Chapter 3 Spatial Domain Image Processing.pdf

Getnet T. Email: getnet6202@gmail.com , College of Informatics , University of Gondar, July 2025
College of Informatics
Department of Computer Science
ComputerVision and Image Processing (CoSc4113)
ChapterThree: Spatial Domain Image Processing
University of Gondar

Spatial Domain Image Processing
Objectives
By the end of this lesson, you will be able to:
Explain the concept and significance of spatial domain image processing
1
2
3
4
Apply basic intensity transformation functions to enhance digital images
Implement spatial filtering techniques for image enhancement
Interpret image histograms to assess image properties
perform histogram processing to improve image contrast and detail
Combine multiple spatial enhancement methods for advanced image
processing
5
6

Spatial Processing of Digital Images
Basic Intensity Transformation Functions
Histogram of Images
Histogram Processing
Spatial Filtering
Combining Spatial Enhancement Methods
Contents
1
2
3
4
5
6

Spatial Processing of Digital Images
1
The Term Spatial domain refers to the image plane itself, and image
processing methods in this category are based on direct
manipulation of pixels in an image
Spatial Domain Technique operate directly on the pixel of an image
Two Principal categories of Spatial Processing are Intensity
transformation and Spatial Filtering
Spatial domain process on image can be described as
g(x,y)=T[f(x,y)]
Where f(x,y) is the input image, g(x,y) is the output image, T is
an operator
T operates on the neighbors of (x, y) (a square or rectangular sub-image centered at
(x,y) to yield the output g(x, y).

Spatial Processing of Digital Images Cont’d
1
The operator T can apply to a single image or to a set of images
such as performing the pixel-by-pixel sum of a sequence of
images for noise reduction
The point (x,y) shown is an arbitrary location in the image and
the small region shown containing the point is a neighborhood
of (x,y)
Spatial domain

1
The process shown in figure illustrates moving the origin of the
neighborhood from pixel to pixel and applying the operator T to
the pixels in the neighborhood to yield the output at the location
Thus, for any specific location (x,y), the value of the output image
g at those coordinates is equal to the result of applying T to the
neighborhood with origin at (x,y) in f.
3 x 3 neighborhood of (x,y)
A 3 x 3 neighborhood about a point (x,y)
in an image in the spatial domain. The
neighborhood is moved from pixel to pixel
in the image to generate an output image

1
The smallest neighborhood (mask) is of size 1 x 1 (pixel)
Here, T is called intensity transformation function or (mapping,
gray level function)
g(x,y) =T[f(x,y)]
S=T(r)
Where s,r, denote the intensity of g and f at any point (x,y)
S r

 For example, if T(r) has the form as shown in
figure, the effect of applying the
transformation to every pixel of f to generate
the corresponding pixels in g would be to
produce an image of higher contrast than the
original by darkening the intensity levels
below k and brightening the levels above k
 This technique, sometimes called as contrast
stretching, values of r lower than k are
compressed by the transformation function
into a narrow range of s, toward black.
 The opposite is true for values of r higher
than k
1

 Function shown in figure, T(r)
produces a two-level (binary) image.
 A mapping of this form is called a
thresholding function.
 Approaches whose results depend
only on the intensity at a point
sometimes are called point processing
techniques, as opposed to the
neighborhood processing techniques.
1

Basic IntensityTransformation Functions
2
 Intensity transformations the values of pixels, before and after processing,
will be denoted by r and s, respectively.
 Expression of the form s = T(r), is used where T is a transformation that
maps a pixel value r into a pixel value s.
 As dealing is in digital quantities, values of a transformation function
typically are stored in a one-dimensional array and the mappings from r to s
are implemented via table lookups.
 For an 8-bit environment, a lookup table containing the values of T will have
256 entries.
 There are three basic gray level transformation
 Linear (Identity and Negative Transformation)
 Logarithmic (log and log inverse)
 Power – law (nth power and nth root transformations)

