Lecture 5. Morphological Image
Processing
Introduction
►
Morphology: a branch of biology that deals with the form
and structure of animals and plants
►
Morphological image processing is used to extract image
components for representation and description of region
shape, such as boundaries, skeletons, and the convex hull
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Preliminaries (1)
►
Reflection
The reflection of a set B, denoted B, is defined as
B  {w | w  b, for b  B}
►
Translation
The translation of a set B by point z  ( z1 , z2 ), denoted ( B ) Z ,
is defined as
( B) Z  {c | c  b  z, for b  B}
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Example: Reflection and Translation
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Preliminaries (2)
►
Structure elements (SE)
Small sets or sub-images used to probe an image under study for
properties of interest
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Examples: Structuring Elements (1)
origin
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Examples: Structuring Elements (2)
Accommodate the
entire structuring
elements when its
origin is on the
border of the
original set A
Origin of B visits
every element of A
At each location of
the origin of B, if B
is completely
contained in A,
then the location is
a member of the
new set, otherwise
it is not a member
of the new set.
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Erosion
With A and B as sets in Z 2 , the erosion of A by B, denoted A B,
defined
A B   z | ( B) Z  A
The set of all points z such that B, translated by z, is contained by A.
A
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B   z | ( B) Z  Ac  
8
Example
of
Erosion
(1)
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Example
of
Erosion
(2)
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Dilation
2
With A and B as sets in Z , the dilation of A by B,
denoted A  B, is defined as
 

A  B= z | B  A  
z
The set of all displacements z, the translated B and A
overlap by at least one element.
 

A  B  z |  B  A  A

z

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Examples of Dilation (1)
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Examples of Dilation (2)
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Duality
►
Erosion and dilation are duals of each other with respect to
set complementation and reflection
 A  B
c
 A B
c
and
c
A

B

A
B


c
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Duality
►
Erosion and dilation are duals of each other with respect to
set complementation and reflection
 A  B
c
  z |  B  Z  A
c
  z |  B  Z  A  
c
c
  z |  B  Z  Ac  
 Ac  B
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Duality
►
Erosion and dilation are duals of each other with respect to
set complementation and reflection
 A  B
c
    A  
  z |  B   A  
c
 z| B
Z
c
Z
 A B
c
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Opening and Closing
►
Opening generally smoothes the contour of an object,
breaks narrow isthmuses, and eliminates thin protrusions
►
Closing tends to smooth sections of contours but it
generates fuses narrow breaks and long thin gulfs,
eliminates small holes, and fills gaps in the contour
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Opening and Closing
The opening of set A by structuring element B,
denoted A B, is defined as
A B   A  B  B
The closing of set A by structuring element B,
denoted A B, is defined as
A B   A  B  B
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Opening
The opening of set A by structuring element B,
denoted A B, is defined as
A B
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 B  |  B 
Z
Z
 A
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Example: Opening
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Example: Closing
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Duality of Opening and Closing
►
Opening and closing are duals of each other with respect
to set complementation and reflection
 A B
c
 (A
c
B)
c
A
B

(
A
B)


