Digital Image Processing

Chapter # 3

Image Enhancement in Spatial Domain

 Image Enhancement
Process an image to make the result more suitable than the
original image for a specific application
–Image enhancement is subjective (problem /application
oriented)

Image enhancement methods:

Spatial domain: Direct manipulation of pixel in an image (on
the image plane)
Frequency domain: Processing the image based on modifying the
Fourier transform of an image

Many techniques are based on various combinations of methods

from these two categories

2
Image Enhancement

3
 Basic Concepts

Spatial domain enhancement methods can be generalized as

g(x,y)=T[f(x,y)]
f(x,y): input image
g(x,y): processed (output) image
T[*]: an operator on f (or a set of input images),
defined over neighborhood of (x,y)

Neighborhood about (x,y): a square or rectangular

sub-image area centered at (x,y)

4
Basic Concepts

5
 Basic Concepts
g(x,y) = T [f(x,y)]
Pixel/point operation:
Neighborhood of size 1x1: g depends only on f at (x,y)
T: a gray-level/intensity transformation/mapping function
Let r = f(x,y) s = g(x,y)
r and s represent gray levels of f and g at (x,y)
Then s = T(r)
Local operations:
g depends on the predefined number of neighbors of f at (x,y)
Implemented by using mask processing or filtering
Masks (filters, windows, kernels, templates) :
a small (e.g. 3×3) 2-D array, in which the values of the
coefficients determine the nature of the process

6
 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

7
 Identity Function

• Output intensities are

identical to input
intensities.

• Is included in the
graph only for
completeness.

8
 Image Negatives
 Reverses the gray level order
 For L gray levels the transformation function is
s =T(r) = (L - 1) - r

9
Log Transformations
Function of s = cLog(1+r)

10
 Log Transformations

Properties of log transformations

–For lower amplitudes of input image the range of gray
levels is expanded
–For higher amplitudes of input image the range of gray
levels is compressed
Application:
– This transformation is suitable for the case when the
dynamic range of a processed image far exceeds the
capability of the display device (e.g. display of the
Fourier spectrum of an image)
– Also called “dynamic-range compression / expansion”
11
Log Transformations

12
 Inverse Log Transformations

• Do opposite to the Log Transformations

• Used to expand the values of high pixels in an
image while compressing the darker-level values.

13
Power-Law Transformation

14
 Power-Law Transformation
For γ < 1: Expands values of dark pixels,
compress values of brighter pixels
For γ > 1: Compresses values of dark pixels,
expand values of brighter pixels
If γ=1 & c=1: Identity transformation (s = r)

A variety of devices (image capture, printing, display) respond

according to power law and need to be corrected

Gamma (γ) correction

The process used to correct the power-law response
phenomena

15
Power-Law Transformation

16
 Gamma correction

• Cathode ray tube (CRT)

devices have an intensity-
Monitor
to-voltage response that is
a power function, with 
varying from 1.8 to 2.5
Gamma  = 2.5 • The picture will become
correction
darker.
• Gamma correction is done
by preprocessing the image
Monitor
before inputting it to the
monitor with s = cr1/

 =1/2.5 = 0.4
17
Power-Law Transformation: Example

18
Power-Law Transformation: Example

19
 Piecewise-Linear Transformation
Contrast Stretching
Goal:
– Increase the dynamic range of the gray levels for low
contrast images

Low-contrast images can result from

–poor illumination
–lack of dynamic range in the imaging sensor
–wrong setting of a lens aperture during image
acquisition

20
Contrast Stretching Example

21
Piecewise-Linear Transformation: 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

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 Piecewise-Linear Transformation: Bit Plane slicing

• Highlighting the contribution made to

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

23
8 bit planes

Bit-plane 7 Bit-plane 6
Bit- Bit- Bit-
plane 5 plane 4 plane 3
Bit- Bit- Bit-
plane 2 plane 1 plane 0

