1. Chapter 3: Image Enhancement in the Spatial Domain
3.1 Background
3.2 Some basic intensity transformation functions
3.3 Histogram processing
3.4 Fundamentals of spatial filtering
3.5 Smoothing spatial filters
3.6 Sharpening spatial filters

2. Preview
Spatial domain processing: direct manipulation of pixels in an image
Two categories of spatial (domain) processing
Intensity transformation:
Operate on single pixels
Contrast manipulation, image thresholding
Spatial filtering
Work in a neighborhood of every pixel in an image
Image smoothing, image sharpening

3. 3.1 Background Spatial domain methods: operate directly on pixels
Spatial domain processing
g(x,y) = T[f(x, y)]
f(x, y): input image
g(x, y): output (processed) image
T: operator
Operator T is defined over a neighborhood of point (x, y)

4. 3.1 Background Spatial filtering
For any location (x,y), output image g(x,y) is equal to the result of applying T to the neighborhood of (x,y) in f
Filter: mask, kernel, template, window

5. 3.1 Background
The simplest form of T: g depends only on the value of f at (x, y)
T becomes intensity (gray-level) transformation function
s = T(r)
r: intensity of f(x,y)
s: intensity of g(x,y)
Point processing: enhancement at any point depends only on the gray level at that point

6. 3.1 Background Point processing (a) Contrast stretching
Values of r below k are compressed into a narrow range of s
(b) Thresholding

7. 3.2 Some Basic Intensity Transformation Functions
r: pixel value before processing
s: pixel value after processing
T: transformation
s = T(r)
3 types
Linear (identity and negative transformations)
Logarithmic (log and inverse-log transformations)
Power-law (nth power and nth root transformations)

8. 3.2 Some Basic Gray Level Transformations

9. 3.2.1 Image Negatives Negative of an image with gray level [0, L-1]
s = L – 1 – r
Enhancing white or gray detail embedded in dark regions of an image

10. 3.2.2 Log Transformations General form of log transformation
s = c log(1+r)
c: constant, r ≥ 0
This transformation maps a narrow range of low gray-level values in the input image into a wider range of output levels
Classical application of log transformation:
Display of Fourier spectrum

11. Log Transformations
(a) Original Fourier spectrum: 0 ~ 1,500,000 range
scaled to 0 ~ 255
(b) Result of log transformation: 0 ~ 6.2 range

12. 3.2.3 Power-Law Transformations

13. 3.2.3 Power-Law Transformations
Basic form of power-law transformations
s = c r γ
c, γ : positive constants
Gamma correction:
process of correcting this power-law response
Example: cathode ray tube (CRT)
Intensity to voltage response is power function with exponent (γ) 1.8 to 2.5
Solution: preprocess the input image by performing transformation
s = r1/2.5 = r0.4

14. 3.2.3 Power-Law Transformations
CRT monitor gamma correction example

15. 3.2.3 Power-Law Transformations
MRI gamma correction example
original
γ = 0.6
γ = 0.4
γ = 0.3

16. 3.2.3 Power-Law Transformations
Arial image gamma correction example
original
γ = 3.0
γ = 4.0
γ = 5.0

17. 3.2.4 Piecewise-Linear Transformation Functions
Contrast stretching
Low contrast image
Piecewise linear
function
Result of contrast stretching
Result of thresholding

18. 3.2.4 Piecewise-Linear Transformation Functions
Contrast stretching
(c) Contrast stretching
(r1, s1) = (rmin, 0), (r2, s2) = (rmax, L-1)
rmin, rmax: minimum, maxmum level of image
(d) Thresholding:
r1 = r2 = m, s1 = 0, s2 = L-1
m: mean gray level

19. 3.2.4 Piecewise-Linear Transformation Functions
Intensity-level slicing

20. 3.2.4 Piecewise-Linear Transformation Functions
Intensity-level slicing
Display a high value for all gray levels in the range of interest, and a low value for all other images
- produces binary image
(b) Brightens the desired range of gray levels
but preserves the background and other parts

21. 3.2.4 Piecewise-Linear Transformation Functions
Intensity-level slicing

22. 3.2.4 Piecewise-Linear Transformation Functions
Bit-plane slicing

23. 3.2.4 Piecewise-Linear Transformation Functions
Bit-plane slicing

24. 3.2.4 Piecewise-Linear Transformation Functions
Bit-plane slicing
Multiply bit plane 8 by 128
Multiply bit plane 7 by 64
Add the results of two planes

25. 3.3 Histogram Processing
The histogram of digital image with gray levels in the range [0, L-1] is a discrete function
h(rk) = nk
rk: kth gray level
nk: number of pixels in image having gray levels rk
Normalized histogram
p(rk) = nk/n
n: total number of pixels in image
n = MN (M: row dimension, N: column dimension)

26. 3.3 Histogram Processing Histogram horizonal axis:
rk (kth intensity value)
vertical axis:
nk: number of pixels, or
nk/n: normalized number

