B-spline Model Research Articles

B-SPLINE BASED STEREO FOR 3D RECONSTRUCTION OF LINE-LIKE OBJECTS USING AFFINE CAMERA MODEL

This paper presents a novel curve based algorithm of stereo vision to reconstruct 3D line-like objects. B-spline approximations of 2D edge curves are selected as primitives for the reconstruction of their corresponding space curves so that, under the assumption of affine camera model, a 3D curve can be derived from reconstructing its control points according to the affine invariant property of B-Spline curves. The superiority of B-spline model in representing free-form curves gives good geometric properties of reconstruction results. Both theoretical analysis and experimental results demonstrate the validity of our approach.

Analyzing fuzzy surface modeling using load-balanced computation

Models that can code and render uncertainty support users doing spatial analysis, because they provide direct access to and visualization of the imprecision induced by data compression. Interval or fuzzy models provide an important alternative to traditional approaches, especially for scientists and decision makers eager to explore different scenarios, possibly using different parameters. We found parallel computing to be a key for using advanced modeling techniques. In many cases, however, the high computational cost of even simple interrogation processes might prohibit using fuzzy models: In this context, we present a load-balanced solution that efficiently uses parallel computing to analyze surface models based on the fuzzy B-spline model. We obtained efficient portability of the algorithm by introducing a dynamic and flexible load-balancing strategy that provides good results on different architectures, including a Cray-T3D.

“True” color surface anatomy: mapping the Visible Human to patient-specific CT data

The mapping of “true” color and texture information into traditional medical imaging modality data can add considerable information and aid in diagnostics. One of the goals of this work has been to create CT color lookup tables for all visually well-defined structures in the Visible Human male cryosection data set which then can be used to color patient-specific CT data. The primary goal has been to develop a method for stripping textures from a volumetric data set for polygonal models and non-uniform rational B-spline (NURBS) models generated from a volumetric data set. It is believed that these methods can eventually be used to provide clinicians with 3D models with physiologically accurate color textures.

Range image measurement and shaded image measurement: Two non-contact coordinate measurement approaches designed for special application tasks

Reproduction of sculptured surfaces is a common and difficult process in the manufacturing industry, such as mould and die cavity making for plastic toys. In this paper, two surface measurement techniques, namely range image measurement (RIM) and shaded image measurement (SIM), were investigated. The principle of the two approaches is based on the techniques of non-contact surface coordinate measurement. In RIM the range data of a surface are measured by laser scanning, while in SIM the coordinates of the surface are obtained by a shape from shading model and the photometric stereo technique. The acquired information from these two different surface feature measuring techniques can be used to reconstruct the surface profile by surface models such as the Bezier and B-spline models. The features of the modelled surface can subsequently be used to produce the surface on a numerically controlled (NC) machine. The validity of the approaches was examined by two machining tests, one being a medical trial and the other an industrial case. The results of the tests showed that the principle of the two approaches is workable, but some technical problems of the SIM technique need to be overcome.

Minimax emission computed tomography using high-resolution anatomical side information and B-spline models

In this paper a minimax methodology is presented for combining information from two imaging modalities having different intrinsic spatial resolutions. The focus application is emission computed tomography (ECT), a low-resolution modality for reconstruction of radionuclide tracer density, when supplemented by high-resolution anatomical boundary information extracted from a magnetic resonance image (MRI) of the same imaging volume. The MRI boundary within the two-dimensional (2-D) slice of interest is parameterized by a closed planar curve. The Cramer-Rao (CR) lower bound is used to analyze estimation errors for different boundary shapes. Under a spatially inhomogeneous Gibbs field model for the tracer density a representation for the minimax MRI-enhanced tracer density estimator is obtained. It is shown that the estimator is asymptotically equivalent to a penalized maximum likelihood (PML) estimator with resolution-selective Gibbs penalty. Quantitative comparisons are presented using the iterative space alternating generalized expectation maximization (SAGE-FM) algorithm to implement the PML estimator with and without minimax weight averaging.

