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). We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P? whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P? while ensuring that the error between P? and P is below a certain threshold. We consider two fundamentally different error measures -- Hausdorff and Frechet error measures. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ? > 0, compute a simplified polygonal curve P? whose error is less than ? and size at most the size of an optimal simplified polygonal curve with error ? 2. We consider monotone curves in the case of Hausdorff error measure and arbitrary curves for the Frechet error measure. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice. The control polygon of a Bezier curve is well-defined and has geometric significance---there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. We propose a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary. We describe a Bayesian method, for fitting curves to data drawn from an exponential family, that uses splines for which the number and locations of knots are free parameters. The method uses reversible jump Markov chain Monte Carlo to change the knot configurations and a locality heuristic to speed up mixing. For nonnormal models, we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the Bayesian information criterion, BIC, under priors that make this approximation accurate. Our technique is based on a marginalised chain on the knot number and locations, but we provide methods for inference about the regression coefficients, and functions of them, in both normal and nonnormal models. Simulation results suggest that the method performs well, and we illustrate the method in two neuroscience applications. Introduction Plane Curves Main Definitions and Facts Some General Classes of Plane Curves Atlas of Plane Curves Structure of the Atlas Atlas of Curves Geometric Indices for Atlas of Curves Space Curves Main Defintions and Facts Some Classes of Space Curves Applications of Curves Dynamic Properties of Plane Curves Plane Curves and Mechanisms Geometric Splines Appendices Smooth Functions of One and Two Variables Vector Functions of Scalar Argument Coordinate Systems. Coordinate Transformations Search of Some Geometric Characteristics Greek Roots of Names for Some Curves English-German-French Indices Bibliography Index of Terms It is of increasing contemporary interest to preserve ambient isotopy during geometric modeling. Bezier curves are pervasive in computer aided geometric design, as one of the fundamental computational representations for geometric modeling. For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge under Hausdorff distance. A natural question is whether subdivision produces topologically reliable PL approximations. Here we focus upon ambient isotopy and prove that sufficiently many subdivisions produce a control polygon ambient isotopic to the Bezier curve. We also derive closed-form formulas to compute the number of subdivision iterations to ensure ambient isotopic equivalence in the resulting approximation. This work relies upon explicitly constructing homeomorphism and ambient isotopy, which provides more algorithmic efficiency than only showing the existence of these equivalence relations. Abstract In many applications, bearings are measured to a moving object with the goal of estimating the object's course of movement. If movement is appropriately modeled as a smooth deterministic curve in the plane, then a cubic spline is a reasonable representation of the curve. Maximum likelihood estimators are presented for parameters of regression splines, assuming that observation errors follow a Von Mises distribution. Location estimates are obtainable even when data are sparse. Path estimation error, number and placement of knots, and outlier detection are discussed. Examples, including both simulated paths and observations from a wildlife radio-tracking study, are presented. Many scientific, financial, data mining and sensor network applications need to work with continuous, rather than discrete data e.g., temperature as a function of location, or stock prices or vehicle trajectories as a function of time. Querying raw or discrete data is unsatisfactory for these applications -- e.g., in a sensor network, it is necessary to interpolate sensor readings to predict values at locations where sensors are not deployed. In other situations, raw data can be inaccurate owing to measurement errors, and it is useful to fit continuous functions to raw data and query the functions, rather than raw data itself -- e.g., fitting a smooth curve to noisy sensor readings, or a smooth trajectory to GPS data containing gaps or outliers. Existing databases do not support storing or querying continuous functions, short of brute-force discretization of functions into a collection of tuples. We present FunctionDB, a novel database system that treats mathematical functions as first-class citizens that can be queried like traditional relations. The key contribution of FunctionDB is an efficient and accurate algebraic query processor - for the broad class of multi-variable polynomial functions, FunctionDB executes queries directly on the algebraic representation of functions without materializing them into discrete points, using symbolic operations: zero finding, variable substitution, and integration. Even when closed form solutions are intractable, FunctionDB leverages symbolic approximation operations to improve performance. We evaluate FunctionDB on real data sets from a temperature sensor network, and on traffic traces from Boston roads. We show that operating in the functional domain has substantial advantages in terms of accuracy (15-30 ) and up to order of magnitude (10x-100x) performance wins over existing approaches that represent models as discrete collections of points. We describe a novel algorithm for extracting a resolution-independent vector representation from pixel art images, which enables magnifying the results by an arbitrary amount without image degradation. Our algorithm resolves pixel-scale features in the input and converts them into regions with smoothly varying shading that are crisply separated by piecewise-smooth contour curves. In the original image, pixels are represented on a square pixel lattice, where diagonal neighbors are only connected through a single point. This causes thin features to become visually disconnected under magnification by conventional means, and creates ambiguities in the connectedness and separation of diagonal neighbors. The key to our algorithm is in resolving these ambiguities. This enables us to reshape the pixel cells so that neighboring pixels belonging to the same feature are connected through edges, thereby preserving the feature connectivity under magnification. We reduce pixel aliasing artifacts and improve smoothness by fitting spline curves to contours in the image and optimizing their control points. Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bezier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C 2 cubic Bezier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper. From a random point O in an infinite strip of width 1, we move in a randomly chosen direction along a curve Г. What shape of Г gives the minimum value to the expectation of the length of the path that reaches the boundary of the strip? After certain arguments suggesting that the desired curve belongs to one of four classes, it is proved that the best curve in those classes consists of four parts: an interval OA of length a, its smooth continuation, an arc AB of radius 1 and small length φ, an interval BD that is smooth continuation of the arc, and an interval DF (with a corner at the point D). If O is the origin and OA is the x axis of a coordinate system, then the coordinates of the above-mentioned points are as follows: A(a, 0), B(a + sin φ, 1 − cos φ), F(a, 1), and @math For the best curve, we have: a ≈ 0.814 and φ ≈ 0.032. Related questions are discussed. Bibliography: 9 titles. In this paper, we study the progressive iteration approximation property of a curve (tensor product surface) generated by blending a given data point set and a set of basis functions. The curve (tensor product surface) has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular. Thus, the B-spline and NURBS curve (surface) have the progressive iteration approximation property, and Bezier curve (surface) also has the property if the corresponding collocation matrix is nonsingular. Abstract Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation. We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic. 1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve Families and Capacity.- 10. Hausdorff Measure.- 11. Local Boundary Behaviour.- References.- Author Index. This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by imaging applications, where such boundary curves represent edges of images. The shearlet approach is able to characterize both locations and orientations of the edge points, including corner points and junctions, where the edge curves exhibit abrupt changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transforms which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature. We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative.