Spline

We study the problem of computing a planar curve, restricted to lie between two given polygonal chains, such that the integral of the square of arc-length derivative of curvature along the curve is minimized. We introduce the minimum variation B-spline problem, which is a linearly constrained optimization problem over curves, defined by B-spline functions only. An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.

Summary form only given. The purpose of this paper is to determine the radial distribution of the emission coefficient from the measured intensity distribution emitted by an extended source of radiation, particularly a plasma source. The source is assumed to be optically thin and axially symmetrical. This problem is solved by inverting Abel's integral equation. A smoothing procedure is made

This work concerns a new kind of neural structure that involves a multidimensional adaptive activation function. The proposed architecture, based on multi-dimensional cubic spline, allows to collect information from the previous network layer in aggregate form. In other words the number of network connections (structural complexity) can be very low respect to the problem complexity. This fact, as experimentally demonstrated in the paper, improve the network generalization capabilities and speed up the convergence of the learning process. A specific learning algorithm is derived and experimental results, demonstrate the effectiveness of the proposed architecture.

This paper presents a new neural architecture suitable for digital signal processing applications. The architecture, based on adaptable multidimensional activation functions, allows to collect information from the previous network layer in aggregate form. In other words the number of network connections (structural complexity) can be very low respect to the problem complexity. This fact, as experimentally demonstrated in the paper, improve the network generalization capabilities and speed up the convergence of the learning process. A specific learning algorithm is derived and experimental results, on channel equalization, demonstrate the effectiveness of the proposed architecture.

A novel Empirical Mode Decomposition (EMD) algorithm, called 2T-EMD, for both mono-and multivariate signals is proposed in this paper. It differs from the other approaches by its computational lightness and its algorithmic simplicity. The method is essentially based on a redefinition of the signal mean envelope, computed thanks to new characteristic points, which offers the possibility to decompose multivariate signals without any projection. The scope of application of the novel algorithm is specified, and a comparison of the 2T-EMD technique with classical methods is performed on various simulated mono-and multivariate signals. The monovariate behaviour of the proposed method on noisy signals is then validated by decomposing a fractional Gaussian noise and an application to real life EEG data is finally presented.

An explicit analytical formula for a short kernel fifth-order polynomial interpolator is obtained. It is also possible to obtain the explicit forms of even higher order interpolation kernels with the method of calculation used, but it is seen that these local kernels become "remainderdominated" as the order increases. The frequency domain properties and the accuracies of the obtained kernel and the known convolution kernels are compared. Frequency domain comparison with the cubic B-spline interpolators is also given. Some cases of proper use of the calculated kernels have been pointed out.

Symbolic computation of NURBS plays an important role in many areas of NURBS-based geometric computation and design. However, any nontrivial symbolic computation, especially when rational B-splines are involved, would typically result in B-splines with high degrees. In this paper we develop degree reduction strategies for NURBS symbolic computation on curves. The specific topics we consider include zero curvatures and critical curvatures of plane curves, various ruled surfaces related to space curves, and point/curve bisectors and curve/curve bisectors.

The space-time least-squares finite element methods are constructed for the advection-diffusion equation by using both linear shape functions and quadratic B-spline shape functions. Two test problems are studied to demonstrate the accuracy of the present methods. Results of the two schemes have been compared.

A rational cubic function with two parameters has been constructed to visualize the positive data. The main focus of the work is the representation of the data in such a way that its view looks smooth and attractive. In the first step simple data dependent constraints are derived on the parameters in the description of the rational cubic function to visualize the shape of positive data then, it is extended to a rational bi-cubic partially blended functions (Coons-patches) and derived constraints on parameters to visualize the shape of positive surface data. The developed scheme is locally positive and economical. The approximation order of rational cubic spline function is O h 3 i .

