ?A

8
Dilation on Binary Image (contd)

• Binary dilation of A by B (contd)
• -- Definition 2 (Minkowsky addition of two
  sets A and B)
• A ? B Ub ?B (A)b
• - Example
• (A)x translation (shift) of a set A by a
  point x
• A a
• (A)x ax a? A

(A)x
A
x
9
Dilation on Binary Image (contd)

• Binary dilation of A by B (contd)
• -- Example of dilation
• A (0,1), (1,1), (2,1), (2,2), (3,0)
• x (0,1)

A
(A)x
x
Origin of domain
10
Dilation on Binary Image (contd)

• Binary dilation of A by B (contd)
• -- Example of dilation

A
A ? B
B
Origin of domain
11
Dilation on Binary Image (contd)

• Binary dilation of A by B (contd)
• -- Example of dilation
• A a, Bb, A ? B Ub ?B (A)b
• B(4,1), (5,1), (5,2), A ? B (A) (4,1) U
  A(5,1) U A(5,2)

A ? B
A
B
12
Dilation on Binary Image (contd)

• Binary dilation of A by B (contd)
• -- Example of dilation

A ? B
d
A
d
B
d/4
d/4
d/8
13
Dilation on Binary Image (contd)

• Property of Binary dilation
• (A)x ? B (A ? B)x
• A? C ? A ? B ? C ? B
• A ? B B ? A
• A ? (B ? C) (A ? B) ? C
• (A ? B) ? C ? (A ? C) ? (B ? C)
• (A ? B) ? C (A ? C) ? (B ? C)

14
Erosion on Binary Image

• Binary erosion of A by B (A and B are
  two sets in space Z2)
• -- Definition 1
• A B Z (B)z ?
  A
• the result is a set of points Z which satisfies
  following condition
• B shifted by Z is contained in A
• - Definition 2
• A B ? b?B (A)-b
• Minkowsky subtraction of two sets A and B
• ? b?B (A)b

15
Erosion on Binary Image (contd)

• Binary erosion of A by B (contd)
• -- Example of Minkowsky subtraction

A
B
16
Erosion on Binary Image (contd)

• Binary erosion of A by B (contd)
• -- Example of erosion

A
A B Z BZ ? A
B
17
Erosion on Binary Image (contd)

• Binary erosion of A by B (contd)
• -- Example of erosion

A B Z BZ ? A
A
B
18
Erosion on Binary Image (contd)

• Binary erosion of A by B (contd)
• -- Example of erosion

A B
A
B
Origin of domain
19
Erosion on Binary Image (contd)

• Binary erosion of A by B (contd)
• -- Example of erosion

A B
d
A
d
B
d/4
d/4
d/8
20
Erosion on Binary Image (contd)

• Property of Binary erosion
• (A)x B (A B)x
• A? C ? A B ? C B
• A? B ? D A ? D B
• (A ? B) C (A C) ? (B C)
• Duality (A B)c (Ac ? B)

21
Opening and Closing

• Dilation
• - expanding image
• Erosion
• - shrinking image
• Opening
• - smoothing contour, removing isolated
  noise, breaking bridge
• Closing
• - fusing breaks, filling gaps, removing
  holes

22
Opening and Closing (contd)

• Definition
• Opening
• A ? B (A B) ? B
• Closing
• A? B (A ? B) B
• Duality (A ? B)c (Ac ? B)
• A ? B (Ac ? B)c

23
Opening and Closing (contd)

• Example
• Opening
• A ? B
  (A B) ? B

A
Translate of B in A
B
24
Opening and Closing (contd)

• Example
• Closing
• 
  A? B (A ? B) B

A
B
25
Opening and Closing (contd)

• Property
• Opening
• A B ? A
• C ? D ? C ? B ? D ? B
• (A B) B A B
• Closing
• A ? A B
• C ? D ? C B ? D B
• (A B) B A B

26
Opening and Closing (contd)
-- Example of opening
A ? B
A
B
27
Hit-or-miss transformation
-- Shape detection -- Using two structure
elements
W - X
At least one pixel width
X
Structure element II complement of X with
respect to W
Structure element I
28
Hit-or-miss transformation (contd)
-- The match (or fit) of B in A is called
hit-or-miss transform, denoted A ? B -- B is
composed of X and W-X A ? B (A
X) ? Ac (W-X)
W - X
X
29
Hit-or-miss transformation (contd)

• General notation
• structure element B (B1, B2)
• e.g., B1 X (object)
• B2 W-X (background)
• A ? B (A B1) ? Ac
  B2
• This set contains all the (origin) points,
  at which, B1 found a match
• (hit) in A and B2 found a match in Ac,
  simultaneously.

