Point Spread Function (PSF)

• Point Spread Function (PSF) is a fundamental concept that describes the response of an imaging system to a point source, encapsulating blurring from physical optics, atmospheric effects, and instrumental aberrations.
• PSF modeling employs analytical functions, data-driven approaches, and hierarchical methods to enable accurate photometry, astrometry, and weak lensing analyses in astronomy and microscopy.
• Engineered and computationally optimized PSFs, using deep learning and phase-controlled designs, advance imaging precision by tailoring responses for specific tasks under challenging conditions.

The point spread function (PSF) is a fundamental concept in optical imaging, astrophysics, and computational imaging, describing the response of an imaging system to a point source. It encapsulates all blurring mechanisms introduced by physical optics, the atmosphere, instrument aberrations, and detector effects. Accurate PSF models are essential for high-precision science, particularly in contexts such as weak gravitational lensing, photometry and astrometry in astronomical surveys, microscopy, and computational imaging.

1. Mathematical Formulations and Physical Origins

The PSF, denoted PSF$(x, y)$ or (in 3D) PSF$(x, y, z)$, is the impulse response of an imaging system, i.e., the image obtained in response to an ideal point emitter. For incoherent imaging (e.g., fluorescence, astronomy), the system is linear in intensity and the observed image $I_{\rm obs}$ is related to the sky (object) intensity $O$ via convolution: $$I_{\rm obs}(x, y) = [O * \mathrm{PSF}](x, y) + N(x, y)$$ where $N$ is noise, and $*$ denotes 2D convolution (Infante-Sainz et al., 2019).

In coherent imaging, the impulse response is a complex function $h(x, y, z)$ and the observed field is linear in the input field $U_{\rm in}$: $$U_{\rm out}(x, y, z) = \iint h(x - x', y - y', z) \, U_{\rm in}(x', y', 0)\;dx'\,dy'$$ For incoherent systems, the observable PSF is given by $|\!h|^2$ (Rahman et al., 9 Feb 2025).

In many physical systems, especially ground-based telescopes, atmospheric turbulence broadens the PSF core and introduces power-law wings. Analytic forms used include the Moffat profile: $\mathrm{Moffat}(r) = [1 + (r/\alpha)^2]^{-\beta}$ where $\alpha$ controls the core width and $\beta$ the power-law tail; as $\beta \rightarrow \infty$, the profile approaches a Gaussian (Li et al., 2016, Infante-Sainz et al., 2019).

In multi-element optics, aberrations and misalignments modulate the effective PSF; wavefront error can be expanded in Zernike polynomials, and the observed PSF is the squared modulus of the Fourier transform of the aberrated pupil function (Jia et al., 2020).

2. PSF Modeling Techniques and Representations

2.1 Analytical Basis Function Expansions

Moffatlet and Gaussianlet Bases

To reconstruct the PSF from stellar images, Li et al. introduced "Moffatlets" and "Gaussianlets," orthonormal radial basis functions constructed from Moffat and Gaussian profiles, respectively. Ellipticity is imparted via affine coordinate transforms. Moffatlet bases provide a more physically realistic match to PSF wings, essential for accurate size and ellipticity recovery in weak-lensing pipelines (Li et al., 2016).

Principal Component Analysis (PCA) and EMPCA

EMPCA performs expectation–maximization PCA on an ensemble of stellar images, finding orthogonal principal components (PCs) capturing PSF variability. Each PSF is reconstructed as a weighted sum of the leading PCs. EMPCA accommodates arbitrary PSF complexity but may be susceptible to contamination of PCs by noise, adversely affecting PSF moment estimates (Li et al., 2016).

2.2 Data-driven and Non-Parametric Approaches

PSF Modeling via Matrix Factorization and Deep Learning

Matrix factorization methods such as MCCD decompose the PSF field over the focal plane into global (across-CCD) and local (per-CCD) components, combining low-rank basis functions with spatially varying coefficients. Deep neural networks, such as encoder–decoder architectures with convolutional and residual blocks (e.g., Tel–Net), learn field-dependent mappings from sparse star stamps to the entire PSF field, outperforming traditional interpolation, especially in the regime of low SNR and sparse sampling (Liaudat et al., 2020, Jia et al., 2020).

