Cardinal Spline Research Articles

Existence and uniqueness of spline reconstruction from local weighted average samples

The sampling theorem is one of the most powerful and useful results in signal analysis. However, most of the results on sampling theorem assumes that the distances between the consecutive sampling points are strictly less than one. In this paper, we consider the case when the distance between the consecutive sampling points is always one and study the reconstruction of cardinal spline functions from their weighted local average samples \(y_{n}=f\star h(n),n\in {\mathbb {Z}}\), where the weight function \(h(t)\) has support in \([-\frac{1}{2}, \frac{1}{2}].\) We prove under certain restrictive conditions on the average function \(h\) that for a given \(y_{n}\), there exists a unique cardinal spline \(f(t)\) of degree d satisfying \(y_{n}=f\star h(n),n\in {\mathbb {Z}}.\)

Parametric Human Body Model for Digital Apparel Design

With the advance in 3D body scanning technology, it opens opportunities for virtual try-on and automatic made-to-measure in apparel products domain. This paper proposed a novel feature-based parametric method of human body shape from the cloud points of 3D body scanner [T2. Firstly, we improved the skeleton construction through adding and adjusting the position of joints. Secondly, automatic extraction approach of semantic feature cross-sections is developed based on the hierarchy. According to the unique distribution of cloud points of each cross-section of each body part, the extraction method of key points on the cross-section is described. Thirdly, we presented an interpolation approach of key points which fit cardinal spline to cross-section for each body part, in which tension parameter is used to represent the simple deformation of body shape. Finally, a connection approach of body part is proposed by sharing a boundary curve. The proposed method has been tested with our virtual human model (VHM) system which is robust and easier to use. The process generally requires about five minutes for generating a full body model that represents the body shape captured by 3D body scanner. The model can be imported in a CAD environment for application to a wide variety of ergonomic analyses.

On Cardinal Spline Interpolation

Abstract. In the present paper it is shown that the interpolation problem for multiple knot cardinal splines subject to general interpolation conditions has a unique solution with polynomial growth if the data grow correspondingly provided a certain determinantal condition is satisfied. An application to Hs error estimates for the interpolation with periodic multiple knot splines is given.

Complex B-splines and Hurwitz zeta functions

AbstractWe characterize nonempty open subsets of the complex plane where the sum $\zeta (s, \alpha )+ {e}^{\pm i\pi s} \hspace{0.167em} \zeta (s, 1- \alpha )$ of Hurwitz zeta functions has no zeros in $s$ for all $0\leq \alpha \leq 1$. This problem is motivated by the construction of fundamental cardinal splines of complex order $s$.

Reconstruction of splines from local average samples

We study the reconstruction of cardinal splines f ( t ) from their average samples y n = f ∗ h ( n ) , n ∈ Z , when the average function h ( t ) has support in [ − 1 / 2 , 1 / 2 ] . We investigate the existence and uniqueness of the solution of the following problem: For given dates y n , find a cardinal spline f ( t ) , of a given degree, satisfying y n = f ∗ h ( n ) , n ∈ Z .

PAPERS PUBLISHED IN ARCHAEOLOGY, ETHNOLOGY & ANTHROPOLOGY OF EURASIA IN 2011

This paper presents innovative contributions to the fields of cardinal spline interpolation and subdivision. In particular, it unifies cardinal Br-spline fundamental functions for interpolation that are made of r=ML+1 (L∈N∪{0}) distinct pieces between each pair of interpolation nodes and are featured by the properties of C2M−2 smoothness, approximation order 2M and support width 2M(r+1)r, with the basic limit functions of a special class of non-stationary subdivision schemes of arity M.After introducing a general result, we focus our attention on the subclass of fourth-order accurate, C2 smooth Br-splines with maximum width of the compact support 6. The binary subdivision scheme yielding these fundamental functions outperforms the existing interpolatory schemes and seems to be the most adequate starting point to obtain compactly supported fundamental (spline) functions for local interpolation over quadrilateral and triangular meshes.