24
Histograms

25
 Example Histogram

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Example Histogram

27
Histogram Examples

28
Contrast Stretching through Histogram
If rmax and rmin are the maximum and minimum gray level
of the input image and L is the total gray levels of output
image The transformation function for contrast stretching
will be

29
Histogram Equalization

30
Histogram Equalization

31
Histogram Equalization

32
 Histogram Equalization
•Spreading out the frequencies in an image (or
equalising the image) is a simple way to improve
dark or washed out images
•The formula for histogram
equalisation is given where sk  T (rk )
– rk: input intensity k

– sk: processed intensity   pr ( r j )

j 1
– k: the intensity range
(e.g 0.0 – 1.0) k nj
– nj: the frequency of intensity j 
– n: the sum of all frequencies j 1 n
33
Histogram Equalization: Example

34
 Histogram Equalization: Example

Initial Image Image After Equalization

35
Notice that the minimum value (52) is now 0 and the maximum value (154) is now 255.
Histogram Equalization: Example

36
 Histogram Equalization: Example

No. of pixels
6
2 3 3 2 5

4 2 4 3 4

3
3 2 3 5
2
2 4 2 4
1
Gray level
4x4 image
0 1 2 3 4 5 6 7 8 9
Gray scale = [0,9]
histogram
37
 Histogram Equalization: Example
Gray
0 1 2 3 4 5 6 7 8 9
Level(j)
No. of
0 0 6 5 4 1 0 0 0 0
pixels
k

n
j 0
j 0 0 6 11 15 16 16 16 16 16

k nj 6 11 15 16 16 16 16 16
s 0 0 / / / / / / / /
j 0 n
16 16 16 16 16 16 16 16
3.3 6.1 8.4
sx9 0 0 9 9 9 9 9
3 6 8
38
 Histogram Equalization: Example

No. of pixels
6
3 6 6 3 5

8 3 8 6 4

3
6 3 6 9
2
3 8 3 8
1

Output image
0 1 2 3 4 5 6 7 8 9
Gray scale = [0,9] Gray level
Histogram equalization39
Mathematical/Logical Operations on Images
 Addition
–Averaging images for noise removal
 Subtraction
–Removal of background from images
–Image enhancement
–Image matching
–Moving/displaced object tracking
 Multiplication
–Superimposing of texture on an image
–Convolution and correlation of images
 And and or operations
–To remove the unnecessary area of an image through
mask operations

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Image Averaging for Noise Reduction

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Image Averaging for Noise Reduction

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Image Averaging for Noise Reduction

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 Image Subtraction
• Takes two images as input and produces a third
image whose pixel values are those of the first
image minus the corresponding pixel values from
the second image
• Variants
– It is also often possible to just use a single image as
input and subtract a constant value from all the pixels
– Just output the absolute difference between pixel
values, rather than the straightforward signed output.

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 Image Subtraction
• The subtraction of two images is performed in a
single pass

• If the operator computes absolute differences between

the two input images then:

• If it is simply desired to subtract a constant value C

from a single image then:

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 Image Subtraction
• If the operator calculates absolute differences, then it is
impossible for the output pixel values to be outside the
range
• In rest of the two cases the pixel value may become
negative
• This is one good reason for using absolute differences.

• How to solve problem of negative pixels?

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 Image Subtraction
• How to solve problem of negative pixels?
• Let we have an 8 bit Grayscale image (Value Range= 0 t0 255)
• The result of image subtraction may come in the range of -255 to
+255
• One scheme can be to add 255 to every pixel and then divide by
2
• Method is easy and fast
• Limitations
– Truncation errors can cause loss of accuracy
– Full range of display may not be utilized

06/13/21 47
 Image Subtraction

• How to solve problem of Negative Pixels?

