harris角点(课程) 2013-3-26

1. Compute x and y derivatives of image
I x G I
x
I y G I
ywk.baidu.com
2. Compute products of derivatives at every pixel
I x2 I x I x
I y2 I y I y
I xy I x I y
E (u)
x 0 W ( p )
2 w ( x ) | I ( x u ) I ( x ) | 0 0 0
| I ( x 0 u ) I ( x 0 ) |2 I I0 I u 0 x
T 2
I u x
T
T
2
I I u u x x uT Mu
Problems of Moravec detector
• Noisy response due to a binary window function • Only a set of shifts at every 45 degree is considered • Only minimum of E is taken into account Harris corner detector (1988) solves these problems.
Harris corner detector
Only a set of shifts at every 45 degree is considered Consider all small shifts by Taylor’s expansion
E(u, v) w( x, y)I ( x u, y v) I ( x, y)
Moravec corner detector
flat
Moravec corner detector
flat
Moravec corner detector
flat
edge
Moravec corner detector
flat
edge
corner isolated point
Moravec corner detector
7.75 3.90 0.50 0.87 1 0 0.50 0.87 A 0 10 0.87 0.50 3 . 90 3 . 25 0 . 87 0 . 50
T
Harris corner detector
?
Desired properties for features
• Distinctive: a single feature can be correctly matched with high probability. • Invariant: invariant to scale, rotation, affine, illumination and noise for robust matching across a substantial range of affine distortion, viewpoint change and so on. That is, it is repeatable.
T
Harris corner detector
Only minimum of E is taken into account A new corner measurement by investigating the shape of the error function
uT Mu represents a quadratic function; Thus, we
w( x, y ) I x u I y v O(u , v )
2 2 x, y
2
x, y


2
E (u , v) Au 2 2Cuv Bv 2 A w( x, y ) I x2 ( x, y )
x, y 2 B w( x, y ) I y ( x, y ) x, y
C w( x, y ) I x ( x, y ) I y ( x, y )
x, y
Harris corner detector
Equivalently, for small shifts [u,v] we have a bilinear approximation:
u E (u, v) u v M v
100 80 60 40 20 0 10 5 0 0 2 4 6 8 10
100 80 60 40 20 0
12
100 80 60 40 20 0
10 5 0 0 2 4 6 8 10 12
10 5 0 0 2 4 6 8 10
12
flat
edge
corner
Harris corner detector
, where M is a 22 matrix computed from image derivatives:
2 Ix M w( x, y ) x, y I x I y
IxI y 2 Iy
Harris corner detector (matrix form)
Intensity change in shifting window: eigenvalue analysis
u E (u, v) u, v M v
Ellipse E(u,v) = const
使方程等于常数，用于绘制 一条等高线。
1, 2 – eigenvalues of M
第5章 视觉图像特征信息提取
5.2 兴趣点检测
内容
• • • • • Features（点特征） Harris corner detector SIFT Extensions Applications
Features
Features
• Also known as interesting points, salient points or keypoints. Points that you can easily point out their correspondences in multiple images using only local information.
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
intensity
Four shifts: (u,v) = (1,0), (1,1), (0,1), (-1, 1) Look for local maxima in min{E}
内容
• • • • • Features（点特征） Harris corner detector SIFT Extensions Applications
Harris corner detector
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
3. Compute the sums of the products of derivatives at each pixel
can analyze E’s shape by looking at the property of M
Harris corner detector
High-level idea: what shape of the error function will we prefer for features?
T
朝各个方向函数的递增 速度均相同
Visualize quadratic functions
4 0 1 0 4 0 1 0 A 0 1 0 1 0 1 0 1
T
在垂直方向上函数的递 增速度大
Visualize quadratic functions
3.25 1.30 0.50 0.87 1 0 0.50 0.87 A 0 4 0.87 0.50 1 . 30 1 . 75 0 . 87 0 . 50
T
Visualize quadratic functions
in all directions
flat
edge 1 >> 2
1
Harris corner detector
a00 a11 (a00 a11 ) 2 4a10 a01 Only for reference, you do not need 2 them to compute R
Measure of corner response:
R detM k trace M
det M 12 traceM 1 2
2
哪来的公式？ 来自于作者的灵感（ 论文中描述）.
(k – empirical constant, k = 0.04-0.06)
Harris corner detector
2 “Edge” R<0 “Corner”
R>0
• R只与M的特征值有关 • 角点：R为大数值正数 • 边缘：R为大数值负数 • 平坦区：R为小数值
“Flat” |R| small
“Edge” R<0 1
Another view
Another view
Another view
Summary of Harris detector
Harris corner detector
Noisy response due to a binary window function Use a Gaussian function
Harris corner detector
Only a set of shifts at every 45 degree is considered Consider all small shifts by Taylor’s expansion
Applications
• • • • • Object or scene recognition Structure from motion Stereo Motion tracking …
Components
• Feature detection locates where they are • Feature description describes what they are • Feature matching decides whether two are the same one
direction of the fastest change
direction of the slowest change
(max)-1/2 (min)-1/2
椭圆的长、短半轴为什么取 这样的值？
Visualize quadratic functions
1 0 1 0 1 0 1 0 A 0 1 0 1 0 1 0 1
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