I came across this blog-post describing an Iterative Least Squares approach for solving a sensor fusion problem. I'm confused why there is a need for an iterative approximation, when I think it can be solved analytically? I am an outsider to this field, so I am hoping someone can point out where I may be misunderstanding.
- The problem is motivated by describing 4 satellites that sense an object's coordinates (illustrated above for 2-D, but abstracted as a 3-D problem).
- The position of the object, sensed by each satellite, is modeled after a multivariate Gaussian distribution $p_{i}(x, y, z) =\left(b^3 / \pi^{3 / 2}\right) \exp \left(-b^2 ((x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i}^{2}\right))$ where $(x_{i}, y_{i}, z_{i})$ is the satellite's position.
- This blog goes on to motivate that because the system is overdetermined there is a need to approximate to recover the most likely position $(x, y, z)$ of the sensed object.
What I'm confused about is, if each satellite models the object's position after a Gaussian, isn't the most likely position of the object the MLE of the product of $n$ multivariate Gaussian PDFs, for $n =$ number of sensors? This also happens to the mean of $N$ multivariate Gaussian PDFs, is this not?
If this is the case, why is there a need to iteratively approximate $(x, y, z)$ when one can find the MLE of $$p_{1}(x, y, z) p_{2}(x, y, z) p_{3}(x, y, z) p_{4}(x, y, z)$$ where $$p_{i}(x, y, z) =\left(b^3 / \pi^{3 / 2}\right) \exp \left(-b^2 ((x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2})\right)$$
The cause of your confusion is that the model of the object's position per satellite is not a multivariate Gaussian. It is a shell around the satellite at a radius $R_i$ determined by the time delay, as can be seen in the image. This shell has an uncertainty which may be modeled by a Gaussian around the radius $R_i$.