Implementação do Método de Intersecção de Quádricas – Novo modelo de projecção para sistemas catadióptricos

This project studies the geometry of image formation, specifically of non-central catadioptric vision systems. Such systems, where the camera assumes an arbitrary position around a curved mirror, remain largely untested and unstudied. Existing models are either based on Snell’s Law or on the Fermat Principle. Nevertheless, because such models involve implicit and non-linear equations, multidimensional equation systems solving is required. Therefore, the core objective of this work is to study a model capable of project tridimensional points to the image plane, using straightforward implicit equations. Such model – Quadrics Intersection Method – is based on the intersection of a curved mirror surface (representing a non-ruled quadric) with an analytical mirror. Since the desired reflection point belongs to the resulting quartic curve, it is function of a single unknown. The assumptions are that the camera (and its intrinsic parameters), the quadric, and the 3D point to be projected are known. The Quadrics Intersection Method is a novel framework to project 3D points through curved mirrors to an image device. This approach provides increased computer efficiency and accuracy when compared to Snell’s Law and Fermat Principle. The project begins with the study of the above-mentioned theoretical models, moving on to their implementation. Among other  reakthroughs, it proposes improved restrictions to reflection point determination, and performance-enhancing techniques. Furthermore, the project features the comparison of the three projection models, in terms of performance and precision. Results demonstrate that the Quadrics Intersection method outperforms the remaining ones, for the same performance level and for any of the following kinds of reflecting curves: hyperbolic, parabolic and spherical mirrors. Finally, a simple graphical application that uses the Quadrics Intersection Method to project 3D points on a spherical mirror is introduced.