This is a question from a physics experiment I am currently working on and I would try to translate it to a mathematical form to explain certain problems/doubts I am facing and ask the community to experts here for a possible solution.
The Problem
The figure above shows two planes. In my experiment these planes denote-
- Pixel Values on hardware device which is a 2d array of pixels which can be individually modified. The individual pixels are shown as boxes that I have drawn using a paint app. (gray plane, bottom one in the figure)
- A detuning plane that shows a constant value of an ideal detuning that we hope to achieve. The detuning values are produced based on the individual pixel values in the bottom plane through some kind of a transformation function $f(z, \rho)$ . Note: In the real experiment this transformation happens through some physical interactions that we are not concerned about at the moment.
The goal is to set the pixel values in the bottom plane in such a way that after passing through the transformation function $f(z, \rho)$ they all land up on the ideal detuning plane with the ideal detuning value given by $\delta^*$.
Limitations
Now, we do not know what exactly the $f(z, \rho)$ function looks like. The transformation function is essentially unknown to us. However we do have access to limited amount of information about the transformation from some very specific measurements that we can do. In the real experiment it translates to having a camera (refer figure) that can do measurements for the detuning along the $z$ axis but has no information regarding the variation for the detuning along the $\rho$ axis as there is only one camera available along the $z$ axis (see fig). One example measurement is shown by a rough hand drawing in the bottom right corner of the figure, for detuning values across the z axis. The ideal detuning is shown by dotted straight line $\delta^*$ and the actual value of the measured detuning is the curvy line on top of it.
The only information we have is that the pixel values (PV)-vs-detuning transformation $f$ is monotonic. If you increase pixel value, the detuning increases and vice versa. The way in which the PVs are set along the z axis is simple interpolation and the steps are shown below.
- Select a particular z value along the $z$ axis.
- Set a random pixel value for that $z$, such that its detuning is less than ideal $<\delta^*$
- Set another value such for that $z$, such that its detuning is more than ideal $>\delta^*$
- Now on this PV-vs-$\delta$ plot for that particular $z$, interpolate between these two points. They will obviously intersect the $\delta^*$ line at some point and the corresponding PV will the ideal PV for that $z$ value.
Now this is not quite true as, the $PV-vs-\delta$ relation is not linear (known from theory) and the we are averaging along the $\rho$ direction as we do not have a camera to directly have detuning measurements along that direction. In effect this translates to having
- all $(z_1,\rho_i)$ where $i=1, 2, 3, ...$ have the same pixel values i.e. PVs are set along different $z$ values and each column of $z$ has same PV for all $\rho$s.
My Question
Now, I want to know, is there any way, without having a camera to make measurements along the $\rho$ direction and only having a camera to do measurements along the $z$ axis and using that detuning-vs-z position as feedback, is there any systematic way to change PVs along the $\rho$ direction and improve the final measurement outcome? For instance, with only doing the $\delta-vs-z$ measurement, can I in some systematic way change $(z_1,\rho_i)$ where $i=1,2,3,...$ so that we have an improvement in the same measurement outcome?
Can we use some similar techniques like interpolation along the $\rho$ direction too so that I can have variation of pixel values along both rows and columns (as opposed to single columns along z having all same $\rho$ values) and improve the outcome?
An example image above showing the current PV variation along the z axis. Note that each column $z_i$ has the same $(z_i,\rho_j)$ pixel values for all $j'$s.
An example pattern that I want to create with radial variation is shown above. If you look closely into columns $z_6$ or $z_7$ you can clearly see that there is some variation along $\rho$ direction too. Here I can created a random pattern where I shifted all PVs in the top 1/3rd of the rectangle by $-10$, kept the middle part same and shifted all PVs in the bottom 1/3rd of the rectangle by $+10$. What I want to know is if there is a systematic way in which I can shift the pixel values along $\rho$ to improve the measurement of $\delta-vs-z$ values.