Fitting details

This page describes the parameterization and fitting machinery built on top of the optical model.

Wavefront parameterization

Single-donut fits

For a single donut, the wavefront is a standard annular Zernike series \(W(u,v) = \sum_j a_j Z_j(u/R, v/R)\), and the fitted parameters are the individual Zernike coefficients \(a_j\).

Multi-sensor fits: double Zernikes

When fitting multiple donuts simultaneously across the focal plane, the wavefront varies with field angle \((\theta_x, \theta_y)\). This field dependence is modelled as a double Zernike (DZ) expansion,

\[ W(u,v;\theta_x,\theta_y) = \sum_{kj} c_{kj}\, Z_k(\theta)\, Z_j(\rho), \]

where \(Z_k(\theta)\) is a Zernike polynomial in field coordinates and \(Z_j(\rho)\) is one in pupil coordinates. DZMultiDonutModel fits the coefficients \(c_{kj}\) directly.

Sensitivity matrix parameterization

DZBasisMultiDonutModel instead fits amplitudes of a user-supplied sensitivity matrix \(S_{mkj}\) (e.g., derived from a detailed optical model), which expresses the wavefront as a linear combination of double Zernike basis functions:

\[ W = \sum_m b_m \sum_{kj} S_{mkj}\, Z_k(\theta)\, Z_j(\rho). \]

Figure 2 illustrates how the accuracy of the TA wavefront approach improves as the Zernike order of the decomposition increases, using quiver plots of the residual deviation from real batoid raytracing.

Atmospheric PSF

The optical donut image is convolved with a Kolmogorov atmospheric kernel parameterized by FWHM (or second-moment tensor \(I_{xx}, I_{xy}, I_{yy}\)). The convolution is performed in pixel space via cv2.filter2D (or scipy.signal.fftconvolve as a fallback).

Loss function

The default loss is the standard chi residual

\[ \chi_i = \frac{d_i - m_i}{\sqrt{\sigma_i^2 + m_i}}, \]

where \(\sigma_i^2\) is the read-noise variance and \(m_i\) is the model value (Poisson shot noise). The systematic_loss variant adds a fractional noise floor \((\alpha \, m_i)^2\) to the denominator, capping the per-pixel SNR and down-weighting pixels where unmodeled correlated residuals dominate.