Image (and Video) Coding and Processing Lecture 5: Point Operations

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Image (and Video) Coding and ProcessingLecture
5 Point Operations

• Wade Trappe

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Lecture Overview

• Todays lecture will focus on
• Point-Operations These are operations that
  basically do not involve any explicit spatial
  memory
• Examples
• Contrast stretching
• Noise Clipping
• Histogram Equalization
• Note Most of this talk is borrowed from a
  lecture by my colleague Min Wu at UMD

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Point Operations / Intensity Transform

• Basic idea
• Zero memory operation
• each output only depend on the input intensity
  at the point
• Map a given gray or color level u to a new level
  v, i.e. v f ( u )
• Doesnt bring in new info.
• But can improve visual appearance or make
  features easier to detect
• Example-1 Color coordinate transformations
• RGB of each pixel ? luminance chrominance
  components ? etc.
• Example-2 Scalar quantization
• quantize pixel luminance/color with fewer bits

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Gamma Characteristics Gamma Correction

• Non-linearity in CRT display
• Voltage U vs. Displayed luminance L
• L U ? where ? 2.0 2.5
• Use preprocessing to compensate ?-distortion
• U L 1/ ?
• log(L) gives similar compensation curve to
  ?-correction
• good for many practical applications
• Camera may have L1/?c capturing distortion with
  ?c 1.0-1.7
• Power-law transformations are also useful for
  general purpose contrast manipulation

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Typical Types of Gray-level Transformation
Figure is from slides at Gonzalez/ Woods DIP book
website (Chapter 3)
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Example Negative Transformation
Figure is from slides at Gonzalez/ Woods DIP book
website (Chapter 3)
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Example Log Transformation
Figure is from slides at Gonzalez/ Woods DIP book
website (Chapter 3)
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Example Effects of Different Gammas
L0
( vectors sample image from Matlab )
L01/2.2
L02.2
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Luminance Histogram

• Represents the relative frequency of occurrence
  of the various gray levels in the image
• For each gray level, count the of pixels having
  that level
• Can group nearby levels to form a big bin count
  pixels in it

( From Matlab Image Toolbox Guide Fig.10-4 )
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Luminance Histogram (contd)

• Interpretation
• Treat pixel values as i.i.d random variables
• Histogram is an estimate of the probability
  distribution of the r.v.
• Unbalanced histogram doesnt fully utilize the
  dynamic range
• Low contrast image histogram concentrating in a
  narrow luminance range
• Under-exposed image histogram concentrating on
  the dark side
• Over-exposed image histogram concentrating on
  the bright side
• Balanced histogram gives more pleasant look and
  reveals rich content

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Example Balanced and Unbalanced Histograms
Figure is from slides at Gonzalez/ Woods DIP book
website (Chapter 3)
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Image with Unbalanced Histogram
( From Matlab Image Toolbox Guide Fig.10-10
10-11 )
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Contrast Stretching for Low-Contrast Images

• Stretch the over-concentrated graylevels in
  histogram via a nonlinear mapping
• Piece-wise linear stretching function
• Assign slopes of the stretching region to be
  greater than 1

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Contrast Stretching Example
original
stretched
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Clipping Thresholding

• Clipping
• Special case of contrast stretching with ? ?
  0
• Useful for noise reduction when interested signal
  mostly lie in range a,b
• Thresholding
• Special case of clipping with a b T
• Useful for binarization of scanned binary images
• documents, signatures, fingerprints

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Examples of Histogram Equalization
( From Matlab Image Toolbox Guide Fig.10-10
10-11 )
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Equalization Example (contd)
original
equalized
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Histogram Equalization

• Goal Map the luminance of each pixel to a new
  value such that the output image has
  approximately uniform distribution of gray levels
• To find what mapping to use first model pixels
  as i.i.d. r.v.
• How to generate r.v. with desired distribution? ?
  Match c.d.f
• Want to transform one r.v. with certain p.d.f. to
  a new r.v. with uniform p.d.f.
• For r.v. U with continuous p.d.f. over 0,1,
  construct a new r.v. V by a monotonically
  increasing mapping v(u) such that
• Can show V is uniformly distributed over 0,1 ?
  FV(v) v
• FV(v) P(V? v) P( FU(u)? v)
• P( U ? F-1U(v) ) FU( F-1U(v)
  ) v
• For u in discrete prob. distribution, the output
  v will be approximately uniform

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How to Do Histogram Equalization?

• Approach map input luminance u to the
  corresponding v
• v will be approximately uniform for u with
  discrete prob. distribution
• b/c all pixels in one bin are mapped to a new
  bin (no splitting)

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Histogram Equalization Algorithm
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Histogram Equalization A Mini-Example

• xi 0 1 2 3 4
  5 6 7 (L8)
• p(xi) 0.1 0.2 0.4 0.15 0.1
  0.05 0 0
• v 0.1 0.3 0.7 0.85 0.95
  1.0 1.0 1.0
• v 0 1.5 4.7 5.8 6.6
  7 7 7 (L-1)/(1-vmin) 7.78
• 0 2 5 6
  7 7 7 7

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Summary Contrast Stretching vs. Histogram Eq.

• What are in common?
• What are different?

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Generalization of Histogram Equalization

• Histogram specification
• Want output v with specified p.d.f. pV(v)
• Use uniformly distributed r.v. W as an
  intermediate step
• W FU(u) FV(v) ? V F-1V (FU(u) )
• Approximation in the intermediate step needed for
  discrete r.v.
• W1 FU(u) , W2 FV(v) ? take v s.t. its w2 is
  equal to or just above w1

Figure is from slides at Gonzalez/ Woods DIP book
website (Chapter 3)
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Generalization of Histogram Equalization

• Histogram modification
• u ? v f(u) ? v unif.-quantized v
• f(u) u1/2 or f(u) u1/3
• f(u) log(1u)
• range compression(good for spectrum
  visualization)
• Histogram specification
• Want output v with specified p.d.f. pV(v)
• Use uniformly distributed r.v. W as an
  intermediate step
• W FU(u) FV(v) ? V F-1V (FU(u) )
• Approximation in the intermediate step needed for
  discrete r.v.
• W1 FU(u) , W2 FV(v) ? take v s.t. its w2 is
  equal to or just above w1

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For Next Time

• Next time we will focus on quantization