Subdivision Rules Research Articles

Representing images using curvilinear feature driven subdivision surfaces.

This paper presents a subdivision-based vector graphics for image representation and creation. The graphics representation is a subdivision surface defined by a triangular mesh augmented with color attribute at vertices and feature attribute at edges. Special cubic B-splines are proposed to describe curvilinear features of an image. New subdivision rules are then designed accordingly, which are applied to the mesh and the color attribute to define the spatial distribution and piecewise-smoothly varying colors of the image. A sharpness factor is introduced to control the color transition across the curvilinear edges. In addition, an automatic algorithm is developed to convert a raster image into such a vector graphics representation. The algorithm first detects the curvilinear features of the image, then constructs a triangulation based on the curvilinear edges and feature attributes, and finally iteratively optimizes the vertex color attributes and updates the triangulation. Compared with existing vector-based image representations, the proposed representation and algorithm have the following advantages in addition to the common merits (such as editability and scalability): 1) they allow flexible mesh topology and handle images or objects with complicated boundaries or features effectively; 2) they are able to faithfully reconstruct curvilinear features, especially in modeling subtle shading effects around feature curves; and 3) they offer a simple way for the user to create images in a freehand style. The effectiveness of the proposed method has been demonstrated in experiments.

A convergent conical algorithm with $$\omega $$ ω -bisection for concave minimization

The conical algorithm is a global optimization algorithm proposed by Tuy in 1964 to solve concave minimization problems. Introducing the concept of pseudo-nonsingularity, we give an alternative proof of convergence of the algorithm with the $$\omega $$ ? -subdivision rule. We also develop a new convergent subdivision rule, named $$\omega $$ ? -bisection, and report numerical results of comparing it with the usual $$\omega $$ ? -subdivision.

Creases and boundary conditions for subdivision curves

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points.The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.

A ‘subdivision regression’ model for data analysis

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. We propose two new models for data analysis and compression based on subdivision schemes:(a) The ‘subdivision regression’ model, which can be viewed as a special multi-resolution decomposition.(b) The ‘tree regression’ model, which allows the identification of certain patterns within the data. The paper focuses on analysis and mentions compression as a byproduct. We suggest applying certain criteria on the output of these models as features for data analysis. Differently from existing multi-resolution analysis methods, these new models and criteria provide data features related to the schemes (the filters) themselves, based on a decomposition of the data into different resolution levels, and they also allow analysing data of non-smooth functions and working with varying-resolution subdivision rules. Finally, applications of these methods for music analysis and other potential usages are mentioned.

Construction and Application of Subdivision Surface Scheme Using Lagrange Interpolation Polynomial

This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.

基于三次B 样条逆向细分的自由曲线的多分辨率表示

The study of reverse subdivision mainly concentrates on the surfaces, and the study of reverse subdivision for the particular curve subdivision is relatively rare. Studies about the multiresolution representation and simplification of freedom curves based on reverse subdivision are also rare. According to the facts that the geometric meaning and the rules of B-spline of degree three are obvious and simple, we present the reverse B-spline of degree three subdivision rules, and on the basis of that we introduce a new method of multiresolution representation of freedom curves. By establishing the error vectors when decomposing the freedom curves and combining the B-spline of degree three subdivision rules, we can achieve decomposition and exact reconstruction of freedom curves. We offer examples of multiresolution representation, decomposition and reconstruction of freedom curves using our method. The presented method is easy to be programmed, and more obvious in geometrical meaning than some existing methods. The experiments suggest that the decomposed curves which are generated by our reverse rules can match the shape of the primary curves. The structure of decomposition and reconstruction filters is very brief, and the amount of computations of decomposition and reconstruction of the curves using our method is less than that using some previous methods.

Non-uniform non-tensor product local interpolatory subdivision surfaces

In this paper we exploit a class of univariate, C1 interpolating four-point subdivision schemes featured by a piecewise uniform parameterization, to define non-tensor product subdivision schemes interpolating regular grids of control points and generating C1 limit surfaces with a better behavior than the well-established tensor product subdivision and spline surfaces. As a result, it is emphasized that subdivision methods can be more effective than splines, not only, as widely acknowledged, for the representation of surfaces of arbitrary topology, but also for the generation of smooth interpolants of regular grids of points.To our aim, the piecewise uniform parameterization of the univariate case is generalized to an augmented parameterization, where the knot intervals of the kth level grid of points are computed from the initial ones by an updating relation that keeps the subdivision algorithm linear. The particular parameters configuration, together with the structure of the subdivision rules, turn out to be crucial to prove the continuity and smoothness of the limit surface.

