Sampling and Quantisation

Continuous to Discrete

The physical world is continuous, but computers work with discrete values. Two processes convert between them:

Sampling

Sampling maps a continuous function to a discrete one:

• Spatial coordinates: continuous → discrete pixel positions
• A continuous image $I(x, y)$ becomes samples $I[i, j]$

Sampling an Image

$I_{\text{sampled}}[i,j] = I(x_i, y_j)$

where $(x_i, y_j)$ are the discrete sample positions on a rectangular grid.

Sampling Rate

The sampling rate (or spatial frequency) determines how densely the continuous signal is sampled:

• Higher rate: more samples, better representation of detail
• Lower rate: fewer samples, risk of aliasing

Quantisation

Quantisation maps continuous values to discrete ones:

• Intensity values: continuous → discrete levels
• Each colour channel typically quantised to 8 bits (256 levels)

Quantisation Levels

For $b$ bits per channel, there are $2^b$ discrete levels:

• 8 bits: 256 levels (0-255)
• 10 bits: 1024 levels
• 12 bits: 4096 levels

Relationship Between Sampling and Quantisation

Process	Domain	Effect
Sampling	Spatial	Determines spatial resolution
Quantisation	Intensity	Determines tonal resolution

Both contribute to image quality and file size.

Summary

• Sampling discretises spatial coordinates
• Quantisation discretises intensity values
• Higher sampling rate captures more detail
• More quantisation levels capture smoother gradients

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Past Paper Questions

2023 Paper 3 Question 3: What is meant by “sampling an image”? How does it differ from quantisation?