Pysersic Profile Fitter: Galaxy Image Modeling

• Pysersic Profile Fitter is a computational tool that decomposes galaxy images by fitting analytical Sérsic profiles to capture diverse morphological features.
• It integrates Bayesian inference with differentiable Fourier-space rendering and symbolic regression to enable rapid, scalable, and accurate parameter estimation.
• The approach supports robust uncertainty quantification, validated on large datasets, making it ideal for current and future deep, large-volume imaging surveys.

The Pysersic Profile Fitter refers to a class of computational tools and algorithms for decomposing and modeling galaxy images, particularly through parametric fitting of their radial surface brightness distributions. Fitting the Sérsic profile—an analytic expression that captures both bulge and disc morphologies—forms the methodological core of most profile fitting packages. Recent innovations in the Pysersic Profile Fitter paradigm, including efficient Fourier-space rendering using symbolic regression, have enabled rapid and differentiable likelihood evaluation, ensuring scalability for future deep, large-volume imaging surveys.

1. Mathematical Formulation of the Sérsic Profile

The Sérsic profile models the surface brightness $I(R)$ as a function of radius $R$ via:

$$I(R) = I_e \exp\left\{ -b_n \left[ \left( \frac{R}{R_e} \right)^{1/n} - 1 \right] \right\}$$

where $I_e$ is the intensity at effective radius $R_e$, $n$ is the Sérsic index (controlling the light concentration), and $b_n$ ensures half the total light falls within $R_e$, typically determined by solving $\Gamma(2n) = 2\gamma(2n, b_n)$ with $\Gamma$ and $\gamma$ denoting the complete and incomplete gamma functions.

Extensions to the Sérsic model include broken exponentials (disc truncations), core-Sérsic (central light deficits), and auxiliary models such as Gaussian, Moffat, and Ferrers profiles for specialized features (rings, spiral arms, bars, point sources).

2. Bayesian Inference and Probabilistic Modeling

Modern implementations such as pysersic (Pasha et al., 2023) employ Bayesian frameworks—via probabilistic programming libraries like numpyro on top of jax—to infer galaxy structural parameters. This paradigm supports uncertainty quantification and efficient posterior sampling using gradient-based methods (Hamiltonian Monte Carlo, No U-Turn Sampler). Additionally, stochastic variational inference (using normalizing flows) allows for rapid, approximate posterior estimation, with inference times for complex models typically measured in minutes on contemporary hardware.

Key advantages of Bayesian inference for galaxy profile fitting:

• Prior selection and parameter constraints
• Full uncertainty propagation
• Flexibility for arbitrary parametric model definitions

3. Fourier-Space Rendering and Symbolic Regression

A central computational challenge—particularly for Bayesian inference and large survey applications—is efficient and differentiable rendering of model images. Direct Fourier transforms of the Sérsic profile lack a closed analytic form, historically necessitating expensive numeric approximations (e.g., mixture-of-Gaussians).

Recent work (Miller et al., 27 Aug 2025) introduces symbolic regression—using tools such as pysr—to discover an approximate analytic emulator for the radial Fourier transform:

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

where $G(k, n)$, constructed by symbolic regression, ensures the correct limiting behavior ($F \to 1$ as $k \to 0$, $F \to 0$ as $k \to \infty$) and differentiability. The symbolic regression operated over a large set of candidate functions, trained using 10,000 numerical Hankel transforms (with $n$ uniformly drawn from $0.4$ to $6.5$, $k$ from an exponential distribution).

This analytic emulator matches the accuracy of numerical methods with mean residuals below $0.5\%$ across injection-recovery tests, and median differences below $1\%$ for $r_e$ and $n$ in real galaxy applications. Critically, it enables a $2.5\times$ reduction in computational runtime for gradient-based inference.

4. Implementation in Software

Open-source software packages such as pysersic (Pasha et al., 2023), Profiler (Ciambur, 2016), and ProFit (Robotham et al., 2016) have integrated advanced methodologies for Sérsic profile fitting:

• Profiler: Focused on 1D isophotal decomposition, with a Tkinter-based GUI and support for a broad suite of analytic profile functions. PSF convolution (Gaussian, Moffat, user-defined vector) performed in 2D, guaranteeing flux conservation and accounting for component ellipticity. Input data typically originates from isophotal measurements, providing key information (ellipticity, position angle, harmonic amplitudes) for accurate light profile reconstruction.
• ProFit: Bayesian 2D photometric fitting leveraging a fast C++ backend for adaptive pixel integration (e.g., recursive oversampling near steep gradients). Separation of optimization and model rendering supports flexible use of optimization algorithms and easy extensibility to new profile forms. Benchmarks demonstrate ProFit surpasses GALFIT in both performance and accuracy for $0.5 < n < 8$.
• pysersic: Pure Python, powered by jax (for JIT compilation and autodiff) and numpyro. Features modular rendering engines (real-space oversampling, pure Fourier, hybrid methods). The latest symbolic regression-based Fourier emulator fully exploits hardware accelerators (CPU, GPU, TPU), drastically speeding up both MCMC and variational inference approaches.

5. Scientific Applications and Validation

Profile fitting is central to extragalactic astronomy—enabling measurement of key morphological parameters (total flux, $r_e$, $n$), bulge-disc decomposition, and component isolation (bars, rings, nuclear sources). Rigorous validation is demonstrated:

• Case Studies (Ciambur, 2016):
  • NGC 3348 (core-Sérsic profile, $R_b \approx 0.43''$, $\gamma \approx 0.09$, $n \approx 4.91$).
  • Pox 52 (nucleated Seyfert, comparison of Moffat vs. vector PSF, showing sensitivity of central parameter estimation).
  • NGC 2549 (multi-component edge-on system, use of Sersic bars and truncated disc models).
• Large Survey Implications (Robotham et al., 2016):
  • Automated multi-component decomposition for SDSS, KiDS, LSST imaging.
  • Data quality greatly influences parameter recovery; moving from SDSS to KiDS offers large gains, while KiDS to LSST yields diminishing returns given similar pixel scales and PSF.
• Injection-Recovery Tests & Real Galaxy Fits (Miller et al., 27 Aug 2025):
  • Model recovery via symbolic regression emulator yields sub-percent residuals and preserves parameter estimates.

6. Performance, Efficiency, and Scalability

The shift to differentiable analytic emulators in Fourier space enables rapid likelihood evaluation and gradient computation, facilitating:

• Bayesian inference on large datasets (scaling to millions of galaxies).
• Hardware-accelerated execution (CPUs, GPUs, TPUs).
• Seamless integration into modern analysis pipelines.

Traditional numeric and mixture-based approaches are increasingly outpaced by symbolic regression emulators in terms of both computational speed ($2.5\times$ faster) and the ability to handle gradient-based optimization or MCMC posterior sampling.

7. Outlook and Extensibility

The modular design of tools such as pysersic allows for future support of arbitrary parametric models, including extensions beyond traditional Sérsic profiles (broken, core, exponential, Gaussian, Moffat, Ferrers, edge-on disc). As extragalactic surveys further expand in scope and depth, these techniques ensure profile fitting algorithms remain both robust and computationally tractable.

In summary, the Pysersic Profile Fitter encapsulates the convergence of analytic modeling, probabilistic inference, and efficient computation for galaxy photometric and morphological analysis. Through innovations such as symbolic regression-based Fourier emulation, these tools are essential for reproducible, scalable, and rigorous exploration of galaxy structure in the era of large astronomical datasets.