Non-uniform B-spline Research Articles

Construction of new multiple knot B-spline wavelets

This paper deals with construction of non-uniform multiple knot B-spline wavelet basis functions (with minimal support). These wavelets are semi-orthogonal on a bounded interval. A large family of multiple knot B-spline wavelets is presented that gives a variety of basis functions with explicit formulas and locally compact supports. Moreover, the structure of this wavelet is conceptually simple and easy to implement. Finally, some examples of multiple knot B-spline wavelets are also presented.

Planar Pythagorean-Hodograph B-Spline curves

We introduce a new class of planar Pythagorean-Hodograph (PH) B-Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) Bézier curves, presented by R. Farouki and T. Sakkalis in 1990, including the latter ones as special cases. Pythagorean-Hodograph B-Spline curves are non-uniform parametric B-Spline curves whose arc length is a B-Spline function as well. An important consequence of this special property is that the offsets of Pythagorean-Hodograph B-Spline curves are non-uniform rational B-Spline (NURBS) curves. Thus, although Pythagorean-Hodograph B-Spline curves have fewer degrees of freedom than general B-Spline curves of the same degree, they offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. After providing a general definition for this new class of planar parametric curves, we present useful formulae for their construction and discuss their remarkable attractive properties. Then we solve the reverse engineering problem consisting of determining the complex pre-image spline of a given PH B-Spline, and we also provide a method to determine within the set of all PH B-Splines the one that is closest to a given reference spline having the same degree and knot partition.

Lane Boundary Detection Algorithm Based on Vector Fuzzy Connectedness

In most actual autonomous guided vehicles (AGV), path finding and navigational control systems are usually implemented using images captured by cameras mounted on the vehicles. This paper presents and discusses a lane boundary detection technique that is necessary for the task of autonomous driving. In this paper, a new method called vector fuzzy connectedness (VFC) is presented to detect and estimate road lane boundaries. First, a preprocessed technique is used to obtain a skeleton image. Based on the result, the curvatures of the left and right lane boundaries are estimated, and the control points are found by the VFC method. Finally, the non-uniform b-spline (NUBS) interpolation method is introduced to construct the road lane boundaries. The proposed VFC method integrates the vector concept and fuzzy connectedness into the lane boundary detection algorithm. As shown in the example results, the proposed method can extract various road lane shapes and types from real road frames even under complex road environments. For navigation tasks, it is necessary to determine the position of the vehicle relative to the road. These results prove that the proposed detection method can assist in a number of actual AGV assistant applications. In the future, some intelligent techniques will be applied to test the AGV system with obstacle avoidance conditions on real world roads.

A unified approach to construct generalized B-splines for isogeometric applications

Generalized B-splines have been employed as geometric modeling and numerical simulation tools for isogeometric analysis (IGA for short). However, the previous models used in IGA, such as trigonometric generalized B-splines or hyperbolic generalized B-splines, are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper, a unified approach to construct the generalized non-uniform B-splines over the space spanned by {α(t), β(t), ξ(t), η(t), 1, t, · · ·, tn−4} is proposed, and the corresponding isogeometric analysis framework for PDE solving is also studied. Compared with the NURBS-IGA method, the proposed frameworks have several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form. Furthermore, with the proposed spline models, isogeometric analysis can be performed on the computational domain bounded by transcendental curves/surfaces, such as the involute of circle, the helix/helicoid, the catenary/catenoid and the cycloid. Several numerical examples for isogeometric heat conduction problems are presented to show the effectiveness of the proposed methods.

A PIA optimization algorithm for non-uniform B-spline curves and surfaces

This paper aims to address an important issue in B-spline interpolation. In order to reduce shape distortion due to parameterization in the iterative interpolation of B-spline curves and surfaces, we consider both the initial iteration direction of the control vertices and the parameter values corresponding to the nearest points on the initial iterative curves or surfaces. Genetic algorithm with a carefully designed objective function is employed in finding the optimal parameter solutions for high order B-spline curves and surfaces. Experiments show that the shape distortion due to parameterization has been significantly reduced.

