# 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.


