3'1 Background

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Chapter 3 Image Enhancement in the Spatial Domain

• 3.1 Background
• Specific applicationproblem oriented
• Trial and error is necessary
• Spatial domain will be denoted by the expression
  g(x,y)Tf(x,y)
• The simplest form of T sT(r)
• Contrast stretching (Fig. 3.2 (a))
• Thresholding function binary image (Fig. 3.2)
• Masks (filters, kernels, templates, windows)
• Enhancement mask processing or filtering
• 3.2 Some gray level transformations
• Three basic types of functions used for image
  enhancement
• Linear
• logarithmic
• Power-law

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• 3.2.1 Image negatives
• Is obtained by using the negative transformation
  sL-1-r
• Produces the equivalent of a photographic
  negative
• Suited for enhancing white or gray detail
  embedded in dark regions of an image
• 3.2.2 Log transformations
• The general form of the log transformation
  sclog(1r)
• Expand the values of dark pixels while
  compressing the high-level values
• Compress the dynamic range of images with large
  variations
• 3.2.3 Power-law transformation
• The basic form
• Gamma correction
• CRT device have an intensity-to-voltage response
  that is a power function
• Produce images that are darker than intended
• Is important if displaying an image accurately on
  a computer screen

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• Low r wash-out in the background (Fig. 3.8
  r0.3)
• High r enhance a wash-out appearance (Fig. 3.9
  r5.0 areas are too dark)
• 3.2.4 Piecewise-linear transformation functions
• Advantage the form of piecewise functions can be
  arbitrary complex over the previous functions
• Disadvantage require considerably more user
  input
• Contrast stretching
• One of the simplest piecewise function
• Increase the dynamic range of the gray levels in
  the image
• A typical transformation control the shape of
  the transformation
• r1r2 s10 and s2L-1
• Gray level slicing
• Highlight a specific range of gray levels
• Display a high value for all gray levels in the
  range of interest and a low value for all other
  gray levels produce a binary image
• Brighten the desired range of gray levels, but
  preserves the background and gray level
  tonalities (Fig. 3.11)
• The higher order bits (especially the top four)
  contain the majority of the visually significant
  data

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• Bit-plane slicing
• Highlight the contributions made to total image
  appearance by specific bits
• higher order bits contain the majority of the
  visually significant bits
• Separating a digital image into its bit plane is
  suitable for analyzing the relative importance
  played by each bit of the image
• Useful for image decomposition
• The binary image for bit-plane 7 like a
  thresholding function

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Chapter 3 Image Enhancement in the Spatial Domain
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3.3 Histogram processing

• Histogram of a digital image--the gray levels in
  the range0, L-1
• The sum of all components of a normalized
  histogram is equal to 1
• Low contrast a narrow histogram, a dull,
  wash-out gray look
• High contrast cover a broader range of the gray
  scale and the distribution of pixels is not too
  far uniform, with very few vertical lines being
  much higher than the others
• A great deal of details and high dynamic range
• 3.3.1 Histogram equalization (Histogram
  linearization)
• Histogram of ST (r) 0? r?1
• Produce a level s for every pixel value in the
  original image, the transformation satisfies the
  following conditions
• (1) T(r) is single-valued and
  monotonically increasing in the interval
• 0? r? 1 and
• (2) 0? T ( r ) ? 1 for
  0? r? 1
• rT-1(s) 0? s? 1

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• The goal of this operation--produce an output
  image that
• has a uniform histogram
• The results are predicable and the method is
  simple to implement
• Probability density function continuous
• For discrete values probability and summation
  instead of density functions Pr(rk)( the
  discrete version of the transformation function
• Spread the histogram of the input image so that
  the levels of the histogram
• Equalized image will span a fuller range
• The results are predicable, and the method is
  easy to implement automatically

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• To enhance the contrast of a monochrome image
• Construct a histogram of the grey levels present
• 256 bins representing 0 through 255
• Each bin is the number of pixels with that grey
  level
• Remap the grey levels so that the histogram is
  (roughly) flat
• To ensure that every collection of N adjacent
  bins has the same pixel count
• All pixels within one bin in the input image will
  be within one (possibly different) bin in the
  output image. That is, two pixels with equal grey
  level in the input image will have equal grey
  level in the output image.
• For some adequately small value of N (i.e., at
  some adequately fine scale), collections of N
  adjacent bins will not have the same pixel count.
  
• The pixels must be dithered to equally fill all
  bins, including those that would not have had any
  pixels at all. Note that dithering adds a
  (pseudo) random value to the pixel's grey level.

