Computational Imaging

SNR of N fused exposures at a fixed signal per view or a fixed total dose, and the synthetic NA, finest period and space-bandwidth gain of Fourier ptychography.

Model: Textbook approximation

Single-view SNR

31.40

S / √(S + B + σ_r²)

Fused SNR, N = 25

157.02

√N × single view

Synthetic NA

0.500

Space-bandwidth gain 25.0× over one coherent image

Finest period, synthetic

1100 nm

Coherent, objective alone: 5500 nm; incoherent widefield: 2750 nm

Number of views NSignal per view (e⁻ per pixel)Background per view (mean e⁻ per pixel)Read noise per exposure (e⁻ rms)Objective NAIllumination NA (largest LED angle)Wavelength (nm)

Fused SNR vs number of views

Finest resolved period vs illumination NA

Key Formulas

One view: SNR₁ = S / √(S + B + σ_r²)

Same signal per view: SNR_N = √N · SNR₁

Same total dose: SNR_N = S / √(S + B + N·σ_r²)

Synthetic aperture: NA_syn = NA_obj + NA_ill, finest period λ / NA_syn

Space-bandwidth gain: (NA_syn / NA_obj)²

The N views are independent and summed with equal weights after registration; B is the mean background under the signal, so it adds its shot noise. Splitting a fixed dose costs only read noise, so fusion is free in the shot-noise limit. The synthetic aperture is that of Fourier ptychography: coherent images under tilted plane-wave illumination stitched in Fourier space, which needs overlapping pupils and a phase-retrieval solver that converges. With NA_ill = NA_obj the coherent synthetic cutoff equals the incoherent widefield limit λ/(2NA_obj).

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Independent exposures summed with equal weights after perfect registration; each adds the shot noise of its signal and mean background and one read noise. The synthetic aperture NA_obj + NA_ill is the coherent Fourier-ptychography cutoff; pupil overlap, phase-retrieval convergence and aberrations are not modelled.

Janesick J. R. (2007). Photon Transfer DN → λ. SPIE Press. Shot, background and read noise of a sensor pixel; N reads add N σ_r². doi:10.1117/3.725073
Zheng G., Horstmeyer R., Yang C. (2013). Wide-field, high-resolution Fourier ptychographic microscopy. Nat. Photonics 7, 739–745. Objective NA 0.08; maximum synthetic NA 0.5, set by the outermost LED angle (illumination NA ≈ 0.42, implied, not stated). doi:10.1038/nphoton.2013.187

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