Extracting roads from dense point clouds in large scale urban environment

This paper describes a method for extracting roads from a large scale unstructured 3D point cloud of an urban environment consisting of many superimposed scans taken at different times. Given a road map and a point cloud, our system automatically separates road surfaces from the rest of the point cloud. Starting with an approximate map of the road network given in the form of 2D intersection locations connected by polylines, we first produce a 3D representation of the map by optimizing Cardinal splines to minimize the distances to points of the cloud under continuity constraints. We then divide the road network into independent patches, making it feasible to process a large point cloud with a small in-memory working set. For each patch, we fit a 2D active contour to an attractor function with peaks at small vertical discontinuities to predict the locations of curbs. Finally, we output a set of labeled points, where points lying within the active contour are tagged as “road” and the others are not. During experiments with a LIDAR point set containing almost a billion points spread over six square kilometers of a city center, our method provides 86% correctness and 94% completeness.

Adaptive image interpolation by cardinal splines in piecewise constant tension

The cardinal spline in tension is modified to allow for different tensions in different sampling intervals. Varying the tension in proportion to an index of sharp change in image brightness, we obtain image interpolation results with less ringing artifacts compared to those by the cubic spline interpolation.

A generalized Taylor factorization for Hermite subdivision schemes

In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so-called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of a generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity.

On the Periodized Square of L 2 Cardinal Splines

We establish properties of and propose a conjecture concerning ∑ m (S(x+m))2, where S is a piecewise polynomial cardinal spline in .

수정된 전역 DWA에 의한 자율이동로봇의 경로계획

The global dynamic window approach (DWA) is widely used to generate the shortest path of mobile robots considering obstacles and kinematic constraints. However, the dynamic constraints of robots should be considered to generate the minimum-time path. We propose a modified global DWA considering the dynamic constraints of robots. The reference path is generated using A* algorithm and smoothed by cardinal spline function. The trajectory is then generated to follows the reference path in the minimum time considering the robot dynamics. Finally, the local path is generated using the dynamic window which includes additional terms of speed and orientation. Simulation and experimental results are presented to verify the performance of the proposed method.

3-D Modelling of Basic Weft-Knitted Fabric Structures

Based on the structural analysis of sample knitted fabrics and the knit geometric theory, we have built an algorithm that can make three-dimensional structure models for weft knitted fabrics such as welt stitches, tuck stitches, cross stitches, rib stitches and purl stitches. In order to reduce computational complexity, we have set the anchoring point on the diametral plane of yarn in the knitting chart. Drawing from the beginning knitting point to the finished knitting point, a smooth knit curve model can be obtained with the cardinal spline curve. Using this algorithm, we have developed the weft knit three-dimensional structure modeling software that can display the three-dimensional model of knitted fabrics including welt stitches; tuck stitches, cross stitches, rib stitches and purl stitches.Our work confirms 3-D modeling of three foundation knits of weft knit (plain stitches, purl stitches, rib stitches), tuck stitches, welt stitches and cross stitches with a commercially available PC. This software can be used by someone with no understanding of knitting to gain an understanding of the knitting process, and determine the structure of a knitted fabric product.

Sampling From a System-Theoretic Viewpoint: Part II—Noncausal Solutions

This paper puts to use concepts and tools introduced in Part I to address a wide spectrum of noncausal sampling and reconstruction problems. Particularly, we follow the system-theoretic paradigm by using systems as signal generators to account for available information and system norms (L2 and L∞) as performance measures. The proposed optimization-based approach recovers many known solutions, derived hitherto by different methods, as special cases under different assumptions about acquisition or reconstructing devices (e.g., polynomial and exponential cardinal splines for fixed samplers and the Sampling Theorem and its modifications in the case when both sampler and interpolator are design parameters). We also derive new results, such as versions of the Sampling Theorem for downsampling and reconstruction from noisy measurements, the continuous-time invariance of a wide class of optimal sampling-and-reconstruction circuits, etcetera.

Smooth Path Planning Method for Autonomous Mobile Robots Using Cardinal Spline

We propose a smooth path planning method for autonomous mobile robots. Due to nonholonomic constraints by obstacle avoidance, the smooth path planning is a complicated one. We generate smooth path that is considered orientation of robot under nonholonomic constraints. The proposed smooth planning method consists of two steps. Firstly, the initial path composed of straight lines is obtained from V-graph by Dijkstra's algorithm. Then the initial path is transformed by changing the curve. We apply the cardinal spline into the stage of curve generation. Simulation results show a performance of proposed smooth path planning method.

