1. Digital Image Processing
Lecture 2
Intensity Transformations and Spatial Filtering
Second Semester
Azad University
Islamshar Branch

2. Lecture Notes
Basic Relationships Between Pixels and Introduction to the Mathematical Tools
Some Basic Intensity Transformation function
Histogram Processing
Fundamentals of Spatial Filtering
Smoothing Spatial Filtering
Sharpening Spatial Filtering
Combining Spatial Enhancement Methods
Using Fuzzy Techniques for Intensity Transformations

3. Basic Relationships Between Pixels
1. Neighbors of pixel
4-neighbors of p, denoted by N4(p):
4 diagonal neighbors of p, denoted by ND(p):
8 neighbors of p, denoted N8(p)

4. 4-neighbors of pixel 4-neighbors of pixel is denoted by N4(p)
It is set of horizontal and vertical neighbors
(x-1)
(x)
(x+1)
f(x, y) is a red circle
f(x,y-1) is top one
f(x-1,y) is left one
f(x+1,y) is right one
f(x,y+1) is bottom one
(y-1)
(y)
(y+1)

5. 4 diagonal neighbors of pixel
Diagonal neighbors of pixel is denoted by ND(p)
It is set of diagonal neighbors
f(x, y) , is a yellow circle
f(x-1,y-1) is top-left one
f(x+1,y-1) is top-right one
f(x-1,y+1) is bottom-left one
f(x+1,y+1)is bottom-right one
(x-1)
(x)
(x+1)
(y-1)
(y)
(y+1)

6. 8-neighbors of pixel f(x,y) is a yellow circle
8-neighbors of pixel is denoted by N8(p)
4-neighbors and Diagonal neighbors of pixel
(x-1)
(x)
(x+1)
f(x,y) is a yellow circle
(x-1,y-1), (x,y-1),(x+1,y-1),
(x-1,y), (x,y), (x+1,y),
(x-1,y+1),(x,y+1), (x+1,y+1)
(y-1)
(y)
(y+1)

7. Basic Relationships Between Pixels 2. Connectivity
Let V is the set of intensity values used to define connectivity
There are three type of connectivity:
Connectivity : 2 pixels (p and q) with value in V are 4-connectivity if q is in the set N4(p)

8. Basic Relationships Between Pixels
2. Connectivity
2. 8-Connectivity : 2 pixels (p and q) with value in V are 8-connectivity if q is in the set N8(p)
3. m-Connectivity : 2 pixels (p and q) with value in V are m-connectivity if
q is in N4(p), or
q is in ND(p) and the set N4(p) ∩ N4(q) has no pixels whose values are from V.

9. Basic Relationships Between Pixels
3.Path
A (digital) path (or curve) from pixel p with coordinates (x0, y0) to pixel q with coordinates (xn, yn) is a sequence of distinct pixels with coordinates
(x0, y0), (x1, y1), …, (xn, yn)
Where (xi, yi) and (xi-1, yi-1) are connected for 1 ≤ i ≤ n.
Here n is the length of the path.
If (x0, y0) = (xn, yn), the path is closed path.
We can define 4-, 8-, and m-paths based on the type of connectivity used.

10. Examples: Adjacency and Path
V = {1, 2}
8-connectivity
m-connectivity

11. Examples: Adjacency and Path
V = {1, 2}

12. Basic Relationships Between Pixels
4. Connected in S
Let S represent a subset of pixels in an image. Two pixels p with coordinates (x0, y0) and q with coordinates (xn, yn) are said to be connected in S if there exists a path
(x0, y0), (x1, y1), …, (xn, yn)
Where:

13. Basic Relationships Between Pixels
Let S represent a subset of pixels in an image
For every pixel p in S, the set of pixels in S that are connected to p is called a connected component of S.
If S has only one connected component, then S is called Connected Set.
We call R a region of the image if R is a connected set
Two regions, Ri and Rj are said to be adjacent if their union forms a connected set.
Regions that are not to be adjacent are said to be disjoint.

14. Basic Relationships Between Pixels
Boundary (or border)
The boundary of the region R is the set of pixels in the region that have one or more neighbors that are not in R.
If R happens to be an entire image, then its boundary is defined as the set of pixels in the first and last rows and columns of the image.
Foreground and background
An image contains K disjoint regions, 𝑅 𝑘 , k = 1, 2, …, K. Let 𝑅 𝑢 denote the union of all the K regions, and let 𝑅 𝑢 𝑐 denote its complement.
All the points in 𝑅 𝑢 is called foreground;
All the points in 𝑅 𝑢 𝑐 is called background.

