1. Digital Image Processing
Chapter 3:
Image Enhancement in the
Spatial Domain
15 June 2007

2. The space where all pixels form an image
Spatial Domain
What is spatial domain
The space where all pixels form an image
In spatial domain we can represent an image by f(x,y)
where x and y are coordinates along x and y axis with
respect to an origin
There is duality between Spatial and Frequency Domains
Using the Fourier
transform, the word
“distance” is lost but the
word “frequency”
becomes alive.
Images in the spatial domain
are pictures in the xy plane
where the word “distance”
is meaningful.

3. Image Enhancement means improvement of images to be
suitable for specific applications.
Example:
Note: each image enhancement technique that is suitable for
one application may not be suitable for other applications.
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

4. Image Enhancement Example
Original image
Enhanced image using
Gamma correction
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

5. Image Enhancement in the Spatial Domain
= Image enhancement using processes performed in the
Spatial domain resulting in images in the Spatial domain.
We can written as
where f(x,y) is an original image, g(x,y) is an output and
T[ ] is a function defined in the area around (x,y)
Note: T[ ] may have one input as a pixel value at (x,y) only or
multiple inputs as pixels in neighbors of (x,y) depending in
each function.
Ex. Contrast enhancement uses a pixel value at (x,y) only
for an input while smoothing filte use several pixels around
(x,y) as inputs.

6. Types of Image Enhancement in the Spatial Domain
- Single pixel methods
- Gray level transformations
Example
- Historgram equalization
- Contrast stretching
- Arithmetic/logic operations
Examples
- Image subtraction
- Image averaging
- Multiple pixel methods
Spatial filtering
- Smoothing filters
- Sharpening filters

7. Gray Level Transformation
Transforms intensity of an original image into intensity of an
output image using a function:
where r = input intensity and s = output intensity
Example:
Contrast
enhancement
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

8. L = the number of gray levels
Image Negative
L-1
Original
digital
mammogram
White
Output intensity
Negative
digital
mammogram
Black
L-1
Black
Input intensity
White
L = the number of gray levels
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

9. Log Transformations Application Fourier spectrum Log Tr. of Fourier
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

10. Power-Law Transformations
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

11. Image Desired displayed image at Monitor Image After displayed at
Power-Law Transformations : Gamma Correction Application
Image
displayed at
Monitor
Desired
image
Image
displayed at
Monitor
After
Gamma
correction
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

12. Power-Law Transformations : Gamma Correction Application
MRI Image after
Gamma Correction
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

13. Power-Law Transformations : Gamma Correction Application
Ariel images
after Gamma
Correction
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

14. Contrast means the difference between the brightest and darkest
Contrast Stretching
Contrast means the difference
between the brightest and darkest
intensities
Before contrast
enhancement
After
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

15. How to know where the contrast is enhanced ?
Notice the slope of T(r)
if Slope > 1  Contrast increases
if Slope < 1  Contrast decrease
if Slope = 1  no change Ds Dr
Smaller Dr yields wider Ds
= increasing Contrast
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

16. Gray Level Slicing (Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

17. Bit-plane Slicing Bit 7 Bit 6 Bit 5 Bit 3 Bit 2 Bit 1
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

18. Histogram = Graph of population frequencies
Grades of the course 178 xxx

19. = graph of no. of pixels vs intensities
Histogram of an Image
= graph of no. of pixels vs intensities
Dark image has
histogram on the left
จำนวน pixel
Bright image has
histogram on the right
จำนวน pixel
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

20. Histogram of an Image (cont.)
low contrast image
has narrow histogram
high contrast image
has wide histogram
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

21. = intensity transformation based on histogram
Histogram Processing
= intensity transformation based on histogram
information to yield desired histogram
- Histogram equalization
To make histogram distributed uniformly
- Histogram matching
To make histogram as the desire

22. Monotonically Increasing Function
= Function that is only increasing or constant
Properties of Histogram
processing function
1. Monotonically increasing
function 2.

23. Probability Density Function
Histogram is analogous to Probability Density Function
(PDF) which represent density of population
Let s and r be Random variables with PDF ps(s) and pr(r ) respectively
and relation between s and r is
We get

24. Histogram Equalization
Let
We get !

25. Histogram Equalization
Formula in the previous slide is for a continuous PDF
For Histogram of Digital Image, we use
nj = the number of pixels with intensity = j
N = the number of total pixels

26. Histogram Equalization Example
Intensity
# pixels 20 1 5 2 25 3 10 4 15 6 7
Total
100
Accumulative Sum of Pr
20/100 = 0.2
(20+5)/100 = 0.25
( )/100 = 0.5
( )/100 = 0.6
( )/100 = 0.75
( )/100 = 0.8
( )/100 = 0.9
( )/100 = 1.0
1.0

