Filtering and Image Enhancements

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Filtering and Image Enhancements

• Dr. Ramprasad Bala
• Computer and Information Science
• UMASS Dartmouth
• CIS 465 Topics in Computer Vision

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Introduction

• Image processing more accurately filtering,
  enhancement or conditioning is the process of
  extracting the signal or structure in the image
  while suppressing unwanted or uninteresting
  variations in the image.
• Operations can be based on local neighborhood or
  on the entire image.
• Correlation or Convolution.

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What needs fixing?
Image Enhancement
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Image Enhancement

• Image enhancement operators improve detectability
  of important image details or objects manually or
  automatically.
• Noise reduction/suppression
• Smoothing
• Contrast stretching
• Edge enhancement

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Image Restoration
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Image Restoration

• Image restoration attempts to restore a degraded
  image to an ideal condition for further
  processing.
• Constrained by the understanding and modeling of
  the ideal image formation and image degradation
  process.
• If successful we can use the model to invert the
  degradation to return the ideal image.

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Gray-level Mapping

• Change the appearance of an image by transforming
  the pixels via a single function that maps an
  input gray-value into a new output value.
• This can be done to the entire image or to use
  different functions for different sub-images.
• Remapping the gray values is called stretching

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Example
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Example
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Gamma correction

• Any function map can be used.
• The square root function (f(x) x ½ ) for
  example boosts, nonlinearly, all intensities, but
  more so the lower intensities than higher ones.
• The function f(x) x 1/? is called Gamma
  correction. Where ? can be varying.
• Image variation will be increased in the
  intensity ranges where the slope of the function
  is greater than 1.

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Point operator

• A point operator applied to an image is an
  operator in which the output pixel is determined
  only by the input pixel,
• Outx,y f(Inx,y)
• function f() may depend on some global
  parameter.
• A contrast stretching operator is a point
  operator that used a piecewise smooth function
  f(Inx,y) of the input gray level to enhance
  important details of the image.

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Disclaimer

• While monotonic point operators work well in
  enhancing images for human consumption (or
  graphics etc), they do change the original
  intensities considerably.
• In applications where the original intensity
  levels play a significant role (eg. mammography
  images) point operators are not good solutions
  for image enhancement.

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Histogram Equalization

• Histogram equalization is often used to enhance
  an image.
• It accomplishes the following
• The output image uses all available gray levels
• The output image has approximately the same
  number of pixels of each gray level.
• While a. makes sense, b. is ad hoc and its
  effectiveness can be judged only empirically.

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Histogram Equalization
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Histogram Equalization
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Mathematically Speaking

• Condition a. implies that all gray levels will be
  representedz z1,z z2z zn
• Condition b. implies that gray level zk is used q
  times where q (RxC)/n, where R,C and n are
  number of rows, number of columns and number of
  intensities in the image respectively.

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Implementation

• Hini is the number of pixels of the input image
  having gray level zi.
• The first gray level threshold t1is found by
  advancing I in the input image histogram until
  approximately q1 pixels are accounted for
  i.e.all input image pixels zk lt t1 1 will be
  mapped to gray level z1 in the output image.

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Assignment 6

• Implement histogram equalization.
• Your program will take as input the number of
  intensities you want represented in the image.
• Display the original image, its intensity
  histogram, the output image and its intensity
  histogram. The assignment should run off of a
  gui.
• Due March 21th.

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Noise Removal

• Salt and Pepper Noise.
• The presence of a single bright pixel in a dark
  region or vise versa is called Salt and Pepper
  noise.
• Typically results from classification errors
  resulting from variation in surface material or
  illumination.
• A (4- or 8 neighbor) mask is usually applied to
  eliminate this noise.
• CC can also be used for removal of small regions.

