Advanced Image-Plane Lens Modeling

• Image-plane lens modeling methodology is a suite of techniques that defines the mapping from source to observed image using analytical, numerical, and perturbative methods.
• It employs recursive perturbation, regularized inversion, and neural field representations to accurately capture lensing complexities and systematic effects.
• The approach is pivotal in gravitational lens analysis, optical calibration, and computational imaging, enabling robust recovery of lens parameters and dark matter substructure.

Image-plane lens modeling methodology encompasses a suite of analytical, numerical, and algorithmic techniques designed to infer, optimize, and interpret the mapping from a source plane to an observed image in optical, astronomical, and computational imaging contexts. These methodologies enable precise prediction or reconstruction of source or lens parameters from observed image-plane data, often leveraging mathematical models based on physical lensing equations, forward and inverse problems, optimization under observational systematics, or neural representations for optics and aberrations. Techniques span gravitational lensing, geometric distortion compensation, system calibration, and high-dimensional blur modeling, unified by the foundational principle that the observed image encapsulates key information about lensing geometry, potential, or transfer functions.

1. Analytical and Perturbative Approaches in Multi-plane Gravitational Lensing

Image-plane modeling of gravitational lensing systems with multiple deflecting components (e.g., galaxies or stars along the line of sight) often relies on direct manipulation of the image plane through perturbation theory. A notable technique is the Taylor-series expansion in small mass ratios as described for multi-plane lens systems (Izumi et al., 2010). The lens equation is expanded as

$$z = \sum_{p_2, p_3, ..., p_N} \nu_2^{p_2}\nu_3^{p_3}\ldots \nu_N^{p_N}z_{p_2p_3...p_N},$$

where $z$ (image position) is a function of the source and lens parameters, $\nu_i$ are mass ratios, and $z_{p_2p_3...p_N}$ are expansion coefficients determined iteratively.

At zeroth order, the method reduces to the canonical single-plane mapping. For two-plane systems, one obtains quadratic equations for seed solutions, with higher-order corrections calculated via recursively solving linear relations of the form

$z_n + a z_n^* = b_n,$

where $a$ and $b_n$ depend on solutions at previous orders and explicit lens/source parameters. Importantly, this recursive framework analytically demonstrates the emergence of $2^N$ images for $N$ lens planes, reflecting the lensed-image counting theorem.

However, the expansion's convergence deteriorates near caustics—regions where image positions change rapidly and the mapping becomes singular—limiting applicability in those domains. Numerical comparisons show robust agreement (relative error below ∼1%) between perturbative and ray-tracing solutions outside caustic neighborhoods, with breakdowns for large mass ratios or for images close to primary lensing masses.

2. Optimization-Based Image-Plane Inversion and Regularization Techniques

In gravitational lens modeling of extended backgrounds, modern image-plane methodologies employ forward modeling with iterative optimization—most notably, the semilinear method and matrix-free approaches (Rogers et al., 2011). The process proceeds as follows:

• The observed image is modeled as the convolution (in the image plane) of a lensed source intensity profile and a point spread function (PSF), parameterized as

$b_j = \sum_i s_i f_{ij},$

where $f_{ij}$ encodes the combined lens-mapping and blurring.

• The best-fit source profile $\mathbf{s}$ is found by solving a regularized least squares problem

$$(F^T F + \lambda H^T H)\mathbf{s} = F^T \hat{\mathbf{d}},$$

with $F_{ij} = f_{ij}/\sigma_j$, and including a regularization matrix $H$, e.g., identity or finite-difference operator.

• To avoid memory bottlenecks and ill-conditioning in very large systems, the process exploits matrix-free iterative solvers (CGLS, steepest descent), embedding ray-tracing and fast Fourier convolution as implicit operators.
• Outer-loop nonlinear optimization (for lens mass parameters) uses global optimizers such as genetic algorithms (GA, e.g., Ferret) or particle swarm optimizers (PSO, e.g., Locust), thoroughly exploring parameter space and model degeneracies. GAs, especially with "linkage learning" for correlated parameters, are reported to map degeneracies more exhaustively than PSOs.
• Regularization parameter selection employs L-curve analysis, with the solution chosen at the point of maximum curvature $\kappa=\frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}$ on the plot of source norm vs. image $\chi^2$.

Practically, this approach balances model fidelity and noise suppression, and robustly enforces positivity on physical source intensities. Determination of model degrees of freedom via Tikhonov filter factors or Monte Carlo noise vectors ensures correct statistical interpretation for comparison.

3. Non-Parametric, Basis Set, and Neural Field Representations

To recover complex or non-analytic lens potentials and source structures in the image plane, methodologies leveraging continuous, non-parametric models have been proposed. Notable is the use of shapelet basis sets (Birrer et al., 2015) and continuous neural fields (Biggio et al., 2022):

Basis Set Approach:

• Both the source and the lensing potential are represented via adaptive basis sets, such as 2D Cartesian shapelets. The approach accommodates a hierarchy of spatial scales with a tractable number of parameters.
• Lens mass distributions incorporate softened power-law elliptical profiles for the main potential, with explicit additions for substructure and potential perturbations.
• High-performance Monte Carlo frameworks combine particle swarm optimization for initial exploration and MCMC for posterior mapping and uncertainty quantification, enabling sensitivity to sub-clump masses down to $10^{-4}$ of the main lens mass.

