Point Operations and The Histogram

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Point OperationsandThe Histogram
Image Processing - Lesson 3
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Image Operations

• Pixel Operations
• Operation depends on Pixel's value.
• Context free.
• Operation can be performed on the Histogram.
• Example
• Geometric Operations
• Operation depend on Pixel's coordinates.
• Context free.
• Independent of pixels value.
• Example
• Spatial Operations
• Operation depends on Pixel's value and
  coordinates.
• Context dependent.
• Binary v.s. Gray-Scale images.
• Spatial v.s. Frequency domain.
• Example

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The Image Histogram

• Point operations depend on pixels value and can
  be performed on image histogram.
• Image Histogram
• Normalized Histogram
• where N is the total number of pixels in the
  image I.
• PI(k) defines the probability to get the value k
  in the image I.
• Accumulated Histogram
• A(k) is the prob. that a pixel has a value less
  or equal to k
• Note that A(k)-A(k-1)P(k)

HI(k) pixels with gray-level k
PI(k)H(k)/N
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Histogram
15000
10000
5000
0
1
2
3
4
5
6
7
8
9
10
gray level
Normalized Histogram
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
7
8
9
10
gray level
Accumulated Histogram
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0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
gray level
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The Image Histogram (Cont.)
PI(k)
1
k
PI(k)
1
0.5
k
PI(k)
0.1
k
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• Decreasing the image contrast
• Increasing the new image average

PI(k)
0.1
k
0.5
PI(k)
0.1
k
PI(k)
0.1
k
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Histogram properties

• The image mean
• The image s.t.d.

mean
std
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Uses of the Histogram

• Digitizing parameters
• Auto focus
• Enhancement
• Histogram equalization
• Histogram stretching
• Threshold selection
• Histogram matching

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Auto-Focus

• In some optical equipment (e.g. slide projectors)
  inappropriate lens position creates a blurred
  (out-of-focus) image.
• We would like to automatically adjust the lens.

Demo
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• Blurred image can be detected by its histogram
• Image mean is not affected by blurring.
• Image s.t.d. is decreased by blurring.
• Algorithm Adjust lens according the changes in
  the histogram s.t.d.

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Thresholding
knew
F(k)
255
kold
255
Threshold value
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Thresholding Value
Original Image
Binary Image
Threshold too high
Threshold too low
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Segmentation using Thresholding
Original
Histogram
50
75
Threshold 75
Threshold 50
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Original
Histogram
21
Threshold 21
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Point Operations

• A point operation can be defined as a function
• knewF(kold)
• F(k) takes any value kold in the source image
  into knew in the destination image.
• Example knewF(kold)kold10
• increasing brightness
• knewF(kold)round(kold0.8)
• decreasing contrastbrightness

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Point Operations

• Simplest case - Linear Mapping

knew
F(k)
q
p
kold
p
q
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• If it is required to map the full gray-level
  range (256 values) to its full range - A
  piecewise linear mapping is required

knew
F(k)
255
stretching
kold
255
contraction
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Clipping Function

• If most of the gray-levels in the image are in
  u1 u2, the following mapping increases the
  image contrast.
• What is F(k)?

knew
F(k)
255
kold
u1
u2
255
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knew
255
F(k)
kold
255
?
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Applying Point Operations on an image

• Given a point operation
• kbF(ka)
• F(ka) takes any value ka in image A into kb in
  image B.

kb
F(k)
ka
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Image Statistics (Histograms)
Ia
Ha
Aa
F(k)
Ib
Ab
Hb
Histogram of Ia differs from that of Ib.
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Is it possible to obtain Hb directly from Ha and
F(k)?
k
kb
F(k)
ka
Hb
Ha
k
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Is it possible to obtain Ab directly from Aa and
F(k)?

• Requirement F is an increasing function
• (F-1 exists).
• Since F(k) is monotonic, the area under Ha
  between 0 and ka is equal to the area under Hb
  between 0 and kb

k
kb
F(k)
ka
Hb
Ha
k
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The histogram of image B can be calculated
Ia
Ha
Aa
F(k)
Ib
Ab
Hb
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If F(k) is not increasing?

• Requirement F is a non-decreasing function
  (F-1 may/may not exists).

k
kb
F(k)
ka
Hb
Ha
k
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• Is it possible to obtain F(k) directly from Ha
  and Hb (Aa and Ab)?

Ia
Ha
Aa
F(k)
Ib
Ab
Hb
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Example Finding F(k) for Gray Level Separation
Visual discrimination between objects depends on
the their gray-level separation. Can we improve
discrimination AFTER image has been quantized?
Hard to discriminate
3 4
Doesnt help
23 24
This is better
23
3
??????
3 4
23 24
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Histogram Equalization

• For better visual discrimination we would like to
  re-assign gray-levels with maximal uniformity.
• Define a gray-level transformation
• such that
• The histogram Hb is as flat as possible.
• The order of Gray-levels is maintained.
• The histogram bars are not fragmented.

Hb
Ha
k
k
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Histogram Equalization
Ha
Hb
k
k
Aa
Ab
k
k
Define
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Histogram Equalization 2 Pointer Algorithm
Method Aa -gt Ab
Original
Goal
Aa
Ab
k
k
j Pnew
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Histogram Equalization 2 Pointer Algorithm
Method Ha -gt Hb
Original
Goal
Pnew
old
0 1 2 3 4 5 6 7 8 9 10 11 0 0 0 1 1 1 3 5 8 9 9
11
new
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25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
old
0 1 2 3 4 5 6 7 8 9 10 11 0 0 0 1 1 1 3 5 8 9 9
11
new
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Histogram Equalization - Example
Original
Equalized
Demo
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2 Pointer Algorithm
2-pointer Histogram Equalization Algorithm
Pold 0 Pnew 0 E_levelsum(Hist)/K
BucketE_level while (Pold ? k) if
Hist(Pold) lt Bucket Bucket
Bucket - Hist(Pold) New_Gray(Pold) Pnew
Pold else Hist(Pold)
Hist(Pold)-Bucket Bucket E_level Pnew
end end
K number of gray-level values. Hist A vector
with the original histogram. New_gray A vector
with the gray-level transformation.
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Histogram Matching

• Transforms an image A so that its histogram
  matches that of another image B.
• Could be used before comparing two images of the
  same scene acquired under different lighting
  condition.

Aa
Ab
D
D
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Application Texture Synthesis(Heeger Bergen
1995)