SLIDE 4 4
Harris Detector
Approximate using 1st order Taylor series expansion of I:
∑ ∑ ∑
= = = + + =
y x y x y x y y x x
y x I y x I y x w C y x I y x w B y x I y x w A Bv Cuv Au v u E
, , 2 , 2 2 2
) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 ) , (
[ ]
( , ) A C u E u v u v C B v ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
x y x I I x ∂ ∂ = / ) , ( y y x I I y ∂ ∂ = / ) , (
[ ]
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ≈ + + ≈ + + v u I I y x I v I u I y x I v y u x I
y x y x
) , ( ) , ( ) , (
Plugging this into previous formula, we get:
where
Harris Corner Detector
In summary, expanding E(u,v) in a Taylor series, we have, for small shifts, [u,v], a bilinear approximation: where M is a 2 x 2 matrix computed from image derivatives:
Note: Sum computed over small neighborhood around given pixel
x y x I I x ∂ ∂ = / ) , ( y y x I I y ∂ ∂ = / ) , (
[ ]
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ≅ v u v u v u E ) , ( M
∑
⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ =
y x y y x y x x
I I I I I I y x w
, 2 2
) , ( M
Harris Corner Detector
Intensity change in shifting window: eigenvalue analysis λmax, λmin – eigenvalues of M
Eigenvector = direction of the slowest change in E Eigenvector associated with λmax = direction of the fastest change in E
(λmax)-1/2 (λmin)-1/2
Ellipse E(u,v) = const
[ ]
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ≅ v u v u v u E ) , ( M
Selec7ng Good Features
λ1 and λ2 both large
Image patch SSD surface
