Edge detection

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Edge detection
Mainly a summary of chapter 8 in Computer
Vision A modern approach, by David A. Forsyth
and Jean Ponce, 2003

• IP/CV Seminar
• Veronica Gidén, Nakajima Laboratory
• May 2006

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What is an edge?

• Points in the image where the brightness changes
  particularly sharply
• We want edge points to be associated with the
  boundaries and other kinds of meaningful changes.
• However, it is hard to tell a meaningful edge
  from a nuisance edge

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A 1-dimensional edge

• We can intuitively say that there should be an
  edge between the 4th and 5th pixels in the
  following 1-dimensional data
• 5 7 6 4 152 148 149
• But where should we set the threshold?
• Edge detection is often non-trivial

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Why do we want to find edges?

• Sharp changes in image properties usually reflect
  important events and changes in properties of the
  world, like
• discontinuities in depth
• discontinuities in surface orientation
• changes in material properties
• variations in scene illumination
• Reduces the amount of data and filters out less
  relevant information and preserves the important
  structural properties of an image.

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Example of an application innon-photo realistic
rendering

• Client-Server Visualization of City Models
• through Non Photorealistic Rendering
• Jean-Charles Quillet Gwenola Thomas Jean-Eudes
  Marvie
• September 23, 2005

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Edges and derivatives

• The definition of a derivative
• Thus, since an image is discrete, we can estimate
  the derivative as a symmetric, finite difference
  
• Edges are fast changes in intensity - they have
  large derivatives!
• Edge detection algorithms generally compute a
  derivative of the intensity change -
  differentiation.

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Edges and derivatives
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Problem Noise

• Image noise is a primary problem, since edge
  detectors are constructed to respond strongly to
  sharp changes
• The noise pixels are typically uncorrelated, they
  can be very different
• Noise IS sharp changes in an image!
• So, we can't just differentiate the image!

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What is noise?

• Image measurements from which we do not know how
  to extract information
• Or from which we do not care to extract
  information
• All the rest is signal

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The solution smoothing

• Smooth the image with a filter before the
  differentiation
• Since differentiation is linear and
  shift-invariant, there is some filter kernel that
  differentiates
• Can obtain the differentiated image I by
  convolution with an appropriate filter K
• First smoothen the image and then differentiate
  it
• Thus convolve with the derivative of the
  smoothing filter!
• Most often, we use a Gaussian smoothing filter

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Filters and convolution

• A filter is an array, in which the values
  determines what it does
• Spacial filtering moving the filter mask from
  point to point in an image while at each pixel,
  the response of the filter is calculated. This is
  called convolution.
• 2D discrete convolution
• Or shorter

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Linear, shift-invariant filters

• Linear filters the response is given by a sum of
  products of the filter coefficients and the
  corresponding image pixels in the area spanned by
  the filter
• For linear filters, the output for the sum of two
  images is equal to the sum of the outputs for the
  images separately
• Shift-invariant filters the value of the output
  depends on the pattern in an image neighbourhood,
  not the position of the neighbourhood

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Gaussian filters

• The Gaussian distribution is also called the
  normal distribution
• In the 2D frequency domain
• Pixels where this distribution is non-zero are
  used to build a convolution matrix, which is
  applied to the original image.
• Each pixel's value is set to a weighted average
  of that pixel's neighborhood smoothing.

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Why does smoothing help?

• E.g. contours of object long chain of points
  where the image derivative is large
• Large derivatives due to noise tends to be a
  local event
• Smoothing tends to suppress noise but not really
  the changes we are actually interested in
  (edges!)

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Why use a Gaussian smoothing filter?

• Convolving a Gaussian with a Gaussian results in
  another Gaussian
• Can get heavily smoothed images by resmoothing
  smoothed images
• This is useful, since discrete convolution can
  be expensive and we often want differently
  smoothed versions of an image

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Why use a Gaussian smoothing filter?

