Concepts and workflow¤

CAMINO performs forward modelling and inference of defocused point-spread functions (PSFs) from telescope exposures. The model connects the optical state of the telescope, represented by the amplitude and phase of a pupil-plane complex field, to the measured detector image through physical optical propagation, detector sampling, and image-formation effects.

This page collects the assumptions, notation, and background needed to understand the CAMINO model and the associated inference workflow.

Physical context¤

The wavefront-sensing and control (WFC&S) problem for a segmented telescope spans a large dynamic range, from the coarse misalignment immediately after deployment to the final diffraction-limited state. For JWST, early deployment stages used geometric and centroid-based techniques to locate and position the segment images. Large coarse piston errors were subsequently corrected using dispersed fringe sensing with the Dispersed Hartman Sensor, which measures relative segment piston through wavelength-dependent interference patterns. Global solutions were then obtained through least-squares reconstruction across the segments.

CAMINO addresses the later, high-precision regime in which the optical state can be inferred from the detailed morphology of defocused PSFs. The weak-lens observations considered here produce extended, approximately annular PSFs rather than sharply peaked in-focus images. This makes robust localisation and physically consistent optical propagation important parts of the inference process.

Optical representation¤

Pupil-plane field¤

The optical field at the entrance pupil is represented as a complex field

\[ U(x,y) = A(x,y)\,\exp\left[i\,\phi(x,y)\right], \]

where:

\(A(x,y)\) is the amplitude transmission of the pupil;
\(\phi(x,y)\) is the optical phase;
\(x\) and \(y\) are transverse coordinates in the pupil plane.

The phase is determined by the optical path difference (OPD). Thus the pupil-plane state contains both the amplitude structure of the aperture and the phase structure of the wavefront.

The forward model propagates this complex field to the detector plane. Because the complex field is retained during propagation, both amplitude and phase information contribute to the resulting intensity pattern.

ABCD matrices and linear canonical transforms¤

CAMINO uses a physical-optics description based on the scalar diffraction approximation. Optical propagation is represented using the Collins diffraction integral in the framework of linear canonical transforms (LCTs).

A first-order optical system is described by an ABCD ray-transfer matrix,

\[ \begin{pmatrix} x_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} x_1 \\ \theta_1 \end{pmatrix}, \]

where \(x\) is the transverse spatial coordinate and \(\theta\) is the propagation angle relative to the optical axis.

For a general ABCD system with \(B\ne0\), the propagated field can be written as the Collins diffraction integral,

\[ U(x') = \frac{1}{i\lambda B} \exp\left(\frac{i\pi D x'^2}{\lambda B}\right) \int U(x) \exp\left(\frac{i\pi A x^2}{\lambda B}\right) \exp\left(-\frac{i2\pi xx'}{\lambda B}\right)\,dx. \]

Here:

\(U(x)\) and \(U(x')\) are the complex fields at the input and output planes;
\(x\) and \(x'\) are the transverse coordinates in those planes;
\(\lambda\) is the wavelength;
\(A\), \(B\), \(C\), and \(D\) are the ABCD matrix elements;
\(B\) has units of length and sets the scale of the diffraction integral.

The quadratic phase factors are the input and output chirps. The cross-term contains the Fourier-transform kernel, with spatial-frequency scaling set by \(\lambda B\).

For the optical system considered here, propagation is represented by a thin lens followed by free-space propagation. The corresponding matrices are

\[ M_{\mathrm{lens}}(f) = \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}, \qquad M_{\mathrm{free}}(L) = \begin{pmatrix} 1 & L \\ 0 & 1 \end{pmatrix}, \]

where \(f\) is the focal length and \(L\) is the propagation distance from the lens to the detector plane. The complete system is therefore

\[ M = M_{\mathrm{free}}(L)\,M_{\mathrm{lens}}(f) = \begin{pmatrix} 1-L/f & L \\ -1/f & 1 \end{pmatrix}. \]

This matrix determines the LCT parameters used for numerical propagation.

Matrix Fourier transform¤

The Collins integral is evaluated numerically using a separable chirp-transform-chirp representation,

\[ U_{\mathrm{out}} = C_{\mathrm{post}}\, \mathcal{F}_{\mathrm{MFT}} \left[ C_{\mathrm{pre}}\,U_{\mathrm{in}} \right], \]

where \(C_{\mathrm{pre}}\) and \(C_{\mathrm{post}}\) are the quadratic phase factors determined by the ABCD coefficients and \(\mathcal{F}_{\mathrm{MFT}}\) is a matrix Fourier transform (MFT).

An MFT is used rather than a fast Fourier transform (FFT) because the input and output sampling rates can be specified independently. This allows the numerical detector grid to be matched directly to the physical detector pixel scale and optical geometry, rather than imposing the sampling relationship associated with an FFT.

The propagation operators are differentiable, allowing derivatives of the forward model with respect to optical and exposure-level parameters.

Defocus¤

Defocus is represented physically by changing the propagation distance rather than by inserting an approximate quadratic phase term into the pupil.