Interpolation over Sparse Sampling

Interpolation schemes include polynomial fitting, radial basis functions, Delaunay triangulation, and Kriging (Gaussian process regression). Kriging uses the spatial covariance of measured coefficients to provide minimum-variance, unbiased estimates at arbitrary field points, and provides the lowest residuals in ellipticity and size correlations but is computationally intensive and may require further development for Stage-IV surveys (Bergé et al., 2011).

2.3 Hierarchical and Multi-layer Methods

Hierarchical PSF reconstruction (e.g., in Hyper Suprime-Cam data) uses a three-layer architecture: (1) per-exposure polynomial fitting, (2) principal component modeling of stacked residuals across all exposures, and (3) global correction of residual patterns correlated with field distortion. This procedure enables removal of stable, instrument-induced residuals beyond the reach of single-exposure techniques, achieving sub-milliarcsecond systematic control in weak-lensing fields (Alonso et al., 24 Apr 2024).

3. PSF Engineering and Computational Design

Recent advances in computational imaging enable direct synthesis and optimization of PSFs for specific tasks:

• Universal 3D PSF Engineering: Cascaded phase-only diffractive surfaces can realize arbitrary spatially and spectrally varying 3D PSFs within the classical diffraction limit. Optimization involves minimizing an $L_2$ loss between desired and simulated PSFs by adjusting phase patterns on each layer, subject to sampling and feature constraints. Multi-layer configurations allow mapping between volumes for snapshot 3D and multispectral imaging (Rahman et al., 9 Feb 2025).
• Implicit Neural Phase Function Representation: Basis-free representation of the pupil phase function $\phi(x, y)$ using a coordinate-based neural network, trained end-to-end to fit a target PSF's spatial characteristics. This enables smooth, resolution-agnostic phase pattern synthesis exceeding the capabilities of pixel-wise or Zernike-limited designs, and achieves higher structural similarity and signal-to-noise for target PSF sets (Valouev, 7 Oct 2024).
• Wavefront-controlled PSF Engineering: Vectorial PSF models, incorporating polarization and emitter orientation, are essential for high-precision single-molecule localization. Deliberate phase engineering with spatial light modulators (SLMs) allows for custom 3D+$\lambda$ PSFs; calibration protocols provide residual wavefront error control at the 10–20 m$\lambda$ level (Siemons et al., 2018, Schneider et al., 2023).

4. Measurement, Calibration, and Empirical Construction

4.1 Laboratory and Astronomical Calibration

• Empirical PSF Stacking: Extended PSFs in astronomical imaging require stacking thousands of stars at different brightnesses, with separate treatments for core and wings, iterative masking, and normalization at junction radii. Resulting two-dimensional PSFs encapsulate both instrumental and atmospheric scattering to large radii (e.g., SDSS PSFs out to 8′), necessary for crowding-corrected photometry and low-surface-brightness studies (Infante-Sainz et al., 2019, Bocchio et al., 2016).
• Defocus PSF Characterization: Defocus-induced PSFs can be described by Gaussian kernels with object-dependent radii parameterized from thin-lens equations. Model parameters are solved by minimization of a weighted sum of image differences, SSIM, Laplacian focus loss, and a defocus-histogram loss between simulated and real defocused image series (He et al., 2022).
• Semi-blind Recovery from Occulted Images: PSFs can be reconstructed from images where parts of the scene are known to be occulted (zero intensity), by solving a multi-linear system for PSF weights in segmented domains, iteratively updating scene and PSF estimates. Central and tail accuracy at the sub-percent level is achievable, with broad applicability to laboratory and astronomical calibration (Hofmeister et al., 2022).

