ComputerVision101_Week1_ImageDefinition_DomainAndRange

university computervision week1 theory

Image Definition, Domain, and Range

Source:

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

Main definition

The lecture defines an image as

$$f:E\to V$$

This means:

• $E$ is the domain
• $V$ is the range

At every position $x\in E$, the image gives a value $f(x)\in V$.

Domain

The domain tells you where you are measuring.

For a flat image:

$$E\subseteq {\mathbb{R}}^2$$

So a point in the image is something like

$$x=(x,y)$$

This is just a position on the camera sensor or image plane.

The lecture notes also point out:

• for ordinary photographs, $E\subseteq {\mathbb{R}}^2$
• for scans / volumetric data, $E\subseteq {\mathbb{R}}^3$
• the coordinate frame can change while the underlying signal stays the same

Range

The range tells you what kind of value is stored at each position.

Examples:

Grayscale:

$$f(x)\in \mathbb{R}$$

Color:

$$f(x)=(r(x),g(x),b(x))\in {\mathbb{R}}^3$$

Binary image / mask:

$$f(x)\in \{0,1\}$$

Segmentation labels:

$$f(x)\in \{0,1,2,\dots ,K\}$$

The handwritten notes phrase the binary case as:

• indicator image
• binary mask / segmentation

with an example such as:

• 0 = background
• 1 = person

and note that semantic segmentation just uses more labels.

Easy way to remember it

• Domain = input = position
• Range = output = measured value

So:

$$\text{position}\mapsto \text{brightness/color/label}$$

Why this matters

This is the foundation for the rest of the lecture:

• If the domain becomes discrete, that is sampling
• If the range becomes discrete, that is quantization

Links

• ComputerVision101_Week1_WhatIsAnImage
• ComputerVision101_Week1_Sampling
• ComputerVision101_Week1_Quantization