Lecture 4 Chapter 5.6 5.8, 10.3.2 10.3.3

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Lecture 4 Chapter 5.6 - 5.8, 10.3.2 10.3.3
Edge detection
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Image edges

• Points of sharp change in an image are
  interesting
• changes in reflectance
• changes in object
• changes in illumination
• noise
• Sometimes called edge points or edge pixels
• We want to find the edges generated by scene
  elements and not by noise

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Edge detection

• Convert a 2D image into a set of curves
• Extracts salient features of the scene
• More compact than pixels

Computer Vision Linear Filters and Edge
Detection Slides by D.A. Forsyth, C.F. Olson,
S.M. Seitz, L.G. Shapiro
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Origin of edges
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity

• Edges are caused by a variety of factors

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Computer Vision Linear Filters and Edge
Detection Slides by D.A. Forsyth, C.F. Olson,
S.M. Seitz, L.G. Shapiro
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Edge detection
How can you tell whether a pixel is on an edge?
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Computer Vision Linear Filters and Edge
Detection Slides by D.A. Forsyth, C.F. Olson,
S.M. Seitz, L.G. Shapiro
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Edge detection

• Basic idea look for a neighborhood with lots of
  change

81 82 26 24 82 33 25 25 81 82 26
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• Questions
• What is the best neighborhood size?
• How should change be detected?

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Finding edges

• General strategy
• Determine image gradients after smoothing
• (gradients are directional derivatives computed
  using finite differences)
• Mark points where the gradient magnitude is large
  with respect to neighboring points
• Ideally this yields curves of edge points.

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Image gradients

• We use the image gradient to determine whether a
  pixel is an edge.
• Two components gx, gy
• Both components use finite differencing to
  approximate derivatives
• Gradients have magnitude and orientation
• Vertical edges respond strongly to the x
  component
• Horizontal edges respond strongly to the y
  component
• Diagonal edges will respond less strongly, but to
  both components
• Overall magnitude should be the same

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Sobel operator

• The Sobel operator is a simple example that is
  common.

-1 0 1
1 2 1 Sx -2 0 2
Sy 0 0 0 -1 0 1
-1 -2 -1

• On a pixel of the image I
• let gx be the response to Sx
• let gy be the response to Sy

Then the gradient is ?I gx gy
T
g (gx gy ) is the
gradient magnitude. ? atan2(gy,gx)
is the gradient direction.
2
2
1/2
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Smoothing and differentiation

• Issue noise
• Need to smooth image before determining image
  gradients
• Should we perform two convolutions (smooth, then
  differentiate)?
• Not necessarily we can use a derivative of
  Gaussian filter
• Differentiation is convolution and convolution is
  associative
• D (G I) (D G) I What are D, G, and I?

Gaussian
Gaussian derivative in x
Plot of Gaussian derivative
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Smoothing and differentiation

• Shape of Gaussian derivative
• Light on one side (positive values)
• Dark on other side (negative values)
• Values fall off from horizontal center line
• After initial peaks, values fall off from
  vertical center line

Gaussian derivative in x
Plot of Gaussian derivative
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Smoothing and differentiation

• Important implementation trick we dont need to
  convolve by a 2D kernel
• A 2D Gaussian function is separable
• Gs(x, y) Gs(x) Gs(y)
• This means we can convolve the image with two 1D
  functions (rather than one 2D function)
• This results in considerable savings for an n x n
  image and k x k kernel
• 1 2D kernel approximately n2k2 multiplications
  and additions
• 2 1D kernels approximately 2n2k
• The gradient operator is convolved with the
  appropriate 1D kernel or applied in succession

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Gradient magnitudes after smoothing
Original
Sigma 1
Sigma 5
As the scale (sigma) increases, finer features
are lost, but diffuse edges are gained. Note
that the gradient magnitude encompasses
horizontal, vertical, and diagonal edges.
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Gradient magnitudes after smoothing
Original
Sigma 1
Sigma 5
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 trail how do we identify the
significant points? 3) How do we link the
points up into curves?
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We wish to mark points along the curve where the
gradient magnitude is largest. We can do this by
looking for a maximum along a slice along the
gradient direction. These points should form a
curve. There are two algorithmic issues at
which point is the maximum, and where is the next
one along the curve?
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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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Non-maxima suppression

• At q, the gradient Gq is a vector perpendicular
  to the edge direction
• The locations p and r are one pixel in the
  direction of the gradient and the opposite
  direction.
• One pixel in the gradient direction is g
  Gx/Gmag, Gy/Gmag
• Recall that Gmag is the length of the gradient
  vector Gx, Gy
• r q g, p q - g

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Non-maxima suppression

• At p and r, the gradient magnitude should be
  interpolated from the surrounding four pixels.
• If the gradient magnitude at q is larger than the
  interpolated value at p and r, then q is marked
  as an edge

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Predicting the next edge point
Assume the marked point is an edge point. Then
we construct the tangent to the edge curve (which
is normal to the gradient at that point) and use
this to predict the next points (here either r or
s).
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Remaining issues

• Must check that the gradient magnitude is
  sufficiently large.
• A common problem is that at some points along the
  curve the gradient magnitude will drop below the
  threshold, but not at others.
• Use hysteresis a high threshold to start edge
  curves and a lower threshold to continue them.
• Performance at corners is poor.

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Canny edge detector

• The Canny edge detector (1986) is still used most
  often in practice. It is essentially what we
  have discussed
• Smooth and differentiate the image using
  derivative of Gaussian filters in x and y
• Detect initial candidates by thresholding the
  gradient magnitude
• Apply non-maxima suppression at the candidates
• Aggregate edge pixels into contours by following
  edges perpendicular to the gradient
• When aggregating, allow contour gradient
  magnitude to fall below initial threshold, but
  must remain above lower threshold
• Note that this detector (and others) is sensitive
  to the parameters used (sigma, thresholds)

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Zero-crossing detectors
Edge detection using the zero-crossing of the 2nd
derivative is historically important. Perf
ormance at corners is poor, but zero-crossings
always form closed contours.
step edge
smoothed
1st derivative
zero crossing
2nd derivative
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original image
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fine scale (sigma1), medium threshold, no
hysteresis
Much detail (and noise) that disappears at
coarser scales
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coarse scale (sigma4), high threshold, no
hysteresis
Curves are often broken, not closed contours
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coarse scale (sigma4), low threshold, no
hysteresis
Additional edges found are questionable.