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.