Digital Image Processing
Chapter 3: Intensity Transformations
and Spatial Filtering
Background

Spatial domain process

g ( x, y )  T [ f ( x, y )]
where f ( x, y ) is the input image, g ( x, y )
is the processed image, and T is an
operator on f, defined over some
neighborhood of ( x, y )

Neighborhood about a point

Gray-level transformation function
s  T (r )

where r is the gray level of f ( x, y ) and
s is the gray level of g ( x, y ) at any
point ( x, y )

Contrast enhancement

For example, a thresholding function

Masks (filters, kernels, templates,
windows)

A small 2-D array in which the values
of the mask coefficients determine the
nature of the process
Some Basic Gray Level
Transformations

Image negatives
s  L 1  r

Enhance white or gray details

Log transformations
s  c log( 1  r )

Compress the dynamic range of images
with large variations in pixel values

From the range 0-1.5  10
0 to 6.2
6
to the range


Power-law transformations
s  cr  or s  c(r   )
   1 maps a narrow range of dark

input values into a wider range of
output values, while   1 maps a
narrow range of bright input values into
a wider range of output values
 : gamma, gamma correction

Monitor,   2.5

Piecewise-linear transformation
functions

The form of piecewise functions can be
arbitrarily complex

Contrast stretching

Gray-level slicing

Bit-plane slicing
Histogram Processing

Histogram
h(rk )  nk


where rk is the kth gray level and
the number of pixels in the image
having gray level rk
Normalized histogram
p(rk )  nk / n
nkis

Histogram equalization
s  T (r ), 0  r  1
r  T 1 (s), 0  s  1

Probability density functions (PDF)
dr
p s ( s )  pr ( r )
ds
r
s  T (r )  ( L  1) pr (w)dw
0
ds dT (r )
d  r
  ( L  1) p (r )

 ( L  1)
p
(
w
)
dw
r
r

dr
dr
dr  0
1
ps ( s ) 
L 1
k
k
nj
j 0
j 0
n
sk  T (rk )  ( L  1) pr (rj ) ( L  1)
, k  0,1,2,..., L  1

Histogram matching (specification)
r
s  T (r )  ( L  1) pr (w)dw
0
z
G( z )  ( L  1) pz (t )dt  s
0
z  G 1 ( s)  G 1[T (r )]
pz (z)
is the desired PDF
k
k
nj
j 0
j 0
n
sk  T (rk )  ( L  1) pr (rj )  ( L  1)
, k  0,1,2,..., L  1
k
vk  G ( z k )  ( L  1) p z ( zi ) sk , k  0,1,2,..., L  1
i 0
zk  G 1[T (rk )], k  0,1,2,..., L  1

Histogram matching





Obtain the histogram of the given
image, T(r)
Precompute a mapped level sk for each
level rk
Obtain the transformation function G
from the given pz (z )
Precompute zk for each value of sk
Map rk to its corresponding level sk ;
then map level sk into the final level zk

Local enhancement

Histogram using a local neighborhood,
for example 7*7 neighborhood

Histogram using a local 3*3
neighborhood

Use of histogram statistics for
image enhancement
 r denotes a discrete random variable


p(ri ) denotes the normalized
histogram component corresponding to
the ith value of r
Mean
L 1
m   ri p (ri )
i 0

The nth moment
L 1
 n (r )   (ri  m) n p(ri )
i 0

The second moment
L 1
 2 (r )   (ri  m) 2 p(ri )
i 0


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



rs ,t
is the gray level at coordinates
(s,t) in the neighborhood
p (rs ,t ) is the neighborhood normalized
histogram component
mean:
mS xy 

r
s ,t
( s ,t )S xy
p(rs ,t )
local variance
 S2 
xy
2
[
r

m
]
 s,t S xy p(rs,t )
( s ,t )S xy




E, k0 , k1 , k2 are specified parameters
M G is the global mean
DG is the global standard deviation
Mapping
if mS xy  k0 M G

E  f ( x, y )
g ( x, y)  
and k1 DG   S xy  k2 DG
 f ( x, y)
otherwise

Fundamentals of Spatial Filtering

The Mechanics of Spatial Filtering
R  w(1,1) f ( x  1, y  1) 
w(1,0) f ( x  1, y )   
w(0,0) f ( x, y )   
w(1,0) f ( x  1, y ) 
w(1,1) f ( x  1, y  1)


Image size:
Mask size:
g ( x, y ) 
a
M N
m n
b
  w(s, t ) f ( x  s, y  t )
s   at   b


a  (m  1) / 2 and b  (n  1) / 2
x  0,1,2,..., M  1 and y  0,1,2,..., N  1

Spatial Correlation and Convolution

Vector Representation of Linear
Filtering
R  w1 z1  w2 z 2  ...  w9 z9
9
  wi zi
i 1
Smoothing Spatial Filters

Smoothing Linear Filters



Noise reduction
Smoothing of false contours
Reduction of irrelevant detail
1 9
R   zi
9 i 1
a
g ( x, y ) 
b
  w(s, t ) f ( x  s, y  t )
s   at   b
a
b
  w(s, t )
s   at   b

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
Sharpening Spatial Filters

Foundation

The first-order derivative
f
 f ( x  1)  f ( x)
x

The second-order derivative
 f
 f ( x  1)  f ( x  1)  2 f ( x)
2
x
2

Use of second derivatives for
enhancement-The Laplacian

Development of the method
 f  f
 f  2  2
x
y
2
2
2
 f
 f ( x  1, y )  f ( x  1, y )  2 f ( x, y )
2
x
2 f
 f ( x, y  1)  f ( x, y  1)  2 f ( x, y)
2
y
2
 2 f  [ f ( x  1, y)  f ( x  1, y )  f ( x, y  1) 
f ( x, y  1)]  4 f ( x, y)
if the center coefficien t

 f ( x, y )   2 f ( x, y ) of the Laplacian mask


is negative
g ( x, y )  
if the center coefficien t

 f ( x, y )   2 f ( x, y ) of the Laplacian mask

is positive


Simplifications
g ( x, y )  f ( x, y )  [ f ( x  1, y )  f ( x  1, y )  f ( x, y  1) 
f ( x, y  1)]  4 f ( x, y )
 5 f ( x, y )  [ f ( x  1, y )  f ( x  1, y )  f ( x, y  1) 
f ( x, y  1)]

Unsharp masking and highboost
filtering

Unsharp masking

Substract a blurred version of an image
from the image itself
g mask ( x, y )  f ( x, y )  f ( x, y )

f ( x, y ) : The image, f ( x, y )
blurred image
: The
g ( x, y)  f ( x, y)  k * g mask ( x, y)
,k 1

High-boost filtering
g ( x, y)  f ( x, y)  k * g mask ( x, y)
,k 1

Using first-order derivatives for
(nonlinear) image sharpening—The
gradient
 f 
Gx   x 
f      f 
G y   
 y 

The magnitude is rotation invariant
(isotropic)

f  mag (f )  G  G
2
x
 f  2  f 
     
 x   y 
f  G x  G y
2



1
2
2
y

1
2

Computing using cross differences,
Roberts cross-gradient operators
Gx  ( z9  z5 )

and
G y  ( z8  z 6 )
f  ( z9  z5 )  ( z8  z6 )
2
f  z9  z5  z8  z6
2

1
2

Sobel operators

A weight value of 2 is to achieve some
smoothing by giving more importance to
the center point
f  ( z7  2 z8  z9 )  ( z1  2 z 2  z3 )
 ( z3  2 z6  z9 )  ( z1  2 z 4  z7 )
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