Digital Images and Pixels

What is a Digital Image?

A digital image can be understood from two perspectives:

• Computing perspective: A 2D array of pixels stored in memory
• Mathematical perspective: A 2D function $I(x, y)$ giving intensity at any coordinate

The Pixel

A pixel (Picture Element) is:

• In computing: A triple of values for red, green, and blue channels, typically stored as bytes (0-255)
• In mathematics: A point sample with no dimension or area

Important: A pixel is not a box, disk, or tiny light. It is a sample point.

Why This Matters

A pixel as a point sample means:

• No inherent size or shape
• Represents the value at an exact location
• Can be resampled to any output resolution

Image as a 2D Function

Mathematically, an image can be viewed as: $I: \mathbb{R}^2 \rightarrow \mathbb{R}^3$

mapping spatial coordinates $(x, y)$ to a colour vector $(r, g, b)$.

For greyscale: $I: \mathbb{R}^2 \rightarrow \mathbb{R}$

mapping to a single intensity value.

Resolution

Spatial resolution: The number of pixels in each dimension (width × height).

Example: A 1920 × 1080 image has 2,073,600 pixels.

Summary

• A digital image is a 2D array of pixel samples
• Pixels are point samples with no area, not little squares
• Images can be viewed mathematically as 2D functions
• Resolution determines the sampling density

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Past Paper Questions

2024 Paper 3 Question 4: Explain why a pixel should be considered a point sample rather than a small square. What implications does this have for image processing operations?