IMAGE ENHANCEMENT IN THE
SPATIAL DOMAIN
Digital Image Processing, Chapter 3 Gonzalez Woods
Digital Image Processing In Life-Science
Sefi addadi 21-3-2012
WHAT DID WE LEARN LAST TIME?
WHAT WILL BE TODAY?
Image enhancement in the spatial domain
• Gray level transformation
– Negative, Log and power law
• Histogram processing
– Equalization
– Matching
– Local enhancement
• Spatial filtering
– Smoothing filters (mean, Gaussian, median)
– Sharpening filters
– Edge detection
– Filter combinations
• Background correction
– Acquisition (a priori) correction
– Retrospective (a posteriori) correction
• Rolling ball background correction
IMAGE ENHANCEMENT IN THE
SPATIAL DOMAIN
• The principal objective of enhancement –
“Process an image so that the result is more suitable than the
original image for a specific application”.
• “Spatial domain refers to the image plane itself, and approaches in
this category are based on direct manipulation of pixels in an
image”.
• There is no general theory of image enhancement.
• When an image is processed for visual interpretation, the viewer is
the ultimate judge of how well a particular method works.
Gonzalez Woods Chapter 3
MATHEMATICAL REPRESENTATION
General expression
Origin
Neighbourhood
y
g(x, y) = T{f(x, y) }
x
Target
x
(x, y)
Image f (x, y)
y
Image g(x, y)
GRAY LEVEL TRANSFORMATIONS
• Among the simplest of all
image enhancement
techniques.
• Generally denoted by
r – values before
transformation
s – Values after transformation
T – the performed
transformation s = T(r)
POWER- LAW TRANSFORMATIONS
• S= cr where 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.
• The opposite being true
for highs.
Many devices used for image capture, printing, and display
respond according to a power law – Gamma correction
HISTOGRAM PROCESSING
• Histogram in digital images is defined as a discreet
function
• Where
h(rk) = nk
rk is the gray level of k and
nk is the number of pixels of that value
Histogram normalization
Performed by dividing each value by the total number of pixels in the
image.
p(rk) = nk / n
• 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.
Dark image
Bright image
Low contrast image
High contrast image
Histogram Equalization
p(rk) = nk / n
histogram equalization
Each pixel with level
rk is mapped into a
corresponding pixel
with level sk in the
output image.
histogram linearization
“Automated” process
does not require
additional input
HISTOGRAM MATCHING
• The method used to generate a processed image that
has a specified histogram shape.
• In contrast to the aim of histogram where we perform
automatic uniform histogram.
HISTOGRAM MATCHING
1. Obtain the histogram of the given
image.
2. Precompute a mapped level sk for each
level rk.
3. Obtain the transformation function G
from the given pz(z)
4. Precompute zk for each value of sk
using the iterative scheme defined in
connection
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 then map level sk into the
final level zk.
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
HISTOGRAM MATCHING
Original
Histogram equalized
image
Using mappings from
curve
HISTOGRAM SUMMARY
• Useful for assessment of acquisition quality.
• Critical for image presentation
• Used as basis for subsequent analysis steps such as
thresholding etc’.
NEIGHBORHOOD OPERATIONS
• Neighborhood is defined as
any area bigger than the
single pixel.
Origin
x
• May be of any size or
shape.
• Most used rectangle are
around the central pixel
• Might be referred to as
filters, kernels, templates,
or windows.
(x, y)
Neighbourhood
y
Image f (x, y)
NEIGHBOURHOOD OPERATIONS
• For each pixel in the origin image, the outcome is
written on the same location at the target image.
Origin
Neighbourhood
y
x
(x, y)
Image f (x, y)
Target
SIMPLE NEIGHBOURHOOD
OPERATIONS
Simple neighbourhood operations example:
• Min: Set the pixel value to the minimum in the
neighbourhood
• Max: Set the pixel value to the maximum in the
neighbourhood
SIMPLE NEIGHBOURHOOD OPERATIONS
Max Filter
Radius = 3
Original
http://www.biologyimagelibrary.com/
Min Filter
Radius = 3
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THE SPATIAL FILTERING PROCESS
Origin
x
Simple 3*3
Neighbourhood
y
e
3*3 Filter
Image f (x, y)
a
b
c
d
e
f
g
h
i
Original
Image Pixels
*
j
k
l
m
n
o
p
q
r
Filter (w)
eprocessed = n*e +
j*a + k*b + l*c +
m*d + o*f +
p*g + q*h + r*i
Performed for each pixel in the original image
to generate the filtered image
SMOOTHING SPATIAL FILTERS
Simple spatial filter
Average all of the pixels in a neighbourhood around a
central value according to selected mask.
Especially useful
in removing noise
from images
Also useful for
highlighting gross
detail
1/
9
1/
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1/
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1/
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1/
9
1/
9
9
1/
9
1/
9
1/
SMOOTHING SPATIAL FILTERING
Origin
x
Simple 3*3
Neighbourhood
1/ 100
1/ 108
1/
104
9
9
9
199
1/
/9 106
9
1/
98
9
195
/9
1/
85
9
1/
90
9
3*3 Smoothing
Filter
104 100 108
1/
99 106 98
95
90
85
Original
Image Pixels
y
*
9
1/
9
1/
9
1/
9
1/
9
1/
9
1/
9
1/
9
1/
9
Filter
Image f (x, y)
e = 1/9*106 + 1/9*104 + 1/9*100 + 1/9*108 + 1/9*99 + 1/9*98 +
1/ *95 + 1/ *90 + 1/ *85 =
98.3333
9
9
9
MEAN FILTER
SMOOTHING
AVERAGING
FILTER
Original
Mean R=1
Mean R=2
Mean R=4
As R grows
Fine details and
noise vanish
WEIGHTED AVERAGE
• Created by defining different weight for different
pixels around the center pixel.