4.2 X-ray and Specialized Optical PSFs

• XMM-Newton 2D PSF Model: A comprehensive 2D (energy, off-axis, azimuth) parametric PSF model for the EPIC telescopes incorporates King+Gaussian envelopes, primary/secondary spokes, and gross shell deformations, rigorously improving both spurious-source suppression and astrometric accuracy. This modular approach underpins standard SAS data analysis and calibration (Read et al., 2011).
• Physical-Optics PSF Simulation: For systems where surface microroughness is significant, direct computation of the PSF as the squared modulus of the Fourier transform of the complex pupil function—including all phase errors—is validated to $<5\%$ against experiment. This approach provides a unified treatment of figure error and microroughness, superseding ray-tracing approximations for high angular resolution imaging (Tayabaly et al., 2016).

5. PSF Modeling in Large Survey Data Pipelines

5.1 Pipeline Structures

• Dark Energy Survey Y6/PIFF: PSFs are modeled per CCD and exposure using a pixel grid (PixelGrid) and BasisPolynomial interpolation in spatial coordinates and color, with moments up to fourth order as diagnostics. Cross-band color-dependent interpolation is critical for controlling differential chromatic refraction and tree-ring-induced size ripples, achieving systematics $\sigma_{\rm sys}^2 \lesssim 10^{-7}$ (Schutt et al., 10 Jan 2025).
• MCCD (Multi-CCD Modeling): Simultaneous modeling of local per-CCD and global focal-plane-wide PSF basis components, with spatially constrained coefficients, enhances the ability to capture cross-CCD features and main field-scale patterns. Regularization, positivity, and sparsity constraints ensure robust, noise-suppressed modeling; empirical validation yields 16–22% lower pixel-level RMSE than per-CCD approaches (Liaudat et al., 2020).
• Interpolation for PSF Modeling:
  • Polynomial, RBF, Delaunay, Kriging: Comparative studies have found that Kriging (Gaussian process) interpolation provides superior performance in minimizing residual PSF-induced systematics, particularly for the ellipticity field correlations critical in weak lensing (Bergé et al., 2011).
• Deep Learning Approaches: Novel methods such as CNNs rapidly estimate PSF parameters from star cutouts to feed forward-modeling pipelines (e.g., MCCL). Deep Wiener Deconvolution Networks leverage known PSFs for efficient, non-blind image restoration, outperforming fixed-PSF or average-PSF approaches and achieving unbiased recovery of galaxy color, flux, and shape statistics (Herbel et al., 2018, Wang et al., 2022).

6. Impact, Limitations, and Future Directions

• Precision Requirements for Cosmology: Weak gravitational lensing cosmology demands PSF size and ellipticity errors below $10^{-3}$ in fractional size and absolute ellipticity. Instrumental systematics and atmospheric variability necessitate PSF models that exploit multi-exposure, multi-band, and cross-instrument calibration, as well as regularization against overfitting to noise (Li et al., 2016, Alonso et al., 24 Apr 2024, Schutt et al., 10 Jan 2025).
• Physical Constraints and Sampling: Diffraction limits, spatial bandwidth, axial and lateral resolution are bounded by the optical system’s aperture and detector sampling. Universal PSF engineering is constrained by the number of input/output degrees of freedom in diffractive layers (Rahman et al., 9 Feb 2025).
• Instrumental and Laboratory Optimization: In beam-diagnostic systems using scintillating screens, the PSF is a convolution of lens-aberration and scintillator-geometry blur, demanding correction down to the micro-meter scale for precision profile recovery (Novokshonov, 5 Nov 2024).
• Future Algorithms: Upcoming surveys (LSST, Euclid) will require composite physically-motivated models combining optical, atmospheric, and detector kernels, global fitting frameworks exploiting temporal and spatial persistence, and potentially hybrid nonparametric–physical-optics approaches for PSF field learning (Liaudat et al., 2020, Schutt et al., 10 Jan 2025).
• Task-specific and Engineered PSFs: In computational imaging, advanced deep-learning and coordinate-network representations of the phase function enable rapid, continuous, and high-fidelity PSF modulation for custom imaging tasks in microscopy and optical information processing (Valouev, 7 Oct 2024, Rahman et al., 9 Feb 2025).

The continued development of data-driven, physically grounded, and computationally efficient PSF modeling strategies remains central to extracting unbiased physical information from current and next-generation imaging systems across scientific domains.