Chapter 3
Image Enhancement
Domain








Background
Some Basic Gray Level Transformations
Histogram Processing
Enhancement Using Arithmetic/Logic Operations
Basics of Spatial Filtering
Smoothing Spatial Filters
Sharpening Spatial Filters
Combining Spatial Enhancement Methods
in
the
Spatial
 Background
 masks
 filters,
 kernels,
 templates
 windows
 Spatial domain processes
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 ) .
 T can operate on a set of input images, such as performing the pixel-by-pixel sum of
K images for noise reduction.
 Point processing
 A gray-level (also called an intensity or mapping) transformation function
s  T (r )
 Contrast stretching,
 Thresholding function
 Some Basic Gray Level Transformations
 Image Negatives
 The negative of an image with gray levels in the range [0, L  1] is obtained by using the
negative transformation shown in Fig. 3.3, which is given by the expression
s  L 1  r
 Log Transformations
 The general form of the log transformation shown in Fig. 3.3 is
s  c log ( 1  r)
 Power-Law Transformations
 Power-law transformations have the basic form
s  cr 
 Piecewise-Linear Transformation Functions
 Contrast stretching
 Gray-level slicing
 Bit-plane slicing
 Histogram Processing
 The histogram of a digital image with
gray levels in the range [0, L  1] is a
discrete function h(rk )  nk , where rk is
the kth gray level and nk is the number of
pixels in the image having gray level rk .
 The right side of the figure shows the
histograms
corresponding
to
these
images. The horizontal axis of each
histogram plot corresponds to gray level
values, rk . The vertical axis corresponds
to values of h(rk )  nk or p(rk )  nk / n
if the values are normalized.
 Histogram Equalization
s  T (r ) 0  r  1
(a) T (r ) is single-valued and monotonically increasing in the interval 0  r  1 ;
and
(b) 0  T (r )  1 for 0 < r < 1.
 Let pr (r ) and p s (s) denote the probability density functions of random variables
r and s, respectively
dr
p s ( s )  pr ( r )
ds
 A transformation function of particular importance in image processing has the form
r
s  T ( r )   pr ( w)dw
0
 We find p s (s) as a uniform probability function by applying
ds dT (r ) d  r


pr ( w)dw  pr (r )


dr
dr
dr  0
p s ( s )  pr ( r )
dr
1
 pr ( r )
1
ds
pr ( r )
0  s 1
 For discrete values,
 The probability of occurrence of gray level rk in an image is approximated by
n
p r (rk )  k k  0,1,2,, L  1
n
 The discrete version of the transformation function is given as
k
k n
j
sk  T (rk )   pr (rj )  
k  0,1,2,, L  1
j 0
j 0 n
 Histogram Matching (Specification)
 Development of the method
r
s  T ( r )   pr ( w)dw
0
z
G( z )   pz (t )dt  s
0
z  G 1 ( s)  G 1 (T (r ))
k
k
nj
j 0
n
sk  T (rk )   pr (rj )  
j 0
k
vk  G ( z k )   p z ( z i )  s k
k  0,1,2,, L  1
k  0,1,2,, L  1
i 0
z k  G 1 ( sk ) k  0,1,2,, L  1
(G( zˆ)  sk )  0 k  0,1,2,, L  1
 Implementation
 The procedure we have just developed for histogram matching may be summarized as follows:
1. Obtain the histogram of the given image.
k
k n
j
s

T
(
r
)

p
(
r
)

