Some more explanations regarding the above article:
As described in the article, I've applied methods related to the Newton method to fit peaks into the EFB03 data. The idea is, to find a local minimum of the square error sum. For differentiable functions, the derivative is zero at the minimum of a function. The Newton method can find such a point efficiently, if the derivative is locally sufficiently similar to a linear function. Unfortunately, the error functions defined immediately by the RMS error of a ("power-law") peak with respect to the actual data isn't that well-behaved at the guessed starting points of the iteration. But by using a sufficiently high power of the RMS error, the resulting function can be made sufficiently well-behaved. The following graphics visualizes the underlying principle for the 1-dimensional case:
Click to view attachmentPDF version:
Click to view attachmentAn explanation of zeta has been pending:
Here two attempts to give the parameter zeta an intuitive meaning, visually (Fourier series and effect on peak)
Click to view attachmentand audible, with each "beat" starting a new damping by increasing zeta in a linear way. Parameter u1 varies with each "beat" in a total of two cycles. The audio version samples only the Fourier series, not the respective derived peak. (The file is packed twice with 7-zip, for effective compression, and to obtain a .zip extension.)
Click to view attachmentAnd back to EFB03:
This graphics shows linear regression data ("raw") of vertical stripes of width 100 pixels obtained from near the left and from near the right side of EFB03,
the best-fit "power-law" peaks with zeta=1, and the residuals (remaining errors).
Click to view attachmentThere appear to be systematic errors with respect to the considered family of peaks. This effect is more distinct near the right of EFB03 than near the left side. I'm presuming, that these deviations can be reduced a little, but not considerably by including zeta as variable peak parameter. Inferring zeta is rather fragile and probably time-consuming, therefore I'm looking for other options first.
My favored approach will be a description by a sum of two peaks, a narrow high and a wide shallow one. Since the horizontal variability of the peak parameters clearly cannot be described by a single peak, I'm considering to investigate multi-peak approaches as the next step to narrow down the 2d-structure of EFB03 stray light.
The additional math needed to feed into some modified Newton method doesn't look difficult at first glance. Although I don't know yet, how well-behaved the approach will be numerically. I guess, that dedicating another about two weeks will return first (continuous) 2d approximates.
Other related tasks appear at the horizon, e.g.
- writing a ray tracer to explain/model the 2d structure of the stray light physically, and
- pinning down the effect of TDI regarding integration over some neighboring color filter (probably responsible for horizontal substripes within framelets).