A time varying filter approach for empirical mode decomposition

A modified version of empirical mode decomposition (EMD) is presented to solve the mode mixing problem. The sifting process is completed using a time varying filter technique. In this paper, the local cut-off frequency is adaptively designed by fully facilitating the instantaneous amplitude and frequency information. Then nonuniform B-spline approximation is adopted as a time varying filter. In order to solve the intermittence problem, a cut-off frequency realignment algorithm is also introduced. Aimed at improving the performance under low sampling rates, a bandwidth criterion for intrinsic mode function (IMF) is proposed. The proposed method is fully adaptive and suitable for the analysis of linear and non-stationary signals. Compared with EMD, the proposed method is able to improve the frequency separation performance, as well as the stability under low sampling rates. Besides, the proposed method is robust against noise interference.

A direct method to solve optimal knots of B-spline curves: An application for non-uniform B-spline curves fitting.

B-spline functions are widely used in many industrial applications such as computer graphic representations, computer aided design, computer aided manufacturing, computer numerical control, etc. Recently, there exist some demands, e.g. in reverse engineering (RE) area, to employ B-spline curves for non-trivial cases that include curves with discontinuous points, cusps or turning points from the sampled data. The most challenging task in these cases is in the identification of the number of knots and their respective locations in non-uniform space in the most efficient computational cost. This paper presents a new strategy for fitting any forms of curve by B-spline functions via local algorithm. A new two-step method for fast knot calculation is proposed. In the first step, the data is split using a bisecting method with predetermined allowable error to obtain coarse knots. Secondly, the knots are optimized, for both locations and continuity levels, by employing a non-linear least squares technique. The B-spline function is, therefore, obtained by solving the ordinary least squares problem. The performance of the proposed method is validated by using various numerical experimental data, with and without simulated noise, which were generated by a B-spline function and deterministic parametric functions. This paper also discusses the benchmarking of the proposed method to the existing methods in literature. The proposed method is shown to be able to reconstruct B-spline functions from sampled data within acceptable tolerance. It is also shown that, the proposed method can be applied for fitting any types of curves ranging from smooth ones to discontinuous ones. In addition, the method does not require excessive computational cost, which allows it to be used in automatic reverse engineering applications.

Cubic B-Spline Curves with Shape Parameter and Their Applications

The present studies on the extension of B-spline mainly focus on Bezier methods and uniform B-spline and are confined to the adjustment role of shape parameters to curves. Researchers pay little attention to nonuniform B-spline. This paper discusses deeply the extension of the quasi-uniform B-spline curves. Firstly, by introducing shape parameters in the basis function, the spline curves are defined in matrix form. Secondly, the application of the shape parameter in shape design is analyzed deeply. With shape parameters, we get another means for adjusting the curves. Meanwhile, some examples are given. Thirdly, we discuss the smooth connection between adjacent B-spline segments in detail and present the adjusting methods. Without moving the control points position, through assigning appropriate value to the shape parameter, C1 continuity of combined spline curves can be realized easily. Results show that the methods are simple and suitable for the engineering application.

Anisotropic spline approximation with non-uniform B-splines

Recently the author and U. Reif introduced the concept of diversification of uniform tensor product B-splines. Based on this concept, we give a new constructive modification of non-uniform B-splines. The resulting spline spaces are perfectly fitted for the approximation of functions defined on domains . We build a bounded quasi-interpolant and prove that for our spline spaces an anisotropic error estimate in the -norm, , is valid. In particular, we show that the constant of the error estimate does not depend on the shape of or the knot grid.

Simultaneous 4D-CBCT reconstruction with sliding motion constraint.