k  0,1,2,, L  1 to precompute a mapped


k
k
r
j
2. Use
n
j 0
j 0
level sk for each level rk .
3. Obtain the transformation function G from the given pz (z) using
k
vk  G ( z k )   p z ( z i )  s k
i 0
k  0,1,2,, L  1 .
4. Precompute zk for each value of sk , where z k  zˆ is the smallest integer such
that (G( zˆ)  sk )  0 k  0,1,2,, L  1 .
5. For each pixel in the original image, if the value of that pixel is rk , map this
value to its corresponding level sk ; then map level sk into the final level zk . Use
the precomputed values from Steps (2) and (4) for these mappings.
 Local Enhancement
 Use of Histogram Statistics for Image Enhancement
 Global mean and variance
L 1
 n (r )   (ri  m) n p(ri )
i 0
L 1
m   ri p ( ri )
i 0
L 1
 (r )   2 (r )   (ri  m) 2 p(ri )
2
i 0
 Local mean and variance
mS xy 
 S2 
xy
r
s ,t
( s ,t )S xy
[ r
( s ,t )S xy
s ,t
p(rs ,t )
 mS xy ] p(rs ,t )
 Summary of the enhancement method
E  f ( x, y ) if mSxy  k0 M G and k1DG   Sxy  k1DG
g ( x, y )  
otherwise
 f ( x, y )
where E , k0 , k1 , and k2 are specified parameters; M G is the global mean of
the input image; and DG is its global standard deviation.
E  4.0 , k0  0.4 , k1  0.02 , and k2  0.4
 Enhancement Using Arithmetic/Logic Operations
 Image Subtraction
g ( x , y )  f ( x , y )  h ( x, y )
 A few comments on implementation
 The values in a difference image can range from a minimum of -255 to a
maximum of 255
 One method is to add 255 to every pixel and then divide by 2
 The other method, at first, the value of the minimum difference is obtained and
its negative added to all the pixels in the difference image. Then, all the pixels in
the image are scaled to the interval [0, 255] by multiplying each pixel by the
quantity 255/Max, where Max is the maximum pixel value in the modified
difference image
 Image Averaging
g ( x, y )  f ( x , y )   ( x , y )
1 K
g ( x, y )   g i ( x, y )
K i 1
E{g ( x, y )}  f ( x, y )
 g2 ( x , y ) 
 g ( x, y ) 
1 2
  ( x, y )
K
1
  ( x, y )
K
 Basics 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)
g ( x, y ) 
a
b
  w(s, t ) f ( x  s, y  t )
s   a t  b
9
R  w1 z1  w2 z 2    w9 z9   wi zi
i 1
mn
R  w1 z1  w2 z 2    wmn z mn   wi zi
i 1
 Smoothing Spatial Filters
 Smoothing Linear Filters
1 9
R   zi
9 i 1
a
g ( x, y ) 
b
  w(s, t ) f ( x  s, y  t )
s  a t  b
a
b
  w(s, t )
s  a t b
 Order-Statistics Filters
 Median filter
 Median filters are particularly effective in the presence of impulse noise, also
called salt-and-pepper noise
 Max filter
 Min filter
 Sharpening Spatial Filters
 Foundation
f
 f ( x  1)  f ( x)
x
2 f
 f ( x  1)  f ( x  1)  2 f ( x)
x 2
 Use of Second Derivatives for Enhancement-The Laplacian
2 f 2 f
 f  2  2
x
y
2
2 f
 f ( x  1, y )  f ( x  1, y )  2 f ( x, y )
x 2
2 f
 f ( x, y  1)  f ( x, y  1)  2 f ( x, y )
y 2
f  [ f ( x  1, y )  f ( x  1, y )  f ( x, y  1)  f ( x, y  1)]  4 f ( x, y )
 f ( x, y )   2 f ( x, y ) if the center coefficien t of the


Laplacian mask is negative
g ( x, y )  
2
 f ( x, y )   f ( x, y ) if the center coefficien t of the

Laplacian mask is positive
(3.7-5)
 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 high-boost filtering
f s ( x , y )  f ( x, y )  f ( x , y )
where f s ( x, y) denotes the sharpened image obtained by unsharp masking, and
f ( x, y ) is a blurred version of f ( x, y )
 High-boost filtering
f hb ( x, y )  Af ( x, y )  f ( x, y )
f hb ( x, y )  ( A  1) f ( x, y )  f ( x, y )  f ( x, y )
f hb ( x, y)  ( A  1) f ( x, y)  f s ( x, y)
f hb
 Af ( x, y )   2 f ( x, y ) if the center coefficien t of the


Laplacian mask is negative

2
 Af ( x, y )   f ( x, y ) if the center coefficien t of the

Laplacian mask is positive
 Use of First Derivatives for Enhancement—The Gradient
 f 
G x   x 
f      f 
G y   
 y 

f  mag(f )  Gx2  Gy2

1/ 2
f  Gx  G y
Gx  ( z 9  z 5 )
G y  ( z8  z 6 )

f  ( z9  z5 ) 2  ( z8  z6 ) 2

1/ 2
Roberts cross-gradient operator
f  z9  z5  z8  z6
Sobel operator
f  ( z7  2z8  z9 )  ( z1  2z2  z3 )  ( z3  2z6  z9 )  ( z1  2z4  z7 )
 Combining Spatial Enhancement Methods