Image enhancement in the spatial domain1

1
Image Enhancement in the
Spatial Domain

Principle Objective of
Enhancement
• Process an image so that the result will be
more suitable than the original image for a
specific application.
• The suitableness is up to each application.
• A method which is quite useful for
enhancing an image may not necessarily be
the best approach for enhancing another
images

2 domains
• Spatial Domain : (image plane)
– Techniques are based on direct manipulation of pixels
in an image
• Frequency Domain :
– Techniques are based on modifying the Fourier
transform of an image
• There are some enhancement techniques based
on various combinations of methods from these
two categories.

Good images
• For human visual
– The visual evaluation of image quality is a highly
subjective process.
– It is hard to standardize the definition of a good
image.
• For machine perception
– The evaluation task is easier.
– A good image is one which gives the best machine
recognition results.
• A certain amount of trial and error usually is
required before a particular image enhancement
approach is selected.

Spatial Domain
• Procedures that operate
directly on pixels.
g(x,y) = T[f(x,y)]
where
– f(x,y) is the input image
– g(x,y) is the processed
image
– T is an operator on f
defined over some
neighborhood of (x,y)

Point Processing
• Neighborhood = 1x1 pixel
• g depends on only the value of f at (x,y)
• T = gray level (or intensity or mapping)
transformation function
s = T(r)
• Where
– r = gray level of f(x,y)
– s = gray level of g(x,y)

3 basic gray-level
transformation functions
• Linear function
– Negative and identity
transformations
• Logarithm function
– Log and inverse-log
transformation
• Power-law function
– nth power and nth root
transformations
Input gray level, r
Negative
Log
nth root
Identity
nth power
Inverse Log

Identity function
• Output intensities are
identical to input
intensities.
• Is included in the
graph only for
completeness.
Input gray level, r
Negative
Log
nth root
Identity
nth power
Inverse Log

Image Negatives
• An image with gray level in the
range [0, L-1]
where L = 2n ; n = 1, 2…
• Negative transformation :
s = L – 1 –r
• Reversing the intensity levels
of an image.
• Suitable for enhancing white
or gray detail embedded in
dark regions of an image,
especially when the black area
dominant in size.
Input gray level, r
Negative
Log
nth root
Identity
nth power
Inverse Log

Example of Negative Image
Original image Negative Image : gives a
better vision to analyze
the image

Log Transformations
s = c log (1+r)
• c is a constant
and r  0
• Log curve maps a narrow
range of low gray-level
values in the input image
into a wider range of
output levels.
• Used to expand the
values of dark pixels in an
image while compressing
the higher-level values.
Input gray level, r
Negative
Log
nth root
Identity
nth power
Inverse Log

Example of Logarithmic Image
Result after apply the log
transformation with c = 1,
range = 0 to 6.2
Fourier Spectrum with
range = 0 to 1.5 x 106

Inverse Logarithmic
Transformations
• Do opposite to the Log Transformations
• Used to expand the values of high pixels
in an image while compressing the darker-
level values.

Power-Law Transformations
s = cr
• c and  are positive
constants
• Power-law curves with
fractional values of  map
a narrow range of dark
input values into a wider
range of output values,
with the opposite being
true for higher values of
input levels.
• c =  = 1  Identity
function
Input gray level, r
Plots of s = cr for various values of 
(c = 1 in all cases)

Another example : MRI
(a) a magnetic resonance image of
an upper thoracic human spine
with a fracture dislocation and
spinal cord impingement
– The picture is predominately dark
– An expansion of gray levels are
desirable  needs  < 1
(b) result after power-law
transformation with  = 0.6, c=1
(c) transformation with  = 0.4
(best result)
(d) transformation with  = 0.3
(under acceptable level)
a b
c d

Effect of decreasing gamma
• When the  is reduced too much, the
image begins to reduce contrast to the
point where the image started to have very
slight “wash-out” look, especially in the
background

Another example
(a) image has a washed-out
appearance, it needs a
compression of gray levels
 needs  > 1
(b) result after power-law
transformation with  = 3.0
(suitable)
(c) transformation with  = 4.0
(suitable)
(d) transformation with  = 5.0
(high contrast, the image has
areas that are too dark,
some detail is lost)
a b
c d

Piecewise-Linear
Transformation Functions
• Advantage:
– The form of piecewise functions can be
arbitrarily complex
• Disadvantage:
– Their specification requires considerably more
user input

