Edges and Contours

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Edges and Contours Chapter 7

• continued

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Edge detection

• All of the previously studied edge detection
  filters worked on finding local gradients (1st
  derivative)
• As such, all had to make decisions regarding the
  actual existence of an edge based on local
  information

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Sobel edges threshold on magnitude
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Sobel edges threshold on magnitude

• The preceding images were edge-detected with a
  Sobel operator then binarized with various
  thresholds
• The overall trend is predictable but the
  individual results are not
• This threshold dependency is the curse of image
  processing
• Difficult to isolate real edges
• Difficult to find accurate edge location
• We need smarter algorithms

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Scale space processing

• One method of dealing with the threshold
  dependency is through scale-space processing
• An image is processed at various scales
  (sizes/resolution) revealing different sets of
  features at each scale
• Features common to all or many sets are deemed
  important

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The Gaussian Pyramid

• Select a set of Gaussian filter parameters (sigma
  and width) and blur the image
• Save the image
• Reduce its size by subsampling (eliminating
  every other row/column)
• Do it again

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The Gaussian Pyramid

• You end up with a series of images (a pyramid)
  each exposing different features
• Coarse features at the top of the pyramid
• Fine features at the bottom
• You can now use the coarse features to select
  where to concentrate your processing of the fine
  features

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Better (different) algorithms

• Another method of dealing with the threshold
  dependency is through more complex algorithms

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A better kernel

• The Sobel mask is a derivative mask (computes the
  weighted difference of neighborhood pixels)
• So is the Laplacian (negative in the middle,
  positive around the rest)
• From calculus the 2nd derivative of a signal is
  zero when the derivative magnitude is extremal

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A better kernel

• Combining these observations we can derive the
  Laplacian-Gaussian filter due to Marr and
  Hildreth
• Also known as a DoG (Difference of Gaussians)
  because it can be approximated by subtracting two
  Gaussian kernels of same width and differing
  sigmas

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The Gaussian kernel

• Sigma 1.6, Filter Width 15

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The Laplacian-Gaussian kernel

• Bypassing the math and jumping to the results of
  applying the rules of calculus

• Similar in form to the Gaussian kernel

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The Laplacian-Gaussian kernel

• In 1 dimension
• Can be approximated by difference of two Gaussians

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The Laplacian-Gaussian kernel

• Convolution with this kernel results in edge
  magnitudes of both positive and negative values
• Were interested in where the sign changes occur
  (the zero crossing)
• These are where the edges are (recall the
  calculus result)

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Sobel vs. Laplacian-Gaussian

• L-G does not detect as much noise as Sobel
  why not?

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Still something missing

• What are we ignoring in the following edge
  detection processes?
• Gaussian filter
• Sobel edge detection
• Threshold based binarization of Sobel magnitude
• And
• Laplacian-Gaussian filter
• Zero crossing detection

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We need a smarter algorithm

• We merely selected each edge based on its
  magnitude and zero-crossing
• We completely ignored its immediate surroundings
• Why is this bad?
• Detects edges we dont want
• Prone to noise
• Prone to clutter
• Doesnt detect edges we do want
• Weak magnitude edges are thresholded (thresheld?)
  away
• We need a smarter algorithm!

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The Canny edge detector

• John Canny MIT 1986
• Developed a process (series of steps) for
  identifying edges in an image
• Gaussian smoothing of input image
• Local derivative operator to detect edge
  candidates
• Removal of edges that are clustered around a
  single location
• Adaptive thresholding algorithm for final edge
  selection

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Concepts

• Gaussian smooth original image
• Reduce the effects of noise
• Local derivative operator
• Simple neighbor pixel subtraction to detect
  gradients (1st derivative)
• Non-maximal suppression
• Thin out the edges around a given point
• Threshold
• Use two thresholds combined with spatial
  continuity

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Spatial continuity

• aka contour following
• Apply the high threshold to get obvious edges
• Search for a neighboring edge above the low
  threshold in the direction of the strong edge
• Were using a strong edge as an anchor point for
  a contour of weak and strong edges known as
  Hysteresis

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Canny results

• With very conservative high/low thresholds

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Results
All edges
Weak edges
Contours
Strong edges
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Summary

• Through more complex processing we arrive at a
  set of edges in which we have confidence
• Minimal reliance on thresholds
• The cost is complex (slow, memory intensive)
  processing
• We must trade resources for accuracy!

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ImageJ

• Download/install plugins for
• Sobel, Prewitt, Roberts, Kirsch, Nevatia-Babu
• Plugins-gtFeatureJ-gtEdges
• Canny implementation

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Edge Sharpening

• An process for making an image look better
• In this case better means sharper
• The general approach is to amplify the
  high-frequency components (the edges) to create a
  perceived sharpness

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Method 1 subtract 2nd derivative

• This causes a step edge to undershoot at the
  bottom and overshoot at the top
• The Laplacian works well for the 2nd derivative

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ImageJ

• Note that this is difficult to do in ImageJ since
  the convolution operation clamps negative pixel
  values to 0

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Unsharp mask

• Steps
• Subtract a Gaussian-smoothed version of the image
  from the original creating a mask
• Add a weighted version of the mask to the
  original image

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ImageJ

• Plugins-gtChapter07-gtFilter UnsharpMask

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