# Surface sampling in Scanbox

Answering a request from colleagues in London, the upcoming version of the Scanbox firmware will allow users to change the depth of imaging on a line-by-line basis.

Up to now, during volumetric imaging, users were allowed to change the setting of the electronically tunable lens (ETL) once per frame (during the transition from one frame to another).

The new version offers the possibility of changing the depth ($z$) as a function of the line number ($y$).  For example, the image of a pollen grain below, was obtained using a sinusoidal modulation in depth.

There are at least two potential uses for this new feature.

First, one can increase the yield of imaged cells by first measuring a z-stack and then designing a (smooth) sampling surface $z = f(y)$ that maximizes the number of cells that can be imaged.

For example, assume the projection of cell bodies within the volume is given by the scatter-plot below. Then sampling with the solid curve line will yield many more cells (red circles) than sampling with any horizontal plane (such as the dashed horizontal lines).

I will explain how to calculate an optimal (smooth) function of depth that maximizes the number of cells sampled in a separate post (the algorithm is too large to fit in the margin here.) [Actually, I added it below].

The surface sampling method offers a compromise between random access and volumetric imaging across planes, many of which may not contain many cells, thereby reducing the temporal resolution unnecessarily.  Have any groups done something like this before?

The second use of this new feature is that it can allow us to correct for the ringing in ETL focal plane that results from a step change.

If you have been doing volumetric imaging, where N frames are sampled at each depth, you might have realized that a few of the lines at the top of the frame on the first frame after a depth transition is screwy.  This is due to the ringing of the ETL. One can potentially use the ability to change ETL command on the fly to compensate for this ringing.

I have not yet implemented this.  Are there any Scanbox users doing volumetric imaging that are willing to help?

Algorithm:

Given a set of cell body projections $(y_i, z_i)$ we want to find a smooth function $z=f(y)$ that intersects as many cell bodies as  possible.

We do this by restricting $z(y)$ to be sum of the first $N$ harmonics, $z(y) =a_0+\sum_{k=1}^N a_k \cos ( 2 \pi k y + \phi_k )$, so it is both smooth and periodic as well, which will prevent ringing in the ETL during fly back to the first line. The function has $2N+1$ parameters.  For simplicity, here we normalize the total number of lines in the frame is normalized to be in the range $[0,2 \pi]$.

Given a set of parameters, denote by $d_i$ the minimum distance between the location of cell $i$ and any of the points on the curve $z(y)$. Our objective function is $J= \sum_{k=1}^M \tanh ( (d_i - r_0) \beta )$.  Here, $r_0$ represents the average radius of a cell and $\beta$ controls the sharpness of the error function near that boundary, $M$ is the total number of cells in the volume. For a give sent of points, we used Matlab’s fminsearch to find the optimal parameters for the curve.

How much improvement in yield can we expect using surface sampling versus just one plane?

We ran a few simulations where the number of points is uniformly distributed within the volume and calculated the fraction of cells we can intersect as a function of the number of harmonics used. Zero harmonics means just the horizontal plane that intersects the maximum number of cells.  The graph on the left shows the results.

Even in the simple case of a uniform distribution one can more than double the yield expected from a single plane by having a few (~5-6) harmonics. In other words, using surface sampling we can double the number of cells with respect to a horizontal plane without any penalty in temporal resolution.  Not bad at all.

We will re-do this analysis with some actual volumetric data from our Lab soon. I suspect this estimate represents a lower bound on what can actually be achieved.