Least Committed Splines in 3D Modelling of Free Form Objects from Intensity Images

Generating 3D models of objects from video sequences is an important problem in many multimedia applications ranging from teleconferencing to virtual reality. In this paper, we present a method of estimating the 3D face model from a monocular image sequence, using a few standard results from the affine camera geometry literature in computer vision, and spline fitting techniques using a modified non parametric regression technique. We use the bicubic spline functions to model the depth map, given a set of observation depth maps computed from frame pairs in a video sequence. The minimal number of splines are chosen on the basis of the Schwartz's Criterion. We extend the spline fitting algorithm to hierarchical splines. Note that the camera calibration parameters and the prior knowledge of the object shape is not required by the algorithm. The system has been successfully demonstrated to extract 3D face structure of humans as well as other objects, starting from their image sequences.     

一类代数空间逼近样条

给定空间不共面的四个有序数据点,可以形成一个四面体。在四面体内,Bernstein-Bézier（B-B）形式定义两类正则实多项式代数曲面片,一类是二次的,一类是三次的。此两类曲面片在四面体内的交集为一条正则曲线段。先固定二次曲面片,并得到其参数形式,然后约简三次曲面片所对应的Bernstein系数,使之为带有三个形状调整的形状因子,其中两个分别代表曲线段端点处的曲率,另外一个作为形状的调整。利用二次曲面的参数形式,由三次曲面片可得到曲线的隐参数约束形式,从而得到曲线的参数形式。对给定的空间点列,利用两个形状因子较容易的拼接出G²-连续的逼近曲线,突破了现行代数曲线生成方法,即空间连续曲线均是通过三角形仿射变换,由B-B形式生成的平面弧拼接而成。     

Multidimensional Spline Interpolation: Theory and Applications

Computing numerical solutions of household’s optimization, one often faces the problem of interpolating functions. As linear interpolation is not very good in fitting functions, various alternatives like polynomial interpolation, Chebyshev polynomials or splines were introduced. Cubic splines are much more flexible than polynomials, since the former are only twice continuously differentiable on the interpolation interval. In this paper, we present a fast algorithm for cubic spline interpolation, which is based on the precondition of equidistant interpolation nodes. Our approach is faster and easier to implement than the often applied B-Spline approach. Furthermore, we will show how to loosen the precondition of equidistant points with strictly monotone, continuous one-to-one mappings. Finally, we present a straightforward generalization to multidimensional cubic spline interpolation.      

A surface reconstruction algorithm for topology optimization

For mechanical structural design, topology optimization is often utilized. During this process, a topologically optimized model must be converted into a parametric CAD solid model. The key point of conversion is that a discretized shape of a topologically optimized model must be smoothed, but features such as creases and corners must be retained. Thus, a surface reconstruction algorithm to produce the parametric CAD solid model from a topologically optimized model is proposed in this paper. Our presented algorithm consists of three parts: (1) an enclosed isosurface geometry from which the topologically optimized model is generated, (2) features detected and (3) the parametric CAD solid model reconstructed as biquartic surface splines. In order to generate an enclosed isosurface model effectively, we propose an algorithm based upon the marching cubes method to detect elements intersected by an isosurface. After generating an enclosed isosurface model, we produce biquartic surface splines. By applying our algorithm to an enclosed isosurface model, it is possible to produce smoothed biquartic surface splines with features retained. Some examples are shown and the effectiveness of our algorithm is discussed in this paper.     

Smoothing of cubic parametric splines

The most common curve representation in CADCAM systems of today is the cubic parametric spline. Unfortunately this curve will sometimes oscillate and cause unwanted inflexions which are difficult to deal with. This paper has developed from the need to eliminate oscillations and remove inflexions from such splines, a need which may occur for example when interpolating data measured from a model. A method for interactive smoothing is outlined and a smoothing algorithm is described which is mathematically comparable to manual smoothing with a physical spline.     