In full-waveform laser scanning, short laser pulses are emitted and travel towards Earth and object surfaces. The sensor samples the waveform of the emitted pulse and its complete backscattered echo as a function of time. This technique allows for the three-dimensional reconstruction of the terrain, natural and man-made objects, and for the derivation of (geo-)physical quantities such as the differential target cross-section. The retrieval of the differential target cross-section requires deconvolution which is an illposed problem. In this study, we present a novel technique for the computation of the differential target cross-section using B-splines. This class of mathematical functions enables a well-posed linear approach for deconvolution. Furthermore, it is not dependent on the symmetry of the temporal profiles of the emitted laser waveform and the received echoes, as approaches previously suggested. In this paper, the algorithm for deconvolution is presented in detail and validated for both synthetic and real-world fullwaveform laser scanner data.

This paper is concerned with a numerical scheme to solve a singularly perturbed convection–diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme ...

This paper describes B-spline interpolation and compares it with other reconstruction methods, especially in three-dimensional space. We rst consider the B-spline bases in the terms of convolution in signal processing. Presented analysis requires careful usage of continuous and discrete representation of B-splines. Emphasis is given to important di erence between B-spline interpolation and approximation. The di erence is shown through frequency domain analysis, so we derive frequency responses of the Bspline interpolation and approximation. We conclude by demonstrating the use of several reconstruction lters and appropriate gradient estimators in the volume rendering. Exact reconstruction in the volume visualization is very important in many industrial applications, such as material cavity control.

The problem of rendering large trimmed NURBS models at interactive frame rates is of great interest for industry, since nearly all their models are designed on the basis of this surface type. Most existing approaches first transform the NURBS surfaces into polygonal representation and subsequently build static levels of detail upon them, as current graphics hardware is optimized for rendering triangles.

The channel estimation techniques for OFDM systems based on pilot arrangement are investigated. The channel estimation based on comb type pilot arrangement is studied through different algorithms for both estimating channel at pilot frequencies and interpolating the channel.

We describe a four-pass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce crossings. The third pass finds optimal coordinates for nodes by constructing and ranking an auxiliary graph. The fourth pass makes splines to draw edges. The algorithm makes good drawings and runs fast.

A two parameter family of 1 C rational cubic spline functions is presented for the graphical representation of shape preserving curve interpolation for shaped data. These parameters have a direct impact on the shape of the curve. Constraints are developed on one family of the parameters to visualize positive, monotone and convex data while other family of parameters can assume any positive values. The problem of visualization of constrained data is also addressed when the data is lying above a straight line and curve is required to lie on the same side of the line. The approximation order of the proposed rational cubic function is also investigated and is found to be   3 i O h .

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Baseline wandering is a classical problem in electrocardiogram (ECG) records that generally produces artifactual data when measuring ECG parameters. The authors present a cascade adaptive filter for removing the baseline wander and preserving the low-frequency components of the ECG. This cascade adaptive filter works in two stages. The first stage is an adaptive notch filter at zero frequency. The second stage is an adaptive impulse correlated filter that, using a QRS detector, estimates the ECG signal correlated with the ...

A new extremal distance tracking algorithm is presented for parametric curves and surfaces undergoing rigid body motion. The essentially geometric extremization problem is transformed into a dynamical control problem by differentiating with respect to time. Extremization is then solved with the design of a stabilizing controller. We use a feedback linearizing controller. The controller simultaneously accounts for the surface shape and motion while asymptotically achieving (and maintaining) the extremal pair. Thus collision detection takes place in a framework fully analogous to the framework used for the simulation of dynamical response.