27. Histogram Equalization Example (cont.)
Intensity
(r)
No. of Pixels
(nj)
Acc Sum of Pr
Output value
Quantized Output (s) 20
0.2
0.2x7 = 1.4 1 5
0.25
0.25*7 = 1.75 2 25
0.5
0.5*7 = 3.5 3 10
0.6
0.6*7 = 4.2 4 15
0.75
0.75*7 = 5.25
0.8
0.8*7 = 5.6 6
0.9
0.9*7 = 6.3 7
1.0
1.0x7 = 7
Total
100

28. Histogram Equalization
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

29. Histogram Equalization (cont.)
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

30. Histogram Equalization (cont.)
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

31. Histogram Equalization (cont.)
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

32. After histogram equalization, the image become a low contrast image
Histogram Equalization (cont.)
Original
image
After histogram equalization, the image
become a low contrast image
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

33. Histogram Matching : Algorithm
To transform image histogram to be a desired histogram
Concept : from Histogram equalization, we have
We get ps(s) = 1
We want an output image to have PDF pz(z)
Apply histogram equalization to pz(z), we get
We get pv(v) = 1
Since ps(s) = pv(v) = 1 therefore s and v are equivalent
Therefore, we can transform r to z by
T( )
G-1( ) r s z

34. Histogram Matching : Algorithm (cont.)
s = T(r)
v = G(z) 2 1 3
z = G-1(v) 4
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

35. Histogram Matching Example
Input image
histogram
Desired Histogram
Example
Intensity
( s )
# pixels 20 1 5 2 25 3 10 4 15 6 7
Total
100
Intensity ( z )
# pixels 5 1 10 2 15 3 20 4 6 7
Total
100
Original
data
User define

36. Histogram Matching Example (cont.)
1. Apply Histogram Equalization to both tables r
(nj)
SPr s 20
0.2 1 5
0.25 2 25
0.5 3 10
0.6 4 15
0.75
0.8 6
0.9 7
1.0 z
(nj)
SPz v 5
0.05 1 10
0.15 2 15
0.3 3 20
0.5 4
0.7
0.85 6
0.95 7
1.0
sk = T(rk)
vk = G(zk)

37. Histogram Matching Example (cont.)
2. Get a map
Actual Output
Histogram
We get
r  s
v  z r s 1 2 3 4 5 6 7
s  v v z 1 2 4 3 5 6 7 r z 1 2 3 4 5 6 7 z
# Pixels 1 20 2 30 3 10 4 15 5 6 7
sk = T(rk)
zk = G-1(vk)

38. Histogram Matching Example (cont.)
Desired histogram
Transfer function
Actual histogram
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

39. Histogram Matching Example (cont.)
After
histogram
equalization
After
histogram
matching
Original
image

40. Local Enhancement : Local Histogram Equalization
Concept: Perform histogram equalization in a small neighborhood
After Local Hist Eq.
In 7x7 neighborhood
Orignal image
After Hist Eq.
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

41. Local Enhancement : Histogram Statistic for Image Enhancement
We can use statistic parameters such as Mean, Variance of Local area
for image enhancement
Image of tungsten filament taken using
An electron microscope
In the lower right corner, there is a
filament in the background which is
very dark and we want this to be
brighter.
We cannot increase the brightness of
the whole image since the white
filament will be too bright.
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

42. Example: Local enhancement for this task
Local Variance
image
Multiplication
factor
Original image
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

43. Local Enhancement Output image
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

44. AND OR Logic Operations Application: Crop areas of interest Original
image
Image mask
Result Region
of Interest
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

45. Arithmetic Operation: Subtraction
Application: Error measurement
Error
image
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

46. Arithmetic Operation: Subtraction (cont.)
Application: Mask mode radiography in angiography work
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

47. Arithmetic Operation: Image Averaging
Application : Noise reduction
Degraded image
(noise)
Image averaging
Averaging results in reduction of
Noise variance
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

48. Arithmetic Operation: Image Averaging (cont.)
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

49. Basics of Spatial Filtering
Sometime we need to manipulate values obtained from
neighboring pixels
Example: How can we compute an average value of pixels
in a 3x3 region center at a pixel z?
Pixel z 2 4 1 2 6 2 9 2 3 4 4 4 7 2 9 7 6 7 5 2 3 6 1 5 7 4 2 5 1 2 2 5 2 3 2 8
Image

50. Basics of Spatial Filtering (cont.)
Step 1. Selected only needed pixels
Pixel z … 2 4 1 2 6 2 9 2 3 4 4 4 3 4 4 … … 7 2 9 7 6 7 9 7 6 5 2 3 6 1 5 3 6 1 7 4 2 5 1 2 … 2 5 2 3 2 8