15. Basic Relationships Between Pixels
Distance Measures
Given pixels p, q and z with coordinates (x, y), (s, t), (u, v) respectively, the distance function D has following properties:
D(p, q) ≥ [D(p, q) = 0, iff p = q]
D(p, q) = D(q, p)
D(p, z) ≤ D(p, q) + D(q, z)

16. Basic Relationships Between Pixels
Distance Measures
The following are the different Distance measures:
a. Euclidean Distance :
De(p, q) = [(x-s)2 + (y-t)2]1/2
b. City Block Distance:
D4(p, q) = |x-s| + |y-t|
c. Chess Board Distance:
D8(p, q) = max(|x-s|, |y-t|)

17. Introduction to Mathematical Tools in DIP
Array vs. Matrix Operation
Array product
Matrix product

18. Introduction to Mathematical Tools in DIP
Linear vs. Nonlinear Operation
H is said to be a linear operator;
H is said to be a nonlinear operator if it does not meet the above qualification.

19. Introduction to Mathematical Tools in DIP
Arithmetic Operations
Arithmetic operations between images are array operations. The four arithmetic operations are denoted as
s(x,y) = f(x,y) + g(x,y)
d(x,y) = f(x,y) – g(x,y)
p(x,y) = f(x,y) × g(x,y)
v(x,y) = f(x,y) ÷ g(x,y)

20. Example: Arithmetic Operations (Subtraction)

21. Example : Image Multiplication

22. Spatial Operation
1. Single-pixel operations
Alter the values of an image’s pixels based on the intensity (For example Negative of an image).

23. Spatial Operation
2. Neighborhood operations

24. Introduction to Mathematical Tools in DIP
Image Transform
Given T(u, v), the original image f(x, y) can be recoverd using the inverse tranformation of T(u, v).

25. Image Transform

26. Example: Image Denoising by Using FFT Transform

27. Intensity Transformation
and
Spatial Filtering

28. Spatial Domain Process

29. 1. Intensity Transformation Function (Point Processing)

30. Example 1 : Image Negatives

31. Example 1 : Image Negatives

32. Example 2 : Log Transformations

33. Example 2 : Log Transformations
Log function has an important characteristic in which compress a dynamic range of images (Fourier Spectrum)

34. Example 3 : Power-Law (Gamma) Transformations
If γ <1 then the function maps a narrow range of dark input values into a wider range of output.
If γ>1 then the function has opposite effect.

35. Example 3 - 1 : Gamma Correction (γ <1 )

36. Example 3 - 2 : Gamma Correction (γ <1 )

37. Example 3 - 3 : Gamma Correction (γ >1 )

38. Intensity-level Slicing
1. Intensity Transformation Function (Piecewise-Linear Transformations)
Contrast Stretching
— is a process that expand the range of intensity levels in an image so that it spans the full intensity range of the recording medium or display device.
Intensity-level Slicing
— Highlighting a specific range of intensities in an image often is of interest.

39. Contrast Stretching

40. Intensity-level Slicing
Can be implemented:
By displaying in one value (white) all the values in the range of interest an in another in another (black) all other intensity.
By transformation the brightness (darkness) the desired range of intensity and unchanged all other intensity level.

41. Example: Intensity-level Slicing
Highlight the major blood vessels and study the shape of the flow of the contrast medium (to detect blockages, etc.)

42. Histogram Processing Histogram Equalization Histogram Matching
Local Histogram Processing
Using Histogram Statistics for Image Enhancement

43. Histogram Processing

44. Some Typical Histograms
· The shape of a histogram provides useful information for
contrast enhancement.
Dark image

45. Some Typical Histograms
Bright image
Low contrast image

46. Some Typical Histograms
High contrast image

47. Histogram Equalization
What is the histogram equalization?
Basic idea: find a map such that the histogram of the modified (equalized) image is flat (uniform).
Let us assume for the moment that the input image to be
enhanced has continuous gray values, with r = 0 representing
black and r = 1 representing white.
We need to design a gray value transformation s = T(r), based on the histogram of the input image, which will enhance the image.

48. Histogram Equalization
T(r) assume to be :
1) a monotonically increasing function for 0 £ r £ 1 (preserves
order from black to white).
2) maps [0,1] into [0,1] (preserves the range of allowed
Gray values).