Contrast Stretching
• Produce higher
contrast than the
original by
– darkening the levels
below m in the original
image
– Brightening the levels
above m in the original
image

Thresholding
• Produce a two-level
(binary) image

Contrast Stretching
• increase the dynamic range of the
gray levels in the image
• (b) a low-contrast image : result
from poor illumination, lack of
dynamic range in the imaging
sensor, or even wrong setting of a
lens aperture of image acquisition
• (c) result of contrast stretching:
(r1,s1) = (rmin,0) and (r2,s2) =
(rmax,L-1)
• (d) result of thresholding

Gray-level slicing
• Highlighting a specific
range of gray levels in an
image
– Display a high value of all
gray levels in the range of
interest and a low value for
all other gray levels
• (a) transformation highlights
range [A,B] of gray level and
reduces all others to a
constant level
• (b) transformation highlights
range [A,B] but preserves all
other levels

Bit-plane slicing
• Highlighting the contribution
made to total image
appearance by specific bits
• Suppose each pixel is
represented by 8 bits
• Higher-order bits contain the
majority of the visually
significant data
• Useful for analyzing the
relative importance played
by each bit of the image
Bit-plane 7
(most significant)
Bit-plane 0
(least significant)
One 8-bit byte

Example
• The (binary) image for bit-
plane 7 can be obtained
by processing the input
image with a thresholding
gray-level transformation.
– Map all levels between 0
and 127 to 0
– Map all levels between 129
and 255 to 255
An 8-bit fractal image

8 bit planes
Bit-plane 7 Bit-plane 6
Bit-
plane 5
Bit-
plane 4
Bit-
plane 3
Bit-
plane 2
Bit-
plane 1
Bit-
plane 0

Histogram Processing
• Histogram of a digital image with gray levels in
the range [0,L-1] is a discrete function
h(rk) = nk
• Where
– rk : the kth gray level
– nk : the number of pixels in the image having gray
level rk
– h(rk) : histogram of a digital image with gray levels rk

Example
2 3 3 2
4 2 4 3
3 2 3 5
2 4 2 4
4x4 image
Gray scale = [0,9]
histogram
0 1
1
2
2
3
3
4
4
5
5
6
6
7 8 9
No. of pixels
Gray level

Normalized Histogram
• dividing each of histogram at gray level rk by the
total number of pixels in the image, n
p(rk) = nk / n
• For k = 0,1,…,L-1
• p(rk) gives an estimate of the probability of
occurrence of gray level rk
• The sum of all components of a normalized
histogram is equal to 1

Histogram Processing
• Basic for numerous spatial domain
processing techniques
• Used effectively for image enhancement
• Information inherent in histograms also is
useful in image compression and
segmentation

Example
rk
h(rk) or p(rk)
Dark image
Bright image
Components of
histogram are
concentrated on the
low side of the gray
scale.
Components of
histogram are
concentrated on the
high side of the gray
scale.

Example
Low-contrast image
High-contrast image
histogram is narrow
and centered toward
the middle of the
gray scale
histogram covers broad
range of the gray scale
and the distribution of
pixels is not too far from
uniform, with very few
vertical lines being much
higher than the others

Histogram Equalization
• As the low-contrast image’s histogram is narrow
and centered toward the middle of the gray
scale, if we distribute the histogram to a wider
range the quality of the image will be improved.
• We can do it by adjusting the probability density
function of the original histogram of the image so
that the probability spread equally

Example
before after Histogram
equalization

Example
before after Histogram
equalization
The quality is
not improved
much because
the original
image already
has a broaden
gray-level scale

Gray
Level(j)
0 1 2 3 4 5 6 7 8 9
No. of
pixels
0 0 6 5 4 1 0 0 0 0
0 0 6 11 15 16 16 16 16 16
0 0
6
/
16
11
/
16
15
/
16
16
/
16
16
/
16
16
/
16
16
/
16
16
/
16
s x 9 0 0
3.3
3
6.1
6
8.4
8
9 9 9 9 9


k
j
j
n
0



k
j
j
n
n
s
0

Example
3 6 6 3
8 3 8 6
6 3 6 9
3 8 3 8
Output image
Gray scale = [0,9]
Histogram equalization
0 1
1
2
2
3
3
4
4
5
5
6
6
7 8 9
No. of pixels
Gray level