51. Basics of Spatial Filtering (cont.)
Step 2. Multiply every pixel by 1/9 and then sum up the values 4 6 7 9 1 3 … X 1
Mask or
Window or
Template

52. Basics of Spatial Filtering (cont.)
Question: How to compute the 3x3 average values at every pixels?
Solution: Imagine that we have
a 3x3 window that can be placed
everywhere on the image 2 4 1 2 6 2 9 2 3 4 4 4 7 2 9 7 6 7 5 2 3 6 1 5 7 4 2 5 1 2
Masking Window

53. Basics of Spatial Filtering (cont.)
Step 1: Move the window to the first location where we want to
compute the average value and then select only pixels
inside the window.
Step 2: Compute
the average value 4 6 7 1 9 2 5 3 4 1 9 2 3 7
Sub image p
Step 3: Place the
result at the pixel
in the output image
Original image
4.3
Step 4: Move the
window to the next
location and go to Step 2
Output image

54. Example : moving averaging
Basics of Spatial Filtering (cont.)
The 3x3 averaging method is one example of the mask
operation or Spatial filtering.
w The mask operation has the corresponding mask (sometimes
called window or template).
w The mask contains coefficients to be multiplied with pixel
values.
Example : moving averaging
w(2,1)
w(3,1)
w(3,3)
w(2,2)
w(3,2)
w(1,1)
w(1,2) 1
Mask coefficients
The mask of the 3x3 moving average
filter has all coefficients = 1/9

55. Basics of Spatial Filtering (cont.)
The mask operation at each point is performed by:
1. Move the reference point (center) of mask to the
location to be computed
2. Compute sum of products between mask coefficients
and pixels in subimage under the mask.
Mask frame
p(2,1)
p(3,2)
p(2,2)
p(2,3)
p(3,3)
p(1,1)
p(1,3)
p(3,1) …
w(2,1)
w(3,1)
w(3,3)
w(2,2)
w(3,2)
w(1,1)
w(1,2)
Mask coefficients
Subimage
The reference point
of the mask

56. Basics of Spatial Filtering (cont.)
The spatial filtering on the whole image is given by:
Move the mask over the image at each location.
Compute sum of products between the mask coefficeints
and pixels inside subimage under the mask.
Store the results at the corresponding pixels of the
output image.
Move the mask to the next location and go to step 2
until all pixel locations have been used.

57. 3x3 moving average filter
Examples of Spatial Filtering Masks
Examples of the masks
Sobel operators
3x3 moving average filter 1 2 -1 -2 -2 -1 1 2 1
3x3 sharpening filter -1 8

58. Smoothing Linear Filter : Moving Average
Application : noise reduction
and image smoothing
Disadvantage: lose sharp details
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

59. Smoothing Linear Filter (cont.)
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

60. Order-Statistic Filters
Original image
subimage
Statistic parameters
Mean, Median, Mode,
Min, Max, Etc.
Moving
window
Output image

61. Order-Statistic Filters: Median Filter
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

62. Sharpening Spatial Filters
There are intensity discontinuities near object edges in an image
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

63. Laplacian Sharpening : How it works
Intensity profile
Edge
p(x)
1st derivative
2nd derivative

64. Laplacian Sharpening : How it works (cont.)
p(x)
Laplacian sharpening results in larger intensity discontinuity
near the edge.

65. Laplacian Sharpening : How it works (cont.)
Before sharpening
p(x)
After sharpening

66. Laplacian Masks
Used for estimating image Laplacian -1 8 -1 4
The center of the mask
is positive or 1 -8 1 -4
The center of the mask
is negative
Application: Enhance edge, line, point
Disadvantage: Enhance noise

67. Laplacian Sharpening Example
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

68. Laplacian Sharpening (cont.)
Mask for -1 5 -1 9 or
Mask for 1 -8 or 1 -4
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

69. Unsharp Masking and High-Boost Filtering-1
k+8 -1
k+4
Equation:
The center of the mask is negative
The center of the mask is positive

70. Unsharp Masking and High-Boost Filtering (cont.)
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

71. First Order Derivative
Intensity profile 20 40 60 80
100
120
140
160
180
200
0.5 1 50
150
-0.2
0.2
0.1
p(x)
Edges
1st derivative
2nd derivative

72. First Order Partial Derivative:
Sobel operators 1 2 -1 -2 -2 -1 1 2 P

73. First Order Partial Derivative: Image Gradient
Gradient magnitude
A gradient image emphasizes edges
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.

74. First Order Partial Derivative: Image Gradient

75. Image Enhancement in the Spatial Domain : Mix things up !B
smooth A - +
Sharpening C
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition. E

76. Image Enhancement in the Spatial Domain : Mix things up !
Power
Law Tr. G C E S F A
Multiplication
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition. H