Spatio-temporal tracking of myocardial deformations with a 4-D B-spline model from tagged MRI

Accurate delineation of the volumetric motion of the left ventricle (LV) of the heart from tagged magnetic resonance imaging (MRI) is an important area of research. We have built a system that takes extracted tag line features from short axis (SA) and long axis (LA) image sequences as input and fits a four-dimensional (4-D) time-varying B-spline model to the data by simultaneously fitting the model knot solids to MRI frames via matching three sequences of solid knot planes to the LV tag planes for 4-D tracking. Important advantages of the model are that reconstruction of tag surfaces, three-dimensional (3-D) material point localization, as well as displacement reconstruction are all achieved in a single step. The generated 3-D displacement fields are validated with a cardiac motion simulator, and 3-D motion fields capturing in vivo deformations in a porcine model with posterolateral myocardial infarction are illustrated.

Designing fuzzy controllers by rapid learning

We propose a learning approach to designing fuzzy controllers based on the B-spline model. Unlike other normalised parameterised set functions for defining fuzzy sets, B-spline basis functions do not necessarily span from membership values zero to one, but possess the property “partition of unity”. B-spline basis functions can be automatically determined after each input is partitioned. Learning of a fuzzy controller based on B-spline basis functions is then equivalent to the adaptation of a B-spline interpolator. Parameters of the controller output of each rule can be adapted by using the gradient descent method. Optimal placements of the B-spline basis functions for specifying each input can be found by an algorithm working similarly to a self-organising neural network. Through comparative examples of function approximation we show that learning of such a fuzzy controller generally converges fast. This approach can be extended to the problems of supervised as well as unsupervised learning.

Parametrically deformed free-form surfaces as part of a variational model

A new approach is described which provides deformation methods for multi-patch tensor based free-form surfaces. The surface deformation generated is controlled by global geometric constraints. For example, the objective can be to deform a free-form surface until it becomes tangent to a pre-defined plane at a given point. This point can be fixed or free to slide on the surface. The parametric deformation of surfaces is dedicated to modifications of free-form surfaces within CAD software and to the design of objects submitted to aesthetic requirements. It is an alternative to previous approaches and it works with multiple surfaces through a simple mechanical model. The deformation method uses an analogy between the control polyhedron of each surface (based on a B-Spline model) and the mechanical equilibrium of a rigid bar network. The user can localize the surface deformation into an arbitrary shaped area through the selection of control polyhedron vertices spread over the entire surface. These vertices are used to automatically construct the associated bar network. The bar network equilibrium parameters are set up to achieve isotropic or anisotropic deformation as required by the designer. The surface deformation is then automatically carried through an optimization process which modifies mechanical parameters to agree with the global geometric constraint set up. The G 1 continuity across the different during the deformation process using a set of geometric constraints in addition to mechanical ones. Parametric free-form surface deformation can be subjected to non-linear geometric constraints such as the tangency of a surface to a pre-defined plane. The resolution of such a problem uses an optimization process which minimizes the variation of the parameters governing the equilibrium of the bar network, namely the external forces applied to the nodes of the network. Several examples illustrate basic deformation types with various sets of constraints.

Optimal boresight error design of radomes of revolving symmetry

An optimisation technique is applied to the design of three-dimensional radomes of revolving symmetry. A design example for an ogive radome is given. The best possible thickness profile of the radome is found and the boresight error is reduced significantly. Both B-spline modelling and polynomial modelling for the thickness profile are considered.

Dynamic NURBS with geometric constraints for interactive sculpting

This article develops a dynamic generalization of the nonuniform rational B-spline (NURBS) model. NURBS have become a defacto standard in commercial modeling systems because of their power to represent free-form shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or D-NURBS, are physics-based models that incorporate mass distributions, internal deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using D-NURBS, a modeler can interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. D-NURBS move and deform in a physically intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and a finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves. We demonstrate D-NURBS models and constraints in applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from cross sections, and free-form deformation. We also introduce a new technique for 2D shape metamorphosis using constrained D-NURBS surfaces.