Current approaches using deformable vector field (DVF) for motion-compensated 4D-cone beam CT (CBCT) reconstruction typically utilize an isotropically smoothed DVF between different respiration phases. Such isotropically smoothed DVF does not work well if sliding motion exists between neighboring organs. This study investigated an anisotropic motion modeling scheme by extracting organ boundary local motions (e.g., sliding) and incorporated them into 4D-CBCT reconstruction to optimize the motion modeling and reconstruction methods. Initially, a modified simultaneous algebraic reconstruction technique (mSART) was applied to reconstruct high quality reference phase CBCT using all phase projections. The initial DVFs were precalculated and subsequently updated to achieve the optimized solution. During the DVF update, sliding motion estimation was performed by matching the measured projections to the forward projection of the deformed reference phase CBCT. In this process, each moving organ boundary was first segmented. The normal vectors of the boundary DVF were then extracted and incorporated for further DVF optimization. The regularization term in the objective function adaptively regularizes the DVF by (1) isotopically smoothing the DVF within each organ; (2) smoothing the DVF at boundary along the normal direction; and (3) leaving the tangent direction of boundary DVF unsmoothed (i.e., allowing for sliding motion). A nonlinear conjugate gradient optimizer was used. The algorithm was validated on a digital cubic tube phantom with sliding motion, nonuniform rotational B-spline based cardiac-torso (NCAT) phantom, and two anonymized patient data. The relative reconstruction error (RE), the motion trajectory's root mean square error (RMSE) together with its maximum error (MaxE), and the Dice coefficient of the lung boundary were calculated to evaluate the algorithm performance. For the cubic tube and NCAT phantom tests, the REs are 10.2% and 7.4% with sliding motion compensation, compared to 13.4% and 8.9% without sliding modeling. The motion trajectory's RMSE and MaxE for NCAT phantom tests are 0.5 and 0.8 mm with sliding motion constraint compared to 3.5 and 7.3 mm without sliding motion modeling. The Dice coefficients for both NCAT phantom and the patients show a consistent trend that sliding motion constraint achieves better similarity for segmented lung boundary compared with the ground truth or patient reference. A sliding motion-compensated 4D-CBCT reconstruction and the motion modeling scheme was developed. Both phantom and patient study demonstrated the improved accuracy and motion modeling accuracy in reconstructed 4D-CBCT.

Contour propagation using non-uniform cubic B-splines for lung tumor delineation in 4D-CT.

Accurate target delineation is a critical step in radiotherapy. In this study, a robust contour propagation method is proposed to help physicians delineate lung tumors in four-dimensional computer tomography (4D-CT) images efficiently and accurately. The proposed method starts with manually delineated contours on the reference phase. Each contour is fitted by a non-uniform cubic B-spline curve, and its deformation on the target phase is achieved by moving its control vertexes such that the intensity similarity between the two contours is maximized. Since contour is usually the boundary of lesion or tissue which may deform quite differently from the tissues outside the boundary, the proposed method treats each contour as a deformable entity, a non-uniform cubic B-spline curve, and focuses on the registration of contour entity instead of the entire image to avoid the deformation of contour to be smoothed by its surrounding tissues, meanwhile to greatly reduce the time consumption while keeping the accuracy of the contour propagation. Eighteen 4D-CT cases with 444 gross tumor volume (GTV) contours manually delineated slice by slice on the maximal inhale and exhale phases are used to verify the proposed method. The Jaccard similarity coefficient (JSC) between the propagated GTV and the manually delineated GTV is 0.885±0.026, and the Hausdorff distance (HD) is [Formula: see text] mm. In addition, the time for propagating GTV to all the phases is 3.67±3.41 minutes. The results are better than fast adaptive stochastic gradient descent (FASGD) B-spline method, 3D+t B-spline method and diffeomorphic Demons method. The proposed method is useful to help physicians delineate target volumes efficiently and accurately.

A new calculating method to perform springback predictions for varied curvature sheet bending based on the B-spline function

This paper presents a new method to calculate the bending moments in varied curvature sheet bending. Compared with traditional bending theories which suppose the curvature radii to be constant, the new formulas of stress and strain consider the geometrical characteristics of the profiles, thus they can reflect the internal force in the sheet more accurately. Also, they are consistent with traditional bending theories when the curvature remains unchanged. Springback predictions are performed based on the calculated bending moments. The final shapes of the sheet after springback are obtained by proper discrete and connecting technologies. Two cases are provided to demonstrate the calculating method. In the first case, a cubic non-uniform B-spline curve is used to interpolate the known die profile. This curvature continuity spline gives a relatively accurate expression of the curvature change phenomenon. In the second case, the math expression of a desired bending shape is already known, and the method we proposed is proved to be generally applicable by predicting the final shape of the sheet. The calculating results can reveal how the bending moments change with the curvature radii. They are in great agreement with experiments and also they have been compared with numerical simulations.

