Image Enhancement in the Spatial Domain

lemonadeviraginityAI and Robotics

Nov 6, 2013 (3 years and 9 months ago)

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Chapter
3

Image Enhancement in the Spatial
Domain





Background



Some Basic Gray Level Transformations



Histogram Processing



Enhancement Using Arithmetic/Logic Operations



Basics of Spatial Filtering



Smoothing Spatial Filters



Sharpening Spatial Filters



Co
mbining Spatial Enhancement Methods







Background


















Spatial domain processes

)]
,
(
[
)
,
(
y
x
f
T
y
x
g




where
)
,
(
y
x
f

is the input image,
)
,
(
y
x
g

is the processed image, and
T

is an
operato
r on
f
, defined over some neighborhood of
)
,
(
y
x
.



T
can operate on a
set

of input images, such as performing the pixel
-
by
-
pixel sum of
K
images for noise reduction.



masks




filters,



kernels,



templates



windows





Point processing




A
gray
-
leve
l

(also called an
intensity

or
mapping) transformation function

)
(
r
T
s





Contrast stretching
,



Thresholding

function

















Some Basic Gray Level Transformations

























Image Negatives



The negative of an image with g
ray levels in the range
]
1
,
0
[

L

is obtained by using the
negative transformation shown in Fig. 3.3, which is given by the expression


r
L
s



1




















Log Transformations



The general form of the log transformation s
hown in Fig. 3.3 is

r)
(
c
s


1
log






















Power
-
Law Transformations



Power
-
law transformations have the basic form


cr
s

























































































Piecewi
se
-
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
]
1
,
0
[

L

is a
discrete function
k
k
n
r
h

)
(
,

where
k
r

is
the

kth

gray level and
k
n
is the number of
pixels in the image having gray level
k
r
.



The right side of the figure shows the
histograms corresponding to these
images. The horizontal axis of each
histogram plot correspon
ds to gray level
values,

k
r
.
The vertical axis corresponds
to values of
k
k
n
r
h

)
(

or
n
n
r
p
k
k
/
)
(


if the values are normalized.




Histogram Equalization

)
(
r
T
s


1
0


r

(a)

)
(
r
T

is single
-
valued and monotonically increasing in the interval
1
0


r
;
and

(b)

1
)
(
0


r
T

for 0 <
r <

1.













Let
)
(
r
p
r

and
)
(
s
p
s

denote the probability density functions of random variables
r

and
s,

respectively

ds
dr
r
p
s
p
r
s
)
(
)
(




A transformation function of particular importance in image processing has the form




r
r
dw
w
p
r
T
s
0
)
(
)
(



W
e find
)
(
s
p
s
as a uniform probability function

by applying

)
(
)
(
)
(
0
r
p
dw
w
p
dr
d
dr
r
dT
dr
ds
r
r
r











1
)
(
1
)
(
)
(
)
(



r
p
r
p
ds
dr
r
p
s
p
r
r
r
s

1
0


s





For discrete values,



T
he probability of occurrence of gray level
k
r

in an image is approximated by

1
,
,
2
,
1
,
0
)
(



L
k
n
n
r
p
k
k
r




The discrete version of the transformation function is given as

1
,
,
2
,
1
,
0
)
(
)
(
0
0









L
k
n
n
r
p
r
T
s
k
j
j
k
j
j
r
k
k





















































Histogram Matching (Specification)



Development of the method




r
r
dw
w
p
r
T
s
0
)
(
)
(




z
z
s
dt
t
p
z
G
0
)
(
)
(

))
(
(
)
(
1
1
r
T
G
s
G
z









1
,
,
2
,
1
,
0
)
(
)
(
0
0









L
k
n
n
r
p
r
T
s
k
j
j
k
j
j
r
k
k

1
,
,
2
,
1
,
0
)
(
)
(
0







L
k
s
z
p
z
G
v
k
k
i
i
z
k
k


1
,
,
2
,
1
,
0
)
(
1




L
k
s
G
z
k
k



1
,
,
2
,
1
,
0
0
)
)
ˆ
(
(




L
k
s
z
G
k






Implementation



The procedure we have just developed for histogram matching may be sum
-
marized as follows:

1. Obtain the hi
stogram of the given image.

2. Use
1
,
,
2
,
1
,
0
)
(
)
(
0
0









L
k
n
n
r
p
r
T
s
k
j
j
k
j
j
r
k
k


to precompute a mapped
level
k
s

for each level
k
r
.

3. Obtain the transformation function
G

from the given
)
(
z
p
z

using
1
,
,
2
,
1
,
0
)
(
)
(
0







L
k
s
z
p
z
G
v
k
k
i
i
z
k
k

.

4. Precompute
k
z

for each value of
k
s
, where
z
z
k
ˆ


is the smallest integer such
that
1
,
,
2
,
1
,
0
0
)
)
ˆ
(
(




L
k
s
z
G
k

.