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3.3.2 Histogram matching (specification)

• Enhancement based on a uniform histogram is not
  the best approach
• It is useful sometimes to specify the shape of
  the histogram that we wish to have
• Generate a processed image that has a specified
  histogram
• Suitable for interactive image enhancement
• Difficulty--build a meaningful histogram
• The smallest integer in the interval 0,L-1 such
  that
• The procedure for histogram matching (Page 99)

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• Two pixels with equal grey level in the input
  image will not necessarily have equal grey level
  in the output image.
• Remapping without dithering seems the most
  appropriate for scientific data
• Improve the visual information as must as
  possible without intentionally adding randomness.
  
• 3.3.3 Local Enhancement
• Enhance details over small areas--- in the
  neighborhood of every pixel in the image
• Local histogram equalization

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• Linear or non-linear transformation
• Based on gray level distribution around the
  neighborhood of a pixel (convolution mask)
• Define a square or rectangular neighborhood and
  move the center of the area
• At each location, the histogram of the points in
  the neighborhood is computed
• A histogram equalization or specification
  function is obtained
• This function is use to map the gray level of the
  pixel centered in the neighborhood
• The center of the neighborhood is then moved to
  an adjacent pixel  
• Use very little a priori knowledge about the
  image contents
• The choice of the transformation, size, and shape
  of the neighborhood depends on the size of objects

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• 3.3.4 Use of histogram statistics for image
  enhancement
• Use statistical parameters obtained from the
  histogram
• Estimate of occurrence of gray level ri
• The nth moment of r about the mean is defined as
• Global mean and variance , local mean and
  variance
• The mean value of the pixel in Sxy is
• The gray level variance of the pixels in the
  region is given by
• The local mean is a measure of average gray level
• The variance is a measure of contrast
• Using Local mean and variance is flexible and
  depends on image appearance

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• Enhance low contrast of an area
• Consider the pixel at a point (x,y) as a
  candidate for processing
• if
• determine whether the contrast if an area makes
  it a candidate for enhancementif
• A summary of the enhancement

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3.4 Enhancement using arithmetic/logic operations

• Image subtraction g(x,y)f(x,y)-h(x,y)
• Masking
• is referred to as ROI (region of interest)
  processing
• Isolate an area for processing
• Arithmetic operations
• Addition
• Subtraction
• Multiplication used to implement gray-level
  rather than binary
• Division
• Logic operations
• And used for masking (Fig. 3.27)
• Orused for masking
• Not operation negative transformation
• Also are used in conjunction with morphological
  operations

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• 3.4.1 Image subtraction
• The difference between two images f(x,y) and
  h(x,y) is expressed as g(x,y)f(x,y)-h(x,y)
• Enhance the difference of two images
• Contrast stretching transformationuseful for
  evaluating the effect of setting to zero the
  lower-order planes (Fig. 3.28(d))
• Mask mode radiography (Fig 3.29)
• Sort of scaling solve image values outside form
  the range 0 to 255 (-255 to 255)
• (1) Add 255 to every pixel and divide by 2 fast
  and simple to implement, but the full rang of the
  display may not be used
• (2) more accuracy and full coverage of the 8-it
  range
• The values of the minimum difference is obtained
  and its negative added to all the pixels in the
  difference image
• All the pixels in the image are scaled to 0,255
  by multiplying 255/Max
• 3.4.2 Image averaging
• g(x,y)f(x,y)?(x,y) (assume every pair of
  coordinates (x,y) the noise is uncorrelated and
  has zero average value)

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• Reduce the noise content by adding a set of noise
  images gi(x,y)
• An image is formed by averaging K different noisy
  images
• As k increases, the variability of the pixel
  values at each location (x,y) decreases
• The image gi(x,y) must be registered in order to
  avoid the introduction of blurring
• Use integrating capabilities of CCD or similar
  sensors for noise reduction by observing the same
  scene over long periods of time
• 3.5 Basics of spatial filtering
• Sub-image (filter, mask, kernel, template or
  window)
• Frequency domain
• Spatial domain
• Linear spatial filtering is give by a sum of
  products of the filter coefficients R
• In general, linear filtering of an image with a
  filter mask of size MxN is given by g(x,y)
• Convolving a mask with an image by pixel-by-pixel
  basis

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• 3.6 Smoothing spatial filters
• Used for blurring and for noise reduction
• Blurring is used for removal of detail and
  bridging of small gaps in lines or curves
• 3.6.1 Smoothing linear filters
• Averaging filter (low pass filter)
• Replace the value of every pixel by the average
  of the gray levels in the neighborhood by the
  filter mask
• Reduce sharp transition (such as random noise)
• Blur edges
• The average of the gray levels in the 3x3
  neighborhoods
• Averaging with limited data validity
• only to pixels in the original image in a
  pre-defined interval of invalid data
• Only if the computed brightness change of a pixel
  is in some pre-defined interval

