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This paper introduces a modified method for construction of a new design-control 6-point approximating refinement scheme. The construction of the new scheme is based on translation of the points of 4-point approximating refinement scheme to the new position according to the linear combination of certain displacement vectors. The initial and terminal points of these vectors are the refinement points of two 4-point approximating binary schemes. The new scheme contains three design-control parameters. These parameters increase the efficiency and flexibility of the new scheme. The mathematical and graphical analysis of the refinement scheme show that this scheme is good choice for curve modeling. [ABSTRACT FROM AUTHOR]

With the wide application of digital media technology and augmented reality, in order to enhance the design experience of mobile visual media design. This paper uses domain processing, point processing, big picture processing and model processing in numerical algorithms, construction of visual media images based on additive and multiplicative models, integrating the SOR model with the EDT model, build PLS model. Depending on the interactivity, patios-temporal fixity and degree of immersion of the virtual object, analyze the results of PLS experience conceptual model construction by combining system characteristics and cognitive elements of mobile visual media design. To explore the mechanisms by which digital media technologies and augmented reality technologies influence the design factors of mobile visual media, and evaluate the measurement model and structural model. The results of the evaluation found that accuracy 0.325, flexibility 0.472 and completeness 0.328 positively and significantly influenced information quality (R=0.697), system quality (R=0.52) and service quality (R=0.741). The system quality, information quality and service quality of the mobile visual media design products are guaranteed.

In order to improve the ability of micromechanical sensors to update data-matching nodes in real-time and speed up the establishment of a neighbor relationship between nodes, this paper proposes a personalized recommendation algorithm for micromechanical sensors based on the cloud model to improve the stability and real-time performance of micromechanical sensors. The algorithm uses the service attribute values of the cloud computing model and cloud clustering method to set feature term labels and establish a cloud service similarity matrix so as to meet the user-matched manufacturing service requirements. The intra-class clustering technique is applied to measure the clustering effect, and the evaluation function of each clustering number is calculated on the basis of considering the time sequence to determine the best clustering result to complete the personalized recommendation path for micromechanical sensors. To verify the application effect of the personalized recommendation algorithm for micromechanical sensors based on the cloud model, experiments are conducted. The results show that the recommendation algorithm in this paper can always control the node energy consumption below 2.5×103, and the average discovery delay is stable between 41-43 seconds. And the sensor response time is 12.1 seconds, and the average absolute deviation value is 0.23, which is nearly 1.3 times smaller than 0.53 and 0.52 of the collaborative recommendation algorithm and hybrid recommendation algorithm. It can be seen that the recommendation algorithm in this paper solves the problem of excessive neighbor discovery delay in the communication process of micromechanical sensors and effectively improves the personalized recommendation performance of micromechanical sensors.

To enrich the content and form of college civic education, this paper constructs a civic education management innovation platform based on big data technology to explore the innovative development path of college civic education. Using the differentiated fusion algorithm, the information processing of college student management data is carried out, and the educational mining information of college students is identified through feature extraction. Bayesian network is introduced in the process of index weight calculation to improve the data mining efficiency to ensure the validity of obtaining the evaluation index of Civic Education. To verify the feasibility of the innovation platform of Civic Education Management based on big data technology, the application effect was tested. The analysis results show that the platform constructed in this paper can make students’ cognitive awareness of course contents reach 0.98, students’ concentration effect reach 0.89, and the overall effect of teaching feedback reach 90% in selecting teaching resources. The average test score of students in the experimental group was 8.5, and the average length range of the independent learning course was 2.78-4.629 hours. Thus, it can be seen that the platform constructed in this paper is conducive to promoting the high-quality development of Civics teaching and enhancing the effectiveness of Civics education in colleges and universities.