Smoothing of bicubic parametric surfaces

Bicubic parametric surfaces are often used to represent complex shapes in systems for computer-aided design and manufacture. Such as surface can be defined by a topologically rectangular mesh of cubic parametric splines, a curve which is an approximate mathematical model of the linear elastic beam.Smoothing a bicubic parametric surface can be done by smoothing the curve net that defines it. This paper describes a method for moving datapoints in a curve net to new ‘smoother’ positions. Different techniques to analyse the result of the smoothing are also discussed.     

Polycube splines

This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains, except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.     

带形状参数的三次三角Hermite插值样条曲线

给出了一种带形状参数的三次三角Hermite插值样条曲线,具有标准三次Hermite插值样条曲线完全相同的性质。给定插值条件时,样条曲线的形状可通过改变形状参数的取值进行调控。在适当条件下,该样条曲线对应的Ferguson曲线可精确表示椭圆、抛物线等工程曲线。通过选择合适的形状参数,该插值样条曲线能达到[C2]连续,而且其整体逼近效果要好于标准三次Hermite插值样条曲线。     

Semiparametric bivariate Archimedean copulas

While parametric copulas often lack expressive capacity to capture the complex dependencies that are usually found in empirical data, non-parametric copulas can have poor generalization performance because of overfitting. A semiparametric copula method based on the family of bivariate Archimedean copulas is introduced as an intermediate approach that aims to provide both accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood and a term that penalizes non-smooth solutions. The performance of the semiparametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semiparametric estimators of Archimedean copulas previously introduced in the literature, two flexible copula methods based on Gaussian kernels and mixtures of Gaussians and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semiparametric Archimedean approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.     

An unsupervised data projection that preserves the cluster structure

In this paper we propose a new unsupervised dimensionality reduction algorithm that looks for a projection that optimally preserves the clustering data structure of the original space. Formally we attempt to find a projection that maximizes the mutual information between data points and clusters in the projected space. In order to compute the mutual information, we neither assume the data are given in terms of distributions nor impose any parametric model on the within-cluster distribution. Instead, we utilize a non-parametric estimation of the average cluster entropies and search for a linear projection and a clustering that maximizes the estimated mutual information between the projected data points and the clusters. The improved performance is demonstrated on both synthetic and real world examples.     

Bivariate hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in R³ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points {(x_i, y_i)}_i=1^N from which none of the pairs (x_i,y_i) and (x_j,y_j) with i≠j coincide, it is proved that the resulting surface (function) is C¹. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is C² if the data are not ‘too irregular’. Finally the approximation properties of the methods are investigated.     

Enhancement of spatially adaptive smoothing splines via parameterization of smoothing parameters

This paper considers the problem of estimating curve and surface functions when the structures of an unknown function vary spatially. Classical approaches such as using smoothing splines, which are controlled by a single smoothing parameter, are inefficient in estimating the underlying function that consists of different spatial structures. In this paper, we propose a blockwise method of fitting smoothing splines wherein the smoothing parameter λ varies spatially, in order to accommodate possible spatial nonhomogeneity of the regression function. A key feature of the proposed blockwise method is the parameterization of a smoothing parameter function λ(x) that produces a continuous spatially adaptive fit over the entire range of design points. The proposed parameterization requires two important ingredients: (1) a blocking scheme that divides the data into several blocks according to the degree of spatial variation of the data; and (2) a method for choosing smoothing parameters of blocks. We propose a block selection approach that is based on the adaptive thinning algorithm and a choice of smoothing parameters that minimize a newly defined blockwise risk. The results obtained from numerical experiments validate the effectiveness of the proposed method.     

L1 approximations of strictly convex functions by means of first degree splines

The L₁ approximation of strictly convex functions by means of first degree splines with a fixed number of knots is studied. The main theoretical results are a system of equations for the knots, which solves the problem, and an estimate of the approximation error. The error estimation allows the determination of bounds for the number of knots needed so that the L₁ approximation error does not exceed a given number. Finally, an algorithm is used, by means of which a solution to the system can be obtained.     