30
Hit-or-miss transformation (contd)

• Hit-or-miss definition by set difference
• A ? B (A B1) - A ? B2
• Note
• Hit-or-miss is the object match plus
  background match

31
Hit-or-miss transformation (contd)

• Example of Hit-or-miss
• X ? T X T1x ? X, T2x ? Xc

X
hit
miss
T
T1 T2
X ? T
X
X ? T
T
T1 T2
32
Applications of Morphological algorithm

• Boundary extraction
• Boundary(A) A
  (A B)
• Region filling
• - given a set A which defines a
  region boundary
• - start with a non-boundary point
  P within the region
• - let X0 P
• - Xk (Xk-1 ? B ) ? Ac, k
  1,2,3,
• - iteration at each step k
• - terminate if Xk Xk-1
• Note A ? Xk includes the filled set and the
  boundary

33
Applications of Morphological algorithm (contd)

• Connected component extraction
• - similar to the region filling
• - start with a point P which is contained
  in A
• - let X0 P
• - Xk (Xk-1 ? B ) ? A, k 1,2,3,
• - iteration at each step k
• - terminate if Xk Xk-1

34
Applications of Morphological algorithm (contd)

• Convex hull extraction
• - set A is convex if any line ab? A (a?A,
  b?A)

a
a
b
b
convex
concave

• convex hull H of an arbitrary set S is the
  smallest convex set
• which contains S

35
Applications of Morphological algorithm (contd)

• Convex hull extraction (contd)
• - Example of detection of convex hull of
  set A
• Given a set A and four structure
  elements B1,B2,B3,B4
• calculate the convex hull region C(A)
  D1 ?D2 ?D3?D4
• where
• Di is derived from Xik (Xik-1 ? Bi
  ) ? A
• (i1,2,3,4), (k1,2,)
• Di Xik when Xik Xik-1
• Initial Xi0 A

36
Applications of Morphological algorithm (contd)

• Thinning
• - peel from outside into inside,
  which is defined in terms of
• the hit-or-miss transform
• A ? B A (A ? B)
• B B1, B2,, Bn
• A ?B (((A ? B1) ? B2)) ? Bn)

37
Applications of Morphological algorithm (contd)

• Thickening
• - The structure element B is similar to
  the structure element for thinning, except that
  1s and 0s are exchanged.
• - morphological dual of thinning
• A ? B A ?(A ? B)
• B B1, B2,, Bn
• - Alternative algorithm
• To thicken a set A, we can also
• - apply thinning algorithm on Ac,
• - obtain region R
• - then take Rc as the thickening
  result

38
Applications of Morphological algorithm (contd)

• Skeletons
• - skeletons can be implemented by the
  operations of erosions and
• openings
• S(A) Uk0K(Sk(A))
• Sk(A) (A kB) (A kB) ? B
• A kB (((A B) B))
  B)
• K maxk (A kB) ??

39
Applications of Morphological algorithm (contd)

• Pruning
• - it is complement to thinning and
  skeletonizing algorithm
• - example hand-writing recognition
• - X A ? B
• - Ending points detection
• - Dilation of ending points ?
  obtain Y
• - X U Y

40
Morphology in gray-scale image

• Dilation
• (f ? b)(s,t) maxf(s-x, t-y) b(x,y)
  (s-x), (t-y) ? Df , (x,y) ? Db
• Where
• Df - domain of f
• Db - domain of b

41
Morphology in gray-scale image (contd)

• Erosion
• (f b)(s,t) minf(sx, ty) - b(x,y)
  (sx), (ty) ? Df , (x,y) ? Db
• Where
• Df - domain of f
• Db - domain of b
• Property
• Dilation will generate brighter image and
  reduce smalldark details
• Erosion will generate darker image and reduce
  smallbright features

42
Morphology in gray-scale image (contd)

• Opening
• f ? b (f b) ? b
• Closing
• f b (f ?b ) b

43
Morphology in gray-scale image (contd)

• Example of opening and closing

f b
f
b
b
f
f b
44
Morphology in gray-scale image (contd)

• Applications
• - Smoothing
• - example opening closing for
  removing noise
• - Morphological gradient
• - example g (f ? b) - (f b)
• - Top-hat transform
• - example h f (f b)
• - Texture segmentation
• - example closing opening
  thresholding