Fundamentals of Computer Vision Analysis

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

1 October 2026

This book aims to provide a reasonably concise introduction to the fundamentals of geometry, algebra, and statistics needed to understand and use the most advanced computer vision techniques. As will become clear, several topics that do not properly belong to image processing have also been included, as they are useful to those interested in developing complex applications based on image processing, involving concepts such as tracking and high-level sensor fusion. To avoid making the discussion unnecessarily cumbersome, wherever possible I have tried not to go into the proofs of the various theorems; instead, with the aim of stimulating curiosity, I have left their treatment to the reader. Indeed, the original goal of this book was never to provide a rigorous and exhaustive treatment, in which one often gets lost in calculations and proofs, risking tiring the reader and distracting attention from some important concepts. Likewise, I have not set myself the goal of discussing every possible topic related to image processing and computer vision; instead, I have limited myself to those topics related to the experiments I have directly addressed in my research activities, with which I am most familiar and on which I can make at least a modest contribution. The writing of this book was in fact strongly influenced by my research areas, which primarily concern applications of Computer Vision to robot perception and the development and control of autonomous vehicles.

Computer Vision is an extremely stimulating field of science, including for those not directly involved in it. The fact that geometry, statistics, and optimization are so closely related in computer vision makes it a comprehensive field of study worthy of interest even to those outside the discipline. However, this broad interconnection among the topics did not make it easier to divide this book into chapters and, consequently, cross-references between chapters, as will become clear, are widespread.

The citations included in the text are very limited, and I refer only to texts that I consider fundamental; whenever possible, I have cited the earliest works that proposed the idea underlying the theory. Reading the articles cited in the bibliography is strongly recommended.

Whenever possible, I have introduced the corresponding English term alongside the Italian term, not out of Anglophilia but to suggest possible keywords to search for on the Internet in order to identify topics related to the one being discussed.

For the organization of this volume, I drew inspiration from several books, whose reading I recommend, 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).

Finally, I would like to emphasize that this book was written over the last 20 years, and some material may be very outdated (especially following the machine-learning revolution that took place in the middle of the last decade), but I like to retain it for historical reasons.

The mathematical notation used will be minimalist:

• matrices will be denoted by uppercase bold letters $\mathbf{A}$, while vectors will be denoted by lowercase bold letters $\mathbf {x}$;
• the transpose of the inverse of a matrix $\mathbf{A}$, that is, $\left(\mathbf{A}^{-1}\right)^{\top} = \left(\mathbf{A}^{\top}\right)^{-1}$, will be written as $\mathbf{A}^{-\top}$;
• in statistical contexts, the notation $\hat{x}$ denotes 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, see 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 2006,2026 Paolo Medici