• Another scheme can be
– first, find the minimum gray value of the subtracted image
– second, find the maximum gray value of the subtracted image
– set the minimum value to be zero and the maximum to be 255
– while the rest are adjusted according to the interval
[0, 255], by timing each value with 255/max

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 Examples of Image Subtraction

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 Examples of Image Subtraction

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Example: Background Removal Using Image Subtraction

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Example: Background Removal Using Image Subtraction

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 Examples of Image Multiplication

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 Logic Operations

• Logic operation performs on gray-level images, the pixel

values are processed as binary numbers

• Light represents a binary 1, and dark represents a binary 0

• NOT operation = negative transformation

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Example of Logical Operations using Masks

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 Neighbourhood Operations
• Neighbourhood operations simply operate on a
larger neighbourhood of pixels than point
operations
Origin x
• Neighbourhoods are
mostly a rectangle
around a central pixel
• Any size rectangle
(x, y)
and any shape filter Neighbourhood
are possible

06/13/21 y Image f (x,56y)

Local Enhancement through Spatial Filtering

 The output intensity value at (x,y) depends not only

on the input intensity value at (x,y) but also on the
specified number of neighboring intensity values
around (x,y)
 Spatial masks (also called window, filter, kernel,
template) are used and convolved over the entire
image for local enhancement (spatial filtering)
 The size of the mask determines the number of
neighboring pixels which influence the output
value at (x,y)
 The values (coefficients) of the mask determine the
nature and properties of enhancing technique

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 Simple Neighbourhood Operations

• Some simple neighbourhood operations include:

– Min: Set the pixel value to the minimum in the
neighbourhood
– Max: Set the pixel value to the maximum in the
neighbourhood
– Median: The median value of a set of numbers is the
midpoint value in that set (e.g. from the set [1, 7, 15,
18, 24] 15 is the median). Sometimes the median works
better than the average

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 Simple Neighbourhood Operations Example

Original Image x Enhanced Image x

123 127 128 119 115 130

140 145 148 153 167 172

133 154 183 192 194 191

194 199 207 210 198 195

164 170 175 162 173 151

y y

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 The Spatial Filtering Process
Origin x
a b c r s t
d
g
e
h
f
i
* u
x
v
y
w
z
Original Image Filter
Simple 3*3 Pixels
e 3*3 Filter
Neighbourhood
eprocessed = v*e +
r*a + s*b + t*c +
u*d + w*f +
y Image f (x, y) x*g + y*h + z*i

• The above is repeated for every pixel in the original image to

generate
06/13/21 the smoothed image 60
Local Enhancement through Spatial Filtering

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 Basics of Spatial Filtering

 Given the 3×3 mask with coefficients: w1, w2,…, w9

 The mask covers the pixels with gray levels: z1, z2,…, z9

 z gives the output intensity value for the processed image

(to be stored in a new array) at the location of z5 in the
input image
06/13/21 62
 Basics of Spatial Filtering
Mask operation near the image border
Problem arises when part of the mask is located outside the
image plane; to handle the problem:
1. Discard the problem pixels (e.g. 512x512input
510x510output, if mask size is 3x3)
2. Zero padding: expand the input image by padding zeros
(512x512input 514x514output)
– Zero padding is not good; creates artificial lines or edges on the
border
3. We normally use the gray levels of border pixels to fill up
the expanded region (for 3x3 mask). For larger masks a
border region equal to half of the mask size is mirrored on
the expanded region.

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Mask Operation Near the Image Border

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 Spatial Filtering for Smoothing
 For blurring/noise reduction;
Blurring is usually used in preprocessing steps,
e.g., to remove small details from an image prior to object
extraction, or to bridge small gaps in lines or curves
 Equivalent to Low-pass spatial filtering in frequency domain
because smaller (high frequency) details are removed based on
neighborhood averaging (averaging filters)
 Implementation: The simplest form of the spatial
filter for averaging is a square
mask (assume m×m mask) with
the same coefficients 1/m2 to
preserve the gray levels
(averaging).
 Applications: Reduce noise; smooth false contours
 Side effect: Edge blurring
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 Smoothing Filters

