Chapter 3 — Pixels - Signals to Transformers

What the Grid of Numbers Actually Means¶

Chapter 2 explained how the sensor produces one integer per photosite. This chapter asks: what is that integer, what are its limits, and what happens when you try to represent too much with too few?

13.1 What a Pixel Is — and Is Not¶

A pixel is a single number: the average brightness measured by one photosite at one grid location. It has:

A position $(i, j)$ in the 2D array ( $i$ = row, $j$ = column)
A value — a non-negative integer representing brightness
No internal structure — it is just a number, not a tiny photograph

When a screen renders an image it draws each pixel as a coloured square. That is a display convention, not a property of the data. The underlying data is a 2D array of integers.

Common misconception: zooming into a digital image reveals more detail. It does not — it only makes the squares bigger. All the detail that exists was fixed at capture time by the sensor’s spatial sampling rate (Chapter 1).

23.2 Spatial Resolution¶

Spatial resolution = the number of pixels: width × height. More pixels means finer spatial sampling, which preserves more scene detail.

But pixels alone say nothing about real-world scale. The same 1000×1000 pixel image could represent a postage stamp or a football field. What matters for computer vision is the ground sampling distance (GSD):

\text{GSD} = \frac{\text{physical scene width (mm)}}{\text{image width (pixels)}}

(1)

GSD is the real-world size each pixel represents. It determines the smallest detectable feature:

\text{minimum detectable feature} \geq 2 \times \text{GSD}

(2)

(The factor of 2 is the Nyquist criterion from Chapter 1, applied to spatial resolution.)

Industrial inspection example:

Setup	GSD	Bolt (20 mm)	Scratch (2 mm)
5×5 pixel grid	20 mm/px	1 pixel	sub-pixel — invisible
25×25 pixel grid	4 mm/px	5 pixels	0.5 pixel — barely detectable
100×100 pixel grid	1 mm/px	20 pixels	2 pixels — detectable

33.3 Quantization¶

After the sensor measures a voltage proportional to electron count, the ADC maps it to a discrete integer. This mapping — rounding a continuous value to the nearest level — is quantization.

For $B$ -bit depth, there are $2^B$ levels. The step size between adjacent levels is:

\Delta = \frac{V_{\max}}{2^B - 1}

(3)

The quantization error is at most $\pm \Delta/2$ .

Bit depth	Levels	Max error	Usage
8-bit	256	0.5 intensity units	JPEG, display
12-bit	4096	~0.06 intensity units	Camera RAW
16-bit	65536	~0.008 intensity units	Scientific imaging

Quantization error is deterministic and irreversible — unlike shot noise (random), it cannot be averaged away. Once the ADC rounds a value, the fractional part is gone.

43.4 False Contours¶

When bit depth is too low, smooth gradients in the scene produce false contours — visible step edges where the true scene varies smoothly.

A 2-bit image has only 4 grey levels (0, 85, 170, 255). A smooth gradient from 0 to 255 jumps in steps of 85 intensity units. Each step is visible as a hard edge — a feature that doesn’t exist in the scene.

False contours are the spatial equivalent of quantization noise. They disappear when bit depth is increased to 8-bit or beyond.

53.5 Two Sources of Error — So Far¶

Source	Type	Scales with	Controllable?
Shot noise (Ch 2)	Random	$\sqrt{S}$	Partially — more light helps
Quantization error (Ch 3)	Deterministic	$\Delta/2$	Yes — more bits eliminates it

Both corrupt pixel values independently of scene content. We have not yet accounted for contrast changes, shading, colour processing, or compression — those come in Chapters 4–5 and dominate the problem in Chapter 6.

Spatial resolution — watch face at different pixel densities

Sampling and quantization — continuous scan line converted to discrete pixels

Continuous image projected onto sensor grid → pixelated output

Quantization artefacts — skull at 8-bit vs fewer bits, false contours appear

Run: uv run python tutorials/00_introduction_to_digital_images/part3_pixels_and_resolution.py to generate the GSD and bit-depth simulations.

6Summary¶

Concept	Key fact
Pixel	Single integer = average brightness at $(i,j)$ ; no internal structure
Resolution	Width × height; more pixels = finer spatial sampling
GSD	Physical size per pixel; sets the smallest detectable feature
Nyquist (spatial)	Minimum 2 pixels per feature period to avoid aliasing
Quantization	Rounding to $2^B$ levels; error ≤ $\Delta/2$ ; irreversible
False contours	Visible step edges from coarse quantization on smooth gradients

Next → Chapter 4 — Contrast and Dynamic Range: we have integer pixel values — what do those values mean, and what happens when the scene’s brightness range exceeds what the sensor can record?