c
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The Properties of Opening and Closing
►
Properties of Opening
(a) A B is a subset (subimage) of A
(b) if C is a subset of D, then C B is a subset of D B
(c) ( A B) B  A B
►
Properties of Closing
(a) A is subset (subimage) of A B
(b) If C is a subset of D, then C B is a subset of D B
(c) ( A B) B  A B
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The Hit-or-Miss
Transformation
if B denotes the set composed of
D and its background,the match
(or set of matches) of B in A,
denoted A  B,
A * B   A  D    Ac  W  D  
B   B1 , B2 
B1 : object
B2 : background
A  B   A  B1   ( Ac  B2 )
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Some Basic Morphological Algorithms (1)
►
Boundary Extraction
The boundary of a set A, can be obtained by first eroding A
by B and then performing the set difference between A
and its erosion.
 ( A)  A   A  B 
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Example 1
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Example 2
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Some Basic Morphological Algorithms (2)
►
Hole Filling
A hole may be defined as a background region surrounded
by a connected border of foreground pixels.
Let A denote a set whose elements are 8-connected
boundaries, each boundary enclosing a background region
(i.e., a hole). Given a point in each hole, the objective is to
fill all the holes with 1s.
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Some Basic Morphological Algorithms (2)
►
Hole Filling
1. Forming an array X0 of 0s (the same size as the array
containing A), except the locations in X0 corresponding to
the given point in each hole, which we set to 1.
2. Xk = (Xk-1 + B) Ac
k=1,2,3,…
Stop the iteration if Xk = Xk-1
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Example
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Some Basic Morphological Algorithms (3)
►
Extraction of Connected Components
Central to many automated image analysis applications.
Let A be a set containing one or more connected
components, and form an array X0 (of the same size as the
array containing A) whose elements are 0s, except at each
location known to correspond to a point in each connected
component in A, which is set to 1.
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Some Basic Morphological Algorithms (3)
►
Extraction of Connected Components
Central to many automated image analysis applications.
X k  ( X k 1  B )  A
B : structuring element
until X k  X k -1
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Some Basic Morphological Algorithms (4)
►
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 or of an arbitrary set S is the smallest
convex set containing S.
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Some Basic Morphological Algorithms (4)
►
Convex Hull
Let B i , i  1, 2, 3, 4, represent the four structuring elements.
The procedure consists of implementing the equation:
X ki  ( X k 1 * B i )  A
i  1, 2,3, 4 and k  1, 2,3,...
with X 0i  A.
When the procedure converges, or X ki  X ki 1 , let D i  X ki ,
the convex hull of A is
4
C ( A)   Di
i 1
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Some Basic Morphological Algorithms (5)
►
Thinning
The thinning of a set A by a structuring element B, defined
A  B  A  ( A * B)
 A  ( A * B)
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Some Basic Morphological Algorithms (5)
►
A more useful expression for thinning A symmetrically is
based on a sequence of structuring elements:
1
2
3
n
B

