Chains are vector-valued signals sampling a curve. They are important to motion signal processing and to many scientific applications including location sensors. We propose a novel measure of smoothness for chains curves by generalizing the scalar-valued concept of monotonicity. Monotonicity can be defined by the connectedness of the inverse image of balls. This definition is coordinate-invariant and can be computed efficiently over chains. Monotone curves can be discontinuous, but continuous monotone curves are differentiable a.e. Over chains, a simple sphere-preserving filter shown to never decrease the degree of monotonicity. It outperforms moving average filters over a synthetic data set. Applications include Time Series Segmentation, chain reconstruction from unordered data points, Optical Character Recognition, and Pattern Matching.
One approach to chain smoothing is to use B-splines and Bezier curves with the @math norm @cite_6 . Correspondingly, we could measure the smoothness'' of a chain by measuring how closely one can fit it to a smooth curve. Our approach differs in that we do not use polygonal approximations or curve fitting: we consider chains to be first-class citizens.
@cite_6
2002667015
We present a new approach to the problem of matching 3-D curves. The approach has a low algorithmic complexity in the number of models, and can operate in the presence of noise and partial occlusions. Our method builds upon the seminal work of (1990), where curves are first smoothed using B-splines, with matching based on hashing using curvature and torsion measures. However, we introduce two enhancements: - We make use of nonuniform B-spline approximations, which permits us to better retain information at high-curvature locations. The spline approximations are controlled (i.e., regularized) by making use of normal vectors to the surface in 3-D on which the curves lie, and by an explicit minimization of a bending energy Defining sharp features in a 3D model facilitates a better understanding of the surface and aids geometric processing and graphics applications, such as reconstruction, filtering, simplification, reverse engineering, visualization, and non-photo realism. We present a robust method that identifies sharp features in a point-based model by returning a set of smooth spline curves aligned along the edges. Our feature extraction leverages the concepts of robust moving least squares to locally project points to potential features. The algorithm processes these points to construct arc-length parameterized spline curves fit using an iterative refinement method, aligning smooth and continuous curves through the feature points. We demonstrate the benefits of our method with three applications: surface segmentation, surface meshing and point-based compression. Abstract Polynomial splines are often used in statistical regression models for smooth response functions. When the number and location of the knots are optimized, the approximating power of the spline is improved and the model is nonparametric with locally determined smoothness. However, finding the optimal knot locations is an historically difficult problem. We present a new estimation approach that improves computational properties by penalizing coalescing knots. The resulting estimator is easier to compute than the unpenalized estimates of knot positions, eliminates unnecessary “corners” in the fitted curve, and in simulation studies, shows no increase in the loss. A number of GCV and AIC type criteria for choosing the number of knots are evaluated via simulation. We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximum-area triangle or convex body ("potato"). Exact polynomial-time algorithms are known for some of these problems, but their time bounds are high (e.g., O(n7) for the largest convex polygon in a simple n-gon). We devise efficient approximation algorithms for these problems. In particular, we give near-linear time algorithms for a (1 - ∈)-approximation of the biggest stick, an O(1)-approximation of the maximum-area convex body, and a (1 - ∈)-approximation of the maximum-area fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).