ComputerVision101_Week1_Discretization

university computervision week1 theory

Discretization

Source:

• HC1a Images and Interpolation
• https://rvdboomgaard.github.io/ComputerVision_LectureNotes/LectureNotes/IP/Images/index.html

Big picture

The lecture starts with a continuous image:

$$f:{\mathbb{R}}^2\to \mathbb{R}$$

But computers cannot store infinitely many positions and infinitely precise values.

So we must make the image discrete in two ways:

1. Discretize the domain: sampling
2. Discretize the range: quantization

One-sentence summary

Sampling answers: “At which positions do we measure?”

Quantization answers: “Which numbers are we allowed to store?”

Mental model

Imagine a smooth brightness surface.

• Sampling puts a grid on top of it
• Quantization rounds the values so the computer can store them

Without these two steps, digital images would not exist.

Formula view

Continuous image:

$$f:{\mathbb{R}}^2\to \mathbb{R}$$

Discrete image:

$$F:{\mathbb{Z}}^2\to \mathbb{R}$$

And in practice, the stored values are often also quantized to integers such as:

$$F(i,j)\in \{0,1,\dots ,255\}$$

Why interpolation appears after discretization

Once you only know the image values on a grid, you often still want the value in between grid points.

That is why interpolation comes next:

$$\text{continuous}\to \text{discrete}\to \text{estimate continuous again}$$

Links

• ComputerVision101_Week1_Sampling
• ComputerVision101_Week1_Quantization
• ComputerVision101_Week1_Interpolation