Shape preserving histopolation using rational quadratic splines

In this paper the area true approximation of histograms by rational quadraticC ¹-splines is considered under constraints like convexity or monotonicity. For the existence of convex or monotone histosplines sufficient and necessary conditions are derived, which always can be satisfied by choosing the rationality parameters appropriately. Since the mentioned problems are in general not uniquely solvable histo-splines with minimal mean curvature areconstructed.     

Modeling with rational biquadratic splines

We develop a rational biquadratic G¹ analogue of the non-uniform C¹ B-spline paradigm. These G¹ splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure. This enables a design process that starts with basic shapes, re-represents them in spline form and uses the spline form to provide shape handles for localized free-form modification that can preserve, in the large, the initial fair, basic shapes.     

Positive interpolation with rational quadratic splines

A necessary and sufficient criterion is presented under which the property of positivity carry over from the data set to rational quadratic spline interpolants. The criterion can always be satisfied if the occuring parameters are properly chosen.     

Peer-to-peer data preservation through storage auctions

Digital archives protect important data collections from failures by making multiple copies at other archives, so that there are always several good copies of a collection. In a cooperative replication network, sites "trade" space, so that each site contributes storage resources to the system and uses storage resources at other sites. Here, we examine bid trading: a mechanism where sites conduct auctions to determine who to trade with. A local site wishing to make a copy of a collection announces how much remote space is needed, and accepts bids for how much of its own space the local site must "pay" to acquire that remote space. We define a spectrum of trading scenarios, ranging from a network of archives and digital libraries that trust each other, to a scenario where sites do as they please, including breaking the rules. Then, we focus on developing techniques for the scenarios where sites trust each other, although we discuss issues that may arise if sites are greedy or malicious. We examine the best policies for determining when to call auctions and how much to bid, as well as the effects of "maverick" sites that behave differently than other sites. Simulations of auction and trading sessions indicate that bid trading can allow sites to achieve higher reliability than the alternative: a system where sites trade equal amounts of space without bidding.     

An automatic adjustment procedure for rational splines

This paper presents the development of an adjustment method for the rational spline. The adjustment of the spline is completed by tensioning each internal section until the spline conforms to user defined data in a way which is both accurate and esthetically pleasing. The tensioning rationale for the rational spline is developed for use in an interactive mode with minicomputer and graphics terminal, but may also be used in a batch environment.     

Shape recognition using fractal geometry

Within this paper fractal transformations are presented as a powerful new shape recognition technique. The motivation behind using fractal transformations is to develop a high speed shape recognition technique which will be scale invariant. A review is given of the most popular existing shape recognition techniques. There then follows a full mathematical analysis of the new technique together with a proof of the authors Fractal Invariance Theorem, the new theorem at the centre of the recognition technique. Through the mathematical analysis it becomes apparent that the fractal recognition technique possesses the remarkable property that it is able to distinguish between similar objects. Details are then given of the practical implementation of the technique together with an algorithm for making the technique rotationally invariant. The technique is then applied to a selection of real world objects and a comparison made with the popular moment invariants technique. This shows that the fractal technique is faster than the technique of moment invariants, and also requires less initial information to be effective. Finally conclusions are drawn and further work detailed.     

Adaptive rational fractal interpolation function for image super-resolution via local fractal analysis

Image super-resolution aims to generate high-resolution image based on the given low-resolution image and to recover the details of the image. The common approaches include reconstruction-based methods and interpolation-based methods. However, these existing methods show difficulty in processing the regions of an image with complicated texture. To tackle such problems, fractal geometry is applied on image super-resolution, which demonstrates its advantages when describing the complicated details in an image. The common fractal-based method regards the whole image as a single fractal set. That is, it does not distinguish the complexity difference of texture across all regions of an image regardless of smooth regions or texture rich regions. Due to such strong presumption, it causes artificial errors while recovering smooth area and texture blurring at the regions with rich texture. In this paper, the proposed method produces rational fractal interpolation model with various setting at different regions to adapt to the local texture complexity. In order to facilitate such mechanism, the proposed method is able to segment the image region according to its complexity which is determined by its local fractal dimension. Thus, the image super-resolution process is cast to an optimization problem where local fractal dimension in each region is further optimized until the optimization convergence is reached. During the optimization (i.e. super-resolution), the overall image complexity (determined by local fractal dimension) is maintained. Compared with state-of-the-art method, the proposed method shows promising performance according to qualitative evaluation and quantitative evaluation.     

