HarrisDetector-Harris角点检测

Corner !
Harris Detector: Some Properties
• Quality of Harris detector for different scale changes
Repeatability rate:
# correspondences # possible correspondences
in all directions
“Flat” region
“Edge” 1 >> 2
1
Harris Detector: Mathematics
Measure of corner response:
特征值的分析转化为易计算的行 列式和迹。
M矩阵的特征值之积等于M行列式， 特征值之和等于对角元素和，也 等于矩阵的迹。 可画出z=xy-0.04*(x+y)^2的曲面图 像，在x,y很小或者某一维很小时 值都很小，只有x,y都很大时，R 值较大。
Classification of image points using eigenvalues of M: 2 “Edge” 2 >> 1 “Corner” 1 and 2 are large, 1 ~ 2 ; E increases in all
directions
1 and 2 are small; E is almost constant
I x2 M w( x, y ) x, y Ix I y
IxI y 2 Iy
Harris Detector: Mathematics
Intensity change in shifting window: eigenvalue analysis
E (u, v) u, v
Harris Detector: Workflow
Take only the points of local maxima of R
Harris Detector: Workflow
Harris Detector: Summary
• Average intensity change in direction [u,v] can be expressed as a bilinear form:
I ( x u, y v) I ( x, y) I x ( x, y) u I y ( x, y) v
E (u, v) w( x, y ) I x ( x, y )u I y ( x, y )v
x, y 2
u w( x, y ) [ I x , I y ] x, y v
Summary of Harris detector
Harris Detector: Workflow
Harris Detector: Workflow
Compute corner response R
Harris Detector: Workflow
Find points with large corner response: R>threshold
– Affine (scale dependent on direction) valid for: orthographic camera, locally planar object
• Photometry
– Affine intensity change (I a I + b)
Rotation Invariant Detection
Inte来自百度文库sity scale: I a I R
threshold
R
x (image coordinate)
x (image coordinate)
Harris Detector: Some Properties
• But: non-invariant to image scale!
All points will be classified as edges
Harris Detector: Mathematics
Change of intensity for the shift [u,v]:
E (u, v) w( x, y) I ( x u, y v) I ( x, y)
x, y
2
Window function
Shifted intensity or
R det M k trace M
2
det M 12 trace M 1 2
(k – empirical constant, k = 0.04-0.06)
Harris Detector: Mathematics
2 • R depends only on eigenvalues of M • R is large for a corner “Edge” R<0 “Corner”
Harris Detector: Some Properties
• Partial invariance to affine intensity change
Only derivatives are used => invariance to intensity shift I I + b
Harris Detector: Basic Idea
“flat” region: no change in all directions
“edge”: no change along the edge direction
“corner”: significant change in all directions
Harris Detector: Some Properties
• Rotation invariance
Ellipse rotates but its shape (i.e. eigenvalues) remains the same Corner response R is invariant to image rotation
Intensity
Window function w(x,y) = 1 in window, 0 outside
Gaussian
Harris Detector: Mathematics
E (u, v) w( x, y) I ( x u, y v) I ( x, y)
x, y 2
Notes on the Harris Detector
Harris corner detector
• C.Harris, M.Stephens. “A Combined Corner and Edge Detector”. 1988
The Basic Idea
• 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
C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000
Models of Image Change
• Geometry
– Rotation – Similarity (rotation + uniform scale)
2
Harris Detector: Mathematics
For small shifts [u,v] we have a bilinear approximation:
E (u, v) u, v
u M v
where M is a 22 matrix computed from image derivatives:
R>0
• R is negative with large magnitude for an edge
• |R| is small for a flat region
“Flat” |R| small
“Edge” R<0
1
Another view
Another view
Another view
E (u, v) u, v u M v
• Describe a point in terms of eigenvalues of M: measure of corner response
R 12 k 1 2
2
• A good (corner) point should have a large intensity change in all directions, i.e. R should be large positive
u M v
1, 2 – eigenvalues of M
direction of the fastest change
Ellipse E(u,v) = const
direction of the slowest change
(max)-1/2
(min)-1/2
Harris Detector: Mathematics
• Harris Corner Detector
C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000