Image Enhancement in the Spatial Domain

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Chapter 3

• Image Enhancement in the Spatial Domain

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Background

• Spatial domain refers to the aggregated of pixels
  composing an image.
• Spatial domain methods are procedures that
  operate directly on these pixels.
• Spatial domain processes will be denoted by the
  expression

g(x,y)T f(x,y)
where f(x,y) is the input image, g(x,y) is the
processed image, and T is an operator of f
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Some Basic Gray Level Transformations

• Discussing gray-level transformation functions.
  The value of pixels, before and after processing
  are related by an expression of the form

s T(r)
Where r is values of pixel before process
s is values of pixel after process
T is a transformation that maps a pixel value r
into a pixel value s.
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Gray Level Transformations

• Image Negatives
• Log Transformations
• Exponential Transformations
• Power-Law Transformations

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Image Negatives

• The negative of an image with gray L levels is
  given by the expression

s L 1 r
Where r is value of input pixel, and
s is value of processed pixel
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Log Transformations

• The general form of the log transformation shown
  in

s c log(1r)
Where c is constant, and it is assumed that r?0
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C 1.0
C 0.8
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Exponential Transformations

• The general form of the log transformation shown
  in

s c exp(r)
Where c is constant, and it is assumed that r?0
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C 1.0
C 0.8
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Power-Law Transformations

• Power-law transformations have the basic form
• Sometime above Equation is written as

Where c and ? are positive constant.
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Piecewise-Linear Transformation Functions

• Translate Mapping
• Contrast stretching
• Gray-level slicing
• Bit-plane slicing

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Translate Mapping
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Contrast stretching

• One of the simplest piecewise linear functions is
  a contrast-stretching transformation.
• The idea behind contrast stretching is to
  increase the dynamic range of the gray levels in
  the image being processed.

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Contrast stretching
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Contrast stretching
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Gray-level slicing

• Highlighting a specific range of gray levels in
  an image often is desired.
• There are several ways of doing level slicing,
  but most of them are variations of two basic
  themes.
• One approach is to display a high value for all
  gray levels in the range of interest and a low
  value for all other gray levels.

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Gray-level slicing
255
255
0
0
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Bit-plane slicing

• Instead of highlighting gray-level ranges,
  highlighting the contribution made to total image
  appearance by specific bits might be desired.
• Suppose that each pixel in an image is
  represented b y 8 bits. Imagine that the image is
  composed of eight 1-bit planes, ranging from
  bit-plane 0, the least significant bit to
  bit-plane 7, the most significant bit.

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Bit-plane 0 (least significant)
1
1
0
0
1
1
0
10110011
1
Bit-plane 7 (most significant)
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Bit-plane slicing
Original Image
Bit-plane 7
Bit-plane 6
Bit-plane 4
Bit-plane 1
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Histogram Processing

• The histogram of a digital image with gray levels
  in the range 0,L-1 is a discrete function

Where rk is the kth gray level and nk
is the number of pixels in the image having gray
level rk
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HISTOGRAM
pixels
130
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36
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level
0
1
2
3
Image 16x14 224 pixels
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Role of Histogram Processing

• The role of histogram processing in image
  enhancement example
• Dark image
• Light image
• Low contrast
• High contrast

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Histogram Processing

• Histogram Equalization
• Histogram Matching(Specification)
• Local Enhancement

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Initial part of discussion

• Let r represent the gray levels of the image to
  be enhanced.
• Normally, we have used pixel values in the
  interval 0,L-1, but now we assume that r has
  been normalized to the interval 0,1, with r0
  representing black and r1 representing white.

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Histogram Equalization

• We focus attention on transformations of the form

That produce a level s for every pixel value r in
the original image.
And we assume that the transformation function
T(r) satisfies the following conditions (a) T(r)
is single-valued and monotonically increasing in
the interval 0 r 1 and (b) 0 T(r) 1 for
0 r 1
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Reason of Condition

• The requirement in (a) that T9r) be single valued
  is needed to guarantee that the inverse
  transformation will exist, and the monotonicity
  condition preserves the increasing order from
  black to white in the output image.
• Condition (b) guarantees that the output gray
  levels will be in the same range as the input
  levels.

