The astigmatism calibration is done in AutoTuning.cpp by acquiring an image at
four astigmatism values, with a positive then negative change in the X
stigmator, then in the Y stigmator.  An initial shot is taken with no
astigmatism change to assess CTF fitting.  The CTF is analyzed with either the
internal ctffind or ctfplotter and the recorded results are the smaller and
larger defocus values in microns and the angle in degrees of the axis with the
smaller defocus value (almost certainly, but this should be checked).

The calibration is not analyzed further - it is stored in the CtfBasedCalib
structure defined in AutoTuning.h:
  int magInd;         Mag index
  bool comaType;      Will always be false
  float amplitude;    Astigmatism change on each axis
  int numFits;        Number of stored values
  float fitValues[3 * MAX_CAL_CTF_FITS];    The 3 values per fit, sequentially

To correct astigmatism, an image is taken, the CTF analyzed, and the results
stored in mCtfCal.  The solution is obtained in CtfBasedNextTask at the lines
starting with

     // Astigmatism measurement: Lookup calibration and get angle that image
     // would need to be rotated

where the lookup routine derives the rotation between the mag where the image
is taken and the mag where the calibration was taken.  This angle is used by
adjusting the angles at which defocus differences are measured in the image
relative to the angle used in measuring differences in the calibration.

Fitting arrays xfit, yfit, zfit are set up containing values for 36 angles.
Three routines are used here:

// Returns the defocus at a particular angle implied by the given CTF fit
// The equation here is from Rohou and Grigorieff, 2015
// 0.5 * (df1 + df2 + (df1 - df2) * cos(2 * (angle - axis)))
 float CAutoTuning::DefocusFromCtfFit(float *fitValues, float angle)

// Returns the difference in defocus at a particular angle between two fits
float CAutoTuning::DefocusDiffFromTwoCtfFits(float *fitValues, int index1, int index2,
  float angle)

The two fits are the ones with positive and negative change to the same
stigmator when this is called by:

// Returns the two coefficients of astigmatism at a particular angle from the 4 fits
void CAutoTuning::AstigCoefficientsFromCtfFits(float *fitValues, float angle, float astig,
  float &xCoeff, float &yCoeff)

Here, the difference in defocus at the given angle are taken between the first
and second set of values from X stigmator changes, and the first and second
set of values from stigmator changes.  The coefficients are obtained by
dividing by the total stigmator change, twice the applied change in each
direction.  xCoeff, yCoeff are thus the defocus difference at a particular angle per
unit of X or Y stigmator change, respectively.  These are used to fill xfit
and yfit.  zfit is filled with the actual defocus differences at the
corresponding angles in the image.  The equation fit is

actualDelFocus = xStig * xCoeff + yStig * yCoeff + const

where xStig, yStig are the stigmator changes that would produce the observed
changes.  These are obtained by calling lsFit2:

  lsFit2(xfit, yfit, zfit, numInDat, &solution[0], &solution[1], &intercept);

and the amount to change the stigmators is the negative of the solution.

The appropriateness of fitting with an intercept is not clear.  Probably the
fit would be less biased if the intercept were omitted, which can be done by
passing NULL instead of &intercept.