Narrow Results

We introduce a new type of subdivision-based multiresolution representation for triangle meshes, kite trees, which has the flexibility to represent arbitrary triangle meshes losslessly. We also develop an algorithm for extracting a balanced kite tree representation of an arbitrary input mesh that preserves mesh topology and regularities in the subdivision structure. Our scheme allows us to combine surface information automatically extracted from input data with algorithmically-generated information in a single multiresolution representation and to represent the results of adaptive refinement of regular subdivision surfaces.

We show that parametric context-sensitive L-systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bézier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L-systems, which were limited to subdivision curves.

In this paper, we propose a new image interpolation algorithm by using geometric subdivision. Similar to image upsampling, the geometric subdivision can supplement unknown data according to a certain rule, but it can only generate smooth data. To preserve the sharp edges of the high-resolution image, we adopt a rational subdivision scheme. By adjusting the weight coefficients of the rational subdivision, we can control the mesh shape near sharp edges. Hence, the image edges are preserved.

Parameterization approaches for non-uniform Doo-Sabin subdivision surfaces are developed in this paper using the eigenstructure of Doo-Sabin subdivision. New methods for evaluation of values and derivatives of Doo-Sabin surfaces at arbitrary parameters by means of non-iterative approaches are presented. Furthermore, generalized basis functions for non-uniform Doo-Sabin surfaces are derived. Thus, many algorithms and analysis techniques developed for parametric surfaces can be extended to Doo-Sabin surfaces.

Conformational searching is a core task in inverse molecular kinematics. Algorithmic improvements affecting either the speed or quality of conformational searching will have a profound impact on applications including ligand-receptor docking, ab initio prediction of protein structure, and protein folding. In this paper, we investigate a specific geometry-constrained conformational searching problem, where some feature atoms have pre-specified target positions. Using Bézier subdivision, we present a method to locate and approximate the solutions of the equations derived from constraints on the feature atoms. The conformations corresponding to these solutions are all the conformations satisfying the target constraints. Three implementations of the subdivision method taking advantage of the sparsity of the coefficients of the polynomial equations are presented and the results are compared and contrasted.

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform, and allow the multiresolution control of objects with control points at any resolution level.

In the first model, the detail is inserted independently of control points, requiring it to be rotated when applying deformations. In contrast, the second method describes details relative to control points, allowing free control point deformations.

Modeling examples of curves and surfaces are presented, showing manipulation facilities of the models.

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

In this article we improve the butterfly and Loop's algorithm. As a result we obtain subdivision algorithms for triangular nets which can be used to generate G¹- and G²-surfaces, respectively.

In this paper we investigate the properties of recursive subdivision from a geometric point of view. We explore the connections between subdivision and such areas of mathematics as spherical trigonometry, inversive geometry, and metric spaces. The methods we develop give new insights to well-known subdivision schemes and they can also be used in the argued construction of new schemes with prescribed properties.

Elsewhere we have introduced a construction to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work presents the construction for regular meshes using diagrams (stencils, masks) and interactions between diagrams to replace matrices and matrix multiplication. Regular triangular meshes with butterfly subdivision and a variant of Loop subdivision due to Litke, et al. are used as examples.

Recently we developed a subdivision scheme for Powell–Sabin splines. It is a triadic scheme and it is general in the sense that it is not restricted to uniform triangles, the vertices must not have valence six and there are no restrictions on the initial triangulation. A sequence of nested spaces or multiresolution analysis can be associated with the base triangulation. In this paper we use the lifting scheme to construct basis functions for the complement space that captures the details that are lost when going to a coarser resolution. The subdivision scheme appears as the first lifting step or prediction step. A second lifting step, the update, is used to achieve certain properties for the complement spaces and the wavelet functions such as orthogonality and vanishing moments. The design of the update step is based on stability considerations. We prove stability for both the scaling functions and the wavelet functions.

In this paper, we design a new family of biorthogonal wavelet transforms that are based on polynomial and discrete splines. The wavelet transforms are constructed via lifting steps, where the prediction and update filters are derived from various types of interpolatory and quasi-interpolatory splines. The transforms use finite and infinite impulse response (IIR) filters and are implemented in a fast lifting mode. We analyze properties of the generated scaling functions and wavelets. In the case when the prediction filter is derived from a polynomial interpolatory spline of even order, the synthesis scaling function and wavelet are splines of the same order. We formulate conditions for the IIR filter to generate an exponentially decaying scaling function.