有理二次插值曲线的形状控制

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题。构造了一种分母为二次的连续有理二次插值函数,这种有理二次插值函数中含有参数,因而给约束控制带来了方便。同时可以通过对参数的控制实现连续的插值。给出了将该种插值曲线约束于给定的折线、二次曲线之上、之下或之间的充分条件及将其约束于给定折线之上、之下或之间的充分必要条件。     

Threading splines through 3D channels

Given a polygonal channel between obstacles in the plane or in space, we present an algorithm for generating a parametric spline curve with few pieces that traverses the channel and stays inside. While the problem without emphasis on few pieces has trivial solutions, the problem for a limited budget of pieces represents a nonlinear and continuous (‘infinite’) feasibility problem. Using tight, two-sided, piecewise linear bounds on the potential solution curves, we reformulate the problem as a finite, linear feasibility problem whose solution, by standard linear programming techniques, is a solution of the channel-fitting problem. The algorithm allows the user to specify the degree and smoothness of the solution curve and to minimize an objective function, for example, to approximately minimize the curvature of the spline. We describe in detail how to formulate and solve the problem, as well as the problem of fitting parallel curves, for a spline in Bernstein-Bézier form.     

加权有理三次样条的形状控制

利用带导数和不带导数的分母为三次的有理三次插值样条构造了一类加权有理三次插值样条函数,由于这种有理三次插值样条中含有参数、调节参数和权系数,因而给约束控制带来了方便。同时只要合适地选择调节参数,就可以使之变成分母为线性的和分母为二次的有理三次插值样条函数。对该样条曲线的区域控制问题进行了研究,给出了将其约束于给定的折线、二次曲线之上、之下或之间的充分条件。最后给出了数值例子。     

Free knot splines for biochemical data

The use of spline functions in the analysis of empirical two-dimensional (2-D) data (y(i), x(i)) is described. Spline functions are excellent empirical functions, which can be used with advantage instead of other ones, such as polynomials or exponentials. The knot location seen as variable value corresponds to classical parameter used to describe oxidation curves. An application on characterization of LDL oxidability shows free knot splines in a regression context.     

保护隐私的有理数科学计算

安全多方计算作为密码学的基本组成部分,是各种密码协议的基础,是国际密码学界的研究热点。近年来,许多学者研究了各种各样的安全多方计算问题,包括保密的信息比较、保密的集合问题和保密的计算几何等,并提出相应的解决方案。而在许多实际应用场景中,安全多方计算问题需要应用有理数进行描述,因此研究有理数域上的安全多方计算问题具有重要的理论与实际意义。但现有的安全多方计算问题的研究成果大多数局限于整数范围,且研究的数据主要是单维度数据。关于有理数域上多维度数据安全多方计算问题的研究较少且无法推广应用。基于有理数的分数表示形式,设计了新的编码方案（有理数编码方案和有理向量编码方案）,可将有理数域上任意维数的数据进行编码,为研究有理数域上其他安全多方计算问题提供了新的解决思路。以该编码方案和单向哈希函数为基础,分别设计了有理数相等、有理向量相等和集合问题的保密判定协议。所设计的协议仅采用基本算术运算和单向哈希函数进行计算,不需要使用公钥加密算法,使得协议的计算效率较高;且协议对研究问题中的数据范围没有限制,适用范围更广。进一步应用模拟范例严格证明了协议在半诚实模型下的安全性;并通过理论分析和模拟实验验证了...     