Kinematic Analysis of Rotary Transplanting Mechanism for Wide-Narrow Row Pot Seedlings

Wide-narrow row pot seedling transplanting (WPST) could obtain high rice yield and quality. In this study, a new type of mechanism for WPST is designed based on analysis of recent advances at home and abroad. The WPST mechanism, which consists of planetary gear trains with seven noncircular bevel gears, is used in an automatic rice pot seedling transplanter. The working trajectory and attitude properties are analyzed according to the agronomy requirements of rice WPST. To establish the kinematics, a non-uniform B-spline curve is used to fit and control the outer pitch curve of the noncircular bevel gears, and D-H transformation matrices are introduced to express the trajectory and attitude of the planting paw in the world frame of the WPST mechanism. Computer-aided analysis and an optimization program of the WPST mechanism are developed. Based on the program, the effects of various parameters on transplanting trajectory and attitude are analyzed. By human-machine interaction, the structural parameters that meet the demands of the WPST mechanism are optimized. Using these parameters, a 3D model of the mechanism is built, and virtual simulation and physical prototype tests are carried out. The simulation and experimental results agree with the theoretical analysis, which validates the theoretical model of the WPST mechanism.

Blade profile optimization of pump as turbine

One major focus on the performance researches of pump as turbine is how to enhance the efficiency of energy recovery. While the key point of increasing the efficiency is to improve the performance of the blade profile which is structural basis of the blade geometry. This article presents an optimization method for the blade profile. It contained the parameterization of blade profile, the Latin Hypercube experimental design, the computational fluid dynamics techniques, the back propagation neural network, and genetic algorithm. Specifically, the nonuniform cubic B-spline curve was used to parameterize the blade profile, the Latin Hypercube experimental design method for the acquirement of the sample points of back propagation neural network. The performance analysis of each sample point was accomplished by the computational fluid dynamics techniques. Then, the learning and training of the back propagation neural network was carried out. Finally, the optimization techniques of combining the back propagation neural network and genetic algorithm were used to solve the optimization problems of the blade profile. Based on the above method, the blade profile of a pump as turbine was optimized and improved. The result shows that the efficiency of the optimized pump as turbine under the optimum operating condition was increased by 2.91%, with the constraint condition to ensure that the difference between the head and the initial head of the pump as turbine is less than the specified value. This proves that using the above method to optimize the blade profile is feasible.

Digitalized accurate modeling of SPCB with multi-spiral surface based on CPC algorithm

The main methods of the existing multi-spiral surface geometry modeling include spatial analytic geometry algorithms, graphical method, interpolation and approximation algorithms. However, there are some shortcomings in these modeling methods, such as large amount of calculation, complex process, visible errors, and so on. The above methods have, to some extent, restricted the design and manufacture of the premium and high-precision products with spiral surface considerably. This paper introduces the concepts of the spatially parallel coupling with multi-spiral surface and spatially parallel coupling body. The typical geometry and topological features of each spiral surface forming the multi-spiral surface body are determined, by using the extraction principle of datum point cluster, the algorithm of coupling point cluster by removing singular point, and the “spatially parallel coupling” principle based on the non-uniform B-spline for each spiral surface. The orientation and quantitative relationships of datum point cluster and coupling point cluster in Euclidean space are determined accurately and in digital description and expression, coupling coalescence of the surfaces with multi-coupling point clusters under the Pro/E environment. The digitally accurate modeling of spatially parallel coupling body with multi-spiral surface is realized. The smooth and fairing processing is done to the three-blade end-milling cutter’s end section area by applying the principle of spatially parallel coupling with multi-spiral surface, and the alternative entity model is processed in the four axis machining center after the end mill is disposed. And the algorithm is verified and then applied effectively to the transition area among the multi-spiral surface. The proposed model and algorithms may be used in design and manufacture of the multi-spiral surface body products, as well as in solving essentially the problems of considerable modeling errors in computer graphics and engineering in multi-spiral surface’s connection available with approximate methods or graphical methods.