Single Basepoint Subdivision Schemes for Manifold-valued Data: Time-Symmetry Without Space-Symmetry

This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of wavelet-like transform for manifold-valued data. This class includes the (single basepoint) Log–Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C 2 scheme, provided the underlying linear scheme is C 2 (this is called “C 2 equivalence”). But when the underlying linear scheme is C 3, Navayazdani and Yu have shown that to guarantee C 3 equivalence, a certain tensor P f associated to f must vanish. They also show that P f vanishes when the underlying manifold is a symmetric space and f is the exponential map. Their analysis is based on certain “C k proximity conditions” which are known to be sufficient for C k equivalence. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P f =0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the third order. In particular, when f is the exponential map of a connection, one recovers the original connection and P f vanishes. It then follows that the condition P f =0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P f implies that the C 4 proximity conditions hold, thus guaranteeing C 4 equivalence. Finally, the analysis in the paper shows that for k≥5, the C k proximity conditions imply vanishing curvature. This suggests that vanishing curvature of the connection associated to f is likely to be a necessary condition for C k equivalence for k≥5.

URBAN FEATURES AND ENERGY CONSUMPTION AT LOCAL LEVEL

There has been a growing interest in discovering human effects on the environment and energy consumption in the recent decades. It is estimated that the shares of energy consumed in transportation and housing systems are around 20 and 30 percent of the total energy consumption. Furthermore, it is believed that the residential greenhouse emissions depend on urban form and structure. This paper explores the effects of urban features on residential energy consumption at neighborhood level using data collected through household questionnaires (n = 140). Two residential districts in the metropolitan Shiraz, located in the south of Iran, were selected as case study areas. Different features of the two areas including building density, typology, housing location, parcel size, floor area and construction materials were compared. Ordinary linear regression was used to discover the impact of explanatory variables on energy consumption. It was found that some physical variables such as parcel size, setback, and the number of floors played significant roles in explaining the variances existing in energy use level. The results can be considered by governmental agencies in order to modify land use policies and subdivision rules in hope of saving energy and achieving a sustainable community.

A New Palladian Shape Grammar

The following paper describes a shape grammar that recreates Palladian villas. A Palladian grammar was originally proposed by Stiny and Mitchell. However, this alternative grammar uses different parametric shape rules and methodology to test the hypothesis that different grammars can generate the same corpus of designs. The formalism is then implemented using a computerised design tool. The grammar includes subdivision rules that allow for a more economical formulation. A new corpus of solutions is explored and the derivation is compared with the original Palladian grammar. The project is part of wider research aimed at formulating a generic housing grammar.

A finite subdivision rule for the n-dimensional torus

Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere extensively, especially those corresponding to 3-manifolds, in an attempt to prove Cannon’s conjecture. There has been a recent interest in generalizing some of their tools, such as extremal length, to higher dimensions. We define finite subdivision rules of dimension n, and find an n − 1-dimensional finite subdivision rule for the n-dimensional torus, using a well-known simplicial decomposition of the hypercube.

An Algorithm Subdividing the Intersection Triangle for Realizing Boolean Operations of STL Models

The efficient implementation of Boolean operatio between STL models is a bottleneck problem to develop excellent RP data processing software, and the subdivision of the intersection is an important step of realization the Boolean operations. An algorithm is proposed in this paper. Based on the closed character of the STL model, the position relationship between the intersection lines and the triangle are classified, and the subdivision rules is summarized. The two data linked list of the intersection points and vertex points are created, a searching process to obtain the subdivision polygon is described.

Constructing subdivision rules from polyhedra with identifications

Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are known. We construct an explicit subdivision rule for many 3-manifolds from polyhedral gluings. The manifolds that satisfy the conditions include all closed manifolds created from right-angled hyperbolic polyhedra, as well as many 3-manifolds with toral or hyperbolic boundary.

A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation

In several applications, such as WENO interpolation and reconstruction [C.W. Shu, SIAM Rev. 51 (2009) 82–126], we are interested in the analytical expression of the weight-functions which allow the representation of the approximating function on a given stencil (Chebyshev-system) as the weighted combination of the corresponding approximating functions on substencils (Chebyshev-subsystems). We show that the weight-functions in such representations [G. Mühlbach, Numer. Math. 31 (1978) 97–110] can be generated by a general recurrence relation based on the existence of a 1-level subdivision rule. As an example of application we apply this recurrence to the computation of the weight-functions for Lagrange interpolation [E. Carlini, R. Ferretti, G. Russo, SIAM J. Sci. Comput. 27 (2005) 1071–1091] for a general subdivision of the stencil {xi-M-,…,xi+M+} of M+1≔M-+M++1 distinct ordered points into Ks+1⩽M≔M-+M+>1 (Neville) substencils {xi-M-+ks,…,xi+M+-Ks+ks} (ks∈{0,…,Ks}) all containing the same number of M-Ks+1 points but each shifted by 1 cell with respect to its neighbour, and give a general proof for the conditions of positivity of the weight-functions (implying convexity of the combination), extending previous results obtained for particular stencils and subdvisions Liu (2009) [Y.Y. Liu, C.W. Shu, M.P. Zhang, Acta Math. Appl. Sin. 25 (2009) 503–538]. Finally, we apply the recurrence relation to the representation by combination of substencils of derivatives of arbitrary order of the Lagrange interpolating polynomial.

