Intensity transformations and spatial filtering techniques can enhance images by modifying pixel intensities or applying neighborhood operations. Key techniques include: 1. Grayscale transformations like logarithms, power laws, and piecewise linear functions which compress or expand tonal ranges. Histogram processing includes equalization to spread intensities uniformly. 2. Spatial filters apply operations to pixels based on neighboring values. Smoothing filters reduce noise while sharpening filters using derivatives enhance edges. The Laplacian is a second derivative filter useful for edge detection. 3. Unsharp masking and high-boost filtering enhance edges by subtracting a blurred version of an image from the original, emphasizing differences.

1. 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