Convexity-preserving Bernstein–Be´ zier quartic scheme

A C1 convex surface data interpolation scheme is presented to preserve the shape of scattered data arranged over a triangular grid. Bernstein–Be´ zier quartic function is used for interpolation. Lower bound of the boundary and inner Be´zier ordinates is determined to guarantee convexity of surface. The developed scheme is flexible and involves more relaxed constraints

The Trigonometric Polynomial Like Bernstein Polynomial

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given

Improved monotone polynomial fitting with applications and variable selection

We investigate existing and new isotonic parameterisations for monotone polynomials, the latter which have been previously unconsidered in the statistical literature. We show that this new parameterisation is faster and more flexible than its alternatives enabling polynomials to be constrained to be monotone over either a compact interval or a semi-compact interval of the form [a;∞), in addition to over the whole real line. Due to the speed and efficiency of algorithms based on our new parameterisation the use of standard bootstrap methodology becomes feasible. We investigate the use of the bootstrap under monotonicity constraints to obtain confidence and prediction bands for the fitted curves and show that an adjustment by using either the ‘m out of n’ bootstrap or a post hoc symmetrisation of the confidence bands is necessary to achieve more uniform coverage probabilities. However, the same such adjustments appear unwarranted for prediction bands. Furthermore, we examine the model selection problem, not only for monotone polynomials, but also in a general sense, with a focus on graphical methods. Specifically, we describe how to visualize measures of description loss and of model complexity to facilitate the model selection problem. We advocate the use of the bootstrap to assess the stability of selected models and to enhance our graphical tools and demonstrate which variables are important using variable inclusion plots, showing that these can be invaluable plots for the model building process. We also describe methods for using the ‘m out of n’ bootstrap to select the degree of the fitted monotone polynomial and demonstrate it’s effectiveness in the specific constrained regression scenario. We demonstrate the effectiveness of all of these methods using numerous case studies, which highlight the necessity and usefulness of our techniques. All algorithms discussed in this thesis are available in the R package MonoPoly (version 0.3-6 or later)

Asteroid taxonomic signatures from photometric phase curves

We explore the correlation between an asteroid's taxonomy and photometric phase curve using the H, G12 photometric phase function, with the shape of the phase function described by the single parameter G12. We explore the usability of G12 in taxonomic classification for individual objects, asteroid families, and dynamical groups. We conclude that the mean values of G12 for the considered taxonomic complexes are statistically different, and also discuss the overall shape of the G12 distribution for each taxonomic complex. Based on the values of G12 for about half a million asteroids, we compute the probabilities of C, S, and X complex membership for each asteroid. For an individual asteroid, these probabilities are rather evenly distributed over all of the complexes, thus preventing meaningful classification. We then present and discuss the G12 distributions for asteroid families, and predict the taxonomic complex preponderance for asteroid families given the distribution of G12 in each family. For certain asteroid families, the probabilistic prediction of taxonomic complex preponderance can clearly be made. The Nysa-Polana family shows two distinct regions in the proper element space with different G12 values dominating in each region. We conclude that the G12-based probabilistic distribution of taxonomic complexes through the main belt agrees with the general view of C complex asteroid proportion increasing towards the outer belt. We conclude that the G12 photometric parameter cannot be used in determining taxonomic complex for individual asteroids, but it can be utilized in the statistical treatment of asteroid families and different regions of the main asteroid belt.Comment: submitted to Icaru

An Algorithm to Recover Generalized Cylinders from a Single Intensity View

Understanding a scene involves the ability to recover the shape of objects in an environment. Generalized cylinders are a flexible, loosely defined class of parametric shapes capable of modeling many real-world objects. Straight homogeneous generalized cylinders are an important subclass of generalized cylinders whose cross sections are scaled versions of a reference curve. In this paper, a general method is presented for recovering straight homogeneous generalized cylinders from monocular intensity images. The algorithm is much more general in scope than any other developed to date. combining constraints derived from both contour and intensity information. We first demonstrate that contour information alone is insufficient to recover a straight homogeneous generalized cylinder uniquely. Next, we show that the sign and magnitude of the Gaussian curvature at a point varies among members of a contour-equivalent class. The image contour fails to constrain two parameters required to recover the shape of a generalized cylinder, the 3D axis location and the object tilt. Next, a method for "ruling" straight homogeneous generalized cylinder images is developed. Once the rulings of the image have been recovered, we show that all parameters derivable from contour alone can be recovered. To recover the two remaining parameters (modulo scale) not constrained by image contour requires incorporating additional information into the recovery process, e.g. intensity information. We derive a method for recovering the tilt of the object using the ruled contour image and intensity values along cross-sectional geodesics. In addition, we derive a method for recovering the location of the object's 3D axis from intensity values along meridians of the surface. Using the different methods outlined in this paper constitutes an algorithm for recovering all the shape parameters (modulo scale) of a straight homogeneous generalized cylinder

Dynamic modelling of articulated figures suitable for the purpose of computer animation

The animation of articulated bodies presents interest in the areas of biomechanics, sports, medicine and the entertainment industry. Traditional motion control methods for these bodies, such as kinematics and rotoscoping are either expensive to use or very laborious. The motion of articulated bodies is complex mostly because of their number of articulations and the diversity of possible motions. This thesis investigates the possibility of using dynamic analysis in order to define the motion of articulated bodies. Dynamic analysis uses physical quantities such as forces, torques and accelerations, to calculate the motion of the body. The method used in this thesis is based upon the inverse Lagrangian dynamics formulation, which, given the accelerations, velocities and positions of each of the articulations of the body, finds the forces or torques that are necessary to generate such motion. Dynamic analysis offers the possibility of generating more realistic motion and also of automating the process of motion control. The Lagrangian formulation was used first in robotics and thus the necessary adaptations for using it in computer animation are presented. An analytical method for the calculation of ground reaction forces is also derived, as these are the most important external forces in the case of humans and the other animals that are of special interest in computer animation. The application of dynamic analysis in bipedal walking is investigated. Two models of increasing complexity are discussed. The issue of motion specification for articulated bodies is also examined. A software environment, Solaris, is described which includes the facility of dynamic and kinematic motion control for articulated bodies. Finally, the advantages and problematics of dynamic analysis with respect to kinematics and other methods are discussed