• Pixels closer to the center pixel
are contribute more to the average
• Size of objects in the image should
taken into account.
a
g ( x, y ) 
b
1/
2/
1/
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2/
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4/
16
2/
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1/
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16
2/
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1/
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  w(s, t ) f ( x  s, y  t )
s   at   b
a
b
  w(s, t )
s   at   b
General annotation for
weighted average filtering
an MxN image
Comparison of Mean and
Median
R=2
Mean Filter
Median Filter
GENERAL FILTERING REMARKS
• There is no such thing as GOOD FILTER
but a SUITABLE filter for a specific need / task.
• Spatial filters can and should be applied in
combination for optimal results.
• Spatial filters change the “numbers” of signal
intensities and might change the area of an object –
should be remembered in quantification
GO BACK TO THE ORIGINAL
SHARPENING SPATIAL FILTERS
“The principal objective of sharpening is to highlight fine
detail in an image or to enhance detail that has been
blurred, either in error or as a natural effect of a
particular method of image acquisition.”
•Sharpening filters are based on spatial differentiation
Averaging ≈ Integration
= Blurring
Gonzalez Woods Chapter 3
Sharpening ≈
spatial differentiation
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
SPATIAL DIFFERENTIATION
Differentiation measures the rate of change of a function.
For a simplified explanation we will use a 1 dimensional
example
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
SPATIAL DIFFERENTIATION
A
B
1ST DERIVATIVE
The formula for the 1st derivative of a function is as
follows:
f
 f ( x  1)  f ( x)
x
Calculates difference between subsequent values and
measures the rate of change of the function
1ST DERIVATIVE (CONT…)
f(x)
8
7
6
5
4
3
2
1
0
5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7
0 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0
8
6
4
f ’(x)
2
0
-2
-4
-6
-8
2ND DERIVATIVE
The formula for the 2nd derivative of a function is as
follows:
 f
 f ( x  1)  f ( x  1)  2 f ( x)
2
 x
2
Takes into account the values both before and after the
current value
1ST AND 2ND DERIVATIVE
8
6
f(x)
4
2
0
8
6
f’(x)
4
2
0
-2
-4
-6
-8
10
f’’(x)
5
0
-5
-10
-15
2ND DERIVATIVES FOR IMAGE
ENHANCEMENT
The 2nd derivative is more useful for image enhancement
 Stronger response to fine detail
 Simpler implementation
The approach
 Define a discrete formulation of the second-order derivative.
 Construct a filter mask based on that formulation.
We are interested in isotropic filters
 Independent of the direction of the discontinuities in the image
 Rotation invariant
Laplacian - the first sharpening filter we will look at
 One of the simplest sharpening filters
 Isotropic
 Linear
THE LAPLACIAN
•The Laplacian is defined as follows:
 f  f
 f  2  2
 x  y
2
2
2
Partial 1st order derivative in the x :
 f
 f ( x  1, y )  f ( x  1, y )  2 f ( x, y )
2
 x
2
In the y direction as follows:
 f
 f ( x, y  1)  f ( x, y  1)  2 f ( x, y )
2
 y
2
THE LAPLACIAN (CONT…)
The sum is given by:
 f  [ f ( x  1, y)  f ( x  1, y)
 f ( x, y  1)  f ( x, y  1)]  4 f ( x, y )
2
A filter built based on this
0
1
0
1
-4
1
0
1
0
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
THE LAPLACIAN (CONT…)
•Applying the Laplacian to an image we get a new
image that highlights edges and other discontinuities
Original
Image
Laplacian
Filtered Image
Laplacian
Filtered Image
Scaled for Display
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
LAPLACIAN IMAGE ENHANCEMENT
Original
Image
=
Laplacian
Filtered Image
Sharpened
Image
In the final sharpened image edges and fine detail are
much more obvious
g ( x, y)  f ( x, y)   f
2
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
LAPLACIAN IMAGE ENHANCEMENT
Original channel
Laplace filter
Subtraction result
http://www.biologyimagelibrary.com/Image - 21481_0_Miller_28_BIL27090
SHARPENING SUMMARY
• “A derivative operator is proportional to the degree of
discontinuity of the image at the point at which the
operator is applied.
• Thus, image differentiation enhances edges and other
discontinuities (such as noise) and deemphasizes
areas with slowly varying gray-level values”
Gonzalez Woods 2nd eddition Chapter 3
EDGE DETECTION AND FILTER
COMBINATIONS
Original
Gaussian blur
Laplace
Gradient
magnitude
X
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
COMBINING SPATIAL
ENHANCEMENT METHODS
• Applying a single spatial
operation is usually not enough
• A combination of a range of
techniques in will usually result
in a better final result.
• This example will focus on
enhancing the bone scan to the
right
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
COMBINING SPATIAL
ENHANCEMENT METHODS (CONT…)
(a)
Laplacian filter of
bone scan (a)
(b)
Sharpened version
of bone scan
(c)
achieved by
subtracting (a) and Sobel filter of bone
scan (a)
(b)
(d)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
COMBINING SPATIAL
ENHANCEMENT METHODS (CONT…)
The product of (c)
and (e) which will
be used as a mask
Result of applying
a power-law trans.
Sharpened image to (g)
which is sum of (a)
(g)
and (f)
(e)
Image (d) smoothed with a
5*5 averaging filter
(f)
(h)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
COMBINING SPATIAL
ENHANCEMENT METHODS (CONT…)
•Compare the original and final images