Adaptive Knot Research Articles

Stochastic occupancy modeling for spaces with irregular occupancy patterns using adaptive B-Spline-based inhomogeneous Markov Chains

This paper presents a discrete time, discrete state-space in-homogeneous Markov Chains model for stochastic occupancy modeling in spaces with irregular occupancy patterns. The goal of the model is to provide accurate predictions of occupancy numbers, enabling appropriate actions to be taken for HVAC system to maintain optimal indoor environment. The proposed Markov Chain model incorporates time in-homogeneity by coupling the time-varying model parameters using a Periodic B-Spline expansion with adaptive knots, which effectively captures patterns in occupancy activity. This method optimizes the distribution of knots based on specific occupancy characteristics observed in different types of rooms. To evaluate the effectiveness of the proposed method, six months of occupancy data collected from a meeting room are utilized. A comprehensive comparison is conducted between the proposed adaptive B-Spline method and other approaches, including the counting method and uniform B-Spline method. The comparison considers both model accuracy and complexity, using metrics such as the Akaike Information Criterion and Bayesian Information Criterion. Results indicate that the proposed model achieves more accurate predictions with fewer model parameters compared to other methods. These forecasts are particularly useful in optimizing the control of HVAC systems, where accurate predictions of future occupancy numbers are essential.

GLS‐PIA: n‐Dimensional Spherical B‐Spline Curve Fitting based on Geodesic Least Square with Adaptive Knot Placement

AbstractDue to the widespread applications of curves on n‐dimensional spheres, fitting curves on n‐dimensional spheres has received increasing attention in recent years. However, due to the non‐Euclidean nature of spheres, curve fitting methods on n‐dimensional spheres often struggle to balance fitting accuracy and curve fairness. In this paper, we propose a new fitting framework, GLS‐PIA, for parameterized point sets on n‐dimensional spheres to address the challenge. Meanwhile, we provide the proof of the method. Firstly, we propose a progressive iterative approximation method based on geodesic least squares which can directly optimize the geodesic least squares loss on the n‐sphere, improving the accuracy of the fitting. Additionally, we use an error allocation method based on contribution coefficients to ensure the fairness of the fitting curve. Secondly, we propose an adaptive knot placement method based on geodesic difference to estimate a more reasonable distribution of control points in the parameter domain, placing more control points in areas with greater detail. This enables B‐spline curves to capture more details with a limited number of control points. Experimental results demonstrate that our framework achieves outstanding performance, especially in handling imbalanced data points. (In this paper, “sphere” refers to n‐sphere (n ≥ 2) unless otherwise specified.)

Concurrent process and feedrate scheduling with convoluted basis functions and its application to fluid jet polishing

Non-traditional laser and fluid jet processes exhibit time-dependent material removal characteristics. The feedrate profile must be planned carefully along the toolpath for accurate surface profile generation while ensuring that the kinematic limits of machine tools are not violated. Conventional methods iteratively solve a deconvolution/convolution problem on the dwell-time density (reciprocal of the feedrate profile) that is computationally heavy, may leave significant residual processing errors, and even generate infeasible feed profiles with the manufacturing equipment. This paper proposes a novel approach that fully addresses the shortcomings above. Dwell-time density is first expressed as a continuous B-spline profile. The associated dwell basis functions (DBF) are convolved with the process influence function (PIF) to generate new process basis functions (PBF). This approach conveniently allows the posing of the problem as a concurrent linear least-squares problem on the control points shared by the DBFs and PBFs while ensuring the numerical stability of the solution and smoothness of the feed profile. To mitigate excessive acceleration peaks and any ringing effect around the edges of the toolpath, this paper also presents methodologies for stabilizing the scheduled feedrate profile by introducing knot vector adjustments (adaptive knot dropping) and linear edge constraints. The effectiveness of the proposed method is demonstrated and validated through simulation case studies and experimentally in fluid jet processing of precision optics. Results indicate that the proposed technique overcomes the limitations of conventional strategies and allows high-frequency surface components beyond the first zero-power frequency of the process footprint to be tracked while still generating a smooth feed profile within the acceleration limits of a machine tool. This ability stems from the localization characteristics associated with the basis functions. By improving the accuracy of high-frequency components, the proposed method exhibits the potential to fabricate topographies with sharper edges, which has been a challenge for conventional techniques.

