It is challenging to characterize the intrinsic geometry of high-degree algebraic curves with lower-degree algebraic curves. The reduction in the curve's degree implies lower computation costs, which is crucial for various practical computer vision systems. In this paper, we develop a characteristic mapping (CM) to recursively degenerate 3n points on a planar curve of n th order to 3(n-1) points on a curve of (n-1) th order. The proposed characteristic mapping enables curve grouping on a line, a curve of the lowest order, that preserves the intrinsic geometric properties of a higher-order curve (ellipse). We prove a necessary condition and derive an efficient arc grouping module that finds valid elliptical arc segments by determining whether the mapped three points are colinear, invoking minimal computation. We embed the module into two latest arc-based ellipse detection methods, which reduces their running time by 25% and 50% on average over five widely used data sets. This yields faster detection than the state-of-the-art algorithms while keeping their precision comparable or even higher. Two CM embedded methods also significantly surpass a deep learning method on all evaluation metrics.

The ultimate goal of self-driving technologies is to offer a safe and human-like driving experience. As one of the most important enabling functionalities, trajectory planning has been extensively studied from the perspective of safety. However, human-like trajectory planning on curved roads has rarely been studied. In this paper, we characterize and model human driving using extensive experimental driving collected on an urban curved road with 30 participants (10 experienced and 20 novice drivers) and five vehicles of different types. Differential global positioning system (GPS) is used to measure vehicle positions in high precision. We study factors that affect the driving trajectory, including vehicle speed, road curvature, and sight distance. We find that the human drivers typically do not follow lane centerline and the human-driven trajectories are very different from planners like rapidly exploring random tree (RRT). To generate human-like driving trajectory, we develop a data-driven trajectory model using general regression neural network (GRNN). The model was validated in various cases with promising performance.

Great attention has been devoted to the development of shape descriptors that is the key to object recognition. Previous works have great success on either relatively simple shapes or limited transformations, e.g., translation, rotation and scaling. We propose a new projective invariant, named characteristic number (CN) that includes more points for complex shapes with rich inner structures. Moreover, we build a novel shape descriptor with CN values calculated on triangles that cover the convex hull of a shape. The matching based on the descriptor also runs fast since only one initial point for the triangular coverage needs to align based on its CN value prior to the matching. The performance of the proposed descriptor is validated by the experiments compared with the classical shape context (SC) and recently developed cross ratio spectrum (CRS) on 32 logos of television networks with a wide range of transformations (512 images in total).