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Transformation function
Level s
skT(rk)
T(r)
Level r
rk
0
1
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Note

• The inverse transformation from s back to r is
  denoted

There are some cases that even if T(r) satisfies
conditions (a) and (b), it is possible that the
corresponding inverse T-1(s) may fail to be
single valued.
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Fundamental of random variable

• PDF (probability density function) is the
  probability of each element
• CDF (cumulative distribution function) is
  summation of the probability of the element that
  value less than or equal this element

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PDF

• The PDF (probability density function) is denoted
  by p(x)

• ???????? ????????????? ???????????????????????????
  ??????????? ???????????????

pdf
1/6
???????????
1
2
3
4
5
6
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CDF

• The CDF (cumulative density function) is denoted
  by P(x)

• ???????? ????????????? ???????????????????????????
  ??????????? ???????????????

cdf
1
1/6
???????????
1
2
3
4
5
6
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Relation between PDF and CDF

• PDF can find from this equation
• CDF can find from this equation

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Idea of Histogram Equalization

• The gray levels in an image may be viewed as
  random variables in the interval 0,1.

Let pr(r) denote the pdf of random variable r and
ps(s) denote the pdf of random variable s if
pr(r) and T(r) are known and T-1(s) satisfies
condition (a), the formula should be
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Idea of Histogram Equalization

• By Leibnizs rule that the derivative of a
  definite integral with respect to its upper limit
  is simply the integrand evaluated at that limit.

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Idea of Histogram Equalization

• Substituting into the first equaltion

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Idea of Histogram Equalization

• The probability of occurrence of gray level rk in
  an image is approximated by

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Histogram Equalization
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Example forHistogram Equalization

• ????????????? ????????????? ??????????????????????
  ???????

????? ??????????? (nj)
0 30
1 50
2 100
3 1500
4 2300
5 4000
6 200
7 20
????
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Example forHistogram Equalization
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Example forHistogram Equalization
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Example forHistogram Equalization(2)
????? (k) ???????????(nj)
0 30 30 0.0037 0.0256
1 50 80 0.0098 0.0683
2 100 180 0.0220 0.1537
3 1500 1680 0.2050 1.4241
4 2300 3980 0.4854 3.3976
5 4000 7980 0.9732 6.8122
6 200 8180 0.9976 6.9830
7 20 8200 1.0000 7.0000
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Histogram Specification

• can called Histogram Matching
• Histogram equalization automatically determines a
  transformation function that seeks to produce an
  output image that has a uniform histogram.
• In particular, it is useful sometimes to be able
  to specify the shape of the histogram that we
  wish the processed image to have.

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MeaningHistogram Specification

• In this notation, r and z denote the gray levels
  of the input and output (processed) images,
  respectively.
• We can estimate pr(r) from the given image
• While pz(z) is the specified probability density
  function that we wish the output image to have

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MeaningHistogram Specification

• Let s be a random variable with the property
• Suppose that we define a random variable z with
  property
• Then two equations imply G(z)T(r)

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ProcedureHistogram Specification

1. Use Eq.(1) to obtain the transformation function
   T(r)
2. Use Eq.(2) to obtain the transformation function
   G(z)
3. Obtain the inverse transformation function G-1
4. Obtain the output image by applying Eq.(3) to all
   the pixels in the input image

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Input Image
Specified histogram
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Result form Histogram Specification
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Local Enhancement

• Normally, Transformation function based on the
  content of an entire image.
• Some cases it is necessary to enhance details
  over small areas in an image.
• The histogram processing techniques are easily
  adaptable to local enhancement.

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Local Enhancement
Nonoverlapping region
Pixel-to-pixel translation
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Local Equalization
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Enhancement Using Arithmetic/Logic Operations

• Are performed on a pixel-by-pixel basis between
  two or more images
• Logic operations are concerned with the ability
  to implement the AND, OR, and NOT logic operators
  because these three operators are functionally
  complete.
• Arithmetic operations are concerned about ,-,,
  / and so on (arithmetic operators)

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Image Subtraction

• The difference between two images f(x,y) and
  h(x,y) expressed as

g(x,y) f(x,y) h(x,y)
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Image Averaging

• Consider a noisy image g(x,y) formed by the
  addition of noise to original image f(x,y)
• The objective of this procedure is to reduce the
  noise content.

Where the assumption is that at every pair of
coordinates(x,y) the noise is uncorrelated and
has zero average value.
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Image Averaging

• Let there are K different noisy images
• If an image is formed by averaging
  K different noisy images

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Image Averaging (Gray Scale)
1 image
2 images
5 images
10 images
20 images
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Image Averaging (Color Image)
(1)
(2)
(3)
Average image