A comparison of four methods for non-linear data modelling

This paper reports results from practical experiments with four different data modelling methods evaluated on five different real or simulated modelling problems. The methods are: partial least squares regression (PLS), back-propagation multilayer perceptron neural networks (MLP), radial basis function neural networks (RBF), and an adaptive B-spline modelling algorithm (ASMOD). The last four methods have the capability of identifying and representing general non-linear dependencies in the data without a priori specifying which specific non-linear dependencies to look for, whereas PLS needs active interaction by the data analyst to identify non-linear dependencies. The real-world modelling problems are: identifying the dynamic actuator characteristics of a hydraulic industrial robot manipulator, modelling carbon consumption as a function of other process variables in a metallurgic industrial process, and estimating water content in fish food products based on NIR spectrometry. In addition the methods have been applied for modelling data generated from a simulation of a chemical catalytic reactor, and to identify a 10-dimensional synthetic function from small samples of noisy data. Quantitative comparisons are made of the prediction accuracy in separate test data sets, and qualitative comparisons are made for some important performance characteristics, such as computation time and simplicity of use.

Comparative Aspects of Neural Network Algorithms for On-Line Modelling of Dynamic Processes

This paper reviews the model structures and learning rules of four commonly used artificial neural networks: the cerebellar model articulation controller (CMAC), B-splines, radial basis functions (RBF) and multi-layered perceptron (MLP) networks. Their dynamic modelling abilities are compared using a two-dimensional non-linear noisy time series. The network performances are evaluated based on their network surface plots, phase/time history plots, learning curves, prediction error autocorrelation functions and finally their short-range prediction error variances. The modelling results suggest that all four networks were able to capture the underlying dynamics of the time series. Also, specific prior knowledge about the time series was incorporated into the B-splines model, and is used to highlight an important trade-off between the model flexibility and high-dimensional modelling ability in the B-splines and CM AC networks. In general, when the network model is well conditioned and linear with respect to its adaptable parameters, simpler on-line learning rules often provide adequate convergence properties. Alternatively, when the model is highly non-linear, complicated learning rules which utilize high-order gradient information are generally required at the expense of increased computational complexity.

Influence of core polarization on the electron affinity of Ca.

We have developed a method, based on the use of B-spline basis sets and model potentials, for determining properties of systems with two or three electrons outside a polarizable closed-shell core. It is applied to the calculation of the electron affinity of Ca and the resulting value of 17.7 meV is in excellent agreement with the most recent experiments. It is found that the dielectronic core-valence interaction reduces the electron affinity by 39.5 meV.

Supercoiled DNA energetics and dynamics by computer simulation

A new formulation is presented for investigating supercoiled DNA configurations by deterministic techniques. Thus far, the computational difficulties involved in applying deterministic methods to supercoiled DNA studies have generally limited computer simulations to stochastic approaches. While stochastic methods, such as simulated annealing and Metropolis-Monte Carlo sampling, are successful at generating a large number of configurations and estimating thermodynamic properties of topoisomer ensembles, deterministic methods offer an accurate characterization of the minima and a systematic following of their dynamics. To make this feasible, we model circular duplex DNA compactly by a B-spline ribbon-like model in terms of a small number of control vertices. We associate an elastic deformation energy composed of bending and twisting integrals and represent intrachain contact by a 6–12 Lennard Jones potential. The latter is parameterized to yield an energy minimum at the observed DNA-helix diameter inclusive of a hydration shell. A penalty term to ensure fixed contour length is also included. First and second partial derivatives of the energy function have been derived by using various mathematical simplifications. First derivatives are essential for Newton-type minimization as well as molecular dynamics, and partial second-derivative information can significantly accelerate minimization convergence through preconditioning. Here we apply a new large-scale truncated-Newton algorithm for minimization and a Langevin/implicit-Euler scheme for molecular dynamics. Our truncated-Newton method exploits the separability of potential energy functions into terms of differing complexity. It relies on a preconditioned conjugate gradient method that is efficient for large-scale problems to solve approximately for the search direction at every step. Our dynamics algorithm is numerically stable over large time steps. It also introduces a frequency-discriminating mechanism so that vibrational modes with frequencies greater than a chosen cutoff frequency are essentially frozen by the method. With these tools, we rapidly identify corresponding circular and interwound energy minima for small DNA rings for a series of imposed linking-number differences. These structures are consistent with available electron microscopy data. The energetic exchange of stability between the circle and the figure-8, in very good agreement with analytical results, is also detailed. Molecular dynamics trajectories at 100 femtosecond time steps then reveal the rapid folding of the unstable circular state into supercoiled forms. Significant bending and twisting motions of the interwound structures are also observed. Such information may be useful for understanding transition states along the folding pathway and the role of enzymes that regulate supercoiling. More generally, new quantitative data obtained by such deterministic approaches may help in interpreting the effects of supercoiling on key biological processes.