The departure from administered interest rate regime to the market determined interest rate regime in 1997 to resurrect the defunct Indian bond market resulted from the attempt to build up domestic bond markets in Asian countries by the joint collaboration of Asia Pacific Economic Co-operation, the International Monetary Fund and the World Bank, and participation therein of the Asian Development Bank Institute. Since then the profit oriented market participants started focusing on yield of the securities available for investment, the Government of India being the major issuer. On the way of India’s progress towards development of a vibrant bond market one of the hindrances is the issue of the methodology of yield curve construction in so far as a constructed yield curve is used in measuring basis risk and yield curve risk in the context of asset liability management and credit default risk in the case of corporate bond. At this juncture there is a need to examine the applicability of different models of yield curve in the government securities market in India. Hence this chapter is divided into the following sections: (a) An introduction to the models of yield curve and their relevance in India (b) Application of above models in current practices of yield curve estimation in India (c) Mismatch between estimates performed by different agencies, (e) Finding a way out – estimation of yield curve based on real time traded data applying spline method and regression method. (f) Examining efficacy of the alternative yield curve in its relative usefulness in fixed income portfolio management.

The aim of the present work is to illustrate how the empirical mode decomposition (EMD) can be associated with demodulation methods to estimate the instantaneous frequency (IF) of multicomponent non stationary signals. The IF estimation of three methods, namely, Hilbert-Huang transform (HHT) [10], EMD-DESA (Discrete Energy Separation Algorithm)also known as Teager-Huang transform (THT) [2] [5], and B-spline version of the

Abstract A rational cubic spline with a family of shape parameters is discussed from the viewpoint of its application in computer graphics. It incorporates both conic sections and parametric cubic curves as special cases. The parameters (weights), in the description of ...

Basic objective of synthesis of cam displacement functions is minimizing the acceleration and jerk of the cam-follower especially in high-speed drives. Here classical splines of 6-, 7-and 8-orders and B-splines of 6-and 8-orders are taken for designing cam displacement functions. Multiple control points are introduced. Acceleration and jerk are minimized by manipulating the control point parameters. A searching procedure is adopted, based on GA and fuzzy membership function. It establishes that introduction of control points largely minimizes acceleration and jerk.

Battle and Lemarie derived independently wavelets by orthonormalizing B-splines. The scaling function Φm (t) corresponding to Battle–Lemarie's wavelet ψm (t) is given by \(\phi _m (t) = \sum\nolimits_{k \in {\text{Z}}} {\alpha _{m,k} B_m (t - k)} \) , where B m(t) is the mth-order central B-spline and the coefficients αm, k satisfy \(\sum\nolimits_{k \in {\text{Z}}} {\alpha _{m,k} e^{ - jk\omega } = {1 \mathord{\left/ {\vphantom {1 {\sqrt {\sum\nolimits_{k \in {\text{Z}}} {B_{2m} (k)e^{ - jk\omega } } } }}} \right. \kern-0em} {\sqrt {\sum\nolimits_{k \in {\text{Z}}} {B_{2m} (k)e^{ - jk\omega } } } }}} \) . In this paper, we propose an FFT-based algorithm for computing the expansion coefficients αm, k and the two-scale relations of the scaling functions and wavelets. The algorithm is very simple and it can be easily implemented. Moreover, the expansion coefficients can be efficiently and accurately obtained via multiple sets of FFT computations. The computational approach presented in this paper here is noniterative and is more efficient than the matrix approach recently proposed in the literature.

Abstract. 1 It is well-known that certain non bandlimited signals such as splines can be reconstructed from uniformly spaced samples similar to bandlimited signals. This usually requires noncausal IIR filters. We revisit this result and consider extensions such as derivative sampling theorems and pulse sampling theorems. It turns out that spline-like signals can often be reconstructed from joint sampling of amplitude and derivative using only FIR filters. We also briefly consider discrete time versions of these results. 1.

lnreractive CAGD sysfems produce faster and more accume curve and surface analysis and design by means of geometric or algebraic interpolation algorithms, where users can directly modify und edit the position and shape of curves or surfaces. For this reason, many claim that ;nterpolation schemata are moreflexible and intuitive than approximation schemata. Also, in engineering and orchitrctural design problems there are position and slope constraints to filfill that, with standard approximation schemata, are harder to produce. However, interpolation methods are limited with respect to the analytic properties of the resulting cwws and surfaces. Piecewise polynomial interpolation methods only guarantees a C' continuity order, which is usually unacceptable. In fhis work we propose an interpolation method named T-Splines or tangent Splines. Starting with interpolating points and derivatives, a T-Spline is the (approximatingi cubic Spline over an ancillary control graph, whose approximation is also a solution of the required interpolation. We present the necessary conditions for finding the a~x&zry control graph from the interpolation points and derivatives. In addition we give a nonuniform parametrization schema such that a nonuniform B-Spline approximation of the ancillary control points is a C2 piecewise polynomial interpolation of the original control points. Some applications are also discussed.