Tooth flank reconstruction and optimizations after simulation process modeling for the spiral bevel gear

Tooth flank reconstruction and optimization methodologies after simulation process modeling are presented, in order to provide accurate model and tooth data for digitized design and manufacture of the spiral bevel gear. Firstly, a simulation process modeling utilizing universal machine settings is developed for an initial solid model. Then, due to its poor accuracy, tooth flank reconstruction exploiting the Non-Uniform B-Spline fitting method is carried out. Finally, some tooth flank optimizations are introduced for higher tooth flank precision and accurate tooth data: (1) Overall tooth flank interpolations based on the Energy method; (2) Tooth flank approximation based on the least square (LSQ); (3) Tooth flank parameterization based on the Newton Iteration method. Results obtained from some numerical examples indicate that validation of the proposed approaches and tooth flank form error is significantly reduced.

Reverse engineering of free-form surface based on the closed-loop theory.

To seek better methods of measurement and more accurate model of reconstruction in the field of reverse engineering has been the focus of researchers. Based on this, a new method of adaptive measurement, real-time reconstruction, and online evaluation of free-form surface was presented in this paper. The coordinates and vectors of the prediction points are calculated according to a Bézier curve which is fitted by measured points. Final measured point cloud distribution is in agreement with the geometric characteristics of the free-form surfaces. Fitting the point cloud to a surface model by the nonuniform B-spline method, extracting some check points from the surface models based on grids and a feature on the surface, review the location of these check points on the surface with CMM and evaluate the model, and then update the surface model to meet the accuracy. Integrated measurement, reconstruction, and evaluation, with the closed-loop reverse process, established an accurate model. The results of example show that the measuring points are distributed over the surface according to curvature, and the reconstruction model can be completely expressed with micron level. Meanwhile, measurement, reconstruction and evaluation are integrated in forms of closed-loop reverse system.

A New Digital Audio Watermarking Algorithm

In this paper a new digital audio watermarking algorithm is proposed based on B-spline approximation. The al- gorithm takes the advantage of non-uniform B-spline wavelet, namely, the number of low-frequency components can be arbitrarily selected, so, it is very flexible. Meanwhile, it avoids the complex calculations of non-uniform B-spline wave- lets. The simulation experiments show that this algorithm guarantees the inaudibility of the watermark, and it is robust against common attacks as well.

A fault diagnosis approach for diesel engine valve train based on improved ITD and SDAG-RVM

Targeting the non-stationary characteristics of the vibration signals of a diesel engine valve train, and the limitation of the autoregressive (AR) model, a novel approach based on the improved intrinsic time-scale decomposition (ITD) and relevance vector machine (RVM) is proposed in this paper for the identification of diesel engine valve train faults. The approach mainly consists of three stages: First, prior to the feature extraction, non-uniform B-spline interpolation is introduced to the ITD method for the fitting of baseline signal, then the improved ITD is used to decompose the non-stationary signals into a set of stationary proper rotation components (PRCs). Second, the AR model is established for each PRC, and the first several AR coefficients together with the remnant variance of all PRCs are regarded as the fault feature vectors. Finally, a new separability based directed acyclic graph (SDAG) method is proposed to determine the structure of multi-class RVM, and the fault feature vectors are classified using the SDAG-RVM classifier to recognize the fault of the diesel engine valve train. The experimental results demonstrate that the proposed fault diagnosis approach can effectively extract the fault features and accurately identify the fault patterns.

Super-resolution reconstruction of remote sensing image based on staggered pixels and non-uniform B-spline curved surface

The adoption of information from two images can improve resolution of images; nevertheless,the method based on interpolation is apt to make fussy boundary or cause partial loss of detailed information. To solve this problem,the authors employed an approach to super- resolution reconstruction based on staggered pixels and non- uniform B- spline surface. Two frames of images with low resolution,which were matched and geometrically registered at sub- pixel level,were adopted,and then these two images were staggered and re- sampled to double times of the origin grid; the positions with non- value were interpolated and filled with tri- B- spline surface interpolation,and non- uniform node parametric method was adopted. And the curved surface was composed of 36 known neighborhood pixels. For solving the value of interpolated points,the authors introduced parallel and golden section methods to iterate searching for the optimal value,which made interpolation more accurate; the last but not unimportant,the interpolated images were restored to reconstruct better visualization high resolution images. The assessment of the results demonstrates that this approach can considerably improve definition, quantity of information,signal- to- noise ratio and resolution.