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features. [ABSTRACT FROM AUTHOR]

Computer-aided diagnosis of lung segmentation is the fundamental requirement to diagnose lung diseases. In this paper, a two-dimensional (2D) Otsu algorithm by Darwinian particle swarm optimization (DPSO) and fractional-order Darwinian particle swarm optimization (FODPSO) is proposed to segment the pulmonary parenchyma from the lung image obtained through computed tomography (CT) scans. The proposed method extracts pulmonary parenchyma from multi-sliced CT. This is a preprocessing step to identify pulmonary diseases such as emphysema, tumor, and lung cancer. Image segmentation plays a significant role in automated pulmonary disease diagnosis. In traditional 2D Otsu, exhaustive search plays an important role in image segmentation. However, the main disadvantage of the 2D Otsu method is its complex computation and processing time. In this paper, the 2D Otsu method optimized by DPSO and FODPSO is developed to reduce complex computations and time. The efficient segmentation is very important in object classification and detection. The particle swarm optimization (PSO) method is widely used to speed up the computation and maintain the same efficiency. In the proposed algorithm, the limitation of PSO of getting trapped in local optimum solutions is overcome. The segmentation technique is assessed and equated with the traditional 2D Otsu method. The test results demonstrate that the proposed strategy gives better results. The algorithm is tested on the Lung Image Database Consortium image collections.

We systematically compare two approaches with the optimal design of multiproduct batch plants that are widely used, e.g., in the chemical industry. Deterministic algorithms like branch-and-bound achieve global optimality, but often require a prohibitively high computation effort. We propose an alternative, hybrid algorithm by combining two metaheuristics: ant colony optimization (ACO) and simulated annealing (SA), in order to find near-optimal solutions in a reasonable time. We develop a parallel implementation of our hybrid approach on graphics processing units using CUDA. We experimentally compare both approaches, and we show that our hybrid metaheuristic approach (ACO + SA) yields almost-optimal solutions, while computing them significantly faster than when using branch-and-bound. [ABSTRACT FROM AUTHOR]

In this paper we revisit the problem of monotonicity preservation of curves and surfaces and we provide some new proofs and open problems. In particular, we prove a new formula for the derivation of rational Bézier curves. We also deal with the rational monotonicity preservation of rational Bézier surfaces and a related conjecture is presented.

We present a new set of basis functions for H ( curl ) -conforming, H ( div ) -conforming, and L 2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H 1 -conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H 1 -conforming Bernstein–Bézier basis functions form part of the H ( curl ) -conforming basis, and the curl of the high-order, non-gradients H ( curl ) -conforming basis functions form part of the H ( div ) -conforming basis, and the divergence of the high-order, non-curl H ( div ) -conforming basis functions form part of the L 2 -conforming basis. Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piecewise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems. [ABSTRACT FROM AUTHOR]

An adapted orthonormal frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent f1(ξ)=r′(ξ)/|r′(ξ)| and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal-plane vectors. Employing the Euler-Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean-hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal-plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal-plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples. [ABSTRACT FROM AUTHOR]

A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties. [ABSTRACT FROM AUTHOR]

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 92d-6c+1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage. [ABSTRACT FROM AUTHOR]

In order to tackle the problem of shape design and shape adjustment of complex surfaces in engineering, continuity conditions between generalized Bézier-like surfaces with multiple shape parameters are studied in this paper. Firstly, the geometric model of the generalized Bézier-like surfaces is built by blending a number of Bézier-like curves with independent shape parameters. Secondly, based on the terminal properties and linear independence of Bernstein-like basis functions, the conditions for G continuity between two adjacent generalized Bézier-like surfaces are derived, and then simplified by choosing appropriate shape parameters. Finally, some properties and applications of the smooth continuity between generalized Bézier-like surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which can greatly improve the ability to construct complex surfaces by using the generalized Bézier-like surfaces. [ABSTRACT FROM AUTHOR]

Composite basis functions for pyramidal elements on the spaces H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) and L 2 ( Ω ) are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart–Thomas–Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, etc.) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions. [ABSTRACT FROM AUTHOR]

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of $\mathrm{G}^{1}$ and $\mathrm{G}^{2}$ between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design. [ABSTRACT FROM AUTHOR]

A rotation-minimizing frame ( f , f , f ) on a space curve r( ξ) defines an orthonormal basis for $\mathbb {R}^{3}$ in which $\mathbf {f}_{1}=\mathbf {r}^{\prime }/|\mathbf {r}^{\prime }|$ is the curve tangent, and the normal-plane vectors f , f exhibit no instantaneous rotation about f . Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form $\mathbf {r}^{\prime }(\xi )=\mathcal {A}(\xi )\,\mathbf{i} \,\mathcal {A}^{*}(\xi )$ for some quaternion polynomial $\mathcal {A}(\xi )$ . By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial $\mathcal {A}(\xi )$ , a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms. [ABSTRACT FROM AUTHOR]

