Outline
• Announcements
• Non-linear filters
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
Input image
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Canny edge map
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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:
I ( x, y, t )  I 0 ( x, y) * G( x, y, t )
– where I ( x, y, t ) is the solution of
I t  I  I xx  I yy
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An Example
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Anisotropic Diffusion
• The anisotropic diffusion equation
I t  div (c( x, y, t )I )  c( x, y, t )I  c  I
– 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
I it,j1  I it, j  
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c
N
 N I
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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
yij  median({ xuv | xuv  N })
• 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 eigenvalues of the gradient matrix
H
{(I )(I )
T
}
window
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