Histogram Matching
(Specification)
• Histogram equalization has a disadvantage
which is that it can generate only one type of
output image.
• With Histogram Specification, we can specify
the shape of the histogram that we wish the
output image to have.
• It doesn’t have to be a uniform histogram

Consider the continuous domain
Let pr(r) denote continuous probability density
function of gray-level of input image, r
Let pz(z) denote desired (specified) continuous
probability density function of gray-level of
output image, z
Let s be a random variable with the property



r
r dw
)
w
(
p
)
r
(
T
s
0
Where w is a dummy variable of integration

Next, we define a random variable z with the property
s = T(r) = G(z)
We can map an input gray level r to output gray level z
thus
s
dt
)
t
(
p
)
z
(
g
z
z 
 
0
Where t is a dummy variable of integration
Therefore, z must satisfy the condition
z = G-1(s) = G-1[T(r)]
Assume G-1 exists and satisfies the condition (a) and (b)

Procedure Conclusion
1. Obtain the transformation function T(r) by
calculating the histogram equalization of the
input image
2. Obtain the transformation function G(z) by
calculating histogram equalization of the
desired density function



r
r dw
)
w
(
p
)
r
(
T
s
0
s
dt
)
t
(
p
)
z
(
G
z
z 
 
0

Procedure Conclusion
3. Obtain the inversed transformation
function G-1
4. Obtain the output image by applying the
processed gray-level from the inversed
transformation function to all the pixels in
the input image
z = G-1(s) = G-1[T(r)]

Example
Assume an image has a gray level probability density
function pr(r) as shown.
0 1 2
1
2
Pr(r)


 




elsewhere
;
0
1
r
;0
2
2r
)
r
(
pr
1
0


r
r dw
)
w
(
p
r

Example
We would like to apply the histogram specification with
the desired probability density function pz(z) as shown.
0 1 2
1
2
Pz(z)
z


 


elsewhere
;
0
1
z
;0
2z
)
z
(
pz
1
0


z
z dw
)
w
(
p

Step 1:
0 1
1
s=T(r)
r
r
r
w
w
dw
)
w
(
dw
)
w
(
p
)
r
(
T
s
r
r
r
r
2
2
2
2
2
0
2
0
0













Obtain the transformation function T(r)
One to one
mapping
function

Step 2:
2
0
2
0
2 z
z
dw
)
w
(
)
z
(
G
z
z


 
Obtain the transformation function G(z)

Step 3:
2
2
2
2
2
r
r
z
r
r
z
)
r
(
T
)
z
(
G






Obtain the inversed transformation function G-1
We can guarantee that 0  z 1 when 0  r 1

Discrete formulation
1
2
1
0
0



 

L
,...,
,
,
k
s
)
z
(
p
)
z
(
G k
k
i
i
z
k
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
1
1






L
,...,
,
,
k
s
G
)
r
(
T
G
z
k
k
k

Example
Image of Mars moon
Image is dominated by large, dark areas,
resulting in a histogram characterized by
a large concentration of pixels in pixels in
the dark end of the gray scale

Image Equalization
Result image
after histogram
equalization
Transformation function
for histogram equalization
Histogram of the result image
The histogram equalization doesn’t make the result image look better than
the original image. Consider the histogram of the result image, the net
effect of this method is to map a very narrow interval of dark pixels into
the upper end of the gray scale of the output image. As a consequence, the
output image is light and has a washed-out appearance.

Histogram Equalization
Histogram Specification
Solve the problem
Since the problem with the
transformation function of the
histogram equalization was
caused by a large concentration
of pixels in the original image with
levels near 0
a reasonable approach is to
modify the histogram of that
image so that it does not have
this property

Histogram Specification
• (1) the transformation
function G(z) obtained
from
• (2) the inverse
transformation G-1(s)
1
2
1
0
0



 

L
,...,
,
,
k
s
)
z
(
p
)
z
(
G k
k
i
i
z
k

Result image and its histogram
Original image
The output image’s histogram
Notice that the output
histogram’s low end has
shifted right toward the
lighter region of the gray
scale as desired.
After applied
the histogram
equalization

Note
• Histogram specification is a trial-and-error
process
• There are no rules for specifying
histograms, and one must resort to
analysis on a case-by-case basis for any
given enhancement task.

Image enhancement in the spatial domain1

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