The injectivity of NURBS curve implies the curve has no self-intersection. In this paper, we propose a geometric condition on the control polygon which guarantees the NURBS curve to be injective for all possible choices of positive weights. The proof is based on the degree elevation algorithm and toric degeneration theory of NURBS curve. [ABSTRACT FROM AUTHOR]

We construct G 1 Hermite interpolating curves in biangular coordinates, and provide sufficient conditions for their convexity. In a biangular coordinate system, the problem reduces to that of choosing suitable functions interpolating the biangular coordinates of the curve at its end points. The simplest linear equations, γ = ( ( 1 − t ) α , t β ) , in biangular coordinates correspond to a sectrix of Maclaurin, which we extend by introducing two shape parameters that pull the curve towards the sides of its triangular envelope. In addition, we consider a class of curves whose biangular coordinates have a constant sum, and we analyze their shape and curvature. [ABSTRACT FROM AUTHOR]

Abstract: We give an efficient algorithm for partitioning the domain of a numeric function into segments. The function is realized as a polynomial in each segment, and a lookup table stores the coefficients of the polynomial. Such an algorithm is an essential part of the design of lookup table methods Ercepovac et al. (2000) , Lee et al. (2003) , Nagayama et al. (2007) , Paul et al. (2007) and Sasao et al. (2004) for realizing numeric functions, such as , , and . Our algorithm requires many fewer steps than a previous algorithm given in Frenzen et al. (2010) and makes tractable the design of numeric function generators based on table lookup for high-accuracy applications. We show that an estimate of segment width based on local derivatives greatly reduces the search needed to determine the exact segment width. We apply the new algorithm to a suite of 15 numeric functions and show that the estimates are sufficiently accurate to produce a minimum or near-minimum number of computational steps. [Copyright &y& Elsevier]

Surface multiresolution processing is an important subject in CAGD. It also poses many challenging problems including the design of multiresolution algorithms. Unlike images which are in general sampled on a regular square or hexagonal lattice, the meshes in surfaces processing could have an arbitrary topology, namely, they consist of not only regular vertices but also extraordinary vertices, which requires the multiresolution algorithms have high symmetry. With the idea of lifting scheme, Bertram (Computing 72(1-2):29-39, ) introduces a novel triangle surface multiresolution algorithm which works for both regular and extraordinary vertices. This method is also successfully used to develop multiresolution algorithms for quad surface and $\sqrt 3$ triangle surface processing in Wang et al. (Vis Comput 22(9-11):874-884, ; IEEE Trans Vis Comput Graph 13(5):914-925, ) respectively. When considering the biorthogonality, these papers do not use the conventional $L^2({{\rm I}\kern-.2em{\rm R}}^2)$ inner product, and they do not consider the corresponding lowpass filter, highpass filters, scaling function and wavelets. Hence, some basic properties such as smoothness and approximation power of the scaling functions and wavelets for regular vertices are unclear. On the other hand, the symmetry of subdivision masks (namely, the lowpass filters of filter banks) for surface subdivision is well studied, while the symmetry of the highpass filters for surface processing is rarely considered in the literature. In this paper we introduce the notion of 4-fold symmetry for biorthogonal filter banks. We demonstrate that 4-fold symmetric filter banks result in multiresolution algorithms with the required symmetry for quad surface processing. In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and $\sqrt 2$ refinements. Furthermore, we show that some filter banks constructed in this paper result in very simple multiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1-2):29-39, ) and Wang et al. (Vis Comput 22(9-11):874-884, ; IEEE Trans Vis Comput Graph 13(5):914-925, ). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al. (Vis Comput 22(9-11):874-884, ) for dyadic multiresolution quad surface processing. Therefore, the properties of the scaling functions and wavelets corresponding to those algorithms can be obtained by analyzing the corresponding filter banks. [ABSTRACT FROM AUTHOR]

