In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it. Geodesics describe the motion of point particles.
Published in Chapter:
Energy Minimizing Active Models in Artificial Vision
Gloria Bueno García (University of Castilla – La Mancha, Spain), Antonio Martínez (University of Castilla – La Mancha, Spain), Roberto González (University of Castilla – La Mancha, Spain), and Manuel Torres (University of Castilla – La Mancha, Spain)
Copyright: © 2009
|Pages: 7
DOI: 10.4018/978-1-59904-849-9.ch084
Abstract
Deformable models are well known examples of artificially intelligent system (AIS). They have played an important role in the challenging problem of extracting useful information about regions and areas of interest (ROIs) imaged through different modalities. The challenge is also in extracting boundary elements belonging to the same ROI and integrate them into a coherent and consistent model of the structure. Traditional low-level image processing techniques that consider only local information can make incorrect assumptions during this integration process and generate unfeasible object boundaries. To solve this problem, deformable models were introduced (Ivins, 1994), (McInerney, 1996), (Wang, 2000). These AI models are currently important tools in many scientific disciplines and engineering applications (Duncan, 2000). Deformable models offer a powerful approach to accommodate the significant variability of structures within a ROI over time and across different individuals. Therefore, they are able to segment, match and track images of structures by exploiting (bottom-up) constraints derived from the image data together with (top-down) a priori knowledge about the location, size, and shape of these structures. The mathematical foundations of deformable models represent the confluence of geometry, physics and approximation theory. Geometry serves to represent object shape, physics imposes constraints on how the shape may vary over space and time, and optimal approximation theory provides the formal mechanisms for fitting the models to data. The physical interpretation views deformable models as elastic bodies which respond to applied force and constraints.