Spectral reconstruction

[Askaniy]

Problem: the reconstructed spectrum (now and after implementing kriging) is not one of the solutions to the inverse problem in the general case.

There are two solutions:

• Write an algorithm that corrects the interpolated spectrum so that it is a solution to the inverse problem (close reference)
  • Advantages: may work with any interpolator
  • Disadvantages: interpolator cannot take into account filter profiles; filter profiles may overlap and spectrum may be strangely distorted
• Use an optimization method where the parameters are the grid step points, and the function under study is the correspondence of the resulting spectrum to the original data (with smoothness regularization).
  • Advantages: spectrum is reconstructed using all available information, not limited to the filter profiles mean points
  • Disadvantages: probably slow and loses the benefits of kriging

The advantages and disadvantages are approximately equal, but the first option is a priority.
All other options listed in the review article are less suitable. We don't have enough spectral cubes to train a neural network using space images.

Extended description of the second solution algorithm:

1. Select number ($n$) of points-parameters (5 nm step → 9 points for visible range, 10 nm → 8, …, 640 nm step → 2 points);
2. First approximation of point values (flat spectrum?);
3. $2n$ interpolations to 5 nm step: for each point, make a spectrum with an upward and a downward deviation by $\Delta y$ (my smoothed linear interpolation algorithm was developed specifically for this);
4. $2n$ convolutions with all filters;
5. Calculate the root-mean-square difference between the expected values and the values for each spectrum variation;
6. Changing point values according to the gradient;
7. Repeat in a loop from step 3 until the resulting interpolated spectrum turns out to be a solution to the inverse problem with sufficient accuracy.