An approximation method combined with conformal transformation for the fast measurement of blade section curves

To realize high-precision machining of free-form blade surfaces, it is necessary to measure and reconstruct the blade morphology several times. Sampling measurements and data compression through free-curve approximation technology are of great benefit in shortening the processing time. A novel B-spline approximation method is proposed in this paper, which can determine a few dominant points based on curvature characteristics to approximate dense and noisy measured points with high precision. We adopt a unique adaptive knot placement strategy combined with a curvature dependent evaluation parameter and a conformal transformation method. This strategy is error-bounded in the initial knot placement, which greatly reduces the time consumed by subsequent iterative insertions. The proposed method shows advantages in approximation accuracy, compression rate, and especially in computational efficiency, in comparison with four traditional knot placement methods. Experiments were carried out and the results show that the dominant point selected by this method can replace the fine measurement results, and the compression rate of the measurement points can reach more than 90%. This method is suitable for the sampling measurement and compression of the data of blades, aircraft wings, and other similar free-form surfaces, and is especially suitable for solving the inefficiency problem caused by repeated measurements made in the process of iterative machining of aero-engine blades.

A new deterministic heuristic knots placement for B-Spline approximation

In this paper, we propose an adaptive knot placement algorithm for B-Spline curve approximation to dense and noisy 2D data points. The proposed algorithm is based on a heuristic rule for knot placement. It consists in constructing a distribution knot function by blending geometric criteria such as discrete derivatives, discrete angular variations and curvature. It has been successfully compared to three well known methods for approximating various noisy functions and sets of data in handwriting context.

Three-dimensional crystals of adaptive knots

Starting with Gauss and Kelvin, knots in fields were postulated to behave like particles, but experimentally they were found only as transient features or required complex boundary conditions to exist and could not self-assemble into three-dimensional crystals. We introduce energetically stable, micrometer-sized knots in helical fields of chiral liquid crystals. While spatially localized and freely diffusing in all directions, they resemble colloidal particles and atoms, self-assembling into crystalline lattices with open and closed structures. These knots are robust and topologically distinct from the host medium, though they can be morphed and reconfigured by weak stimuli under conditions such as those in displays. A combination of energy-minimizing numerical modeling and optical imaging uncovers the internal structure and topology of individual helical field knots and the various hierarchical crystalline organizations that they form.

A Novel Adaptive GA-based B-spline Curve Interpolation Method

Background: Curve interpolation is very important in engineering such as computer aided design, image analysis and NC machining. Many patents on curve interpolation have been invented. Objective: Since different knot vector configuration and data point parameterization can generate different shapes of an interpolated B-spline curve, the goal of this paper is to propose a novel adaptive genetic algorithm (GA) based interpolation method of B-spline curve. Method: Relying on geometric features owned by the data points and the idea of genetic algorithm which liberalizes the knots of B-spline curve and the data point parameters, a new interpolation method of B-spline curve is proposed. In addition, the constraint of a tangent vector is also added to ensure that the obtained B-spline curve can approximately satisfy the tangential constraint while ensuring strict interpolation. Results: Compared with the traditional method, this method realizes the adaptive knot vector selection and data point parameterization. Therefore, the interpolation result was better than the traditional method to some extent, and the obtained curve was more natural. Conclusion: The proposed method is effective for the curve reconstruction of any scanned data point set under tangent constraints. Meanwhile, this paper put forward a kind of tangent calculation method of discrete data points, where users can also set the tangent of each data point in order to get more perfect interpolation results.

