GaussianPSF

The GaussianPSF model represents the microscope point spread function as an isotropic 2D Gaussian function. While this is a mathematical approximation rather than a physical model derived from diffraction theory, it provides excellent computational efficiency for rapid prototyping and performance-critical algorithms.

Mathematical Model

The GaussianPSF is defined as:

\[I(x, y) = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{x^2 + y^2}{2\sigma^2}\right)\]

where:

  • $x, y$ are coordinates in physical units (microns)
  • $\sigma$ is the standard deviation in the same units

This function is normalized to integrate to 1 over the entire domain, ensuring energy conservation.

Constructor and Parameters

GaussianPSF(σ::Real)
  • σ: Standard deviation in microns, representing the width of the PSF

Alternative Constructor

GaussianPSF(psf::AiryPSF)  # Create from an Airy PSF

Key Features

  • Computational Efficiency: Fastest PSF model in the package, using a simple closed-form expression
  • Simplicity: Simple mathematical form makes it ideal for prototyping
  • Approximation: Provides a reasonable approximation of the central peak of diffraction-limited PSFs

Examples

Creating a Gaussian PSF:

# Create a PSF with 150nm standard deviation
psf = GaussianPSF(0.15)

# Create a Gaussian approximation of an Airy disk
airy_psf = AiryPSF(1.4, 0.532)  # NA=1.4, wavelength=532nm
gaussian_approximation = GaussianPSF(airy_psf)  # Automatically sets appropriate σ

Relationship to Airy Function

The Gaussian2D model can approximate the Airy disk pattern using the empirical relationship:

\[\sigma \approx 0.22 \frac{\lambda}{\text{NA}}\]

where λ is the wavelength and NA is the numerical aperture. This approximation works best near the center of the PSF.

Limitations

  • No Diffraction Rings: Doesn't capture the diffraction rings present in real microscope PSFs
  • No Defocus Modeling: Can't model effects of defocus or 3D imaging
  • No Aberrations: Doesn't account for optical aberrations
  • Simplified Physics: Mathematical approximation rather than physically derived model
  • Less Accurate at Edges: Diverges from physical PSFs at larger distances from the center

For standard usage patterns, camera integration, and comparison with other PSF types, see the PSF Overview.