27. 3.3.1 Histogram Equalization
r: intensities of the image to be enhanced
r is in the range [0, L-1]
r = 0: black, r = L-1: white
s: processed gray levels for every pixel value r
s = T(r), 0 ≤ r ≤ L-1
Requirements of transformation function T
T(r) is a (strictly) monotonically increasing in the interval 0 ≤ r ≤ L-1
0 ≤ T(r) ≤ L-1 for 0 ≤ r ≤ L-1
Inverse transformation
r = T-1(s), 0 ≤s ≤L-1

28. 3.3.1 Histogram Equalization
Intensity transformation function

29. 3.3.1 Histogram Equalization
Intensity levels: random variable in interval [0, L-1]
probability density function (PDF)
cumulative distribution function (CDF)
Uniform probability density function

30. 3.3.1 Histogram Equalization

31. 3.3.1 Histogram Equalization
MN: total number of pixels in image
nk: number of pixels having gray level rk
L: total number of possible gray levels
histogram equalization (histogram linearization):
Processed image is obtained by mapping each pixel rk (input image)
into corresponding level sk (output image)

32. 3.3.1 Histogram Equalization

33. 3.3.1 Histogram Equalization
from dark image (1)
Histogram equalization
from light image (2)
Histogram equalization
from low contrast image (3)
Histogram equalization
from high contrast image (4)

34. 3.3.1 Histogram Equalization
Transformation functions

35. 3.3.3 Local Histogram Processing
Histogram processing methods in previous section are global
Global methods are suitable for overall enhancement
Histogram processing techniques are easily adapted to local enhancement
Example
(b) Global histogram equalization Considerable enhancement of noise
(c) Local histogram equalization using 7x7 neighborhood
Reveals (enhances) the small squares inside the dark squares
Contains finer noise texture

36. 3.3.3 Local Histogram Processing
Global histogram
equalized image
Local histogram
equalized image
Original image

37. 3.3.4 Use of Histogram Statistics for Image Enhancement
r: discrete random variable representing intensity values in the range [0, L-1]
p(ri): normalized histogram component corresponding to value ri
nth moment of r
mean (average) value of r
variance of r, б2(r)
sample mean
sample variance

38. 3.3.4 Use of Histogram Statistics for Image Enhancement
Global mean and variance are measured over entire image
Used for gross adjustment of overall intensity and contrast
Local mean and variance are measured locally
Used for local adjustment of local intensity and contrast
(x,y): coordinate of a pixel
Sxy: neighborhood (subimage), centered on (x,y)
r0, …, rL-1: L possible intensity values
pSxy: histogram of pixels in region Sxy
local mean
local variance

39. 3.3.4 Use of Histogram Statistics for Image Enhancement
A measure whether an area is relatively light or dark at (x,y)
Compare the local average gray level mSxy to the global mean mG
(x,y) is a candidate for enhancement if mSxy ≤ k0mG
Enhance areas that have low contrast
Compare the local standard deviation бSxy to the global standard
deviation бG
(x,y) is a candidate for enhancement if бSxy ≤ k2 бG
Restrict lowest values of contrast
(x,y) is a candidate for enhancement if k1 бG ≤ бSxy
Enhancement is processed simply multiplying the gray level
by a constant E

40. 3.3.4 Use of Histogram Statistics for Image Enhancement
Problem: enhance dark areas while leaving the light area
as unchanged as possible
E = 4.0, k0 = 0.4, k1 = 0.02, k2 = 0.4, Local region (neighborhood) size: 3x3

41. 3.4 Fundamentals of Spatial Filtering
Operations with the values of the image pixels in the neighborhood
and the corresponding values of subimage
Subimage: filter, mask, kernel, template, window
Values in the filter subimage: coefficient
Spatial filtering operations are performed directly on the pixels
of an image
One-to-one correspondence between
linear spatial filters and filters in the frequency domain

42. 3.4.1 Mechanics of Spatial Filtering
A spatial filter consists of
(1) a neighborhood (typically a small rectangle)
(2) a predefined operation
A processed (filtered) image is generated as the center of
the filter visits each pixel in the input image
Linear spatial filtering using 3x3 neighborhood
At any point (x,y), the response, g(x,y), of the filter
g(x,y) = w(-1,-1)f(x-1,y-1) + w(-1,0)f(x-1,y) + …
+ w(0,0)f(x,y) + …+ w(1,0)f(x+1,y) + w(1,1)f(x+1,y+1)

43. 3.4.1 Mechanics of Spatial Filtering

44. 3.4.1 Mechanics of Spatial Filtering
Filtering of an image f with a filter w of size m x n
a = (m-1) / 2, b = (n-1) / 2 or m = 2a+1, n = 2b+1
(a, b: positive integer)

45. 3.4.2 Spatial Correlation and Convolution

46. 3.4.2 Spatial Correlation and Convolution

47. 3.4.2 Spatial Correlation and Convolution
Correlation of a filter w(x,y) of size m x n with an image f(x,y)
Convolution of w(x,y) and f(x,y)