In this paper, the relation between B-splines and FIR (Finite Impulse Response) filters is demonstrated and exploited to design a digital filter for trajectory planning, combining the very simple structure and computational efficiency of FIR filters with the flexibility of splines. In particular, the trajectory generator consists of two main elements. The former is devoted to the solution of an optimization problem that, given a set of points to be interpolated (or approximated), provides the control points defining the spline. The latter, a cascade of moving average filters, gives the trajectory profile at each sampling time on the basis of such points. The proposed method has been applied to several robotic and industrial applications, and in this paper two case studies are reported as examples:an industrial robot performing a welding operation and a mobile robot moving in an environment with obstacles. With respect to these tasks, the main features of the trajectory generator are shown: the possibility of planning trajectories with high degree of smoothness (continuity of the derivatives), the possibility of easily changing the duration of the trajectory (and therefore the velocity, acceleration, jerk, etc. of the trajectory) maintaining the same geometric path, the possibility of modifying locally the pre-planned path.

Respiratory motion can cause artifacts in myocardial SPECT. We incorporate respiratory mechanics into the current 4D MCAT and into the next generation spline-based MCAT phantoms. In order to simulate respiratory motion in the current MCAT phantom, the geometric solids for the diaphragm, heart, ribs, and lungs were altered through manipulation of parameters defining them. Affine transformations were applied to the control points defining the same respiratory structures in the spline-based MCAT phantom to simulate respiratory motion. The NURBS surfaces for the lungs and body outline were constructed in such a way as to be linked to the surrounding ribs. Expansion and contraction of the thoracic cage then coincided with expansion and contraction of the lungs and body. The changes both phantoms underwent were splined over time to create time continuous 4D respiratory models. We conclude that both respiratory models are effective tools for researching effects of respiratory motion.

B-Splines in general, and Non-Uniform Rational B-Splines in particular, have become indispensable modeling primitives in computer graphics and geometric modeling applications. In this paper a novel high-performance architecture for the computation of uniform, non-uniform, rational and non-rational B-Spline curves and surfaces is presented. This architecture has been derived through a sequence of steps. First, a systolic architecture for the computation of the basis function values, the basis function evaluation array (the BFEA), is developed. Using the BFEA as its core, an architecture for the computation of non-uniform rational B-Spline curves is constructed. This architecture is then extended to compute NURBS surfaces. Finally, this architecture is augmented to compute the surface normals so that the output from this architecture can be directly used for rendering the NURBS surface. The overall linear structure of the architecture, its small I/O requirements, its non-dependence on the size of the problem (in terms of the number of control points and the number of points on the curve/surface that has to be computed), and its very high throughput make this architecture highly suitable for integration into the standard graphics pipeline of high-end workstations. Results of the timing analysis indicate a potential throughput of one triangle with the normal vectors at its vertices, every two clock cycles.

In this paper, we propose an algorithm to construct 3-D surface model from a set of range data, based on non-uniform rational B-splines (NURBS) surface-fitting technique. It is assumed that the range data is initially unorganized and scattered 3-D points, while their connectivity is also unknown. The proposed algorithm consists of three stages: initial model approximation employing -means clustering, hierarchical decomposition of the initial model, and construction of NURBS surface patch network. The initial model is approximated by both polyhedral and triangular model. Then, the initial model is represented by a hierarchical graph, which is efficiently used to construct the 1 continuous NURBS patch network of the whole object. Experiments are carried out on synthetic and real range data to evaluate the performance of the proposed algorithm. It is shown that the initial model as well as the NURBS patch network are constructed automatically with tolerable computation. The modeling error of the NURBS model is reduced to 10%, compared with the initial mesh model.