A generic technique for construction of ternary interpolatory subdivision schemes, which is based on polynomial and discrete splines, is presented. These schemes have rational symbols. The symbols are explicitly presented in the paper. This is accompanied by a detailed description of the design of the refinement masks and by algorithms that verify the convergence of these schemes. In addition, the smoothness of the limit functions is investigated. The ternary subdivision schemes, whose construction is based on continuous splines, become tools for fast computation ofıory s of arbitrary order at triadic rational points.

The aim of this study is to compute the edge metric dimension of some subdivision of the wheel graphs. In particular, we determine and compare the metric and edge metric dimensions of the graphs obtained after the cycle, spoke and barycentric subdivisions of the wheel graph. Furthermore, some families of graphs have been constructed through subdivision process for which $\mathrm{\text{dim}}({{𝒲}{𝒞}{𝒫}}_{n,1})<{\mathrm{\text{dim}}}_e({{𝒲}{𝒞}{𝒫}}_{n,1})$, and also $\mathrm{\text{dim}}({{𝒲}{𝒞}{𝒴}}_{n,1})={\mathrm{\text{dim}}}_e({{𝒲}{𝒞}{𝒴}}_{n,1})$ which partially answer a question in [A. Kelenc, N. Tratnik and I. G. Yero, Uniquely identifying the edges of a graph: The edge metric dimension, Discrete Appl. Math.251 (2018) 204–220].

The planarity of the direct product of two graphs has been widely studied in the past. Surprisingly, the missing part is the product with K₂, which seems to be less predictible. In this piece of work, we characterize which subdivisions of multipartite complete graphs, have their direct product with K₂ planar. This can be seen as a step towards the characterization of all such graphs.

Let $G$ be a graph and $k$ be a positive integer. The $k$-subdivision $S_k(G)$ of $G$ is the graph obtained from $G$ by replacing each edge by a path of length $k$. The $k$-power $G^k$ of $G$ is the graph with vertex set $V(G)$ in which two vertices $u$ and $v$ are adjacent if and only if the distance $d_G(u,v)$ between $u$ and $v$ in $G$ is at most $k$. Note that $S_2{(G)}^2$ is the total graph $T(G)$ of $G$. The chromatic number $\chi (G)$ of $G$ is the minimum integer $l$ for which $G$ has a proper $l$-coloring. The total chromatic number of $G$, denoted by ${\chi }^{″}(G)$, is the chromatic number of $T(G)$. Rosenfeld [On the total coloring of certain graphs, Israel J. Math.9 (1971) 396–402] and independently, Vijayaditya [On total chromatic number of a graph, J. London Math. Soc.2 (1971) 405–408] showed that for a subcubic graph $G$, ${\chi }^{″}(G)\le 5$. In this note, we prove that for a subcubic graph $G$, $\chi (S_3{(G)}^3)\le 7$.

Let $G=(V,E)$ be a graph of order $n$ and size $m.$ A dominating set $S$ of $G$ is called a secure dominating set if for each $u\in V\setminus S$ there exists $v\in S$ such that $v$ is adjacent to $u$ and $(S\setminus \{v\})\cup \{u\}$ is a dominating set of $G.$ In this case, we say that $u$ is $S$-defended by $v$ or $v$$S$-defends $u.$ The secure domination number ${\gamma }_s(G)$ is the minimum cardinality of a secure dominating set of $G.$ The secure domination subdivision number of $G$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the secure domination number. In this paper, we present several results on this parameter.

C-B-spline curves are an extension of cubic uniform B-spline curves. In addition to having many properties that B-spline curves have, they have an adjustable shape parameter α, and can represent arcs precisely. This paper presents the subdivision algorithm and G¹ condition of C-B-spline curves based on the analysis of C-B-spline basis functions and terminal properties. The G¹ and G² continuous conditions between C-B-spline curves and cubic uniform B-spline curves are given. The results have definite geometric meanings that are useful for the shape modification and representation of C-B-spline curves, and they can also be extended to surface modeling with C-B-spline.

This manuscript focuses on a dissemination method with specified platforms which is designed for a weighing robot produced by SIMT. It is suitable for both OIML R111 and Chinese customary usage. Dissemination method with specified platforms is a kind of subdivision. We don’t change the principle of subdivision but make some changes in weighing method. Dissemination method with the specified platforms makes the automatic weighing robot reach no weights drop. With the analysis of the uncertainty and the accuracy, this method is proved to reach the requirements of E₂ class mg weightgroups. Three characteristics of SIMT’s weighing robot with a dissemination method are standard applicability, method variety and no weights drop. Dissemination method with the specified platforms is a bold attempt in mg weightgroups’ calibration. Since the error of the specified platforms can’t be corrected completely, this method is only suitable to mg weightgroups (E₂ class and down). We can’t calibrate the E₁ class mg group weights with this method.