06/13/21 66
 Smoothing Spatial Filters
• One of the simplest spatial filtering operations we
can perform is a smoothing operation
– Simply average all of the pixels in a neighbourhood
around a central value
– Especially useful
in removing noise
from images
– Also useful for 1
/9 1
/9 1
/9
highlighting gross
detail
1
/9 1
/9 1
/9

1
/9 1
/9 1
/9
67
Simple Averaging Filter
 Smoothing Spatial Filtering
Origin x
104 100 108 1
/9 1
/9 1
/9

* /9 /9 /9
1 1 1
99 106 98

95 90 85 1
/9 1
/9 1
/9
1
/9 100
104 1
/9 108
1
/9
Original Image Filter
Simple 3*3 /9 106
1
99 1
/9 198
/9
3*3 Smoothing Pixels
Neighbourhood /9 190
1
95 /9 185
/9 Filter
e = 1/9*106 +
1
/9*104 + 1/9*100 + 1/9*108 +
1
/9*99 + 1/9*98 +
y Image f (x, y) 1
/9*95 + 1/9*90 + 1/9*85
= 98.3333
• The above is repeated for every pixel in the original image to generate the
smoothed image
06/13/21 68
Spatial Filtering for Smoothing (Example)

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Spatial Filtering for Smoothing (Example)

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 Weighted Smoothing Filters
• More effective smoothing filters can be
generated by allowing different pixels in the
neighbourhood different weights in the
averaging function
– Pixels closer to the 1
/16 2
/16 1
/16
central pixel are more
important 2
/16 4
/16 2
/16
– Often referred to as a
weighted averaging 1
/16 2
/16 1
/16
Weighted Averaging Filter

06/13/21 71
 Order-Statistics Filtering
 Nonlinear spatial filters
 Output is based on order of gray levels in the masked area
(sub-image)
 Examples: Median filtering, Max & Min filtering

Median filtering
 Assigns the mid value of all the gray levels in the mask to the
center of mask;
 Particularly effective when
– the noise pattern consists of strong, spiky components (impulse
noise, salt-and-pepper)
– edges are to be preserved
– Force points with distinct gray levels to be more like their neighbors

06/13/21 72
 Median Filtering

06/13/21 73
 Median Filtering (Example)

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Strange Things Happen At The Edges!
• At the edges of an image we are missing pixels
to form a neighbourhood
Origin x
e e

e e e Image f (x, y)
06/13/21 y 75
 Strange Things Happen At The Edges!
• There are a few approaches to dealing with
missing edge pixels:
– Pad the image
• Typically with either all white or all black pixels

– Replicate border pixels

– Truncate the image

06/13/21 76
 Simple Neighbourhood Operations
Example
x
123 127 128 119 115 130

140 145 148 153 167 172

133 154 183 192 194 191

194 199 207 210 198 195

164 170 175 162 173 151

y
06/13/21 77
 Correlation & Convolution
• The filtering we have been talking about so far
is referred to as correlation with the filter itself
referred to as the correlation kernel
• Convolution is a similar operation, with just
one subtle difference
a b c r s t eprocessed = v*e +
d
f
e
g
e
h
* u
x
v
y
w
z
z*a + y*b + x*c +
w*d + u*e +
t*f + s*g + r*h
Original Image Filter
Pixels
06/13/21 78
• For symmetric filters it makes no difference
Spatial Filtering for Image Sharpening

Background: to highlight fine detail in an image or to

enhance blurred detail
Applications: electronic printing, medical imaging, industrial

inspection, autonomous target detection (smart

weapons)......
Foundation (Blurring vs Sharpening):
 Blurring/smoothing is performed by spatial averaging
(equivalent to integration)
 Sharpening is performed by noting only the gray level
changes in the image that is the differentiation

