Sigma 35/1.4 LoCA on D850 revisited

This is the 30th post in a series of Nikon D850 tests. The series starts here.

There was some discussion in the comments to the previous post about the way I’m presenting the results of the longitudinal chromatic aberration (LoCA) testing when I use the Nikon D850 Focus Shift Shooting (FSS) feature to step the focal plane. Even at the minimum step size, the FSS makes steps that are too large to adequately sample the way that MTF50 of a sharp lens varies with focused distance. 

I have been using the Excel spline feature when I presented the results, but, as Jack Hogan quite properly pointed out, that results in not-credible curves. So yesterday I spend some time coming up with a way to fit more-likely-accurate curves to the data. It was somewhat of a challenge, because the FSS minimum step size is so large that I only have 10 or 11 even marginally useful data points, and I therefore need the dimensionality of the vector that controls the fitting function to be quite small to avoid overfitting the data. Five dimensions is about as large as I’d like to go, and three or four would be better. I tried simple Gaussian PDF functions, but they weren’t quite the right shape. I also tried polynomials with complete lack of success, and rational polynomials (thanks, Frans!) with even worse results. I finally found what I was looking for: two Gaussian PDFs summed, with a fixed y-axis offset. The control space has four dimensions: the mean (mu) of each Gaussian PDF (they are forced to have the same mu), the standard deviation of each, and a scale number that is multiplied by the sum of the two Gaussian PDFs before the offset is added in. You would think that it would be necessary to separately weight each PDF before summing them, but I’m getting good fits without doing that, and that would add a dimension.

I’m now showing the data points to which the curves are fitted and well as the curves, so that you can judge the goodness of fit. You are of course free to ignore the curves and just look at the data points.

Now I’ll re-present the Sigma 35 mm f/1.4 results of the last post with the new methodology.

As before, the vertical axis is MTF50 in cycles per picture height. The horizontal axis is the FSS step number, and is completely arbitrary, since the FSS does not return the lens to the starting focussed distance at the end of the sequence (Nikon: please consider that for a firmware update.) and therefore the lens needs to be manually refocussed before each sequence. I constrained the plots to 11 data points, but I can see that the x-axis spans are not all the same, probably due a rounding error in the code; I’ll fix that in the future. The data points are the x’s.

Is that a better way to look at things?