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• Weighted average (Fig. 3.34)
• Blur an image for the purpose getting a gross
  representation of objects of interest
• The intensity of smaller objects blends with the
  background and larger objects become blob-like
  and easy to detect (Fig. 3.36)
• 3.6.1 Order Statistics filters (rank filters)
• Nonlinear spatial filter based on ordering
  (ranking)
• Median filter
• Remove impulse noises (salt and pepper noises)
• Represent 50 percent of a ranked set
• Large clusters are affected considerably less
• Min filter
• Max filter--useful in finding the brightest
  points
• Non-linear mean filter
• Arithmetic mean
• Harmonic mean
• Geometric mean

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• 3.7 Sharpening spatial filter
• Highlight fine detail or enhance detail
• Enhance detail that has been blurred
• Application ranging from electronic printing and
  medical imaging to industrial inspection
• Can be accomplished by digital differentiation
• 3.7.1 Foundation
• Sharpening filter based on first- and
  second-order derivatives
• Definition of first derivatives
• Must be zero in flat segment
• Muse be nonzero at the onset of a gray level step
  or ramp
• Must be nonzero along the entire ramp (thick
  edge)
• Nature of first Derivate
• Produce thick edges
• Has a strong response to gray-level step

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• Definition of second derivatives is better
  suited than the first-derivative for image
  enhancement
• Must be zero in flat areas
• Muse be nonzero at the onset and end of a gray
  level step or ramp
• Must be zero along ramps of constant slope
• Nature of a second order derivate
• Produces finer edges
• Enhance fine detail (including noise) much more
  than a first order derivate for example a thin
  line
• The response at an isolated point is stronger
  than first Der.
• Has a transition form positive back to negative
• Produce a double response to a gray-level step
• Highlight the fundamental similarities and
  differences between first- and second- order
  derivatives (Fig. 3.38)

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3.7.2 Use of second derivatives for enhancement
The Laplacian

• Consists of defining a discrete formulation of
  the second-derivative and then construct a filter
  mask
• Isotropic filter (rotation invariant)
  independent of the direction of the
  discontinuities in the image
• Development of the method (Laplacian)
• Is the simplest isotropic derivative operator
  (linear operator)
• A function of f(x,y) of two variables is defined
  as
• Filter mask used to implement the Laplacian (Fig.
  3.39)
• Diagonal or no diagonal
• Highlight gray-level discontinuities and
  de-emphasizes regions with slowly varying gray
  levels
• Shortcoming Produces images that grayish edge
  lines and other discontinuities, all superimpose
  on a dark, featureless background image

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• To overcome the shortcoming of the operation
• Enhance small detail and preserve background
  tonality recover background features while
  preserving the sharpening effect
• By adding the original and Laplacian images (Fig.
  3.40)
• A negative center coefficient--subtract sharpen
  result
• A positive center coefficientadd sharpen result
• Simplification
• Composite Laplacian mask
• no diagonal neighbors
• Diagonal neighborssharper than no diagonal
  neighbors

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• Unsharp masking and high-boost filtering
• Un-sharp masking expression-subtract a blurred
  version of an image from the image itself
• The dark room photography
• A further generalization of un-sharp masking
  high-boost filtering
• Expression for computing a high-boost filtered
  image
• The center coefficient of the Laplacian
  masknegative or positive
• Can be implemented with either one of these mask,
  with A?1
• 3.7.3 Use of first derivatives for
  enhancement-The Gradient
• The gradient of f at coordinates (x,y) is

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• The components of the vector are linear operator
• The magnitude of this vector is
  or
• Preserve relative changes in gray levels, but the
  isotropic property is lost
• Is not a linear operator
• The partial derivatives of the gradient vector
  are not rotation invariant
• Give the same result only for vertical and
  vertical edges
• The magnitude of the gradient vector is rotation
  invariant

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• Approximate the magnitude of the gradient by
  using absolute values
• Lost isotropic feature property
• Vertical and horizontal edges preserve the
  isotropic properties only for multiples of 90
• Mask of odd sizes
• Robert operator
• Robert Ross-gradient operators
• An approximation using absolute values (3.7-18)
• Sobel operator
• Use a weight value of 2 to achieve some smoothing
  by giving more importance to the center point
• Constant or slowly varying shades are eliminated
• Prewitt operator

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3.8 Combining spatial enhancement methods

• Application of several complementary enhancement
  techniques
• Laplacian
• Adv. highlight fine detail
• Dis-adv produce noisier results than the
  gradient
• Gradient
• Enhance prominent edges
• Has a stronger response in area of significant
  gray-level transitions
• Further lower the response of fine detail and
  noise for the gradient
• Mask the Laplacian image with a smoothed version
  of the gradient image
• Smooth the gradient and multiply it by the
  Laplacian image
• Preserve details in the strong area while
  reducing noise in the relatively flat areas

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• Enhance the image by sharpening it and bringing
  out more detail (Fig. 3.46)
• Nature of the image the narrow dynamic range of
  the gray levels and high noise content