Fitting a 3D line to a 3D line point cloud

Question

I want to fit a 3D line to a 3D line point cloud using numerical optimization. Currently, I'm using Steepest Descent and the error function is a function of $\alpha$, $\beta$, $\gamma$; that is, 3 angles defining the fitting line.

This raises the question of how to validate the result. I want to visualize the function so that I can check the objective function's minima & maxima. Can I do this with MATLAB and should I? Is this a common approach to analyze an objective function when there is no math formation for it?

Answer 1

So, "3d line pointcloud" is apparently a set of points $P_j \in {\mathbb R}^3$ that are close to a line. I'm not sure what you mean by "3 angles defining the fitting line". A straight line in ${\mathbb R}^3$ in general has 4 degrees of freedom, not 3. Three angles determine a rotation, not at all the same thing.

But what I would do is find the average $\overline{P}$ of your points and an eigenvector $V$ of the covariance matrix for its largest eigenvalue. The line can then be represented as $\overline{P} + t V$.

Answer 2

Let your data points be $x_1,\ldots,x_n\in\mathbb{R}^3$. First, de-mean your data points, i.e. calculate $\bar{x}=\frac1n\sum_i x_i$ and set $x_i\leftarrow x_i-\bar{x}$. Then form the covariance matrix $M=\frac1n\sum_{i=1}^n x_ix_i^T$. Find an eigenvector $u$ corresponding to the largest eigenvalue of $M$. Then the desired 3D line is $\{\bar{x}+tu: t\in\mathbb{R}\}$.