ECE 549CS 543: COMPUTER VISION LECTURE 26

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ECE 549/CS 543 COMPUTER VISION LECTURE 26 EDGE
DETECTION
The edge detection problem Filtering and edge
detection in one dimension Convolution,
smoothing, and noise Gradient-based edge
detection Canny edge detector (1983)
Marr-Hildreth edge detector (1980)

• Reading Chapters 7 and 8
• Homework, due tomorrow
• http//www-cvr.ai.uiuc.edu/ponce/hw5/hw5.pdf

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The edge detection problem
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Convolution
Linear filters Weighted averages

• Represent the weights by a rectangular array F.
• Applying the filter to an image G is equivalent
  to
• performing a convolution
• Rij (FG)ij ?u,v Gi-u, j-v Fu, v

• In the continuous case
• (f g) (x,y) su,v f(x-u,y-v) g(u,v) du dv
• Note fggf.

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Example Smoothing by Averaging
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Smoothing with a Gaussian
f(x,y)(1/2??2)exp-(x2y2)/2?2
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Smoothing with a Gaussian
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Noise

• Issues
• this model allows noise values that could be
  greater than maximum camera output or less than
  zero
• for small standard deviations, this isnt too
  much of a problem it is a fairly good model
• independence may not be justified (e.g. damage to
  lens)
• may not be stationary (e.g. thermal gradients in
  the ccd).

• Simplest noise model
• independent stationary additive Gaussian noise
• the noise value at each pixel is given by an
  independent draw from the same normal probability
  distribution.

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?1
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?16
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Finite differences
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Finite differences and noise

• What is to be done? Intuitively, most pixels look
  a lot like their neighbours
• this is true even at an edge along the edge they
  are similar, across the edge they arent
• this suggests that smoothing the image should
  help, by forcing pixels different from their
  neighbors (noise pixels?) to look more like
  neighbors.

• Finite difference filters respond strongly to
  noise
• obvious reason image noise results in pixels
  that look very different from their neighbors
• generally, the larger the noise the stronger the
  response.

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Finite differences responding to noise
Increasing zero-mean Gaussian noise
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Smoothing and Differentiation

• Smooth before differentiation
• Two convolutions to smooth, then differentiate?
• Actually, no - we can use a derivative of
  Gaussian filter
• ( fg )f g

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1 pixel
3 pixels
7 pixels
The scale of the smoothing filter affects
derivative estimates, and also the semantics of
the edges recovered.
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Gradient-based edge detection
There are three major issues 1) The gradient
magnitude at different scales is different which
should we choose? 2) The gradient
magnitude is large along thick trails how
do we identify the significant points? 3) How
do we link the relevant points up into curves?
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We wish to mark points along the curve where the
magnitude is biggest. We can do this by looking
for a maximum along a slice normal to the
curve (non-maximum suppression). These points
should form a curve. There are then two main
issues at which point is the maximum, and which
minima should we keep?
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Non-maximum suppression
At q, we have a maximum if the value is larger
than those at both p and at r. Interpolate to get
these values.
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Which local maxima should we keep?

• Those above a threshold T
• Those above L that are connected to a
• local maximum above H (thresholding
• with hysteresis)

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The edge detection problem
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The Laplacian of Gaussian (Marr-Hildreth 80)

• Bad idea to apply a Laplacian without smoothing
• Smooth with Gaussian, apply Laplacian.
• This is the same as filtering with a Laplacian of
  Gaussian filter.
• Now mark the zero points where there is a
  sufficiently large derivative, and enough
  contrast.

• Another way to detect an extremal first
  derivative is to look for a zero second
  derivative.
• Appropriate 2D analogy is rotation invariant
• the Laplacian
• r2 f?2f/?x2?2f/?y2

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The Laplacian of a Gaussian
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sigma4
contrast1
contrast4
LOG zero crossings
sigma2
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We have unfortunate behaviour at corners