Outline

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Outline

• Announcements
• Non-linear filters

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Announcements

• Homework 1 is due today
• Please turn in your homework

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Comments on Edge Detectors

• Edge detectors produce good edge maps when the
  images are piece-wise constant
• This is because that edge detectors are assuming
  that edges are step edges

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An Example
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Another Example
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Real/Natural Images

• However, edges may not be very meaningful/useful
  for real/natural images
• Textures
• Objects with inhomogeneous colors
• Corners

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An Example
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Need for Multiple Scales
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Need for Multiple Scales cont.
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Edge Tracking

• Edge tracking
• Most edges found at large scales tend to be
  associated with large, high contrast image events
• However, the localization is poor at large scales
  due to smoothing
• At fine scales, there are many edges
• Edge tracking
• Track edges across scales and accept only the
  fine scale edges that have identifiable parents
  at a larger scale

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Problems with Linear Smoothing

• To reduce noise, we need to apply some of
  smoothing
• While the smoothing reduces noise, at the same
  time it also blurs the edges and other important
  features in the image
• At the extreme case, if we apply a Gaussian
  smoothing filter with a very large ?, everything
  will disappear

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An Example
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Linear Scale Space

• The linear scale space based on the Gaussian
  kernel can be understood as follows
• where is the solution of

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An Example
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Anisotropic Diffusion

• The anisotropic diffusion equation
• Conductance factor is not uniform any more
• Ideally, we would want to encourage smoothing
  within a region in preference to smoothing across
  boundaries

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Anisotropic Diffusion cont.

• The diffusion depends on the local gradient

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Robust Statistics

• Non-linear filter as statistical estimator
• The goal is to estimate the true value of the
  pixel in the presence of noisy measurements
• This class of filters is extremely useful but
  very difficult to analyze
• Robust estimates
• Outliers

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Median Filters

• Given a local neighborhood, the output of the
  filter is the median of all the values within the
  neighborhood
• Multi-stage filters
• The filter responds with the median of a set of
  different medians, obtained in different
  neighborhoods

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Corners and Orientation Representations

• Edge detectors fail at corners
• The assumption that estimates of the partial
  derivatives in the x and y direction suffice to
  estimate an oriented gradient becomes
  unsupportable
• Four types of local windows
• Constant windows
• Edge windows
• Flow windows
• 2D windows

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Corners and Orientation Representations cont.

• Characterization of windows through eigen-values
  of the gradient matrix