In the absence of defocus, the detector plane is at the nominal focal plane, so

\[ L=f. \]

A defocus displacement \(\Delta z\) is represented by

\[ L=f+\Delta z. \]

The corresponding optical system matrix is

\[ M(\Delta z) = M_{\mathrm{free}}(f+\Delta z)\, M_{\mathrm{lens}}(f) = \begin{pmatrix} 1-(f+\Delta z)/f & f+\Delta z \\ -1/f & 1 \end{pmatrix}. \]

This treats defocus as a displacement of the observation plane and preserves the full propagation operator. It therefore does not rely on the near-focus approximations associated with representing defocus solely as a quadratic pupil-plane phase.

For weak-lensing NIRCam observations, the nominal defocus states are discrete pupil configurations such as WLP8 and WLM8. The corresponding \(\Delta z\) values are nevertheless exposure-dependent model parameters and can be refined during inference to account for uncertainty in the actual optical configuration.

Detector sampling and image formation¤

Detector-plane sampling¤

The propagated field is evaluated on a discrete grid representing the physical detector sampling.

Let:

\(\theta\) be an angular coordinate on the sky;
\(x\) be the corresponding physical coordinate in the focal plane;
\(f\) be the effective focal length.

Under the paraxial approximation,

\[ x=f\,\theta. \]

For a detector with angular pixel scale \(\Delta\theta\) in radians per pixel, the corresponding physical sampling is

\[ \Delta x=f\,\Delta\theta. \]

CAMINO uses an oversampling factor of \(s=4\), giving an effective numerical sampling of

\[ \Delta x_{\mathrm{eff}}=\frac{\Delta x}{s}. \]

The propagated field is consequently evaluated on a grid with

\[ N_{\mathrm{out}}=s\,N_{\mathrm{psf}}, \]

and spacing \(\Delta x_{\mathrm{eff}}\) in each dimension.

The distinction between the optical propagation grid and the native detector grid is important: propagation can be performed at an oversampled resolution before detector-level transformations are applied.

PSF and normalisation¤

The detector measures intensity corresponding to the squared modulus of the propagated complex field,

\[ I_{\mathrm{model}}(x,y)=\left|U(x,y)\right|^2. \]

The resulting PSF is normalised to unit flux over the valid detector region,

\[ P(x,y) = \frac{I_{\mathrm{model}}(x,y)} {\sum_{x,y} I_{\mathrm{model}}(x,y)}. \]

The normalised PSF therefore describes the spatial distribution of the source photons independently of the total source flux.

The observed image is modelled as

\[ I_{\mathrm{obs}}(x,y) = f\,P(x,y)+b+\epsilon(x,y), \]

where:

\(f\) is the total source flux;
\(b\) is a spatially uniform background level;
\(\epsilon(x,y)\) represents measurement noise.

For a fixed PSF, the model is linear in \(f\) and \(b\). These two quantities can therefore be determined analytically for each exposure using weighted linear least squares,

\[ \sum_{x,y} \frac{ \left[I_{\mathrm{obs}}(x,y)-fP(x,y)-b\right]^2 }{ \sigma^2(x,y) }, \]

where \(\sigma(x,y)\) is the per-pixel uncertainty from the calibrated exposure's ERR array.

Analytically solving for flux and background reduces the dimensionality of the optimisation problem. The solution is conditioned on the current PSF model and is therefore recomputed as the forward model changes.

Sub-pixel positioning¤

Each exposure also has a source position relative to the detector grid, represented by a sub-pixel offset

\[ (\delta x,\delta y). \]

Rather than translating the final image as a post-processing operation, this offset is incorporated into the pupil-plane field as a phase tilt before propagation. Under Fourier optics, this corresponds to a shift of the PSF in the image plane.

This keeps source positioning inside the differentiable forward model and ensures that the positional parameters are treated consistently with the optical propagation.

Detector response¤

After propagation and PSF normalisation, detector effects are represented as modular pixel-level transformations. Depending on the instrument model, these can include:

geometric rotations;
intra-pixel sensitivity variations;
downsampling to the native detector resolution.

The resulting forward model maps the pupil-plane optical state to detector intensity through a sequence of propagation, sampling, and detector-response operations.

PSF localisation and cutout extraction¤

The inference operates on fixed-size PSF cutouts extracted from calibrated full-frame exposures. The localisation procedure is model-independent and is designed for extended, defocused PSFs in crowded fields.

Background and source mask¤

For each calibrated science image \(I_{ij}\), the background level is estimated using the median,

\[ m=\operatorname{median}(I_{ij}), \]

and the dispersion is estimated from the median absolute deviation (MAD),

\[ \sigma = 1.4826\, \operatorname{median}\left(|I_{ij}-m|\right). \]

Pixels containing significant source flux are identified using an adaptive threshold. The usual threshold is approximately \(5\sigma\), reduced to about \(3\sigma\) if there is insufficient source support.