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Image Smoothing

• Smoothing an image by equally weighting a
  rectangular neighborhood of pixels is called
  using a box filter.
• Mean filter use the average of it neighbors

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Original image modified with some gaussian noise.
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Gaussian Filter

• Instead of weighting the neighbors equally you
  could weight them based on their distance from
  the center pixel.
• When a Gaussian filter is used, pixel x,y is
  weighted according to
• where d sqrt((x-xc)2 (y-yc)2).
• More on this later

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Median Filter

• The Mean filter usually works if the regions tend
  to be homogenous.
• The Median Filter replaces the center pixel with
  the median of the neighboring values.
• Works well for many types of noise.
• Preserves edges, unlike the mean filter.
• Computationally expensive, due to sorting.

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5x5 Median Filter
7x7 Median Filter
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Another Example
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Detecting Edges Using Differencing Masks

• Image points of high contrast can be detected by
  computing intensity differences in local image
  regions.
• Differencing in the 1-D signal
• The derivative would be discretely computed by
  subtracting successive values.
• In the form a mask it will look like this
• M

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Difference masks can be used to obtain different
results. I only the presence of an edge is
important, you can simply compute the absolute.
If direction of the edge is important then leave
the sign in there. In the former case -1 1
would be equivalent mask to 1 1. Similarly for
the second derivative, The mask 1 2 1 would
be equivalent to -1 2 1. The first derivative
gives you information about sharp changes. The
second derivative gives you information on
zero-crossing.
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First derivative mask
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Second derivative mask
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Smoothing versus differencing
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Properties of Differencing

• Coordinates of derivative masks have opposite
  signs in order to obtain a high response in
  signal regions of high contrast.
• The sum of coordinates of derivative masks is
  zero so that zero response is obtained on
  constant regions.
• First derivative masks produce high absolute
  values at points of high contrast.
• Second derivative masks produce zero-crossing at
  points of high contrasts.

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Properties of Smoothing Masks

• Coordinates of smoothing masks are positive and
  sum to one so that output on constant regions is
  the same as the input.
• The amount of smoothing and noise reduction is
  proportional to the mask size.
• Step edges are blurred in proportion to the mask
  size.

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Different operators on 2D signals.

• From calculus we know that the derivative of a 2D
  function is given by
• df/dx df/dy and the direction is given by
  tan-1 of these two values.
• Since an edge might cut across an edge we may
  want to use more than one point to find the
  difference.

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So the difference is defined as
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Intuitively
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Transforming this into a mask...
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Applying this mask to an image
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The Sobel Operator
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Analysis
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Notes on using these operators

• The Roberts mask is more efficient due to smaller
  neighborhood operation.
• All these local operators miss a lot of edges.
• Computing square roots and squares can become
  quite expensive other variations include,
  absolute, maximum.

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Zero crossings of the second derivative

• The main disadvantage of these edge detectors is
  their dependence on the size of objects and
  sensitivity to noise.
• An edge detection technique, based on the zero
  crossings of the second derivative explores the
  fact that a step edge corresponds to an abrupt
  change in the image function.
• The first derivative of the image function should
  have an extrema at the position corresponding to
  the edge in the image, and so the second
  derivative should be zero at the same position.

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Gaussian Filtering and LoG

• A Gaussian function of one variable with spread s
  is of the following form
• A Gaussian function in two variables is shown
  below

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Laplacian of the Gaussian

• Robust calculation of the 2nd derivative
• smooth an image first (to reduce noise) and then
  compute second derivatives.
• the 2D Gaussian smoothing operator G(x,y)

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LoG

• After returning to the original co-ordinates x, y
  and introducing a normalizing multiplicative
  coefficient c (that includes 1/ 2), we get a
  convolution mask of a zero crossing detector
• where c normalizes the sum of mask elements to
  zero.

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Some Properties

• The area under the curve of g(x) making it
  suitable for being used as a filter.
• g(x) is g(x) multiplied by the function x and
  one over s2.
• g(x) is the difference of two even functions
  and the central lobe will be negative when x is
  approximately 0. The zero crossings occur when x
  s2.

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Examples of G-masks
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LoG
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A large LoG filter
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LOG Roberts Prewitt Sobel
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Neural Networks
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Marr-Hildreth Theory
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The Canny Edge Detector

• Smoothes the intensity image and then produces
  extended contour segments by following high
  gradient magnitudes from one neighborhood to
  another.
• Details in chapter 10.

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• Color and Shading