High quality refinable G‐splines for locally quad‐dominant meshes with T‐gons

AbstractPolyhedral modeling and re‐meshing algorithms use T‐junctions to add or remove feature lines in a quadrilateral mesh. In many ways this is akin to adaptive knot insertion in a tensor‐product spline, but differs in that the designer or meshing algorithm does not necessarily protect the consistent combinatorial structure that is required to interpret the resulting quad‐dominant mesh as the control net of a hierarchical spline – and so associate a smooth surface with the mesh as in the popular tensor‐product spline paradigm. While G‐splines for multi‐sided holes or generalized subdivision can, in principle, convert quad‐dominant meshes with T‐junctions into smooth surfaces, they do not preserve the two preferred directions and so cause visible shape artifacts. Only recently have n‐gons with T‐junctions (T‐gons) in unstructured quad‐dominant meshes been recognized as a distinct challenge for generalized splines. This paper makes precise the notion of locally quad‐dominant mesh as quad‐meshes including τ‐nets, i.e. T‐gons surrounded by quads; and presents the first high‐quality G‐spline construction that can use τ‐nets as control nets for spline surfaces suitable, e.g., for automobile outer surfaces. Remarkably, T‐gons can be neighbors, separated by only one quad, both of T‐gons and of points where many quads meet. A τ‐net surface cap consists of 16 polynomial pieces of degree (3,5) and is refinable in a way that is consistent with the surrounding surface. An alternative, everywhere bi‐3 cap is not formally smooth, but achieves the same high‐quality highlight line distribution.

Surface reconstruction using simplex splines on feature-sensitive configurations

Recently, new non-tensor product splines, called as triangle configuration B-splines (TCB-splines), have been proposed, which generalize the Delaunay configuration B-splines (DCB-splines) by introducing more flexibility in knot geometry while preserving the most attractive properties of univariate B-splines (Liu and Snoeyink, 2007). In this paper, we present a surface reconstruction framework that explores the flexibility of this spline for its use in shape modeling. Starting from a carefully designed feature-sensitive triangulation, we apply the so-call link triangulation procedure to obtain a feature-sensitive configuration family and then construct simplex spline bases as well as spline surface on it. The approximation quality of the surface is then progressively improved by adaptively adding more knots onto the parametric domain according to the fitting errors obtained in the previous fit procedure and updating the feature-sensitive configurations. Conditions to ensure the C1 continuity of the whole result surface are also provided. Our framework is tested on several models to demonstrate its efficacy and ability in preserving geometric features. Compared with DCB-splines, our splines can generate visually pleasant surfaces with smaller fitting errors by using the same number of knots and iterations. With almost the same number of control points, our framework produces more accurate and visually pleasant results than the classical B-spline surface fitting method based on adaptive knot placement strategy (Park, 2011). In addition, the resultant surface provides a control net, which enables an intuitive user interactive design.

A baseline correction algorithm for Raman spectroscopy by adaptive knots B-spline

The Raman spectroscopy technique is a powerful and non-invasive technique for molecular fingerprint detection which has been widely used in many areas, such as food safety, drug safety, and environmental testing. But Raman signals can be easily corrupted by a fluorescent background, therefore we presented a baseline correction algorithm to suppress the fluorescent background in this paper. In this algorithm, the background of the Raman signal was suppressed by fitting a curve called a baseline using a cyclic approximation method. Instead of the traditional polynomial fitting, we used the B-spline as the fitting algorithm due to its advantages of low-order and smoothness, which can avoid under-fitting and over-fitting effectively. In addition, we also presented an automatic adaptive knot generation method to replace traditional uniform knots. This algorithm can obtain the desired performance for most Raman spectra with varying baselines without any user input or preprocessing step. In the simulation, three kinds of fluorescent background lines were introduced to test the effectiveness of the proposed method. We showed that two real Raman spectra (parathion-methyl and colza oil) can be detected and their baselines were also corrected by the proposed method.

Adaptive B-spline volume representation of measured BRDF data for photorealistic rendering

Abstract Measured bidirectional reflectance distribution function (BRDF) data have been used to represent complex interaction between lights and surface materials for photorealistic rendering. However, their massive size makes it hard to adopt them in practical rendering applications. In this paper, we propose an adaptive method for B-spline volume representation of measured BRDF data. It basically performs approximate B-spline volume lofting, which decomposes the problem into three sub-problems of multiple B-spline curve fitting along u-, v-, and w-parametric directions. Especially, it makes the efficient use of knots in the multiple B-spline curve fitting and thereby accomplishes adaptive knot placement along each parametric direction of a resulting B-spline volume. The proposed method is quite useful to realize efficient data reduction while smoothing out the noises and keeping the overall features of BRDF data well. By applying the B-spline volume models of real materials for rendering, we show that the B-spline volume models are effective in preserving the features of material appearance and are suitable for representing BRDF data.

