Background
I am working with imaging data and I have computed what are called Haralick features that measure the "texture" of an image. It is common in my field to measure the features in the x direction (left/right) and the y direction (up/down), but for this you get different answers if you rotate the image and then measure again. I am investigating this "rotational invariance".
I have simulated measuring these haralick features on images and rotations of those images. It looks like each haralick feature has a "true" value, just when I measure it in the x direction I get the "true" value plus a sine wave and in the y direction I get the true value plus a cosine wave. I am trying to work out if given just the values of a haralick feature in the x and y direction at one specific rotation, can I calculate the "true" value?
The Problem
The problem simplifies to something like this. I have two equations
$$x = \mu + \alpha\sin(\theta)$$ $$y = \mu + \alpha\cos(\theta)$$
If I know the value of $x$ and $y$ but I don't know $\mu$, $\alpha$ or $\theta$, is it possible to use $x$ and $y$ to work out just $\mu$?
Normally you can't solve 2 equations with 3 unknowns but I only want to solve for one of them. I am guessing the solution will involve transforming the equations to be in terms of $\mu$ and some dummy variable $\phi$, where $\phi$ is a function of $\alpha$ and $\theta$. Then it would be possible to solve for 2 unknowns with 2 equations but I can't work out how to do this in this case.