可变参数的有理分形插值曲线建模

为了有效地处理复杂真实现象中的不规则数据,提出一种利用有理分形插值进行分形曲线建模的方法.首先,基于传统的具有形状参数的有理样条,构造了一类具有函数尺度因子的有理迭代函数系统,并定义了有理分形插值曲线.然后,研究了有理分形曲线的一些重要性质,包括光滑性、稳定性以及收敛性.最后,估计了有理分形曲线计盒维数的上下界.提出的...     

Modeling of fractal data structures

Execution of iterated procedure is analyzed for constructing a fractal set. The procedure is proven convergent, and fractality of the resulting set is demonstrated. Following the research outcomes, it is proposed to modify the mode of the procedure execution.     

Reconstruction of volume data with quadratic super splines

We propose a new approach to reconstruct nondiscrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition. We discuss the smoothness and approximation properties of our model and compare to alternative piecewise polynomial constructions. We observe, as a nonstandard phenomenon, that the derivatives of our splines yield optimal approximation order for smooth data, while the theoretical error of the values is nearly optimal due to the averaging rules. Our approach enables efficient reconstruction and visualization of the data. As the piecewise polynomials are of the lowest possible total degree two, we can efficiently determine exact ray intersections with an isosurface for ray-casting. Moreover, the optimal approximation properties of the derivatives allow us to simply sample the necessary gradients directly from the polynomial pieces of the splines. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.     

多尺度有理分形的图像插值算法

目的图像插值是图像处理中的重要问题,为了提高纹理图像的放大质量,结合以往的有理函数的插值算法,提出一种新的基于有理分形函数的图像插值算法。方法对于输入图像,首先,运用中值滤波和直方图均衡化对输入图像预处理;其次,通过毯子覆盖法求出图像的多尺度分形特征值,进行纹理区域和平滑区域的划分;最后,在纹理区域采用有理分形插值函数,在平滑区域采用有理插值函数。结果对于一般图像,本文算法与NARM(nonlocal autoregressive model),NEDI(new edge-directed interpolation)相当,在纹理区域较多的图像中,本文算法在峰值信噪比(PSNR)和结构相似性(SSIM)数值上较对比算法进一步提高,在视觉效果上,图像对比度明显增强,在Barbara,Truck等的对比图像中,峰值信噪比均提高了0.5 1 dB。结论本文插值算法利用多尺度分形特征将图像划分区域,在不同区域采用不同的插值模型。优化模型参数使得插值质量进一步提高。实验表明本文算法能够对纹理和非纹理区域有效划分对纹理的信息保持优于传统算法,获得了较好的主客观效果。     

Restricted trivariate polycube splines for volumetric data modeling

This paper presents a volumetric modeling framework to construct a novel spline scheme called restricted trivariate polycube splines (RTP-splines). The RTP-spline aims to generalize both trivariate T-splines and tensor-product B-splines; it uses solid polycube structure as underlying parametric domains and strictly bounds blending functions within such domains. We construct volumetric RTP-splines in a top-down fashion in four steps: 1) Extending the polycube domain to its bounding volume via space filling; 2) building the B-spline volume over the extended domain with restricted boundaries; 3) inserting duplicate knots by adding anchor points and performing local refinement; and 4) removing exterior cells and anchors. Besides local refinement inherited from general T-splines, the RTP-splines have a few attractive properties as follows: 1) They naturally model solid objects with complicated topologies/bifurcations using a one-piece continuous representation without domain trimming/patching/merging. 2) They have guaranteed semistandardness so that the functions and derivatives evaluation is very efficient. 3) Their restricted support regions of blending functions prevent control points from influencing other nearby domain regions that stay opposite to the immediate boundaries. These features are highly desirable for certain applications such as isogeometric analysis. We conduct extensive experiments on converting complicated solid models into RTP-splines, and demonstrate the proposed spline to be a powerful and promising tool for volumetric modeling and other scientific/engineering applications where data sets with multiattributes are prevalent.