Evolutionary computation for optimal knots allocation in smoothing splines

In this paper, a novel methodology is presented for optimal placement and selections of knots, for approximating or fitting curves to data, using smoothing splines. It is well-known that the placement of the knots in smoothing spline approximation has an important and considerable effect on the behavior of the final approximation [1]. However, as pointed out in [2], although spline for approximation is well understood, the knot placement problem has not been dealt with adequately. In the specialized bibliography, several methodologies have been presented for selection and optimization of parameters within B-spline, using techniques based on selecting knots called dominant points, adaptive knots placement, by data selection process, optimal control over the knots, and recently, by using paradigms from computational intelligent, and Bayesian model for automatically determining knot placement in spline modeling. However, a common two-step knot selection strategy, frequently used in the bibliography, is an homogeneous distribution of the knots or equally spaced approach [3].In order to optimize the placement and numbers of knots required for approximation using smoothing splines, an Evolutionary Computation Paradigms (ECP) based on a Multi-Objective Genetic Algorithm has been developed, with the main purpose of avoiding the large number of local minima (in terms of approximation error for different system complexity or number of knots) existing in the problem of knots placement. The accuracy, computationally efficient and robustness of the algorithm presented will be compared by different experimental results, with other approaches presented in the bibliography, showing the main advantages of the proposed methodology.

Spherical DCB-Spline Surfaces with Hierarchical and Adaptive Knot Insertion

This paper develops a novel surface fitting scheme for automatically reconstructing a genus-0 object into a continuous parametric spline surface. A key contribution for making such a fitting method both practical and accurate is our spherical generalization of the Delaunay configuration B-spline (DCB-spline), a new non-tensor-product spline. In this framework, we efficiently compute Delaunay configurations on sphere by the union of two planar Delaunay configurations. Also, we develop a hierarchical and adaptive method that progressively improves the fitting quality by new knot-insertion strategies guided by surface geometry and fitting error. Within our framework, a genus-0 model can be converted to a single spherical spline representation whose root mean square error is tightly bounded within a user-specified tolerance. The reconstructed continuous representation has many attractive properties such as global smoothness and no auxiliary knots. We conduct several experiments to demonstrate the efficacy of our new approach for reverse engineering and shape modeling.

Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation

One of the key problems in using B-splines successfully to approximate an object contour is to determine good knots. In this paper, the knots of a parametric B-spline curve were treated as variables, and the initial location of every knot was generated using the Monte Carlo method in its solution domain. The best km knot vectors among the initial candidates were searched according to the fitness. Based on the initial parameters estimated by an improved k -means algorithm, the Gaussian Mixture Model (GMM) for every knot was built according to the best km knot vectors. Then, the new generation of the population was generated according to the Gaussian mixture probabilistic models. An iterative procedure repeating these steps was carried out until a termination criterion was met. The GMM-based continuous optimization algorithm could determine the appropriate location of knots automatically. A set of experiments was then implemented to evaluate the performance of the new algorithm. The results show that the proposed method achieves better approximation accuracy than methods based on artificial immune system, genetic algorithm or squared distance minimization (SDM).

B-spline surface fitting based on adaptive knot placement using dominant columns

By expanding the idea of B-spline curve fitting using dominant points (Park and Lee 2007 [13]), we propose a new approach to B-spline surface fitting to rectangular grid points, which is based on adaptive knot placement using dominant columns along u - and v -directions. The approach basically takes approximate B-spline surface lofting which performs adaptive multiple B-spline curve fitting along and across rows of the grid points to construct a resulting B-spline surface. In multiple B-spline curve fitting, rows of points are fitted by compatible B-spline curves with a common knot vector whose knots are computed by averaging the parameter values of dominant columns selected from the points. We address how to select dominant columns which play a key role in realizing adaptive knot placement and thereby yielding better surface fitting. Some examples demonstrate the usefulness and quality of the proposed approach.