49. Histogram Equalization
Key motivation: cumulative probability function (CDF) of a random variable approximates a uniform distribution
We consider the gray values in the input image and output
image as random variables in the interval [0, 1].
Let 𝑝 𝑖𝑛 (𝑟) and 𝑝 𝑜𝑢𝑡 (𝑠) denote the probability density of the
gray values in the input and output images.
We consider the transformation:
Thus, using a transformation function equal to the CDF of input gray values r, we can obtain an image with uniform gray values.

50. Histogram Equalization
How to implement histogram equalization?
Step 1: For images with discrete gray values, compute:
𝐿 : Total number of gray levels
𝑛 𝑘 : Number of pixels with gray value 𝑟 𝑘
𝑛 : Total number of pixels in the image
Step 2: Based on CDF, compute the discrete version of the previous
transformation :

51. Histogram Equalization
Example 1:
Consider an 8-level 64 x 64 image with gray values (0, 1, …,7). The normalized gray values are (0, 1/7, 2/7, …, 1). The normalized histogram is given below:

52. Histogram Equalization
Example 1:
Normalized gray value
Fraction of # pixels
Gray value
# pixels

53. Histogram Equalization
Example 1:

54. Histogram Equalization
Example 1:
· Notice that there are only five distinct gray levels --- (1/7, 3/7, 5/7, 6/7, 1)
in the output image. We will relabelethem as ( 𝑠 0 , 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ).
· With this transformation, the output image will have histogram

55. Histogram Equalization
Example 1:
Histogram of output image
# pixels
Gray values

56. Histogram Equalization
Example 2:
before
after

57. Is histogram equalization always good?No

58. Histogram Matching (histogram specification)
Generate a processed image that has a specified histogram

59. Histogram Matching (histogram specification)

60. Histogram Matching (Procedure)
Obtain pr(r) from the input image and then obtain the values of s (scaled histogram-equalized)
Use the specified PDF and obtain the transformation function G(z)
Mapping from s to z

61. Histogram Matching (example)
Assuming continuous intensity values, suppose that an image has the intensity PDF
Find the transformation function that will produce an image whose intensity PDF is

62. Histogram Matching (example)
Find the histogram equalization transformation for the input image
Find the histogram equalization transformation for the specified histogram
The transformation function

63. Histogram Matching (Discrete Cases)
Obtain pr(rj) from the input image and then obtain the values of sk, round the value to the integer range [0, L-1].
Use the specified PDF and obtain the transformation function G(zq), round the value to the integer range [0, L-1].
Mapping from sk to zq

64. Example: Histogram Matching
Suppose that a 3-bit image (L=8) of size 64 × 64 pixels (MN = 4096) has the intensity distribution shown in the following table (on the left). Get the histogram transformation function and make the output image with the specified histogram, listed in the table on the right.

65. Example: Histogram Matching
Obtain the scaled histogram-equalized values,
Compute all the values of the transformation function G,

66. Example: Histogram Matching

67. Example: Histogram Matching

68. Example: Histogram Matching

69. Example: Histogram Matching

70. Example: Histogram Matching

71. Example: Histogram Matching

72. Local Histogram Processing
Define a neighborhood and move its center from pixel to
pixel
At each location, the histogram of the points in the
neighborhood is computed. Either histogram equalization or
histogram specification transformation function is obtained
Map the intensity of the pixel centered in the neighborhood
Move to the next location and repeat the procedure

73. Local Histogram Processing

74. Using Histogram Statistics for Image Enhancement

75. Using Histogram Statistics for Image Enhancement
Average Intensity
Variance

76. Using Histogram Statistics for Image Enhancement(Example)

77. Spatial Filtering Introduction
Some neighborhood operations work with the values of the
image pixels in the neighborhood and the corresponding values
of a sub-image that has the same dimensions as the neighborhood.
The sub-image is called a filter, mask, kernel, template, or
window. The values in a filter sub-image are referred to as
coefficients rather than pixels.
These neighborhood operations consists of:
Defining a center point, (x,y)
2. Performing an operation that involves only the pixels in a
predefined neighborhood about that center point.
3. Letting the result of that operation be the “response” of the
process at that point.
4. Repeating the process for every point in the image.

78. Spatial Filtering Introduction
The process of moving the center point creates new
neighborhoods, one for each pixel in the input image. The two
principal terms used to identify this operation are
neighborhood processing or spatial filtering, with the second
term being more common.
If the computations performed on the pixels of the
neighborhoods are linear, the operation is called linear spatial
filtering; otherwise it is called nonlinear spatial filtering.