Coarse-to-fine localisation¤

An initial flux-weighted centroid is calculated from the full detector frame. The centroid is then iteratively refined using successively smaller square windows with side lengths

\[ L=1024,\quad512,\quad256\ {\rm pixels}. \]

At each stage, the updated centroid defines the centre of the next window. This coarse-to-fine procedure reduces sensitivity to neighbouring sources and large-scale background structure while retaining the extended morphology of the defocused PSF.

The final centroid is calculated from the masked flux,

\[ (i_0,j_0) = \frac{ \sum_{i,j}M_{ij}I_{ij}(i,j) }{ \sum_{i,j}M_{ij}I_{ij} }, \]

where \(M_{ij}\) is the thresholded source mask.

Cutout convention¤

Once the PSF centre has been determined, its coordinates are rounded to the nearest detector pixel and a fixed \(128\times128\) pixel cutout is extracted around that location.

No additional sub-pixel centring is performed at this stage. This avoids introducing a translation that could interact with the asymmetric aberration pattern of the PSF.

If the extraction window reaches the detector edge, missing pixels are padded with the previously estimated background value \(m\).

The resulting cutouts therefore have a common geometry while retaining the complete defocused PSF morphology required by the inference model.

System parameterisation¤

The forward model separates parameters describing the underlying optical state from exposure-dependent image-formation parameters.

Global optical parameters¤

The pupil-plane phase, and hence the underlying OPD, is treated as a global parameter shared across the exposures in an inference set. This assumes that the telescope's underlying optical state does not vary significantly over the time span represented by those observations.

The pupil amplitude can likewise be parameterised to allow small deviations from the nominal aperture model. Such variations represent residual throughput inhomogeneities and can be regularised to favour physically plausible structure consistent with the segmented telescope aperture.

Exposure-dependent parameters¤

Each exposure may have its own nuisance and configuration parameters, including:

total source flux \(f\);
background level \(b\);
sub-pixel source position \((\delta x,\delta y)\);
defocus offset \(\Delta z\).

This separation allows observations with different image-formation conditions to constrain a common underlying wavefront while retaining exposure-specific quantities where required.

OPD and exposure association¤

The optical state used by the model can be associated with archived observations through the telescope wavefront-sensing data.

A uniform Coordinated Universal Time (UTC) grid can be used to identify the nearest OPD solution at a given observation time, using load_wss_opd_by_date. OPD metadata can retain the observation ID, filename, detector, and optical configuration, allowing the inferred wavefront state to be correlated directly with the corresponding archived exposure.

Duplicate OPD realisations are removed, and OPD values outside the desired epoch are excluded.

For the NIRCam weak-lens configuration considered here, candidate exposures are selected from imaging observations using the F212N filter and the WLP8 and WLM8 pupil elements. Exposures are grouped by pupil configuration and sorted by acquisition order. A representative weak-lens pair can then be selected by acquisition date, providing a self-consistent, minimally redundant pair of focus-diverse images for each sampled wavefront state.

End-to-end workflow¤

The concepts above combine into the following forward-modelling and inference workflow:

Load and validate an exposure.
Read the calibrated exposure and its associated metadata, including the detector configuration, filter, pupil state, and uncertainty information.
Localise the PSF.
Estimate the background and source mask, then locate the defocused PSF using the coarse-to-fine centroiding procedure.
Extract the cutout.
Round the final centre to the nearest detector pixel and extract the standard \(128\times128\) pixel region, padding detector-edge regions with the estimated background where necessary.
Construct the pupil-plane model.
Combine the aperture amplitude and the inferred global phase/OPD to form the complex pupil-plane field.
Apply exposure-specific parameters.
Incorporate the relevant defocus state and sub-pixel source position for the exposure.
Propagate the field.
Apply the ABCD/LCT propagation, using the MFT implementation to map the pupil-plane sampling onto the detector-plane sampling.
Form and normalise the PSF.
Compute \(|U|^2\) and normalise the result to unit flux.
Apply detector response.
Apply the relevant detector-level transformations, including any geometric, intra-pixel, and downsampling operations.
Fit exposure-level flux and background.
Given the current PSF, solve analytically for the source flux and background using the calibrated per-pixel uncertainties.
Compare model and data.
Evaluate the residuals or objective function between the forward-modelled image and the observed PSF cutout.
Optimise or diagnose the optical parameters.
Update the parameters describing the shared optical state and exposure-dependent quantities, or inspect the current model without optimisation.
Inspect convergence and residuals.
Examine the inferred parameters, model PSFs, and residual structure to diagnose whether the model adequately describes the observed exposures.

Parameter keying¤

CAMINO stores model parameters in a way that distinguishes both their physical meaning and the exposures to which they apply.

A parameter can therefore be associated with an individual exposure or shared across multiple exposures. The get_key and map_param mechanisms provide the interface for defining and accessing these relationships.

Conceptually, parameter keying separates two questions:

What physical quantity does this parameter represent?
Which observations share the same value of that quantity?

Global optical quantities such as the pupil-plane phase can therefore be shared across a focus-diverse observation set, while quantities such as flux, background, position, and residual defocus can remain exposure-specific.

This parameter-sharing structure is part of the inference model rather than a post-processing convention: it determines which observations jointly constrain each physical parameter.