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In this paper, the problem of constructing geometric models from data provided by 3-D imaging sensors is addressed. Such techniques allow for rapid modeling of sculptured free-form shapes and generation of geometric models for existing parts. In order for a complete data set to be obtained, multiple images, each from a different viewpoint, have to be merged. A technique stemming from the Iterative Closest Point (ICP) method for estimating the relative transformations among the viewpoints is developed. Computational solutions are provided for estimating shape from noisy sensory measurements using representations that conform with commonly used representations from Computer Aided Geometric Design (CAGD). In particular, NURBS and triangular surface representations are applied in shape estimation. The surface approximations are refined by the algorithms to meet a user-defined tolerance value.

There are a variety of surface types (such as meshes and implicit surfaces) that lack a natural parameterization. We believe that manifolds are a natural method for representing parameterizations because of their ability to handle arbitrary topology and represent smooth surfaces. Manifolds provide a formal mechanism for creating local, overlapping parameterization and defining the functions that map between them. In this paper we present specific manifolds for several genus types (sphere, plane, n-holed tori, and cylinder) and an algorithm for establishing a bijective function between an input mesh and the manifold of the appropriate genus. This bijective function is used to define a smooth embedding of the manifold that approximates or interpolates the mesh. The smooth embedding is used to calculate analytical quantities, such as curvature and area, which can then be mapped back to the mesh.

Possible applications include texture mapping, surface fitting, arbitrary topology surface modeling, feature detection, and surface comparison.

We suggest a general method for one-dimensional interpolation in general manifolds, which allows to produce interpolation of arbitrary smoothness; interpolation in the group of orthogonal 3 × 3 matrices and the group of isometries of the three dimensional space using the exponential mapping is described as an example of application of this general method.

A number of techniques have been proposed for estimating the term structure, yet solid theoretical foundations and a comparative assessment of the results produced by these techniques are not available. In the present paper we prove, within a well defined mathematical setting, how the existence of discount factors is possible only if the condition of absence of static arbitrage is fulfilled. Besides that we report the results of an extensive set of tests whose scope is to show how the most widely used approaches in this field behave on both real and synthetic data.

Surface representations are utilized in a multitude of applications such as computer vision, medical imaging and computer graphics. B-spline surfaces have a number of desirable properties for representing surfaces, however, the complexity of knot placement strategies has prevented their widespread use in high-level vision environments. A solution to this problem is formulated within the reversible jump Markov chain Monte Carlo framework, whereby a derived posterior distribution may be sampled to calculate expected values for the number of knots required, their expected positions and a maximum likelihood estimate for the resulting control net of a given surface. Recognition of individual models may then be achieved using a hash table constructed using the principal components of the model control nets. Results of the fitting procedure, in terms of estimated knot vectors and spline surface errors, and the recognition of objects are provided for a set of free-form objects.

The aim of this paper is to describe a new five-dimensional, variable degree polynomial space. This space has the same properties of P₄ and possesses generalized tension properties, in the sense that it is close to P₂ for large values of the degrees. Its applications in the construction of spline functions will also be discussed.

In this paper, a novel statistical test is introduced to compare two locally stationary time series. The proposed approach is a Wald test considering time-varying autoregressive modeling and function projections in adequate spaces. The covariance structure of the innovations may be also time-varying. In order to obtain function estimators for the time-varying autoregressive parameters, we consider function expansions in splines and wavelet bases. Simulation studies provide evidence that the proposed test has a good performance. We also assess its usefulness when applied to a financial time series.

The goal of the paper is to establish quadrature formulas on combinatorial graphs. Three types of quadrature formulas are developed. Quadrature formulas of the first type are obtained through interpolation by variational splines. This set of formulas is exact on spaces of variational splines on graphs. Since bandlimited functions can be obtained as limits of variational splines we obtain quadrature formulas which are approximately exact on spaces of bandlimited functions. Accuracy of this type of quadrature formulas is given in terms of geometry of the set of nodes of splines and in terms of smoothness of functions which is measured by means of the combinatorial Laplace operator. Quadrature formulas of the second type are obtained through point-wise sampling for bandlimited functions and based on existence of certain frames in appropriate subspaces of bandlimited functions. The third type quadrature formulas are based on the average sampling over subgraphs. Our quadrature formulas which are based on sampling are exact on a relevant subspaces of bandlimited functions. The results of the paper have potential applications to problems that arise in data mining.

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on ${𝔲}^{\ast }(n)$ which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.

The field of computational statistics refers to statistical methods or tools that are computationally intensive. Due to the recent advances in computing power, some of these methods have become prominent and central to modern data analysis. In this paper, we focus on several of the main methods including density estimation, kernel smoothing, smoothing splines, and additive models. While the field of computational statistics includes many more methods, this paper serves as a brief introduction to selected popular topics.

A numerical local optimization of pen-down motion in writing a string of letters subject to fixed via points and pause points is presented. The square integral of jerk is minimized by following a conditional gradient method.

A linear time invariant state space model is proposed for the production and decay of two epimers (R) and (S) of a hopane released from oil bearing rock during laboratory pyrolysis. Concentrations of R and S are measured over time. The parameters to be estimated are: the initial amounts of precursors X, for R, and Y, for S; the rate constants for the production of R and S; the rate constants for the decay of R and S; and the rate constants for the two-way epimerization between both X and Y and R and S. It is shown that the model is locally identifiable. The parameters are estimated by numerical integration of the rate equations, alternated with a derivative free least squares constrained, optimisation routine. Asymptotic standard errors and covariances of parameters are given and compared with those obtained from a resampling approach (parametric bootstrap). An alternative fitting procedure, based on estimating derivatives of the concentrations of R and S by fitting splines, is implemented and compared with that based on integration of the rate equations. The rate constants are important for elucidation of the reaction pathways, and the estimates of initial concentrations of X and Y have potential for inferring the yields of oil bearing rock.

In this paper, the problem of constructing geometric models from data provided by 3-D imaging sensors is addressed. Such techniques allow for rapid modeling of sculptured free-form shapes and generation of geometric models for existing parts. In order for a complete data set to be obtained, multiple images, each from a different viewpoint, have to be merged. A technique stemming from the Iterative Closest Point (ICP) method for estimating the relative transformations among the viewpoints is developed. Computational solutions are provided for estimating shape from noisy sensory measurements using representations that conform with commonly used representations from Computer Aided Geometric Design (CAGD). In particular, NURBS and triangular surface representations are applied in shape estimation. The surface approximations are refined by the algorithms to meet a user-defined tolerance value.

The field of computational statistics refers to statistical methods or tools that are computationally intensive. Due to the recent advances in computing power, some of these methods have become prominent and central to modern data analysis. In this paper, we focus on several of the main methods including density estimation, kernel smoothing, smoothing splines, and additive models. While the field of computational statistics includes many more methods, this paper serves as a brief introduction to selected popular topics.