Computer Vision: Introduction

Image and Camera Fundamentals

Overview

Image Fundamentals Pixels, Color Spaces RGB, HSV, Grayscale Camera Model Pinhole Camera Focal Length, Field of View Coordinate Transforms Homogeneous Coordinates Rotation, Translation, Scale Perspective Projection 3D to 2D Projection depth information is lost

The Introduction covers the foundations for studying computer vision: the structure of digital images, mathematical camera models, and the basics of coordinate transforms.

Learning Objectives

  • Understand the structure of digital images and color spaces
  • Understand the pinhole camera model
  • Master homogeneous coordinates and transformation matrices
  • Understand projection from 3D to 2D

Table of Contents

  1. Digital Image Fundamentals

    Pixels, resolution, bit depth

  2. Color Spaces

    RGB, HSV, YCbCr, Lab, color conversion

  3. Pinhole Camera Model

    Focal length, field of view, perspective projection

  4. Homogeneous Coordinates and Transformation Matrices

    Unified representation of translation, rotation, and scaling

  5. Camera Matrix

    Intrinsic parameters, extrinsic parameters

  6. Exercises

    Summary exercises for the Introduction

Prerequisites

  • Basics of linear algebra (matrices, vectors)
  • High school mathematics (trigonometric functions)

Key Concepts

Pinhole Camera Model

Projection from a 3D point $(X, Y, Z)$ in camera coordinates ($Z > 0$) to image coordinates $(u, v)$ ($K$ with zero skew):

$$\begin{pmatrix} u \\ v \\ 1 \end{pmatrix} = \dfrac{1}{Z} K \begin{pmatrix} X \\ Y \\ Z \end{pmatrix}, \quad K = \begin{pmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{pmatrix}$$

Homogeneous Coordinates

A 2D point $(x, y)$ is represented as $(x, y, 1)$, and a 3D point $(X, Y, Z)$ as $(X, Y, Z, 1)$. Nonzero multiples (by $k \neq 0$) represent the same point, and dividing by the last component returns the ordinary coordinates.

This allows translation to be expressed as matrix multiplication.

Camera Matrix

$$P = K[R | t]$$

$K$: intrinsic parameters, $R$: rotation matrix, $t$: translation vector (in the world-to-camera direction $X_c = R X_w + t$). Multiplying the homogeneous coordinates of a world point by $P$ and dividing by the third component gives the image coordinates.

Reading

  • History of Computer Vision — From the 1966 MIT Summer Vision Project to today's foundation models and generative AI: a short overview of 60 years of computer vision.