Seismological studies at Parkfield. I. Simultaneous inversion for velocity structure and hypocenters using cubic B-splines parameterization

Abstract A model for the 3D P- and S-wave velocity structure at Parkfield has been developed by the systematic inversion of arrival times for microearthquakes and surface Vibroseis sources, recorded by nearby CALNET stations and by the local borehole-installed 10-station High-Resolution Seismic Network there. A modification of the linear B-splines model parameterization method of Thurber (1983) to a cubic B-splines formulation, through tests inverting synthetic data sets for the Parkfield configuration, is found to produce smooth velocity distributions that still retain some small-scale features. Application of the method to 953 P-wave and 288 S-wave arrivals from 111 microearthquakes and 8 Vibroseis locations throughout the study area yields a velocity structure having variable resolution. A low-velocity zone appears to coincide with the creeping segment of the San Andreas fault zone northwest of the 1966 Parkfield earthquake epicenter. An increase of about 15 per cent in seismic velocities is seen from the northeast to southwest across this segment of the fault zone. The higher velocities to the southwest apparently correspond to the granitic basement. The contrast across the fault is subdued to the southeast along the 1966 rupture zone. A high Vp/Vs ratio of 1.9 is seen in a 2-km-wide volume near the location of the 1966 main shock hypocenter, presumably the nucleation zone for the expected M = 6 Parkfield earthquake.

Physical modeling with B-spline surfaces for interactive design and animation

Physically based modeling has recently become a subject of widespread interest. This paper defines a model of elastic and plastic B-spline surfaces which supports both animation and design operations, including the interference detection, within the "Eplastics"/Alpha_1 modeling environment. We develop "refinement" operations for spring and hinge B-spline models which are compatible with the physics and the mathematics of B-spline models, including refinement. It is also a step towards a more continuous physical representation of a physical model than the more standard discretized geometry point mass models. We present some interactive operations currently in real-time and some which could be real-time with further optimization and use of multiple processors on machines such as the SGI Iris 4D/240GTX. A movie is used to illustrate the results.

Photon correlation spectroscopy of particle polydispersity: A cubic B-spline analysis

A cubic B-spline model has been applied to the analysis of photon correlation data from heterogeneous suspensions of particles. The analysis procedure has been tested with simulated data corresponding to unimodal and bimodal size distributions and used in experimental studies of aqueous suspensions of polystyrene latex microspheres.

A Well-conditioned Cubic B-spline Model for Processing Laser Anemometry Data

A well-conditioned method for processing laser anemometry data is presented. It involves the use of a cubic b-spline model which is suitably arranged to take into account the effects of ill-conditioning as well as data sampling and truncation. The problem of ill-conditioning is analysed in terms of the eigenfunctions and eigenvalues of the equation and the resulting loss of information is expressed in terms of resolution limitations which are easily incorporated within such a model. The method is applied to various sets of simulated and experimentally determined data and the results are discussed. The range of application includes the analysis of data obtained from anemometry experiments which employ frequency shifting techniques.