An investigation of temporal regularization techniques for dynamic PET reconstructions using temporal splines

The use of a temporal B-spline basis for the reconstruction of dynamic positron emission tomography data was investigated. Maximum likelihood (ML) reconstructions using an expectation maximization framework and maximum A-posteriori (MAP) reconstructions using the generalized expectation maximization framework were evaluated. Different parameters of the B-spline basis of such as order, number of basis functions and knot placing were investigated in a reconstruction task using simulated dynamic list-mode data. We found that a higher order basis reduced both the bias and variance. Using a higher number of basis functions in the modeling of the time activity curves (TACs) allowed the algorithm to model faster changes of the TACs, however, the TACs became noisier. We have compared ML, Gaussian postsmoothed ML and MAP reconstructions. The noise level in the ML reconstructions was controlled by varying the number of basis functions. The MAP algorithm penalized the integrated squared curvature of the reconstructed TAC. The postsmoothed ML was always outperformed in terms of bias and variance properties by the MAP and ML reconstructions. A simple adaptive knot placing strategy was also developed and evaluated. It is based on an arc length redistribution scheme during the reconstruction. The free knot reconstruction allowed a more accurate reconstruction while reducing the noise level especially for fast changing TACs such as blood input functions. Limiting the number of temporal basis functions combined with the adaptive knot placing strategy is in this case advantageous for regularization purposes when compared to the other regularization techniques.

Adaptive knot placement in B-spline curve approximation

An adaptive knot placement algorithm for B-spline curve approximation to dense and noisy data points is presented in this paper. In this algorithm, the discrete curvature of the points is smoothed to expose the characteristics of the underlying curve. With respect to the integration of the smoothed curvature, knots are placed adaptively to satisfy a heuristic rule. Several experimental results are included to demonstrate the validity of this algorithm.

Free-knot Splines and Adaptive Knot Selection

Conventional spline procedures have proven to be effective and useful for estimating smooth functions. However, these procedures find piecewise and inhomogeneous smooth functions difficult to handle. Conventional spline procedures often yield an overly-smooth curve in regions where the true function or its derivative are discontinuous. To fully realize the potential of the spline methodology, this article proposes a function estimation procedure that uses adaptive free-knot splines and allows for multiple knots to be replaced at the same location and between design points. The article also proposes an associated algorithm for implementation in non-parametric regression. The proposed knot selection scheme uses a data-adaptive model selection criterion and an evolutionary algorithm that incorporates certain key features of simulated annealing. The evolutionary algorithm accurately locates the optimal knots, while the data-driven penalty guards against selection errors when searching through a large candidate knot space. The algorithm stochastically yields the globally optimal knots. The simulations suggest that the procedure performs competitively well against alternative methods and has a substantial advantage in relation to nonsmooth and piecewise smooth functions.

The continuous non-linear approximation of procedurally defined curves using integral B-splines

This paper outlines an algorithm for the continuous non-linear approximation of procedurally defined curves. Unlike conventional approximation methods using the discrete L_2 form metric with sampling points, this algorithm uses the continuous L_2 form metric based on minimizing the integral of the least square error metric between the original and approximate curves. Expressions for the optimality criteria are derived based on exact B-spline integration. Although numerical integration may be necessary for some complicated curves, the use of numerical integration is minimized by a priori explicit evaluations. Plane or space curves with high curvatures and/or discontinuities can also be handled by means of an adaptive knot placement strategy. It has been found that the proposed scheme is more efficient and accurate compared to currently existing interpolation and approximation methods.

Modelling childhood malnutrition in Zambia: an adaptive Bayesian splines approach

Nutritional status of a child is an important determinant factor of his/her health and survival. Evaluation of child nutritional status is usually based on three anthropometric indices of height-for-age (stunting), weight-for-height (wasting) and weight-for-age (underweight). This paper is a case study that focuses on a new approach of estimating nonlinear effects. It models the dependence of probability of underweight children in Zambia on some covariates, some of which are metrical whose effects are assumed to be nonlinear and estimated nonparametrically through a Bayesian B-spline basis function approach with adaptive knot selection using the data set from the 1992 Zambia Demographic and Health Survey (ZDHS). For all the unknown functions, the number and location of knots as well as the unknown coefficients of the basis functions are estimated simultaneously using reversible jump Markov chain Monte Carlo (RJMCMC) techniques.