Basic IntensityTransformation Functions
2
 In Identity transformation, each value of the input image is directly mapped
to each other value of output image. That results in the same input image
and output image
Identity Transformation

Basic IntensityTransformation Functions Cont’d
2
 The negative of an image with intensity
levels in the range [0, L - 1] is obtained by
using the negative transformation shown in
figure
 The expression is
s=L-1-r
 Reversing the intensity levels of an image in
this manner produces the equivalent of a
photographic negative
 each value of the input image is subtracted
from the L-1 and mapped onto the output
image
 Since the input image of Einstein is an 8bpp
image, so the number of levels in this image
are 256. Putting 256 in the equation, we get
s = 255 – r
Image Negatives:

2
 Such processing is suited for enhancing white or gray level details embedded in dark
regions of an image, especially when the black areas are dominant in size
a. Original digital mammogram b. Negative image obtained using the negative
transformation
 The original image is a digital mammogram showing a small lesion.
 In spite of the fact that the visual content is the same in both images, Analysis of
breast tissue in the negative image is much easier.
Image Negatives:

2
 The general form of the log transformation in
figure is
s=c log(1+r)
where ¢ is a constant, and it is assumed
that r > =0
 The shape of the log curve in figure shows
that this transformation maps a narrow range
of low intensity values in the input into a
wider range of output levels
 This transformation is used to expand the
value of dark pixels in an image compressing
the higher level-values
Log Transformations:

2
 The log function compresses the dynamic range of
images with large variations in pixel values.
 In Fourier spectrum pixel values have a large dynamic
range
 Fourier spectrum with values in the range 0 to 1.5 x106 is
shown in first figure
 When these values are scaled linearly for display in an 8-
bit system, the brightest pixels will dominate the display.
 The effect of this dominance is relatively small area of
the image is not perceived as black
 After applying s = ¢ log (1 + r) with ¢ = 1 to the spectrum
values, then the range of values of the result becomes 0
to 6.2, which is more manageable.
 Second figure shows the result of scaling this new range
linearly and displaying the spectrum in the same 8-bit
display.
Log Transformations:

2
 Power-law transformations have the basic form S
=c.r y
where ¢ and y are positive constants.
 Sometimes above equation is written as S = c(r+) y
for an offset
 Plots of s versus r for various values of are shown
in figure
 Here, possible transformation curves obtained
simply by varying y. Curves generated with values of
y > 1 have exactly the opposite effect as those
generated with values of y < 1
 A variety of devices used for image capture,
printing, and display respond according to a power
law.
 The exponent in the power-law equation is referred
to as gamma and the process used to correct these
power-law response phenomena is called gamma
correction
Power-Law (Gamma) Transformations:
Plots of the equation s =
¢.r7 for various values of y
(C = 1 for all cases)

Histogram of an Image
3

Histogram of an Image cont’d
3

Histogram Processing
4

Histogram Processing cont’d
4

4 Histogram Processing cont’d

Spatial Filtering
5
Remember that types of neighborhood:
 intensity transformation: neighborhood of size 1x1
 spatial filter (or mask ,kernel, template or window): neighborhood of
larger size , like 3*3 mask
 The spatial filter mask is moved from point to point in an
image. At each point (x,y), the response of the filter is
calculated
Origin x
y Image f (x, y)
(x, y)
Neighbourhood

Spatial Filtering cont’d
5
Neighbourhood Operations
 For each pixel in the origin image, the outcome is
written on the same location at the target image.
Origin x
y Image f (x, y)
(x, y)
Neighbourhood
Target

5
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

5
The Spatial Filtering Process
j k l
m n o
p q r
Origin x
y Image f (x, y)
eprocessed = n*e + j*a + k*b +
l*c + m*d + o*f + p*g + q*h +
r*i
Filter (w)
Simple 3*3
Neighbourhood
e 3*3 Filter
a b c
d e f
g h i
Original
Image
Pixels
*
The above is repeated for every pixel in the original image to
generate the filtered image

5
Spatial filters
1.Smoothing Spatial filters [low pass].
for Smoothing / Blurring
2. Sharpening Spatial Filters[high pass].
for Edge Detection / Sharpening