B
,
B
,
B
,...,
B
  

where B i is a rotated version of B i -1
The thinning of A by a sequence of structuring element {B}
A  {B}  ((...(( A  B1 )  B 2 )...)  B n )
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Some Basic Morphological Algorithms (6)
►
Thickening:
The thickening is defined by the expression
A
B  A   A* B
The thickening of A by a sequence of structuring element {B}
A {B}  ((...(( A
B1 )
B 2 )...)
Bn )
In practice, the usual procedure is to thin the background of the set
and then complement the result.
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Some Basic Morphological Algorithms (6)
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Some Basic Morphological Algorithms (7)
►
Skeletons
A skeleton, S ( A) of a set A has the following properties
a. if z is a point of S ( A) and ( D) z is the largest disk
centered at z and contained in A, one cannot find a
larger disk containing ( D) z and included in A.
The disk ( D) z is called a maximum disk.
b. The disk ( D) z touches the boundary of A at two or
more different places.
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Some Basic Morphological Algorithms (7)
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Some Basic Morphological Algorithms (7)
The skeleton of A can be expressed in terms of
erosion and openings.
K
S ( A)   Sk ( A)
k 0
with K  max{k | A  kB  };
Sk ( A)  ( A  kB)  ( A  kB) B
where B is a structuring element, and
( A  kB)  ((..(( A  B)  B)  ...)  B)
k successive erosions of A.
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Some Basic Morphological Algorithms (7)
A can be reconstructed from the subsets by using
K
A   ( S k ( A)  kB)
k 0
where S k ( A)  kB denotes k successive dilations of A.
( S k ( A)  kB)  ((...(( S k ( A)  B)  B)...  B)
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Some Basic Morphological Algorithms (8)
►
Pruning
a. Thinning and skeletonizing tend to leave parasitic components
b. Pruning methods are essential complement to thinning and
skeletonizing procedures
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Pruning:
Example
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X1  A  {B}
55
Pruning:
Example
X 2    X1 * B k 
8
k 1
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Pruning:
Example
X3   X2  H   A
H : 3  3 structuring element
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Pruning:
Example
X 4  X1  X 3
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Pruning:
Example
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Some Basic Morphological Algorithms (9)
►
Morphological Reconstruction
It involves two images and a structuring element
a. One image contains the starting points for the
transformation (The image is called marker)
b. Another image (mask) constrains the transformation
c. The structuring element is used to define connectivity
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Morphological Reconstruction: Geodesic
Dilation
Let F denote the marker image and G the mask image,
F  G. The geodesic dilation of size 1 of the marker image
with respect to the mask, denoted by DG(1) ( F ), is defined as
DG(1) ( F )   F  B   G
The geodesic dilation of size n of the marker image F
with respect to G, denoted by DG( n ) ( F ), is defined as
DG( n ) ( F )  DG(1) ( F )  DG( n 1) ( F ) 
with DG(0) ( F )  F .
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Morphological Reconstruction: Geodesic
Erosion
Let F denote the marker image and G the mask image.
The geodesic erosion of size 1 of the marker image with
respect to the mask, denoted by EG(1) ( F ), is defined as
EG(1) ( F )   F  B   G
The geodesic erosion of size n of the marker image F
with respect to G, denoted by EG( n ) ( F ), is defined as
EG( n ) ( F )  EG(1) ( F )  EG( n 1) ( F ) 
with EG(0) ( F )  F .
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Morphological Reconstruction by Dilation
Morphological reconstruction by dialtion of a mask image
G from a marker image F , denoted RGD ( F ), is defined as
the geodesic dilation of F with respect to G, iterated until
stability is achieved; that is,
RGD ( F )  DG( k ) ( F )
with k such that DG( k ) ( F )  DG( k 1) ( F ).
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Morphological Reconstruction by Erosion
Morphological reconstruction by erosion of a mask image
G from a marker image F , denoted RGE ( F ), is defined as
the geodesic erosion of F with respect to G, iterated until
stability is achieved; that is,
RGE ( F )  EG( k ) ( F )
with k such that EG( k ) ( F )  EG( k 1) ( F ).
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Opening by Reconstruction
The opening by reconstruction of size n of an image F is
defined as the reconstruction by dilation of F from the
erosion of size n of F ; that is
OR( n ) ( F )  RFD  F  nB  
where  F  nB  indicates n erosions of F by B.
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Filling Holes
Let I ( x, y ) denote a binary image and suppose that we
form a marker image F that is 0 everywhere, except at
the image border, where it is set to 1- I ; that is
1  I ( x, y )
F ( x, y )  
0

then
if ( x, y ) is on the border of I
otherwise
H   R ( F )
D
Ic
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SE : 3 3 1s.
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H   R ( F ) 
D
Ic
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Border Clearing
It can be used to screen images so that only complete objects
remain for further processing; it can be used as a singal that
partial objects are present in the field of view.
The original image is used as the mask and the following
marker image:
 I ( x, y )
F ( x, y)  
 0
X  I  RID ( F )
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if ( x, y ) is on the border of I
otherwise
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Summary (1)
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Summary (2)
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Gray-Scale Morphology
f ( x, y ) : gray-scale image
b( x, y ): structuring element
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Gray-Scale Morphology: Erosion and Dilation
by Flat Structuring
f ( x  s, y  t )

 f  b ( x, y)  (min
s ,t )b
f ( x  s, y  t )

 f  b ( x, y)  max
( s ,t )b
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Gray-Scale Morphology: Erosion and Dilation
by Nonflat Structuring
f ( x  s, y  t )  bN ( s, t )

 f  bN  ( x, y)  (min
s ,t )b
f ( x  s, y  t )  bN ( s, t )

 f  bN  ( x, y)  max
( s ,t )b
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Duality: Erosion and Dilation
 f b 
c
 c 
( x, y )   f  b  ( x, y )



where f c   f ( x, y ) and b  b( x,  y )
 f b 
c
 f  b
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c
 c 
  f b



