Image processing mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjoooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

1. Digital Image Processing
Chapter 3: Intensity Transformations
and Spatial Filtering

2. Background
 Spatial domain process
 where is the input image,
is the processed image, and T is an
operator on f, defined over some
neighborhood of
)]
,
(
[
)
,
( y
x
f
T
y
x
g 
)
,
( y
x
f )
,
( y
x
g
)
,
( y
x

3.  Neighborhood about a point

4.  Gray-level transformation function
 where r is the gray level of and
s is the gray level of at any
point
)
(r
T
s 
)
,
( y
x
f
)
,
( y
x
g
)
,
( y
x

5.  Contrast enhancement
 For example, a thresholding function

6.  Masks (filters, kernels, templates,
windows)
 A small 2-D array in which the values
of the mask coefficients determine the
nature of the process

7. Some Basic Gray Level
Transformations

8.  Image negatives
 Enhance white or gray details
r
L
s 

 1

9.  Log transformations
 Compress the dynamic range of images
with large variations in pixel values
)
1
log( r
c
s 


10.  From the range 0- to the range
0 to 6.2
6
10
5
.
1 

11.  Power-law transformations
 or
 maps a narrow range of dark
input values into a wider range of
output values, while maps a
narrow range of bright input values into
a wider range of output values
 : gamma, gamma correction

cr
s  
)
( 
 r
c
s
1


1




13.  Monitor, 5
.
2



16.  Piecewise-linear transformation
functions
 The form of piecewise functions can be
arbitrarily complex

17.  Contrast stretching

18.  Gray-level slicing

20.  Bit-plane slicing

25. Histogram Processing
 Histogram
 where is the kth gray level and is
the number of pixels in the image
having gray level
 Normalized histogram
k
k n
r
h 
)
(
k
r k
n
k
r
n
n
r
p k
k /
)
( 

27.  Histogram equalization
1
0
),
( 

 r
r
T
s
1
0
),
(
1


 
s
s
T
r

29.  Probability density functions (PDF)
ds
dr
r
p
s
p r
s )
(
)
( 




r
r dw
w
p
L
r
T
s
0
)
(
)
1
(
)
(
)
(
)
1
(
)
(
)
1
(
)
(
0
r
p
L
dw
w
p
dr
d
L
dr
r
dT
dr
ds
r
r
r 









 
1
1
)
(


L
s
ps

30. 1
,...,
2
,
1
,
0
,
)
1
(
)
(
)
1
(
)
(
0
0






 
 

L
k
n
n
L
r
p
L
r
T
s
k
j
j
k
j
j
r
k
k

37.  Histogram matching (specification)




r
r dw
w
p
L
r
T
s
0
)
(
)
1
(
)
(
 


z
z s
dt
t
p
L
z
G
0
)
(
)
1
(
)
(
)]
(
[
)
( 1
1
r
T
G
s
G
z 



)
(z
pz
is the desired PDF

38. 1
,...,
2
,
1
,
0
,
)
1
(
)
(
)
1
(
)
(
0
0






 
 

L
k
n
n
L
r
p
L
r
T
s
k
j
j
k
j
j
r
k
k
1
,...,
2
,
1
,
0
,
)
(
)
1
(
)
(
0





 

L
k
s
z
p
L
z
G
v k
k
i
i
z
k
k
1
,...,
2
,
1
,
0
)],
(
[
1


 
L
k
r
T
G
z k
k

40.  Histogram matching
 Obtain the histogram of the given
image, T(r)
 Precompute a mapped level for each
level
 Obtain the transformation function G
from the given
 Precompute for each value of
 Map to its corresponding level ;
then map level into the final level
)
(z
pz
k
s
k
r
k
z k
s
k
r k
s
k
s k
z

48.  Local enhancement
 Histogram using a local neighborhood,
for example 7*7 neighborhood

49.  Histogram using a local 3*3
neighborhood

50.  Use of histogram statistics for
image enhancement
 denotes a discrete random variable
 denotes the normalized
histogram component corresponding to
the ith value of
 Mean
)
( i
r
p
r
r




1
0
)
(
L
i
i
i r
p
r
m

51.  The nth moment
 The second moment





1
0
)
(
)
(
)
(
L
i
i
n
i
n r
p
m
r
r






1
0
2
2 )
(
)
(
)
(
L
i
i
i r
p
m
r
r


52.  Global enhancement: The global mean
and variance are measured over an
entire image
 Local enhancement: The local mean
and variance are used as the basis for
making changes