Continuous time domain properties of causal cubic splines

The paper studies the continuous time domain properties of causal cubic splines with equidistant knots. Causality is achieved by truncating the impulse response of the ideal interpolator to a chosen length. The introduced error affects the final interpolation differently depending on the considered output filter formulation. For the first formulation it is the regularity that is sacrificed while for the second it is the interpolation and splitting of unit delay properties. The analysis of the casual cardinal splines’ spectra also demonstrates different manifestation of the causality induced error that affects either the pass-band region only or both the pass-band and the stop-band region. The frequency kernel is used for objective comparison of the causal and the non-causal interpolators including the causal piecewise cubic MOMS interpolator. Their relative interpolation performance was determined both for wide-band as well as for oversampled narrow-band input signals. The transparent delay is identified for which the causal and the non-causal interpolators achieve identical band-limited reconstruction accuracy.

Skeleton‐based active catheter navigation

The emergence of the active catheter has prompted the development of catheterization in minimally invasive surgery. However, it is still operated using only the physician's vision; information supplied by the guiding image and tracking sensors has not been fully utilized. In order to supply the active catheter with more useful information for automatic navigation, we extract the skeleton of blood vessels by means of an improved distance transform method, and then present the crucial geometric information determining navigation. With the help of tracking sensors' position and pose information, two operations, advancement in the proximal end and direction selection in the distal end, are alternately implemented to insert the active catheter into a target blood vessel. The skeleton of the aortic arch reconstructed from slice images is extracted fast and automatically. A navigation path is generated on the skeleton by manually selecting the start and target points, and smoothed with the cubic cardinal spline curve. Crucial geometric information determining navigation is presented, as well as requirements for the catheter entering the target blood vessel. Using a shape memory alloy active catheter integrated with magnetic sensors, an experiment is carried out in a vascular model, in which the catheter is successfully inserted from the ascending aorta, via the aortic arch, into the brachiocephalic trunk. The navigation strategy proposed in this paper is feasible and has the advantage of increasing the automation of catheterization, enhancing the manoeuvrability of the active catheter and providing the guiding image with desirable interactivity.

Statistical Encounters with Complex B-Splines

Complex B-splines as introduced in Forster et al. (Appl. Comput. Harmon. Anal. 20:281-282, 2006) are an extension of Schoenberg's cardinal splines to include complex orders. We exhibit relationships between these complex B-splines and the complex analogues of the classical difference and divided difference operators and prove a generalization of the Hermite-Genocchi formula. This generalized Hermite- Genocchi formula then gives rise to a more general class of complex B-splines that allows for some interesting stochastic interpretations.

Cardinal splines in nonparametric regression

We discuss a nonparametric regression model on an equidistant grid of the real line. A class of kernel type estimates based on the so-called fundamental cardinal splines will be introduced. Asymptotic optimality of these estimates will be established for certain functional classes. This model explains the often mentioned heuristic fact that cubic splines are adequate for most practical applications.

Cardinal interpolation with periodic polysplines on strips

We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in $C^{2p-2}\left( \mathbb{R}^{n+1}\right) $ , polyharmonic of order p on each strip $\left( j,j+1\right) \times\mathbb{R}^{n}$ , $j\in\mathbb{Z}$ , and periodic in its last n variables, whose restriction to the parallel hyperplanes $\left\{ j\right\} \times\mathbb{R}^{n}$ , $j\in\mathbb{Z}$ , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in $\left\vert j\right\vert $ . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal $\mathcal{L}$ -splines. Keywords: cardinal $\mathcal{L}$ -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63

Signal Reconstruction from Multiple Level Crossings Using Asymmetric Constructing Functions

Signal reconstruction is one of the main tasks in signal processing, where a discrete signal has to be transformed into an analog form. Commonly used analog-to-digital converters sample signals uniformly. The sampling rate is determined by maximal frequency in signal spectrum. A level-crossing sampling approach differs from traditional sampling scheme and results in nonuniformly spaced samples with sampling density depending on the signal's local properties. The method is proposed for signal reconstruction from its level-crossing samples, using asymmetric cubic cardinal splines. Achieved results are demonstrated by simulations. Ill. 10, bibl. 6 (in English; summaries in English, Russian and Lithuanian).