Fundamentals of Computer Vision Analysis

Paolo Medici
Department of Information Engineering, University of Parma / VISLAB srl

22 October 2025

This book aims to provide a reasonably concise introduction to the fundamentals of geometry, algebra, and statistics necessary for understanding and applying the more advanced techniques in computer vision. As you will see, it includes several elements not strictly related to image processing, but that prove useful for anyone interested in developing complex applications based on image analysis, involving concepts such as tracking and high-level sensor fusion.

To keep the exposition concise, I have generally avoided delving into the proofs of the various theorems, but, with the aim of sparking curiosity, I have left them for the reader to explore." The original goal of this book has never been to provide a rigorous and exhaustive treatment, which often leads to getting lost in calculations and proofs, risking to tire the reader and distract attention from the truly important concepts. Similarly, I did not aim to cover every possible topic related to image processing and computer vision; instead, I have focused only on those subjects directly related to the experiments I have personally conducted in my research activities areas in which I feel most confident and where I can offer some meaningful contribution. The writing of this book has been strongly influenced by my research areas, which focus primarily on applications of computer vision to robotic perception and the development and control of autonomous vehicles.

Computer Vision is an extremely stimulating field of science, even for those outside the discipline. The very fact that geometry, statistics, and optimization are so closely “intertwined” in computer vision makes it a comprehensive area of study, worthy of interest even for outsiders. However, this broad interconnection among topics has not made the task of dividing this book into chapters any easier, and consequently, as will be seen, cross-references between chapters are widespread.

The citations included in the text are very limited; I refer only to works that I consider fundamental, and whenever possible, I have cited the original authors who first proposed the ideas underlying the theory. Reading the articles listed in the bibliography is highly recommended.

For the organization of this volume, I drew inspiration from several books, which I recommend reading, including Multiple View Geometry” (HZ04) by Hartley and Zisserman, “Pattern Recognition and Machine Learning” (Bis06), and “Emerging Topics In Computer Vision” (MK04) edited by Medioni and Kang. For topics more closely related to image processing, an excellent book, also available online, is “Computer Vision: Algorithms and Applications” by Szeliski (Sze10).

Lastly, I would like to note that this book has been written over the past 20 years, and some content may be quite outdated (particularly in the wake of the machine learning revolution that took place in the middle of the last decade) but I have chosen to retain it for historical reasons.

The mathematical notation used in this book is minimalist:

• Matrices are denoted by uppercase bold letters $\mathbf{A}$, while vectors are denoted by lowercase bold letters $\mathbf {x}$;
• The transpose of the inverse of a matrix $\mathbf{A}$, that is, $\left(\mathbf{A}^{-1}\right)^{\top} = \left(\mathbf{A}^{\top}\right)^{-1}$, is written as $\mathbf{A}^{-\top}$;
• In statistical contexts, the notation $\hat{x}$ indicates the estimated value of the quantity $x$;
• Unless otherwise specified, the notation $\prescript{b}{}{\mathbf{R}_{a}}$ denotes a change-of-basis matrix that transforms reference frame $a$ into reference frame $b$, while $\prescript{a}{}{\mathbf{v}}$ denotes a vector expressed in reference frame $a$.

Finally, refer to Appendix B for a brief overview of the meaning of the symbols used in this book.

This document is a brief introduction to the fundamentals of geometry, algebra and statistics needed to understand and use computer vision techniques. You can find the latest version of this document at http://www.ce.unipr.it/medici. This manual aim to give technical elements about image elaboration and artificial vision. Demonstrations are usually not provided in order to stimulate the reader and left to him. This work may be distributed and/or modified under the conditions of the Creative Commons 4.0. The latest version of the license is in https://creativecommons.org/licenses/by-nc-sa/4.0/.

Copyright 2006,2025 Paolo Medici