53.  is the gray level at coordinates
(s,t) in the neighborhood
 is the neighborhood normalized
histogram component
 mean:
 local variance
t
s
r ,
)
( ,t
s
r
p



xy
xy
S
t
s
t
s
t
s
S r
p
r
m
)
,
(
,
, )
(




xy
xy
xy
S
t
s
t
s
S
t
s
S r
p
m
r
)
,
(
,
2
,
2
)
(
]
[


54.  are specified parameters
 is the global mean
 is the global standard deviation
 Mapping
2
1
0 ,
,
, k
k
k
E
G
M
G
D










otherwise
)
,
(
and
if
)
,
(
)
,
( 2
1
0
y
x
f
D
k
D
k
M
k
m
y
x
f
E
y
x
g G
S
G
G
S
xy
xy


58. Fundamentals of Spatial Filtering
 The Mechanics of Spatial Filtering
)
1
,
1
(
)
1
,
1
(
)
,
1
(
)
0
,
1
(
)
,
(
)
0
,
0
(
)
,
1
(
)
0
,
1
(
)
1
,
1
(
)
1
,
1
(
















y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
y
x
f
w
R



59.  Image size:
 Mask size:
 and
 and
N
M 
n
m


 




a
a
s
b
b
t
t
y
s
x
f
t
s
w
y
x
g )
,
(
)
,
(
)
,
(
2
/
)
1
( 
 m
a 2
/
)
1
( 
 n
b
1
,...,
2
,
1
,
0 
 M
x 1
,...,
2
,
1
,
0 
 N
y

62.  Spatial Correlation and Convolution

64. 






9
1
9
9
2
2
1
1 ...
i
i
i z
w
z
w
z
w
z
w
R
 Vector Representation of Linear
Filtering

65. Smoothing Spatial Filters
 Smoothing Linear Filters
 Noise reduction
 Smoothing of false contours
 Reduction of irrelevant detail

66. 


9
1
9
1
i
i
z
R

67. 


 


 



 a
a
s
b
b
t
a
a
s
b
b
t
t
s
w
t
y
s
x
f
t
s
w
y
x
g
)
,
(
)
,
(
)
,
(
)
,
(

70.  Order-statistic filters
 median filter: Replace the value of a
pixel by the median of the gray levels
in the neighborhood of that pixel
 Noise-reduction

72. Sharpening Spatial Filters
 Foundation
 The first-order derivative
 The second-order derivative
)
(
)
1
( x
f
x
f
x
f





)
(
2
)
1
(
)
1
(
2
2
x
f
x
f
x
f
x
f








75.  Use of second derivatives for
enhancement-The Laplacian
 Development of the method
)
,
(
2
)
,
1
(
)
,
1
(
2
2
y
x
f
y
x
f
y
x
f
x
f







2
2
2
2
2
y
f
x
f
f







)
,
(
2
)
1
,
(
)
1
,
(
2
2
y
x
f
y
x
f
y
x
f
y
f








76. )
,
(
4
)]
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
[
2
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
f
























positive
is
mask
Laplacian
the
of
t
coefficien
center
the
if
)
,
(
)
,
(
negative
is
mask
Laplacian
the
of
t
coefficien
center
the
if
)
,
(
)
,
(
)
,
(
2
2
y
x
f
y
x
f
y
x
f
y
x
f
y
x
g

79.  Simplifications
)]
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
[
)
,
(
5
)
,
(
4
)]
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
[
)
,
(
)
,
(



















y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
f
y
x
g

81.  Unsharp masking and highboost
filtering
 Unsharp masking
 Substract a blurred version of an image
from the image itself
 : The image, : The
blurred image
)
,
(
)
,
(
)
,
( y
x
f
y
x
f
y
x
gmask 

)
,
( y
x
f )
,
( y
x
f
)
,
(
*
)
,
(
)
,
( y
x
g
k
y
x
f
y
x
g mask

 1
, 
k

82.  High-boost filtering
)
,
(
*
)
,
(
)
,
( y
x
g
k
y
x
f
y
x
g mask

 1
, 
k

85.  Using first-order derivatives for
(nonlinear) image sharpening—The
gradient

























y
f
x
f
G
G
y
x
f

86.  The magnitude is rotation invariant
(isotropic)
 
2
1
2
2
2
1
2
2
)
(
mag

































y
f
x
f
G
G
f y
x
f
y
x G
G
f 



87.  Computing using cross differences,
Roberts cross-gradient operators
)
( 5
9 z
z
Gx 
 )
( 6
8 z
z
Gy 

and
  2
1
2
6
8
2
5
9 )
(
)
( z
z
z
z
f 




6
8
5
9 z
z
z
z
f 





88.  Sobel operators
 A weight value of 2 is to achieve some
smoothing by giving more importance to
the center point
)
2
(
)
2
(
)
2
(
)
2
(
7
4
1
9
6
3
3
2
1
9
8
7
z
z
z
z
z
z
z
z
z
z
z
z
f














91. Combining Spatial Enhancement
Methods
 An example
 Laplacian to highlight fine detail
 Gradient to enhance prominent edges
 Smoothed version of the gradient
image used to mask the Laplacian
image
 Increase the dynamic range of the gray
levels by using a gray-level
transformation