An introduction to dynamical camera analysis with analytical mechanics and surface path motion - Andrade, Gabriel Conceição (2024)
Structure from motion (SfM) is a computer vision problem that aims to obtain three-dimensional scenes from a set of images captured from different viewpoints without prior knowledge of the camera configuration. SfM involves two primary steps: camera pose estimation and geometric recovery. In the initial phase, algorithms are employed to determine the position of a camera based on a single image and 3D scene, or using two or three images where the objective is to utilize one image to determine the positions of the other two cameras relative to it. The current investigation commenced by implementing a three-view-based reconstruction within a widely-used structure from motion system and examining the regions deemed problematic in the dynamical system of the model arising from fast polynomial homotopy continuation. Due to the high complexity of the model, locating these regions was deemed impractical at the time. The model itself was constructed using generic formulations adapted to the problem, necessitating an alternative approach. The work of Fabbri and Kimia provides foundational concepts of differential geometry for multiview reconstruction and basic notions on continuous time approaches for cameras, focusing on the velocity and acceleration of curves. Analyzing the camera structure, similarities with Lagrangian and Hamiltonian mechanics were identified. This work extends the concepts of Fabbri and Kimia by incorporating Lagrangian and Hamiltonian mechanics and introducing a model of camera path constrained by a surface. This constraint is utilized not only because it replicates practical applications but also simplifies both camera estimation and phase space analysis, which is employed to study the problematic regions and their implications for the formed image. Ultimately, a streamlined procedure for camera pose estimation for one or more cameras is proposed, along with conclusions and directions for future research.