Chapter 2 — The Sensor

From Photons to Electron Counts¶

Chapter 1 established what sampling means mathematically. This chapter asks: what does the hardware physically do at each sample point? The answer involves photons, quantum mechanics, and irreducible randomness.

12.1 The Photosite¶

A camera sensor is a grid of photosites — tiny light-sensitive elements, one per pixel. Each photosite executes the sampling operation from Chapter 1 at its spatial location. During the exposure time it:

1. Collects photons arriving from the focused scene

2. Converts photons to electrons via the photoelectric effect (one photon → at most one electron)

3. Accumulates electrons in a potential well until readout

4. Reads out the charge as a voltage, then converts it to a digital integer via an ADC

The integer that emerges is the pixel value. It is a count of electrons — a proxy for the number of photons that arrived, which is itself a proxy for scene brightness.

22.2 Photon Counting Is Inherently Random¶

Photon arrivals are a Poisson process: independent random events occurring at some average rate $\lambda$ (photons per second per photosite area). Even if the scene is perfectly uniform and the sensor is perfect, the count in any one exposure is random.

For a Poisson process with mean $\lambda$:

The standard deviation equals the square root of the mean. This is shot noise — not an instrument defect, but a fundamental consequence of the discrete nature of light.

Implication: even a perfectly flat, perfectly lit, perfectly uniform scene produces pixel values that vary from photosite to photosite. The variation scales as $\sqrt{\lambda}$.

32.3 Signal-to-Noise Ratio¶

Signal is the mean electron count $S$. Noise has two contributions:

• Shot noise: $\sigma_{shot} = \sqrt{S}$ — from photon statistics

• Read noise: $\sigma_r$ — from the amplifier and ADC circuitry

Total noise in quadrature:

At high signal ($S \gg \sigma_r^2$) this simplifies to:

Doubling the signal improves SNR by $\sqrt{2} \approx 41\%$.

42.4 Key Sensor Parameters¶

Mean electron signal: $S = \phi \cdot A \cdot \text{QE} \cdot t_{exp}$ where $\phi$ is photon flux and $t_{exp}$ is exposure time.

52.5 Phone vs DSLR — Why Sensor Size Matters¶

A DSLR photosite is roughly 25× larger than a phone photosite. Same scene, same exposure time → DSLR collects 25× more photons → $\text{SNR}_{DSLR} \approx 5\times \text{SNR}_{phone}$.

This is not about optics quality or processing — it is pure geometry. A larger bucket catches more rain.

Critical consequence for computer vision: pixel values from a phone and a DSLR of the same scene are not numerically comparable, even at the same exposure settings. The pixel value encodes sensor physics, not just scene brightness.

62.6 The Pixel Value Equation¶

where $f$ is the ADC mapping and $\sigma_{dark}$ is dark current noise. Every term except $\phi$ (photon flux from the scene) is a camera-specific nuisance. Two cameras imaging the same scene produce different pixel values — not because the scene differs, but because their hardware parameters differ.

This is the root of the problem explored in Chapter 6.

Run: uv run python tutorials/00_introduction_to_digital_images/part2_sensor_physics.py to generate SNR vs brightness curves and the shot noise simulation.

7Summary¶

Next → Chapter 3 — Pixels: the sensor produces a grid of integers — what does that grid mean, and what are its limits?