48. 3.4.3 Vector Representation of Linear Filtering
Linear spatial filtering by m x n filter
Linear spatial filtering by 3 x 3 filter

49. 3.4.3 Vector Representation of Linear Filtering
Spatial filtering at the border of an image
Limit the center of the mask no less than (n-1)/2 pixels
from the border -> Smaller filtered image
Padding -> Effect near the border
Adding rows and columns of 0’s
Replicating rows and columns

50. 3.4.4 Generating Spatial Filter Masks
Linear spatial filtering by 3 x 3 filter
Average value in 3 x 3 neighborhood
Gaussian function

51. 3.5 Smoothing Spatial Filters
Linear spatial filters for smoothing:
averaging filters, lowpass filters
Noise reduction
Undesirable side effect: blur edges
standard
average
weighted
average

52. 3.5.1 Smoothing Linear Filters
Standard averaging by 3x3 filter
Weighted averaging
Reduce blurring compared to standard averaging
General implementation for filtering with a weighted
averaging filter of size m x n (m=2a+1, n=2b+1)

53. 3.5.1 Smoothing Linear Filters
Result of smoothing with square averaging filter masks
Original
n=3
n=5
n=9
n=15
n=35

54. 3.5.1 Smoothing Linear Filters
Application example of spatial averaging
Original
15x15 averaging
Result of thresholding

55. 3.5.2 Order-Statistic (Nonlinear) Filters
Order-statistic filters are nonlinear spatial filters
whose response is based on ordering (ranking) the pixels
Median filter
Replaces the pixel value by the median of the gray levels
in the neighborhood of that pixel
Effective for impulse noise (salt-and-pepper noise)
Isolated clusters of pixels that are light or dark
with respect to their neighbors, and whose area is less
than n2/2, are eliminated by an n x n median filter
Median
3x3 neighborhood: 5th largest value
5x5 neighborhood: 13th largest value
Max filter: select maximum value in the neighborhood
Min filter: select minimum value in the neighborhood

56. 3.5.2 Order-Statistic Filters

57. 3.6 Sharpening Spatial Filters
Objective of sharpening:
highlight fine detail to enhance detail that has been blurred
Image blurring can be accomplished by digital averaging
Digital averaging is similar to spatial integration
Image sharpening can be done by digital differentiation
Digital differentiation is similar to spatial derivative
Image differentiation enhances edges and other discontinuities

58. Foundation
Image sharpening by first- and second-order derivatives
Derivatives are defined in terms of differences
Requirement of first derivative
Must be zero in flat areas
Must be nonzero at the onset (start) of step and ramp
Must be nonzero along ramps
Requirement of second derivative
Must be zero along ramps of constant slope

59. Foundation
first-order derivative
second-order derivative

60. Foundation

61. Foundation
At the ramp
First-order derivative is nonzero along the ramp
Second-order derivative is zero along the ramp
Second-order derivative is nonzero only at the onset and end
of the ramp
At the step
Both the first- and second-order derivatives are nonzero
Second-order derivative has a transition from positive to negative
(zero crossing)
Some conclusions
First-order derivatives generally produce thicker edges
Second-order derivatives have stronger response to fine detail
First-order derivatives generally produce stronger response
to gray-level step
Second-order derivatives produce a double response at step

62. 3.6.2 Use of Second Derivatives for Enhancement
Isotropic filters: rotation invariant
Simplest isotropic second-order derivative operator: Laplacian
2-D Laplacian operation
x-direction
y-direction

63. 3.6.2 Use of Second Derivatives for Enhancement
4 neighbors
negative center coefficient
8 neighbors
negative center coefficient
4 neighbors
positive center coefficient
8 neighbors
positive center coefficient

64. 3.6.2 Use of Second Derivatives for Enhancement
Image enhancement (sharpening) by Laplacian operation
Simplification

65. (original-Laplacian)
Use of Second Derivatives for Enhancement
Original image
Laplacian image
Enhanced image
(original-Laplacian)

66. 3.6.2 Use of Second Derivatives for Enhancement
Original image
Enhanced image
4 neighbors
Enhanced image
8 neighbors

67. 3.6.3 Unsharp Masking and Highboost Filtering
original image – blurred image
When k=1, unsharp masking
When k > 1, highboost filtering

68. 3.6.3 Unsharp Masking and Highboost Filtering

69. 3.6.3 Unsharp Masking and Highboost Filtering

70. 3.6.4 Using First-Order Derivatives for Image Sharpening
First derivatives in image processing is implemented
using the magnitude of the gradient
gradient of f at (x,y)
magnitude of gradient
Approximation of magnitude
of gradient by absolute values

71. 3.6.4 Using First-Order Derivatives for Image Sharpening
3x3 region
Roberts operators
Sobel operators

72. 3.6.4 Using First-Order Derivatives for Image Sharpening
Simplest approximation to first-order derivative
Roberts cross-gradient operators

73. 3.6.4 Using First-Order Derivatives for Image Sharpening
Sobel operators

74. 3.7.3 Use of First Derivatives for Enhancement
Optical image of
contact lens
Sobel gradient