 ( f  b)
c
83
Opening and Closing
f b   f  b  b
f  b   f  b  b
 f b
f
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 f
c


bf b
c


b  f b   f b
c
c
84
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Properties of Gray-scale Opening
(a) f bf
(b) if f1f 2 , then  f1 b    f 2 b 
(c )
f
b b  f b
where er denotes e is a subset of r and also
e( x, y )  r ( x, y ).
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Properties of Gray-scale Closing
(a) f f b
(b) if f1f 2 , then  f1 b    f 2 b 
(c )
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 f b b 
f b
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Morphological Smoothing
►
Opening suppresses bright details smaller than the
specified SE, and closing suppresses dark details.
►
Opening and closing are used often in combination as
morphological filters for image smoothing and noise
removal.
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Morphological Smoothing
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Morphological Gradient
►
Dilation and erosion can be used in combination with
image subtraction to obtain the morphological gradient of
an image, denoted by g,
g  ( f  b)  ( f  b)
►
The edges are enhanced and the contribution of the
homogeneous areas are suppressed, thus producing a
“derivative-like” (gradient) effect.
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Morphological Gradient
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Top-hat and Bottom-hat Transformations
►
The top-hat transformation of a grayscale image f is
defined as f minus its opening:
That ( f )  f  ( f b)
►
The bottom-hat transformation of a grayscale image f is
defined as its closing minus f:
Bhat ( f )  ( f  b)  f
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Top-hat and Bottom-hat Transformations
►
One of the principal applications of these transformations
is in removing objects from an image by using structuring
element in the opening or closing operation
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Example of Using Top-hat Transformation in
Segmentation
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Granulometry
►
Granulometry deals with determining the size of
distribution of particles in an image
►
Opening operations of a particular size should have the
most effect on regions of the input image that contain
particles of similar size
►
For each opening, the sum (surface area) of the pixel
values in the opening is computed
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Example
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Textual Segmentation
►
Segmentation: the process of subdividing an image into
regions.
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Textual Segmentation
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Gray-Scale Morphological Reconstruction (1)
►
Let f and g denote the marker and mask image with the
same size, respectively and f ≤ g.
The geodesic dilation of size 1 of f with respect to g is
defined as
Dg(1) ( f )   f  g   g
where  denotes the point-wise minimum operator.
The geodesic dilation of size n of f with respect to g is
defined as
Dg( n ) ( f )  Dg(1)  Dg( n 1) ( f )  with Dg(0) ( f )  f
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Gray-Scale Morphological Reconstruction (2)
►
The geodesic erosion of size 1 of f with respect to g is
defined as
Eg(1) ( f )   f  g   g
where  denotes the point-wise maximum operator.
The geodesic erosion of size n of f with respect to g is
defined as
Eg( n ) ( f )  Eg(1)  Eg( n 1) ( f )  with Eg(0) ( f )  f
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Gray-Scale Morphological Reconstruction (3)
►
The morphological reconstruction by dilation of a grayscale mask image g by a gray-scale marker image f, is
defined as the geodesic dilation of f with respect to g,
iterated until stability is reached, that is,
RgD ( f )  Dg( k )  f 
with k such that Dg( k )  f   Dg( k 1)  f 
The morphological reconstruction by erosion of g by f is
defined as
E
(k )
f
with k such that Eg( k )  f   Eg( k 1)  f 
Rg ( f )  Eg
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Gray-Scale Morphological Reconstruction (4)
►
The opening by reconstruction of size n of an image f is
defined as the reconstruction by dilation of f from the
erosion of size n of f; that is,
OR( n ) ( f )  R Df  f  nb
The closing by reconstruction of size n of an image f is
defined as the reconstruction by erosion of f from the
dilation of size n of f; that is,
CR( n) ( f )  R Ef  f  nb
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Steps in the Example
1.
Opening by reconstruction of the original image using a horizontal
line of size 1x71 pixels in the erosion operation
OR( n ) ( f )  R Df  f  nb
2.
Subtract the opening by reconstruction from original image
f '  f  OR( n ) ( f )
3.
Opening by reconstruction of the f’ using a vertical line of size 11x1
pixels
f 1  OR( n) ( f ')  R Df  f ' nb '
4.
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Dilate f1 with a line SE of size 1x21, get f2.
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Steps in the Example
5.
Calculate the minimum between the dilated image f2
and and f’, get f3.
6.
By using f3 as a marker and the dilated image f2 as
the mask,
R Df 2 ( f 3)  D (f k2)  f 3
with k such that D (f k2)  f 3  D (f k21)  f 3
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