• When convolving with a kernel with very small
  standard deviation, the values are very small on
  a large area outside of the center - we can use a
  small array!
• For a large standard deviation we need a big
  array, but we prefer to smooth repeatedly with a
  much smaller array
• A smoothed image is redundant and therefore, some
  pixels can be discarded
• We can smooth, subsample and so on
• Results in an image that has the same info as the
  heavily smoothed image but is much smaller and
  easier to obtain.

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Why use a Gaussian smoothing filter?

• Gaussians are separable kernels
• Convolving with a 2D kernel gives the same
  result as convolving with two 1D kernels (x and
  y direction)

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Main strategies for detecting edges

• Two main strategies, and both model edges as fast
  changes in brightness
• 1) Zero-crossing-based
• Look for zero crossings in the second derivative
  of the image
• 2)Search-based
• Detect edges by searching for maxima and minima
  in the first derivative of the image

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Zero-crossing-basedUsing the Laplacian to
detect edges

• In 1D, the 2'd derivative magnitude is 0 when the
  1'st derivative magnitude is extremal edges!
• Extending this to 2D, we have the Laplacian
  operator
• Linear and rotationally invariant
• Smoothing and applying the Laplacian
• Method Convolve the image with the Laplacian of
  a Gaussian (LoG), then mark the points where the
  result is 0.

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Results
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Usage, advantages and drawbacks

• Usage Adding some percentage of the result back
  to the image gives a a picture with sharpened
  edges and where details are more easily seen
• Advantage Get thin edges
• Drawbacks
• the Laplacian of a Gaussian filter is not
  oriented behaves strangely at corners
• Spaghetti loops
• Due to noise, all 0-crossings may not lie on an
  edge

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Search-basedGradient-based edge-detectors

• More frequently used than zero-crossing-based
• Compute some estimate of the gradient magnitude
  and use it to find edge points
• Problem Get thicker edges than with the
  Laplacian
• To solve this, we look for points where the
  gradient magnitude reaches a maximum along the
  direction perpendicular to the edge. Estimate
  this direction using the direction of the
  gradient.

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The gradient

• First derivatives in image processing are
  implemented using the magnitude of the gradient
• For a function f(x,y) the gradient of f at
  coordinate (x,y) is defined as the 2D vector f
  Gx Gy df/dx df/dy
• The magnitude is the length of the vector
• Usually computed asGxGy

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Gradient-based edge-detectors
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Gradient-based edge-detectors

• Algorithm for gradient-based edge-detectors
• Form an estimate of the image gradient
• Obtain the gradient magnitude from this estimate
• Identify image points where the value of the
  gradient magnitude is maximal in the direction
  perpendicular to the edge and also large these
  points are edge points

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Non-maximum suppressionand edge-following

• Selecting local maxima of the gradient magnitude,
  we get isolated points, but we want an edge!
• Select the point with the maximum gradient
  magnitude along the gradient direction get a
  chain.
• Expect edge points to occur along curve like
  chains
• The important steps
• Determine whether a given point is an edge
  point
• If it is, find the next edge point

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Non-maximum suppression
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Hysteresis

• Problem We get too many curves, not just object
  boundaries
• Reason We mark maxima without regarding how big
  they are
• Solution Use a threshold test so that
  maximagtthreshold
• New problem We get broken edge curves
• Solution Use two thresholds, the larger when
  starting an edge chain and the smaller while
  following it hysteresis

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Problems
Standard-deviation of the Gaussian
Threshold
s1 pixel
High threshold
s4 pixel
High threshold
s4 pixel
Low threshold
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Problem corners

• Edge detectors fail at corners
• The partial derivatives can't describe oriented
  corners, since they cross there
• Special corner detectors look for neighbourhoods
  where the gradient swings sharply

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Problem object boundaries

• Object boundaries are NOT the same as sharp
  changes in image values!
• Objects may not have a strong contrast with the
  backgrounds
• Textures may generate edges of their own
• Shadows may generate edges
• The solution
• Control the illumination
• Large smoothing parameters and high contrast
  thresholds

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Thank you for listening!