Abstract: This paper is about the construction of univariate wavelet bi-frames with each framelet being symmetric. As bivariate filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices. For open surfaces, special multiresolution algorithms are designed to process boundary vertices. When the multiresolution algorithms derived from univariate wavelet bi-frames are used as the boundary algorithms, it is desired that not only the scaling functions but also all framelets be symmetric. In addition, the algorithms for curve/surface multiresolution processing should be given by templates so that they can be easily implemented. In this paper, first, by appropriately associating the lowpass and highpass outputs to the nodes of , we show that both biorthogonal wavelet multiresolution algorithms and bi-frame multiresolution algorithms can be represented by templates. Then, using the idea of the lifting scheme, we provide frame algorithms given by several iterative steps with each step represented by a symmetric template. Finally, with the given templates of algorithms, we obtain the corresponding filter banks and construct bi-frames based on their smoothness and vanishing moments. Two types of symmetric bi-frames are studied in this paper. In order to provide a clearer picture on the template-based procedure for bi-frame construction, in this paper we also consider the template-based construction of biorthogonal wavelets. The approach of the template-based bi-frame construction introduced in this paper can be extended easily to the construction of bivariate bi-frames with high symmetry for surface multiresolution processing. [ABSTRACT FROM AUTHOR]

Abstract: When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function or both have big supports in general. In particular, when the synthesis low-pass filter is a commonly used scheme such as Loop’s scheme or Catmull–Clark’s scheme, the corresponding analysis low-pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words big templates of multiresolution algorithms, are undesirable for surface processing. On the other hand, a frame provides flexibility for the construction of “basis” systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry. In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the refinement. Namely, with either the dyadic or refinement, a frame system constructed in this paper has only one more generator than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them. [Copyright &y& Elsevier]

Simple criteria for the existence of rational rotation-minimizing frames (RRMFs) on quintic space curves are determined, in terms of both the quaternion and Hopf map representations for Pythagorean-hodograph (PH) curves in ℝ. In both cases, these criteria amount to satisfaction of three scalar constraints that are quadratic in the curve coefficients, and are thus much simpler than previous criteria. In quaternion form, RRMF quintics can be characterized by just a single quadratic (vector) constraint on the three quaternion coefficients. In the Hopf map form, the characterization is in terms of one real and one complex quadratic constraint on the six complex coefficients. The identification of these constraints is based on introducing a “canonical form” for spatial PH curves and judicious transformations between the quaternion and Hopf map descriptions. The simplicity of these new characterizations for the RRMF quintics should help facilitate the development of algorithms for their construction, analysis, and practical use in applications such as animation, spatial motion planning, and swept surface constructions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove irreducible offsets to rational plane curves are hyperelliptic in general and compute the genus of them. We also give a criterion for deciding the irreducibility of offsets to rational plane curves. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the problem of partitioning a continuous curve into n parts with equal successive chords, the curve EquiPartition problem (EP). The goal is to locate consecutive curve points, so that the curve can be divided into n segments with equal chords under a distance function. We adopt a level set approach to prove that for any continuous injective curve in a metric space and any number n there always exists at least one n-equipartition (EP). A new approximate algorithm, that is the first EP algorithm, inspired from the level set approach is proposed for finding all solutions with high accuracy. Finally, EP based applications are presented and special properties of their solutions are discussed. [Copyright &y& Elsevier]

Abstract: In this paper we consider a 6-point interpolating binary non-stationary subdivision scheme and present some of its important properties. We highlight some advantages of the scheme and demonstrate its performance by some examples. [Copyright &y& Elsevier]

Abstract: A one-parameter family of spirals that can match planar, two-point Hermite data is presented. These spirals can be used as an alternative to the biarc, which is also a one-parameter family of curves that can match two-point Hermite data. Some suggestions on choosing the free parameter of the family of spirals is given. It is shown that there is a unique Hermite interpolating spiral that passes through a given point in an allowable region. Three examples of the use of these spirals are given: curve completion with spirals, design with spirals, and approximation of the clothoid by spirals. [Copyright &y& Elsevier]

Abstract: Rationally parameterized hypersurfaces can be classified with respect to their RC properties (Rational Convolutions) with the help of the Gröbner bases theory. This classification focuses on special classes of rational parameterizations which provide a rational description of convolution hypersurfaces generally (GRC parameterizations), or just in some special cases (SRC parameterizations). The main aim of this paper is to bring the theory of the so-called PN surfaces (surfaces with Pythagorean Normal vectors) and their PN parameterizations (parameterizations fulfilling the PN condition) in relation to the theory of SRC parameterizations and to show that this type of parameterizations can be further classified with respect to the degree of the construction of convolution surfaces. The connection of SRC PN parameterizations to the well-known concepts of proper and square-root parameterizations is also investigated. [Copyright &y& Elsevier]