Fitting smoothing splines to time-indexed,noisy points on nonlinear manifolds

We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smoothing splines in Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais metric-based steepest-decent algorithm developed in Samir et al. [38]. Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendall's shape space for video-based activity recognition, we demonstrate the effectiveness of the proposed algorithm for optimal curve fitting. This paper derives certain geometrical elements, namely the exponential map and its inverse, parallel transport of tangents, and the curvature tensor, on these manifolds, that are needed in the gradient-based search for smoothing splines. These ideas are illustrated using experimental results involving both simulated and real data, and comparing the results to some current algorithms such as piecewise geodesic curves and splines on tangent spaces, including the method by Kume et al. [24].     

有理三次样条的误差分析及空间闭曲线插值

给出了具有线性分母的有理三次样条函数的误差估计,并在柱面坐标系下对一类空间闭曲线的插值问题进行了研究;通过将柱面展开,把空间闭曲线的插值问题转化为平面中的插值问题,利用具有线性分母的有理三次样条函数进行插值;最终得到的空间曲线能达到曲率连续.对该方法的误差进行了分析,数值例子显示插值效果较好.     

Local hybrid approximation for scattered data fitting with bivariate splines

We suggest a local hybrid approximation scheme based on polynomials and radial basis functions, and use it to improve the scattered data fitting algorithm of (Davydov, O., Zeilfelder, F., 2004. Scattered data fitting by direct extension of local polynomials to bivariate splines. Adv. Comp. Math. 21, 223–271). Similar to that algorithm, the new method has linear computational complexity and is therefore suitable for large real world data. Numerical examples suggest that it can produce high quality artifact-free approximations that are more accurate than those given by the original method where pure polynomial local approximations are used.     

集多种特性的三次三角伪B样条

目的 为了同时解决传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,提出了一类集多种特性的三次三角伪B样条。方法 首先构造了一组带两个参数的三次三角伪B样条基函数,然后在此基础上定义了相应的参数伪B样条曲线,并讨论了该曲线的特性及光顺性问题,最后研究了相应的代数伪B样条,并给出了最优代数伪B样条的确定方法。结果 参数伪B样条曲线不仅满足C²连续,而且无需求解方程系统即可自动插值于给定的型值点。当型值点保持不变时,插值曲线的形状还可通过自带的两个参数进行调控。在适当条件下,该参数伪B样条曲线可精确表示圆弧、椭圆弧、星形线等常见的工程曲线。相应的代数伪B样条具有参数伪B样条曲线类似的性质,利用最优代数伪B样条可获得满意的插值效果。结论 所提出的伪B样条同时解决了传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,是一种实用的曲线造型方法。     

A convex version of multivariate adaptive regression splines

Multivariate adaptive regression splines (MARS) provide a flexible statistical modeling method that employs forward and backward search algorithms to identify the combination of basis functions that best fits the data and simultaneously conduct variable selection. In optimization, MARS has been used successfully to estimate the unknown functions in stochastic dynamic programming (SDP), stochastic programming, and a Markov decision process, and MARS could be potentially useful in many real world optimization problems where objective (or other) functions need to be estimated from data, such as in surrogate optimization. Many optimization methods depend on convexity, but a non-convex MARS approximation is inherently possible because interaction terms are products of univariate terms. In this paper a convex MARS modeling algorithm is described. In order to ensure MARS convexity, two major modifications are made: (1) coefficients are constrained, such that pairs of basis functions are guaranteed to jointly form convex functions and (2) the form of interaction terms is altered to eliminate the inherent non-convexity. Finally, MARS convexity can be achieved by the fact that the sum of convex functions is convex. Convex-MARS is applied to inventory forecasting SDP problems with four and nine dimensions and to an air quality ground-level ozone problem.     

Manifold splines with a single extraordinary point

This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g−2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.