harris角点(课程) 2013-3-26

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?
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
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
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
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
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
Moravec corner detector
flat
Moravec corner detector
flat
Moravec corner detector
flat
edge
Moravec corner detector
flapoint
Moravec corner detector
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
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
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
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
, 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)
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
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
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
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}
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
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
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.
1. Compute x and y derivatives of image
I x G I
x
I y G I
y
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
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)