One of the most fundamental sources of noise in digital imaging sensors is photon shot noise, arising from the discrete and probabilistic nature of light. Even under perfectly stable illumination, the number of photons arriving at a sensor site over a fixed integration time fluctuates due to quantum randomness. This phenomenon obeys Poisson statistics.
In this post, we’ll explore how photon noise behaves under different quantum efficiencies (QE), first assuming an ideal sensor with QE = 1 and then relaxing that assumption to model more realistic cases.
Photon Noise with QE = 1
In an ideal sensor, every incident photon generates a photoelectron. If the average number of photons arriving during the exposure is μ, the actual number of detected photoelectrons n follows a Poisson distribution:
P(n; μ) = (μⁿ * e^(−μ)) / n!
This distribution is discrete, defined only for non-negative integers, and characterized by:
- Mean = μ
- Variance = μ
- Standard deviation (photon noise) = √μ
This model underlies many noise simulations in image sensor analysis. It implies that the signal-to-noise ratio (SNR) due to photon noise alone is:
SNR_photon = μ / √μ = √μ
Photon Noise with QE < 1
In real-world sensors, not every incoming photon gets detected. If QE < 1, each photon has a probability q = QE of generating a photoelectron. In this case, the situation becomes a two-stage random process:
- The number of incident photons N still follows a Poisson distribution:
P(N; μ) = (μᴺ * e^(−μ)) / N! - Given N photons, the number of detected photoelectrons n follows a Binomial distribution:
P(n | N) = C(N, n) * qⁿ * (1−q)^(N−n)
The overall probability distribution for n is then a compound distribution known as the Poisson Binomial, but in this special case, it simplifies to another Poisson distribution:
P(n; μ·q) = ((μ·q)ⁿ * e^(−μ·q)) / n!
However, while the number of detected photoelectrons is still Poisson-distributed (with mean μ·q), the total variance of the system includes an additional scaling due to QE. The key point is that:
- Mean detected signal = μ·q
- Photon noise variance = μ·q
- SNR_photon = √(μ·q)
This result assumes that the QE is independent of wavelength and angle, and that no additional noise sources (e.g., read noise, PRNU) are included.
Why This Matters
- Low QE sensors have inherently worse SNR due to fewer detected photons.
- Photon noise is signal-dependent, which means that bright areas are relatively less noisy than dark ones.
- When comparing sensors, QE directly affects the effective photon-limited dynamic range.
Photon noise is unavoidable, but its magnitude depends on both the average light level and the sensor’s quantum efficiency. While the math is cleanest in the ideal QE = 1 case, real sensors behave similarly, but with fewer electrons per photon, and correspondingly worse SNR.
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