79. Spatial Filtering Introduction
The mechanic of linear spatial filtering
R = w (-1,-1) f(x-1, y-1) + w(-1,0) f (x-1, y) + … + w(0,0) f(0,0) + … + w(1,0) f(x+1, y) + w(1,1) f(x+1, y+1)

80. Spatial Filtering Introduction
There are two closely related concepts that must be understood
clearly when performing linear spatial filtering. One is
correlation; the other is convolution.
Correlation is the process of passing the mask w by the image
array f.
Mechanically, convolution is the same process, except that w is
rotated by 180 degrees prior to passing it by f.

81. Spatial Filtering
Introduction

82. Spatial Filtering
Introduction (correlation Vs. convolution)

83. Spatial Filtering Smoothing Spatial Filters
Smoothing filters are used for blurring and for noise
reduction.
Blurring is used in removal of small details and bridging
of small gaps in lines or curves.
Smoothing spatial filters include linear filters and
nonlinear filters.
The output (response) of a smoothing and linear spatial
filter is simply the average of the pixels contained in the
neighborhood of the filter mask

84. Spatial Filtering
Smoothing Spatial Filters

85. Spatial Filtering
Smoothing Spatial Filters

86. Spatial Filtering
Smoothing Spatial Filters (Example)

87. Spatial Filtering Smoothing Spatial Filters (Example)
Using a spatial averaging filter blends small objects with the background and larger objects become easy to detect .

88. Spatial Filtering Order-statistic (Nonlinear) Filters — Nonlinear
— Based on ordering (ranking) the pixels contained in the
filter mask
— Replacing the value of the center pixel with the value
determined by the ranking result
— E.g., median filter, max filter, min filter
— Median filter provides excellent noise reduction capabilities with
considerably less blurring than linear smoothing filter of similar
size.

89. Spatial Filtering
Order-statistic (Nonlinear) Filters (Example)

90. Spatial Filtering Sharpening Spatial Filters
The principal objective of sharpening is to highlight transitions in
intensity.
Sharpening filter is based on first and second derivatives
First derivative:
1) Must be zero in flat segment
2) Must be nonzero along ramps.
3) Must be nonzero at the onset
of a gray-level step or ramp
Second derivative:
2) Must be zero along ramps.
and end of a gray-level step or ramp

91. Spatial Filtering
Sharpening Spatial Filters

92. Spatial Filtering Sharpening Spatial Filters
Comparing the response between first- and second-ordered
derivatives:
First-order derivative produce thicker edge
2) Second-order derivative have a stronger response to fine detail,
such as thin lines and isolated points.
3) First-order derivatives generally have a stronger response to a
gray-level step.
4) Second-order derivatives produce a double response at step
changes in gray level.
In general the second derivative is better than the first derivative for image enhancement. The principle use of first derivative is for edge extraction.

93. Spatial Filtering Sharpening Spatial Filters (Laplace Operator)
The second-order isotropic derivative operator is the Laplacian for a function (image) f(x,y)

94. Spatial Filtering
Sharpening Spatial Filters (Laplace Operator)

95. Spatial Filtering Sharpening Spatial Filters (Laplace Operator)
The basic way to sharpening an image using the Laplacian is as the following:

96. Spatial Filtering
Sharpening Spatial Filters (Laplace Operator)

97. Spatial Filtering Un-sharp Masking and High-boost Filtering
A process for sharpening images that consists of
subtracting an un-sharp (smoothed) version of an image
from the original image.
Consists of the following steps:
1. Blur the original image
2. Subtract the blurred image from the original
3. Add the mask to the original

98. Spatial Filtering
Un-sharp Masking and High-boost Filtering

99. Spatial Filtering
Un-sharp Masking and High-boost Filtering

100. Spatial Filtering
Un-sharp Masking and High-boost Filtering

101. Spatial Filtering

102. Homework 2 :
Take a grey level image and find the histogram of the image and plot the histogram.
Calculate the mean, variance and histogram of your image. Show the histogram as a bar plot next to your image (use the subplot command). Indicate the size of your image.
A 4x4 image is given as follow.
The image is transformed using the point transform shown. Find the pixel
values of the output image.

103. Homework 3 :
1- a) Equalize the histogram of the 8 × 8 image below. The image has
grey levels 0, 1, , 7.
1-b) Show the original image and the histogram – equalized image
2- a) Perform histogram equalization given the following histogram.
( r=Gray level, n=number of occurrences)

104. Homework 3 :
2 – b) Perform histogram specification of the previous histogram using
the specified histogram shown in the following table. (r=Gray
level, p=probability of occurrences)