5. For each pixel in the original image, if the value of that pixel is
k
r
, map this
value to its corresponding level
k
s
; then map level
k
s
into the final level
k
z
. Use
the precomputed values from Steps (2) and (4) for these mappings.



















































Local Enhancement
























Use of Histogram Statistics for Image Enhancement



G
lobal

mean and variance






1
0
)
(
)
(
)
(
L
i
i
n
i
n
r
p
m
r
r






1
0
)
(
L
i
i
i
r
p
r
m







1
0
2
2
2
)
(
)
(
)
(
)
(
L
i
i
i
r
p
m
r
r
r







L
ocal

mean and variance




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
)
(
]
[
























S
ummary of the enhancement method









otherwise
y
x
f
D
k
D
k
and
M
k
m
if
y
x
f
E
y
x
g
G
S
G
G
S
xy
xy
)
,
(
)
,
(
)
,
(
1
1
0


where
E
,
0
k
,
1
k
,

and
2
k

are specified parameters;
G
M

is the global mean of

the input image; and
G
D

is its global standard deviation.

0
.
4

E
,
4
.
0
0

k
,
02
.
0
1

k
,

and
4
.
0
2

k














































Enhancement Using Arithmetic/Logic Operations
























Image Subtraction

)
,
(
)
,
(
)
,
(
y
x
h
y
x
f
y
x
g





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 t
o add 255 to every pixel and then divide by 2



The other method, at
first, the value of the minimum difference is ob
tained 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, 25
5] by multiplying each pixel by the
quantity
255/Max,

where Max is the maximum pixel value in the modi
fied
difference image





































Image Averaging



)
,
(
)
,
(
)
,
(
y
x
y
x
f
y
x
g









K
i
i
y
x
g
K
y
x
g
1
)
,
(
1
)
,
(

)
,
(
)}
,
(
{
y
x
f
y
x
g
E




2
)
,
(
2
)
,
(
1
y
x
y
x
g
K





)
,
(
)
,
(
1
y
x
y
x
g
K







































































Basics 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
















a
a
s
b
b
t
t
y
s
x
f
t
s
w
y
x
g
)
,
(
)
,
(
)
,
(


















9
1
9
9
2
2
1
1
i
i
i
z
w
z
w
z
w
z
w
R









mn
i
i
i
mn
mn
z
w
z
w
z
w
z
w
R
1
2
2
1
1






Smoothing Spatial Filters



Smoothing Linear Filters














9
1
9
1
i
i
z
R

















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
)
,
(
)
,
(
)
,
(
)
,
(

















































Order
-
Statistics Filters



Median filter



Median filters are particularly effectiv
e in the presence of
impulse noise
,

also
called
salt
-
and
-
pepper

noise



Max filter



Min filter



















Sharpening Spatial Filters



Foundation















)
(
)
1
(
x
f
x
f
x
f






)
(
2
)
1
(
)
1
(
2
2
x
f
x
f
x
f
x
f











Use of Second Derivatives for
Enhancement
-
The

Laplacian

2
2
2
2
2
y
f
x
f
f








)
,
(
2
)
,
1
(
)
,
1
(
2
2
y
x
f
y
x
f
y
x
f
x
f








)
,
(
2
)
1
,
(
)
1
,
(
2
2
y
x
f
y
x
f
y
x
f
y
f
























)
,
(
4
)]
1
,
(
)
1
,
(
)
,
1
(
)
,
1
(
[
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

(3.7
-
5)





















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






















Unsharp

mas
king and high
-
boost filtering


)
,
(
)
,
(
)
,
(
y
x
f
y
x
f
y
x
f
s



where
)
,
(
y
x
f
s

denotes the sharpened image obtained by unsharp masking, and
)
,
(
y
x
f

is a blurred version of

)
,
(
y
x
f







High
-
boost filterin
g


)
,
(
)
,
(
)
,
(
y
x
f
y
x
Af
y
x
f
hb



)
,
(
)
,
(
)
,
(
)
1
(
)
,
(
y
x
f
y
x
f
y
x
f
A
y
x
f
hb





)
,
(
)
,
(
)
1
(
)
,
(
y
x
f
y
x
f
A
y
x
f
s
hb





















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
Af
y
x
f
y
x
Af
f
hb









































Use of First Derivatives for Enhancement

The Gradient


























y
f
x
f
G
G
f
y
x




2
/
1
2
2
)
(
y
x
G
G
f
mag
f







y
x
G
G
f




)
(
5
9
z
z
G
x



)
(
6
8
z
z
G
y





2
/
1
2
6
8
2
5
9
)
(
)
(
z
z
z
z
f






Roberts cross
-
gradient operator

6
8
5
9
z
z
z
z
f







Sobel operator

)
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


































Combining Spatial Enhancement Methods