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CSE559A Lecture 18

Continue on Harris Corner Detector

Goal: Descriptor distinctiveness

• We want to be able to reliably determine which point goes with which.
• Must provide some invariance to geometric and photometric differences.

Harris corner detector:

Other existing variants:

• Hessian & Harris: [Beaudet '78], [Harris '88]
• Laplacian, DoG: [Lindeberg '98], [Lowe 1999]
• Harris-/Hessian-Laplace: [Mikolajczyk & Schmid '01]
• Harris-/Hessian-Affine: [Mikolajczyk & Schmid '04]
• EBR and IBR: [Tuytelaars & Van Gool '04]
• MSER: [Matas '02]
• Salient Regions: [Kadir & Brady '01]
• Others…

Deriving a corner detection criterion

• 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

Corner is the point where the intensity changes in all directions.

Criterion:

Change in appearance of window W for the shift (u,v):

E(u,v) = \sum_{x,y\in W} [I(x+u,y+v) - I(x,y)]^2

First-order Taylor approximation for small shifts (u,v):

I(x+u,y+v) \approx I(x,y) + I_x u + I_y v

plug into E(u,v):

\begin{aligned}
E(u,v) &= \sum_{(x,y)\in W} [I(x+u,y+v) - I(x,y)]^2 \\
&\approx \sum_{(x,y)\in W} [I(x,y) + I_x u + I_y v - I(x,y)]^2 \\
&= \sum_{(x,y)\in W} [I_x u + I_y v]^2 \\
&= \sum_{(x,y)\in W} [I_x^2 u^2 + 2 I_x I_y u v + I_y^2 v^2]
\end{aligned}

Consider the second moment matrix:

M = \begin{bmatrix}
I_x^2 & I_x I_y \\
I_x I_y & I_y^2
\end{bmatrix}=\begin{bmatrix}
a & 0 \\
0 & b
\end{bmatrix}

If either a or b is small, then the window is not a corner.