Introduction to Computer Vision

Course Details

• The lecturer is Prof. Andrew Zisserman
• Teaching Assistants include Dr. Iro Laina, Dr. Johannes Linder, Oisin Mac Aodha, and Simon Hadfield
• Lectures are on Tuesdays from 14:00-16:00 in the Rex Thompson Room
• Practicals are on Fridays from 14:00-16:00 in the Thom Building
• The course website is available on the department website
• Recommended books are "Computer Vision: Algorithms and Applications" by Richard Szeliski, and "Computer Vision: Models, Learning, and Inference" by Simon Prince
• Online discussions will be held on Piazza

Course Overview

• An introduction to computer vision will be provided
• Image formation principles described
• Feature detection and matching algorithms explained
• Image segmentation techniques discussed
• Object recognition methods covered
• 3D vision explored
• Motion analysis techniques investigated

Assessment

• Practicals are worth 50% of the final grade
• Practical assessments are based on correctness, efficiency, and code clarity
• Late practical submissions are penalized
• The exam is worth 50% of the final grade
• The exam is closed book
• Exams test understanding of concepts and algorithms
• Past exam papers are not available

What is Computer Vision?

• Computer vision involves enabling computers to "see"
• It includes extracting meaningful information from images
• Computer vision aims to mimic human vision capabilities
• Applications include image search, robotics, medical imaging, security, and autonomous driving

Image Formation

• How images are formed is studied
• Camera models are used
• Geometric transformations are implemented
• Radiometry concepts are introduced
• Color science is used

Pinhole Camera Model

• This camera model is a simple representation
• It has no lenses and images are inverted
• It uses perspective projection to map 3D world points to 2D image points
• The pinhole camera model equation: $x = f \frac{X}{Z}$, $y = f \frac{Y}{Z}$, where $(X, Y, Z)$ is a 3D point, $(x, y)$ is the 2D projection, and $f$ is the focal length
• The Matrix form (including camera center C) $\begin{bmatrix} x \ y \ 1 \end{bmatrix} = \begin{bmatrix} f & 0 & 0 \ 0 & f & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} R & -RC \end{bmatrix} \begin{bmatrix} X \ Y \ Z \ 1 \end{bmatrix}$, where $R$ is the rotation matrix and $C$ is the camera center

Radial Lens Distortion

• Lenses introduce distortion in images
• Radial distortion, including barrel and pincushion distortion, is common
• Correction for distortion: $x_{corrected} = x(1 + k_1r^2 + k_2r^4 +...)$, $y_{corrected} = y(1 + k_1r^2 + k_2r^4 +...)$, where $(x, y)$ is the original point, $(x_{corrected}, y_{corrected})$ is the corrected point, $k_1, k_2$ are distortion coefficients, and $r = \sqrt{x^2 + y^2}$

Homogeneous Coordinates

• Points are represented in projective space
• An extra coordinate is added for representation
• A 2D point $(x, y)$ becomes $(x, y, 1)$
• A 3D point $(x, y, z)$ becomes $(x, y, z, 1)$
• Transformations can be represented as matrices using this system
• Scaling equivalence: $(x, y, z) = k(x, y, z)$
• A point at infinity: $(x, y, 0)$
• Original line equation: $ax + by + c = 0$ becomes $\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix} = 0$

Coordinate Transformations

• Translation transformation: $\begin{bmatrix} x' \ y' \ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \ 0 & 1 & t_y \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}$
• Rotation transformation: $\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} cos\theta & -sin\theta \ sin\theta & cos\theta \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$
• Scaling transformation: $\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} s_x & 0 \ 0 & s_y \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$
• Affine transformation: $\begin{bmatrix} x' \ y' \ 1 \end{bmatrix} = \begin{bmatrix} a & b & c \ d & e & f \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}$
• Projective transformation (Homography): $\begin{bmatrix} x' \ y' \ w \end{bmatrix} = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}$, where $x = x'/w$ and $y = y'/w$

Radiometry

• Radiometry is the measurement of light
• Irradiance: power measured per unit area
• Radiance: power measured per unit area per unit solid angle
• BRDF (Bidirectional Reflectance Distribution Function) is important
• A Lambertian surface reflects light uniformly in all directions
• A Specular surface reflects light in a mirror-like direction

Color

• Color perception is a key concept
• Color spaces include RGB, HSV, and Lab
• Color constancy is an important feature
• White balance is considered