Continuous Neural Fields:

• The lensing potential $\psi(\mathbf{x})$ is modeled as the output of an MLP $\Phi(\gamma(\mathbf{x}); \theta)$, where $\gamma$ applies Fourier features for spectral richness.
• Unlike parametric or pixel grid representations, this yields an arbitrarily high-resolution, differentiable field over the image plane, capturing both smooth bulk structure and localized perturbations.
• Integrated into fully differentiable pipelines (e.g., Herculens, based on JAX/Flax), this method allows for end-to-end optimization of the potential (and, with care, the source), fitting imaging data with or without explicit priors on form and scale (Biggio et al., 2022).

These approaches, by eschewing rigid functional forms, bridge the gap between parametric and pixel-based fitting and provide the flexibility required for automation in large-sample lens modeling.

4. Self-Calibration, Invariant Geometry, and Systematic Control

Robustness to observational systematics and calibration errors is critical in high-fidelity image-plane modeling. Image-plane self-calibration (IPSC) exploits geometric invariants in observed data, notably in interferometric imaging. The Shape–Orientation–Size (SOS) conservation principle states that, for a 3-element system, element-based phase errors induce only a rigid image translation, preserving the triangle defined by salient features in the image (Carilli et al., 2022, Carilli et al., 20 May 2024).

The IPSC process involves:

• Iteratively aligning image-plane interferograms (using Airy disk centroids and cross-correlation with a model image) to correct tip–tilt errors.
• Summing the realigned frames to recover calibrated images with correct visibility amplitudes, while avoiding uv-plane phase correction.
• Demonstrating that, for systems with more elements (e.g., 5-hole masks), phase errors introduce decoherence and higher-order distortions not addressable by simple translations.

These insights are transferable to lens modeling—specifically, by isolating and correcting global shifts in lensed images or source representations without distorting intrinsic morphologies.

5. Geometric and Blur Field Modeling in Computational Imaging

Beyond gravitational and optical lensing, image-plane modeling encompasses geometric deformation and spatially varying blur in computational photography.

3D Geometric Deformation:

• 3D mesh lifting transforms 2D images into 3D, applies localized deformations (by height mappings via Gaussian or sphere functions) in regions of interest (ROI), and flattens back for display (Li et al., 2013).
• Flattening minimizes local distortion metrics per triangle using iterative energy minimization, adjusting both focus and context regions with tunable metric blending.
• This approach excels in preserving shape features across the ROI and context, outperforming classical 2D focus+context lenses in distortion metrics.

High-Dimensional Blur Fields:

• The lens PSF is modeled as a multilayer perceptron (MLP) $\text{PSF}_\theta(x, d, f, u)$, where $x$ is sensor position, $d$ is target distance, $f$ is focus, and $u$ is displacement (Lin et al., 2023).
• The MLP is trained by minimizing a non-blind deconvolution loss over focal stacks of calibration patterns, capturing spatial, defocus, and sensor (pixel-type, microlens) dependencies.
• This unified representation captures variations in defocus, diffraction, aberration, and sensor effects in a parametric, device-specific manner.

Such models permit accurate rendering, deconvolution, and diagnosis of imaging system variations, down to device-specific "blur signatures" even among nominally identical hardware.

6. Systematic Effects, Constraint Selection, and Reliability

The fidelity of image-plane lens modeling depends critically on the selection and distribution of modeling constraints. For strong gravitational lenses, systematic errors in mass, magnification, and image predictability are largely determined by the number and nature of image-plane constraints—specifically, the inclusion of spectroscopic redshifts and their spatial/redshift coverage (Johnson et al., 2016). Key findings include:

• For simulated cluster-lens models, systematic errors in mass and magnification near the Einstein radius drop below 2% when at least $\sim$25 image systems, including several with spectroscopic redshifts, are used.
• Adding further constraints plateaus in effect, with the specific selection (not just quantity) dictating model reliability.
• Absence of spectroscopic redshifts leads to increased systematic biases (e.g., underpredicted mass and magnification). Even a modest number of spectroscopic redshifts substantially increases predictive reliability.
• Reported image-plane rms may understate prediction capability when best-fit redshifts are left entirely free.

This emphasizes that both the absolute number and the distribution of image-plane constraints are essential for minimizing systematics and achieving robust, physically meaningful lens models.

7. Applications and Future Directions

Image-plane lens modeling methodologies underpin a broad range of scientific and engineering applications, including:

• Recovery and interpretation of dark matter substructure and complex gravitational lensing configurations in galaxy/cluster-scale strong lenses (Birrer et al., 2015, Biggio et al., 2022).
• High-SNR recovery of underlying source properties in interferometric or PSF-limited regimes by geometric or neural field calibration techniques (Carilli et al., 2022, Carilli et al., 20 May 2024, Lin et al., 2023).
• Accurate camera calibration and unified system modeling for computer vision, using invertible neural networks for lens distortion and vignetting correction, and jointly optimizing geometric and photometric parameters (Xian et al., 2023).
• Predicting and controlling systematic errors for large-scale lensing surveys by optimizing constraint selection, enforcing positivity and regularization, and precisely mapping uncertainty (Meneghetti et al., 2016, Johnson et al., 2016).
• Novel designs for flat, multicomponent, or meta-optical systems by criteria such as small-angle phase for flat lenses (Ott et al., 2015), and analytic, inverse design using 3D vector Snell's law and surface PDEs (Lu et al., 2016).

Continued development is expected in integrating neural and physical models, automating model complexity selection, and leveraging geometric invariants and self-calibration strategies for even more diverse imaging and lensing applications.