5
Spatial filters : Smoothing ( low pass)
Use: for blurring and noise reduction.
Type of smoothing filters:
1.Standard average
2. weighted average.
3. Median filter
linear
Order statistics

5
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
highlighting gross
detail
 All weights are equal
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
Simple
averaging
filter

5
Smoothing Spatial Filtering
Origin x
y Image f (x, y)
e = 1/9*106 + 1/9*104 + 1/9*100 +
1/9*108 + 1/9*99 + 1/9*98 + 1/9*95 +
1/9*90 + 1/9*85 = 98.3333
Filter
Simple 3*3
Neighbourhood
106
104
99
95
100108
98
90 85
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
3*3 Smoothing
Filter
104 100 108
99 106 98
95 90 85
Original
Image
Pixels
*
The above is repeated for every pixel in the original image to
generate the smoothed image

5
Spatial filters : Smoothing
linear smoothing : averaging kernels
Standard average

5
Standard and weighted Average- example
130
90
120
110
200
98
94
91
100
99
91
90
90
85
96
82
Standard averaging filter:
(110 +120+90+91+94+98+90+91+99)/9 =883/9 = 98.1
The mask is moved
from point to point in
an image. At each
point (x,y), the
response of the filter
is calculated

5
What happens when the Values of the Kernel Fall
Outside the Image??!

5
First solution :Zero padding,
-ve: black border

5
border padding

5
Averaging effects: blurring + reducing noise
Original image 3 x 3 averaging
5 x 5 averaging 9 x 9 averaging
15 x 15 averaging 35 x 35 averaging
Notice how detail
begins to disappear

5
linear smoothing : averaging kernels
weighted average.
Used to reduce blurring more.

5
Standard and weighted Average- example
130
90
120
110
200
98
94
91
100
99
91
90
90
85
96
82
:Weighted averaging filter:
(110 +2 x 120+90+2 x 91+4 x 94+2 x 98+90+2 x 91+99)/16 =
The mask is moved
from point to point in
an image. At each
point (x,y), the
response of the filter
is calculated

5
Spatial filters: Order-Statistic Filters :
130
90
120
110
200
98
94
91
100
99
95
90
90
85
96
82
Steps:
1. Sort the pixels in ascending order:
90,90, 91, 94, 95, 98, 99, 110, 120
2. replace the original pixel value by the
median :95
95
becomes
 These are non-linear spatial filters whose response is based on ordering
(increasing /decreasing) the pixel contained in the area encompassed by the filter
 Then replacing the value of the center with the middle value determined by
ranking result
 E.g. Median Filter, Max Filter, Min filter
I. Median Filter
 Popular with certain random
noise and impulse noise (salt
& paper noise)
 They provide excellent noise
reduction

5
order statistics: Median filter
use : blurring + reduce salt and pepper noise
The original
image with salt
and pepper noise
The smoothed
image using
averaging
The smoothed image
using median

5
Smoothing Filters: Median Filtering
(non-linear)
 Very effective for removing “salt and pepper” noise).
averaging
median
filtering

5
Spatial filters: Order-Statistic Filters :
ii. Max Filter
 It used to find the brightest points in an image
 Response of a 3 x 3 max filter is given by
R = 𝑀𝑎𝑥 𝑧𝑘 𝐾 = 1, 2, 3, 4, , ,, , 9
 E.g. Given a 3×3 neighborhood
 Then the center pixel (50) will be replaced with the maximum value, which is 90
iii. Min Filter
 Used to find the darkest points in an image

Combining Spatial Enhancing Methods
6
Reading Assignment

End of Spatial Domain Image Processing
Thank You

Chapter 3 Spatial Domain Image Processing.pdf

More Related Content

Similar to Chapter 3 Spatial Domain Image Processing.pdf (20)

More from Getnet Tigabie Askale -(GM) (20)

Recently uploaded (20)

Chapter 3 Spatial Domain Image Processing.pdf