Error Functions

Choose a parametric model to represent a set of features.

For example: line detection

Issues

• Noise int eh measured feature locations
• Extraneous data: clutter (outliers), multiple lines
• Missing data: occlusion

Vertical Distance

Standard least squares approximation. But the problem with vertical least-square error is that it causes huge discrepancies as

Total Least Squares

A line is represented as: $$ax + by = d$$ Unit Normal: $$\hat{n} = [a, b]$$ The error metric between a point and a line: $$E = \sum_{i = 1}^n (ax_i + by_i - d)^2$$

Find (a, b, d) to minimize the sum of squared perpendicular distances

$$\frac{de}{dh} = 2 (U^T U) h = 0$$

where $h = [a, b]$ and $U$ is a matrix of differences of averages for each point.

We can solve this with the SVD trick.

Least Squares as likelihood maximization

Generative Model

Assume the line points are corrupted by Gaussian noise in the direction perpendicular to the line. $$\begin{pmatrix} x \\ y\end{pmatrix} = \begin{pmatrix} u \\ v \end{pmatrix} + \epsilon \begin{pmatrix} a\\ b \end{pmatrix}$$

Robust Estimator

General approach: minimize $\sum_i \rho(r_i (x_i, \theta); \sigma)$

where $r_i (x_i, \theta)$: residual of $i^{th}$ point w.r.t. model parameter $\theta$

$\rho$: robust function with scale parameter $\sigma$

For example: $$\rho(u;\sigma) = \frac{u^2}{\sigma^2 + u^2}$$ For a small residual $u$ (i.e. small error) relative to $\sigma$, this behaves much like the squared distance: we get $\sim u^2 / \sigma^2$. As $u$ grows, we get $\sim 1$

The error function we defined is very sensitive to scale.