Estimating multivariate normal distribution by observing variance in different directions

Question

Assume I am looking for a normal distribution $\mathcal{N}(\mu,\Sigma)$. For simplicity let's say we only have 2 random variables $x$ and $y$ and a known $\mu=0$.

Is it possible to estimate $\Sigma$ by observing the variance along multiple directions?

For example, I measure the variance $\sigma_1$ along the vector $\mathbb{v}_1 = (x_1,y_1)^T$. In another step I obtain a different variance $\sigma_2$ from a different direction $\mathbb{v}_2 = (x_2,y_2)^T$. Ideally one would continue to observe these single variations in different directions and combine them in one multivariate normal distribution.

Does this make sense?

EDIT: Some additional background information might be useful: I have a sensor device with known position and orientation in 2D space (in a future step both may have an uncertainty). The sensor is able to measure only the distance of a point along its orientation. I'm also given the sensor model. So for each distance measure $d_i$, I obtain the standard error $\sigma(d_i)$ which depends on the distance.

Since I'm not able to manipulate the sensor position to my advantage or perform a large number of measurements, I'd like to combine these variances into one covariance matrix in order to make a more reliable prediction of the position of the measured point.

This is just a thought that is still under development with no guaranty to work out correctly. Hence my question of "making sense"...

Answer 1

You could, but it doesn't make too much sense to do it this way. Since there are ${n+1}\choose 2$ independent dimensions in a covariance matrix, you'd need to measure the variance in ${n+1}\choose 2$ different directions, and then do some linear algebra to reconstruct the covariance matrix. It's more straightforward just to measure the covariance matrix all at once.

For example, in $2$ dimensions, you could measure the variance along $(0,1)$, $(0,1)$ and $(1,1)$, then since $v_{(1,1)}=\Sigma_{11}+\Sigma_{22}+2\Sigma_{12}$, you would end up with $$\Sigma=\left(\array{v_{(1,0)} & \frac{v_{(1,1)}-v_{(1,0)}-v_{(0,1)}}{2}\\ \frac{v_{(1,1)}-v_{(1,0)}-v_{(0,1)}}{2} & v_{(0,1)}}\right)$$

I suppose if somehow you were constrained in such a way that you couldn't sample from the entire distribution, but you were only able to sample from projections onto a single dimension at a time, then this approach could be useful.