Edges and Contours

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Edges and Contours Chapter 7
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Visual perception

• We dont need to see all the color detail to
  recognize the scene content of an image
• That is, some data provides critical information
  for recognition, other data provides information
  that just makes things look good

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Visual perception

• Sometimes we see things that are not really
  there!!!

Kanizsa Triangle (and variants)
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Edges

• Edges (single points) and contours (chains of
  edges) play a dominant role in (various)
  biological vision systems
• Edges are spatial positions in the image where
  the intensity changes along some orientation
  (direction)
• The larger the change in intensity, the stronger
  the edge
• Basis of edge detection is the first derivative
  of the image intensity function

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First derivative continuous f(x)

• Slope of the line at a point tangent to the
  function

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First derivative discrete f(u)

• Slope of the line joining two adjacent (to the
  selected point) point

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Discrete edge detection

• Formulated as two partial derivatives
• Horizontal gradients yield vertical edges
• Vertical gradients yield horizontal edges
• Upon detection we can learn the magnitude
  (strength) and orientation of the edge
• More in a minute

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NOTE

• In the following images, only the positive
  magnitude edges are shown
• This is an artifact of ImageJ
• Process-gtFilters-gtConvolve command
• Implemented as an edge operator, the code would
  have to compensate for this

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Detecting edges sharp image
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Detecting edges blurry image
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The problem

• Localized (small neighborhood) detectors are
  susceptible to noise

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The solution

• Extend the neighborhood covered by the filter
• Make the filter 2 dimensional
• Perform a smoothing step prior to the derivative
• Since the operators are linear filters, we can
  combine the smoothing and derivative operations
  into a single convolution

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Edge operator

• The following edge operators produce two results
• A magnitude edge map (image)
• An orientation edge map (image)

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Prewitt operator

• 3x3 neighborhood
• Equivalent to averaging followed by derivative
• Note that these are convolutions, not matrix
  multiplications

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Prewitt sharp image
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Prewitt blurry image
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Prewitt noisy image

• Clearly this is not a good solutionwhat went
  wrong?
• The smoothing just smeared out the noise
• How could you fix it?
• Perform non-linear noise removal first

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Prewitt magnitude and direction
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Prewitt magnitude and direction
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Sobel operator

• 3x3 neighborhood
• Equivalent to averaging followed by derivative
• Note that these are convolutions, not matrix
  multiplications
• Same as Prewitt but the center row/column is
  weighted heavier

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Sobel sharp image
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Sobel blurry image
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Sobel noisy image

• Clearly this is not a good solutionwhat went
  wrong?
• The smoothing just smeared out the noise
• How could you fix it?
• Perform non-linear noise removal first

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Sobel magnitude and direction
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Sobel magnitude and direction
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Sobel magnitude and direction

• Still not goodhow could we fix this now?
• Using the information of the direction (lots of
  randomly oriented, non-homogeneous directions)
  can help to eliminate edged due to noise
• This is a higher level (intelligent) function

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Roberts operator

• Looks for diagonal gradients rather than
  horizontal/vertical
• Everything else is similar to Prewitt and Sobel
  operators

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Roberts magnitude and direction
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Roberts magnitude and direction
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Roberts magnitude and direction
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Compass operators

• An alternative to computing edge orientation as
  an estimate derived from two oriented filters
  (horizontal and vertical)
• Compass operators employ multiple oriented
  filters
• To most famous are
• Kirsch
• Nevatia-Babu

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Kirsch Filter

• Eight 3x3 kernel
• Theoretically must perform eight convolutions
• Realistically, only compute four convolutions,
  the other four are merely sign changes
• The kernel that produces the maximum response is
  deemed the winner
• Choose its magnitude
• Choose its direction

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Kirsch filter kernels
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Kirsch filter
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Nevatia-Babu Filter

• Twelve 5x5 kernel
• Theoretically must perform twelve convolutions
• Increments of approximately 30
• Realistically, only compute six convolutions, the
  other six are merely sign changes
• The kernel that produces the maximum response is
  deemed the winner
• Choose its magnitude
• Choose its direction

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Nevatia-Babu filter