The channel estimation techniques for OFDM systems based on pilot arrangement are investigated. The channel estimation based on comb type pilot arrangement is studied through different algorithms for both estimating channel at pilot frequencies and interpolating the channel. The estimation of channel at pilot frequencies is based on LS and LMS while the channel interpolation is done using linear interpolation, second order interpolation, low-pass interpolation, spline cubic interpolation, and time domain interpolation. Time-domain interpolation is obtained by passing to time domain through IDFT (Inverse Discrete Fourier Transform), zero padding and going back to frequency domain through DFT (Discrete Fourier Transform). In addition, the channel estimation based on block type pilot arrangement is performed by sending pilots at every sub-channel and using this estimation for a specific number of following symbols. We have also implemented decision feedback equalizer for all sub-channels followed by periodic block-type pilots. We have compared the performances of all schemes by measuring bit error rate with 16QAM, QPSK, DQPSK and BPSK as modulation schemes, and multi-path Rayleigh fading and AR based fading channels as channel models.

Many real-world tasks require fast planning of highly dynamic movements for their execution in realtime. The success often hinges on quickly finding one of the few plans that can achieve the task at all. A further challenge is to quickly find a plan which optimizes a desired cost. In this paper, we will discuss this problem in the context of catching small flying targets efficiently. This can be formulated as a non-linear optimization problem where the desired trajectory is encoded by an adequate parametric representation. The optimizer generates an energy-optimal trajectory by efficiently using the robot kinematic redundancy while taking into account maximal joint motion, collision avoidance and local minima. To enable the resulting method to work in real-time, examples of the global planner are generalized using nearest neighbour approaches, Support Vector Machines and Gaussian process regression, which are compared in this context. Evaluations indicate that the presented method is highly efficient in complex tasks such as ball-catching.

I would like to sincerely acknowledge my supervisor Professor John S. Zelek for all the guidance, kindness and support he has provided me throughout this study. I cannot imagine completing my thesis without his insightful supervision and continuous support. I would also like to express my appreciation to my thesis readers, Professor Paul Fieguth and Professor Otman Basir, for providing me with their valuable comments. In addition, I would like to thank the Defense Research and Development Canada (DRDC) for partially supporting this research.

In this paper, we design a new family of biorthogonal wavelet transforms that are based on polynomial and discrete splines. The wavelet transforms are constructed via lifting steps, where the prediction and update filters are derived from various types of interpolatory and quasi-interpolatory splines. The transforms use finite and infinite impulse response (IIR) filters and are implemented in a fast lifting mode. We analyze properties of the generated scaling functions and wavelets. In the case when the prediction filter is derived from a polynomial interpolatory spline of even order, the synthesis scaling function and wavelet are splines of the same order. We formulate conditions for the IIR filter to generate an exponentially decaying scaling function.

In this paper, we consider the estimation of isotonic, convex or related functions by means of splines. It is shown that certain classes of isotone or convex functions can be represented as a positive cone embedded in a Hilbert space. Using this representation, we give an existence and characterization theorem for isotonic or convex splines. Two special cases are examined showing the existence of a globally monotone cubic smoothing spline and a globally convex quintic smoothing spline. Finally, we examine a regression problem and show that the isotonic-type of spline provides a strongly consistent solution. We also point out several other statistical applications.

Various models have been developed, for the designing of distinct objects, for applications like font designing, Computer Aided Design (CAD), Computer Aided Engineering (CAE}, etc. Some methods are better suited for controlling the shape of the curve on an interval, while others are better suitedfor controlling the shape at the individual control points. This work reviews C2 rational splines with interval tension and extends this work for the modelling of interpolatory curves and surfaces through B-spline formulation.