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Harris corner detector
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Moravec corner detector (1980)• We should easily recognize the point by looking through a small window• Shifting a window in any direction should give a large change in intensity
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Moravec corner detector
flat
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Moravec corner detector
flat
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Moravec corner detector
flat edge
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Moravec corner detector
flat edge cornerisolated point
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Moravec corner detectorChange of intensity for the shift [u,v]:
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y
IntensityShifted intensityWindow function
Four shifts: (u,v) = (1,0), (1,1), (0,1), (-1, 1)Look for local maxima in min{E}
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Problems of Moravec detector• Noisy response due to a binary window function• Only a set of shifts at every 45 degree is considered• Responds too strong for edges because only minimum of E is taken into accountHarris corner detector (1988) solves these problems.
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Harris corner detectorNoisy response due to a binary window functionUse a Gaussian function
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Harris corner detectorOnly a set of shifts at every 45 degree is considered Consider all small shifts by Taylor’s expansion
yxyx
yxy
yxx
yxIyxIyxwC
yxIyxwB
yxIyxwA
BvCuvAuvuE
,
,
2
,
2
22
),(),(),(
),(),(
),(),(
2),(
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Harris corner detector
( , ) ,u
E u v u v Mv
Equivalently, for small shifts [u,v] we have a bilinear approximation:
2
2,
( , ) x x y
x y x y y
I I IM w x y
I I I
, where M is a 22 matrix computed from image derivatives:
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Harris corner detectorResponds too strong for edges because only minimum of E is taken into accountA new corner measurement
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Harris corner detector
( , ) ,u
E u v u v Mv
Intensity change in shifting window: eigenvalue analysis1, 2 – eigenvalues of M
direction of the slowest change
direction of the fastest change
(max)-1/2
(min)-1/2
Ellipse E(u,v) = const
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Harris corner detector
1
2
Corner1 and 2 are large,
1 ~ 2;
E increases in all directions
1 and 2 are small;
E is almost constant in all directions
edge 1 >> 2
edge 2 >> 1
flat
Classification of image points using eigenvalues of M:
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Harris corner detectorMeasure of corner response:
2det traceR M k M
1 2
1 2
dettrace
MM
(k – empirical constant, k = 0.04-0.06)
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Another view
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Another view
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Another view
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Summary of Harris detector
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Harris corner detector (input)
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Corner response R
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Threshold on R
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Local maximum of R
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Harris corner detector
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Harris Detector: Summary• Average intensity change in direction [u,v] can be expressed as a bilinear form:
• Describe a point in terms of eigenvalues of M:measure of corner response
• A good (corner) point should have a large intensity change in all directions, i.e. R should be large positive
( , ) ,u
E u v u v Mv
21 2 1 2R k