Unstructured Light Fields

AbstractWe present a system for interactively acquiring and rendering light fields using a hand‐held commodity camera. The main challenge we address is assisting a user in achieving good coverage of the 4D domain despite the challenges of hand‐held acquisition. We define coverage by bounding reprojection error between viewpoints, which accounts for all 4 dimensions of the light field. We use this criterion together with a recent Simultaneous Localization and Mapping technique to compute a coverage map on the space of viewpoints. We provide users with real‐time feedback and direct them toward under‐sampled parts of the light field. Our system is lightweight and has allowed us to capture hundreds of light fields. We further present a new rendering algorithm that is tailored to the unstructured yet dense data we capture. Our method can achieve piecewise‐bicubic reconstruction using a triangulation of the captured viewpoints and subdivision rules applied to reconstruction weights.

An alternative method for constructing interpolatory subdivision from approximating subdivision

This paper presents a new perspective for constructing interpolatory subdivision from primal approximating subdivision. The basic idea is constructing the subdivision rule for new inserted vertices of a new interpolatory subdivision scheme based on an approximating subdivision algorithm applied to a local configuration of the mesh with one vertex updated for interpolation of the vertex. This idea is demonstrated by presenting two new interpolatory subdivision schemes based on Catmull–Clark subdivision for an arbitrary polygonal mesh and Loop subdivision for a triangular mesh, respectively. These algorithms are simple and have a small stencil for computing new points. The new perspective also shows a link between those classic approximating and interpolatory subdivision algorithms such as cubic B-spline curve subdivision and the four-point interpolatory subdivision, Catmull–Clark subdivision and Kobbeltʼs interpolatory scheme, and Loop subdivision and the butterfly algorithm.

Interpolating an arbitrary number of joint B-spline curves by Loop surfaces

In a recent paper (Ma and Wang, 2009), it was found that the limit curve corresponding to a regular edge path of a Loop subdivision surface reduces to a uniform cubic B-spline curve (CBSC) under a degeneration condition. One can thus define a Loop subdivision surface interpolating a set of input CBSCs with various topological structures that can be mapped to regular edge paths of the underlying surface. This paper presents a new solution for defining a Loop subdivision surface interpolating an arbitrary number of CBSCs meeting at an extraordinary point. The solution is based on the concept of a polygonal complex method previously used for Catmull–Clark surface interpolation and is built upon an extended set of constraints of the control vertices under which local edge paths meeting at an extraordinary point reduces to a set of endpoint interpolating CBSCs. As a result, the local subdivision rules near an extraordinary point can be modified such that the resulting Loop subdivision surface exactly interpolates a set of input endpoint interpolating CBSCs meeting at the extraordinary point. If the given endpoint interpolating CBSCs have a common tangent plane at the meeting point, the resulting Loop surface will be G1 continuous. The proposed method of curve interpolation provides an important alternative solution in curve-based subdivision surface design.

Study on the Asymmetric Sharp Feature Construct Method in Combined Subdivision Surface

Compound curve network is built to defined asymmetric sharp features in combined subdivision surface. The basic idea is to make the construction of sharp features separate from the definition of global surface and build compound curve networks consisting of base curve network and feature curve network. The curve’s subdivision rules and correct operator applying on common vertexes directly connected with curve vertexes are redefined to generate asymmetric sharp feature, and subdivision times is selected as sharp feature factor to control the sharp degree of features. When different sharp feature factors are defined at the two sides of interpolated curves, asymmetric features are generated on them. Curves of base curve network and curves of feature curve network take part in the subdivision of the interpolated surface in turn. Examples show that our method is effective and can get high quality interpolated surface.

Multibeam echosounder data cleaning through a hierarchic adaptive and robust local surfacing

Multibeam echo sounders (MBES) datasets generally contain sporadic outlier points. The huge volumes of MBES datasets in a hydrographic framework require the use of semi-automatic techniques. In very shallow waters depth, data cleaning becomes a challenging task when potential dangers to navigation have to be carefully checked. The aim of our paper is to attempt this goal by combining two well-known techniques. The seafloor is constructed as an assemblage of surface elements with the help of a robust statistical approach. The local parameters model is a priori chosen, its scale is driven through a quadtree descending approach using subdivision rules based on both statistical and spatio-temporal inferences. Our multi resolution approach provides, with the algorithm outputs, a classification map that notes areas of concern.

Interpolatory Multiscale Representation for Functions between Manifolds

We investigate interpolatory multiscale transformations for functions between manifolds which are based on interpolatory subdivision rules. We characterize the Hölder–Zygmund smoothness of a function between manifolds in terms of the coefficient decay w.r.t. this multiscale transform.