06/13/21 79
Spatial Filtering for Image Sharpening

Operation of Image Differentiation

 Enhance edges and discontinuities (magnitude of
output gray level >>0)
 De-emphasize areas with slowly varying gray-level
values (output gray level: 0)

Mathematical Basis of Filtering for Image Sharpening

 First-order and second-order derivatives
 Gradients
 Implementation by mask filtering

06/13/21 80
 Derivatives
First Order Derivative
 A basic definition of the first-order derivative of a one-
dimensional function f(x) is the difference

f
 f ( x  1)  f ( x )
x

Second Order Derivative

 Ssimilarly, we define the second-order derivative of a
one-dimensional function f(x) is the difference

2 f
 f ( x  1)  f ( x  1)  2 f ( x )
x 2

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 First and Second Order Derivatives

06/13/21 82
 Example for Discrete Derivatives

06/13/21 83
 Comparison between f" and f´

 f´ generally produces thicker edges in an image

 f" has a stronger response to fine detail
 f´ generally has a stronger response to a gray-level step
 f" produces a double response at step changes in gray level
 For image enhancement, f" is generally better suited
than f´
 Major application of f´ is for edge extraction; f´ used
together with f" results in impressive enhancement effect

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 Laplacian for Image Enhancement

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 Laplacian for Image Enhancement

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 Laplacian for Image Enhancement

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Laplacian for Image Enhancement (Example)

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Laplacian for Image Enhancement (Example)

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Image Sharpening Based on Unsharp Masking

06/13/21 90
 High Boost Filtering

Principal application:
Boost filtering is used when input image is darker than desired,
high-boost filter makes the image lighter and more natural

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 High Boost Filtering Masks

06/13/21 92
 1st Derivative Filtering
• Implementing 1st derivative filters is difficult in practice

• For a function f (x, y) the gradient of f at coordinates (x, y) is

given as the column vector:

 f 
Gx   x 
f      f 
G y   
 y 
 1st Derivative Filtering (cont…)
• The magnitude of this vector is given by:

f  mag (f )

 G G2
x
2
y  1
2

1
 f   f  
2 2 2

      
 x   y  
• For practical reasons this can be simplified as:

f  G x  G y
 1st Derivative Filtering (cont…)
• Now we want to define digital approximations and their
Filter Masks
• For simplicity we use a 3x3 region
• For example z5 denotes f(x,y), z1 denotes f(x-1,y-1)
• A simple approximation for First Derivative is
z1 z2 z3

z4 z5 z6

z7 z8 z9
 1st Derivative Filtering (cont…)
A simple approximation for First Derivative is

Two other definitions proposed by Roberts use cross- difference

z1 z2 z3
If we use
z4 z5 z6

z7 z8 z9
 1st Derivative Filtering (cont…)

If we use absolute values then

z1 z2 z3

z4 z5 z6

z7 z8 z9
The Masks corresponding to these equations are:

Roberts Cross-Gradient Operators

 Gradient Operators

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 Gradient Operators
Normally the smallest mask used is of size 3 x 3
Based on the concept of approximating the gradient several
spatial masks have been proposed:

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 Gradient Operators

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 Gradient Processing (Example)

06/13/21 101
 NOTE


The summation of coefficients in all masks
equals 0, indicating that they would give a
response of 0 in an area of constant gray level.

06/13/21 102
Mask used to estimate the Gradient
 Combining Spatial Enhancement
Methods (cont…)

(a)
Laplacian filter of
bone scan (a)
(b)
Sharpened version of
bone scan achieved (c)
by subtracting (a)
and (b) Sobel filter of bone
scan (a) (d)
Combining Spatial Enhancement Methods
(cont…)
Result of applying a (h)
power-law trans. to
Sharpened image (g)
which is sum of (a)
and (f) (g)
The product of (c)
and (e) which will be (f)
used as a mask
(e)

Image (d) smoothed with

a 5*5 averaging filter
Combining Spatial Enhancement Methods (cont…)
• Compare the Original and Final Images