Fourier Emulator Rendering Technique

• The Fourier emulator rendering technique is a data-driven, differentiable method that efficiently models the Fourier transform of the Sérsic profile in galaxy fitting.
• It employs symbolic regression to derive a closed-form surrogate, reducing computational cost by up to 2.5x compared to traditional approaches.
• The technique integrates with gradient-based optimization workflows, enhancing accuracy and scalability in large-scale astronomical surveys.

The Fourier emulator rendering technique for galaxy profile fitting provides an efficient, accurate, and fully differentiable approach to modeling the Fourier transform of the Sérsic profile, which is a fundamental tool in photometric and morphological analysis of galaxies. Traditional approximations—often relying on computationally expensive oversampling and multi-Gaussian expansions—face limitations due to the absence of a closed-form expression for the Fourier (Hankel) transform of the Sérsic profile. The emulator overcomes these barriers by constructing a symbolic, data-driven formula that can be rapidly evaluated and differentiated, thus facilitating large-scale fitting operations in the context of contemporary and future astronomical surveys (Miller et al., 27 Aug 2025).

1. Sérsic Profile and Challenges in Fourier Representation

The Sérsic profile, parameterized by an index $n$, is a standard description for the surface brightness distribution of galaxies. Its real-space form $I(R)$ is widely used due to its ability to capture a range of light distributions from exponential disks ($n = 1$) to de Vaucouleurs bulges ($n = 4$). The radially symmetric Fourier (Hankel) transform is defined as:

$$\mathcal{F}_r(k) = 2\pi \int_0^\infty I(R) J_0(kR) R\,dR$$

where $J_0$ is the Bessel function of the first kind (order zero), and $k$ is the spatial frequency. No closed-form solution exists for arbitrary $n$, necessitating either numerical integration for each $(k, n)$ or reliance on slow, approximation-heavy expansions, impeding real-time fitting and gradient-based inference.

2. Numerical Computation of the Radial Transform

The methodology initiates by densely sampling the numerically derived values of $\mathcal{F}_r(k)$ using high-precision quadrature (bin width $h = 10^{-4}$), implemented via the “hankel” Python package. The training set comprises 10,000 random $(k, n)$ pairs, with profiles normalized to unity for both total flux and effective radius. Empirical investigation confirms that the Fourier transform varies smoothly as a function of both Sérsic index $n$ and spatial frequency $k$, which is critical to enabling reliable regression-based emulation.

3. Symbolic Regression and Emulator Construction

To obtain a fast and differentiable surrogate, symbolic regression is applied using the “pysr” package. The search process is constrained by a functional template that yields sigmoid-like output, guaranteed to be within $[0,1]$ and to approach the expected limits:

$$\tilde{\mathcal{F}}_r(k, n) = \frac{1}{1 + \exp(G(k, n))}$$

where $G(k, n)$ is discovered by regression and comprises a combination of exponentials, logarithms, powers, and arithmetic operators. The regression optimizes both mean squared error (L2 norm versus numerically integrated “ground truth”) and formula complexity, ensuring that the resulting emulator is robust against overfitting or pathological behavior and compatible with modern autodifferentiation frameworks.

4. Integration and Computational Efficiency in Profile Fitting

The emulator is incorporated into the “pysersic” fitting package, replacing conventional mixture-of-Gaussians or oversampled integrations. Its differentiable and analytic formulation enables direct use in gradient-based optimizers, variational inference, and Markov Chain Monte Carlo (MCMC) workflows. By reconstructing $\mathcal{F}_r(k, n)$ as a succinct expression, each image model rendering becomes faster: expensive operations are replaced with a handful of arithmetic and elementary function evaluations. Comparative runtime assessments indicate a median speed increase of $2.5\times$ over baseline rendering methods.

5. Validation: Accuracy and Reliability

Performance characterization was achieved through injection-recovery experiments: simulated galaxy images (rendered via high-resolution ground truth) were fitted using the emulator, with parameters for $r_e$ and $n$ recovered to within $0.4\%$ fractional bias and $<1.7\%$ scatter. Tests against real HSC-SSP galaxy imaging on 100 GAMA survey galaxies yielded fractional differences below $1\%$ (with scatter below $2.6\%$ for $r_e$ and $4.9\%$ for $n$) compared to the mixture-of-Gaussians hybrid renderer. Inference runtimes—measured for MAP and SVI samplers—were reduced by factors of $2$ to $3$, confirming both efficiency and preservation of fit quality.

6. Limitations and Prospects for Further Development

The emulator maintains high fidelity for moderate Sérsic indices ($n \lesssim 6$), which covers the bulk of extragalactic morphological profiles, but expansion to higher $n$ values remains an objective for future generalization. The symbolic regression approach could further benefit from cost-sensitive evolution of formulas, potentially reducing computation time even more. Alias management, particularly for high-$k$ rendering, may require hybridization with real-space corrections. Future pipelines may also adopt alternative loss functions directly tied to rendered image fidelity rather than only the Fourier domain residuals.

7. Impact and Future Directions in Survey-Scale Astronomy

The Fourier emulator rendering technique directly enables rapid, accurate, and scalable galaxy profile fitting. Its differentiability and computational speed support the shift towards Bayesian inference and uncertainty quantification in large data volumes anticipated from next-generation surveys. The technique streamlines morphological analysis workflows and improves the efficiency of science return, ensuring the sustainability of profile fitting as imaging datasets increase in size and complexity. A plausible implication is that symbolic regression-based emulation of other astrophysical transforms may soon displace brute-force numerical integration across a wide spectrum of model-fitting procedures.