Séamus T. Tuohy

Approximation of measured data with interval B-splines

The objective of this paper is to provide an efficient and reliable method for interpolating or approximating a set of measured data with an interval B-spline curve or surface. In general, measured data possess uncertainty, arising from sensor precision and measurement registration, which can be represented as an interval. Both the interpolation and approximation techniques presented in the paper produce interval bounding geometries that strictly enclose the intervals of the original data; in the case of our interpolation method, the achieved fit is extremely tight, and in the case of our approximation technique the achieved fit depends on the number of control points one is willing to allow. Examples using measured data illustrate our method.

Approximation of high-degree and procedural curves

The objective of this paper is to present an efficient adaptive algorithm to approximate high-degree and procedural continuous parametric curves by integral B-splines. This approximation algorithm covers nonperiodic and periodic curves. The approximation algorithm is motivated and accompanied by an extensive discussion on approximation errors for position and derivatives accuracies. This discussion includes the derivation of local error bounds for position and derivative errors. The practicality and efficiency of the algorithm is demonstrated for a variety of examples in geometric modeling of engineering structures. The approximation algorithm for the curve forms the basis of a similar surface approximation procedure.

Curvature computations for degenerate surface patches

Abstract This paper presents a method to compute curvatures of a surface patch S obtained from a degenerate representation defined over a rectangular domain. To compute curvatures at a point q in S where the surface representation is degenerate one has to assume that the point set S has a tangent plane and well defined curvatures at the point q where the curvatures are to be computed. Our computation method employs a local height function representation of S in a neighborhood of the point q where the height function h is defined over the tangent plane of S at the point q . In our method the second order partial derivatives of the height function h at q are computed by using second order derivatives of any three surfaces curves on S which end up in q with pairwise linearly independent tangent directions. The curvature entities of S at q are then computed by using the second order partial derivatives of the height function h at q . We also show how the method is extended to compute partial derivatives of any order of the function h at q . For this purpose we show (in Theorem 1) how the partial derivatives up to n th order of a surface at a point q can be computed using derivatives up to order n of n + 1 surface curves emanating from q . We also present a definition of a concept of generalized surface curvatures under weaker assumptions appropriate for degenerate surfaces. Under those weaker assumptions one does not require a locally C 2 -smooth height function representation of the surface over the tangent plane at the degenerate point q , nor a locally well defined (single-valued) height function representation of the surface in a neighborhood of the point q . One only needs the existence of a unique local second order approximation of the degenerate surface with a quadratic function defined over the tangent plane at q . The surface curvatures of the corresponding approximating quadratic surface then define the curvatures of the approximated degenerate surface in the contact point q .

Representation of Geophysical Maps with Uncertainty

This paper proposes a method for the reconstruction of surfaces from spatially distributed geophysical data with uncertainties. For efficient interrogation and storage and the ability to represent uncertainty, bi-quadratic uniform integral enveloping or interval B-spline surfaces are fit to uniformly distributed cellular data using linear programming methods. The cellular data function is characterized by an upper and lower bound of the measured values for a range (box) of the independent variables. Interior patch boundaries are formed using a quadtree segmentation of the data set and interior knots are placed according to an algorithm for edge detection. C 1 continuity across patch boundaries is maintained by initially fitting auxiliary patches which then define the boundary conditions for the interior patches. A map from measured geophysical data illustrates the method.

Low-degree approximation of high-degree B-spline surfaces

In this paper, a method for approximate conversion of high degree Bezier and B-spline surfaces to lower degree representations is presented to facilitate the exchange of surface geometry between different geometric modeling systems. Building on previous work on curve approximation, the method uses adaptive sampling to compute approximation error and lofting of isoparametric curves to produce the approximating surface. In addition, a bound for the approximation accuracy is computed using convex hulls.

Scattered data fitting with simplex splines in two and three dimensional spaces

We present a method for scattered data fitting in 2D and 3D euclidean spaces that uses simplex splines. The fitting process includes establishing the domain of the definition of a set of 2D or 3D data with the ahull method, segmentation for generating a proper initial triangulation, knot generation, least squares fitting with simplex splines, and adaptive domain subdivision. This technique has applications in scientific visualization and databases, computer vision, GIS, and CAD/CAM.

Computational methods for blending surface approximation

Blending surfaces form a smooth transition between two distinct, intersecting surfaces or smoothly join two or more disconnected surfaces and are normally procedural surfaces which are difficult to exchange and to interrogate in a reliable and efficient manner. In this paper, an approximation method for blending surfaces which are curvature continuous to the underlying surfaces with a non-uniform rational B-spline (NURBS) surface is presented. The use of NURBS is important since it facilitates the exchange of geometric information between various computer aided design and manufacturing systems. In the method, linkage curves on the underlying surfaces are approximated to within a specified tolerance and cross-link curves are created using the linkage curves, a directional curve and the parametric partial derivatives of the underlying surfaces. Cross-link curves are lofted to form the blending surface and an adaptive sampling procedure is used to test the blending surface against specified tolerances. Cross-link curves are added, where necessary, and the surface relofted until the continuity conditions are satisfied to within specified tolerances. Examples illustrate the applicability of the method.

Experiments in remote monitoring and control of autonomous underwater vehicles

This paper describes research on remote monitoring and control of autonomous underwater vehicles (AUVs). This work is part of a larger effort to create autonomous ocean sampling networks (AOSN), a new concept for collecting synoptic oceanographic data. AOSN is based on the operation of small, low cost autonomous underwater vehicles (AUVs) within an array of moorings that provide communication and power-supply nodes. Critical to the realization of AOSN will be the ability to control these assets and monitor the data collection process remotely. During the time in which our AOSN partners have been developing acoustic communications, our laboratory has performed a variety of remote monitoring and control experiments to test different types of network distributed control software. Experimental results are presented for runs in the Charles River in which a radio modem towed on a float was used to monitor and control the vehicle from a workstation on the Internet. In addition, we have used our acoustic navigation system to send simple command messages to an Odyssey vehicle, achieving simultaneous control of multiple AUVs operating in a high current environment in the Haro Strait.

Reliable interrogation of 3D non-linear geophysical databases

Publisher Summary This chapter describes the reliable interrogation of 3D nonlinear geophysical databases. The representation of implicit surfaces within a rectangular box by triple product B-splines is discussed. The algorithm for evaluating implicit algebraic surfaces based on the computation of critical points is an extension of the algorithm presented in for tracing planar algebraic curves expressed in terms of the bivariate Bernstein basis. Computation of the critical points is performed so that the topology of the surface may be discovered, and the domain subdivided accordingly. Polygonalization of an implicit surface bounded by a subdomain involves the discretization of the space into rectangular elements, and the faceting of these elements to form the surface model. In order to do this, a resolution in each coordinate direction is selected, and represents the finest grain size of the elements that are later to be faceted.

Trivariate parametric B-splines for visualization of ocean data

Describes methods developed in the MIT Design Laboratory for the representation, fitting, interrogation and visualization of trivariate water column data. Features of this work include: reduced dimensional fitting algorithms for efficient data representation and reliable methods for visualizing continuous, trivariate parametric representations of data.