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Example 6: a complex triangular mesh: ( a ) the original mesh; ( b ) interpolation surface by method 1 with λ ij = μ F = 0 . 5; ( c ) interpolation surface by
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Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics.
In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh.
The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the gi...
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Citations
... The process involves a topological modification of the control mesh in the first phase, followed by a Catmull-Clark subdivision in the second phase. Deng et al. [14] proposed an efficient algorithm for constructing a Catmull-Clark surface that interpolates a given mesh. The algorithm involves constructing a new control mesh derived from one Catmull-Clark subdivision step with improved geometric rules. ...
... Equation (1), called the formula of the limit point of Catmull-Clark subdivision, is used to make the Catmull-Clark surface to interpolate part or all vertices of the initial meshes [15][16][17]. On this basis, Deng [14] proposed two local methods called "push-back operation based method" and "normal-based method" to get new control meshM i+1 . Figs. 10d and 11d are the limit surfaces of the control meshes generated using Deng's method. ...
... In contrast, our method employs examples with j = 7 for comparison. We compared some classical interpolating algorithms of Catmull-Clark scheme, including Halstead's [11] method, Deng's [14] method (a set of shape handles in Deng's method is uniformly taken as 0.5), and Chen's [12] method (PIA). Table 1 presents a comparison of the surface energy obtained by varying the value of j in Formula (8). ...
We propose an efficient method with energy constraints for constructing a Catmull–Clark surface that interpolates a given mesh. We approximate the surface energy of Catmull–Clark surfaces near extraordinary points by summing their finite subpatches and then represent the energy of the subpatches as linear combinations of the vertices of control mesh. By minimizing the surface energy as a constraint, we generate a new control mesh whose limit surfaces interpolate a given mesh. Numerous examples and comparisons demonstrate that our method has the following characteristics: (1) The limit surfaces are fairer, reducing unnecessary undulations and having minimal surface energy, and (2) the approximation process is simple and intuitive, requiring only a small number of computational steps and avoiding complex parameterization processes.
... In [10], the authors put forward the preconditioned progressive iterative approximation for the triangular Bézier patches. Based on the different fitting representation, the PIA method in [1,2] is extended to H. Wang the subdivision surface fittings [11][12][13][14][15] and the volumetric subdivision fitting [16]. ...
... , m can be selected arbitrarily in theory. The control points {P k i } n i=0 are updated by the outer iteration (9) with the global relaxation parameter ν and the inner iteration (10) with the local relaxation parameter ω and the different vector (11). The parameters ν and ω selected rationally can speed up the convergence rate of the ELSPIA method. ...
... , n, are defined by (9) and (8), respectively. Substituting (11) and (9) into (10) and (8), respectively, we rewrite (10) and (8) as ...
The progressive and iterative approximation method for least square fitting (LSPIA) (Deng and Lin in Comput Aided Des 47:32–44, 2014) is an efficient method for fitting a large number of data points. By introducing the global and local relaxation parameters, we develop an extended LSPIA (ELSPIA) method which includes the LSPIA method as its special case. The ELSPIA method constructs the sequence of curves and surfaces by adjusting the control points with the outer and inner iteration. It is proved that the sequence of curves and surfaces converges to the least square fitting curve and surface, respectively, even when the collocation matrix is not of full column rank. The ELSPIA method is flexible to allow the local adjustment of the control points. Moreover, the convergence rate of the ELSPIA method can be faster than that of the LSPIA method under the same assumption. Numerical results verify this phenomenon.
... Zheng and Cai [26] proposed a Two-Phase Subdivision (TPS) scheme to construct a smooth surface interpolating a mesh with arbitrary topology. By modifying the geometric rules of the first step of Catmull-Clark subdivision, Deng and Yang [9] used a simple and efficient method to derive interpolation surfaces. Similar methods are also used to √ 3 [7] and Loop subdivision [8]. ...
We propose Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation by combining the Gauss–Seidel iterative method for linear systems and progressive iterative approximation (PIA) for free-form curve and surface interpolation. We address the details of GS-PIA for Loop and Catmull–Clark surface interpolation and prove that they are convergent. In addition, GS-PIA may also be applied to surface interpolation for other stationary approximating subdivision schemes with explicit limit position formula/masks. GS-PIA inherits many good properties of PIA, such as having intuitive geometric meaning and being easy to implement. Compared with some other existing interpolation methods by approximating subdivision schemes, GS-PIA has three main advantages. First, it has a faster convergence rate than PIA and weighted progressive iterative approximation (W-PIA). Second, GS-PIA does not need to compute optimal weights while W-PIA does. Third, GS-PIA does not modify the mesh topology but some methods with fairness measures do. Numerical examples for Loop and Catmull–Clark subdivision surface interpolation illustrated in this paper show the efficiency and effectiveness of GS-PIA.
... The push-back method was firstly proposed in [Maillot and Stam (2001)] to deal with the shrinkage issue of approximating subdivision schemes. Deng and Yang [Deng and Yang (2010)] applied this idea to the Catmull-Clark surface subdivision scheme to derive an interpolation scheme based on the explicit limit point formula. Progressive-iterative approximation (PIA for short) and weighted PIA have been used for constructing interpolatory subdivision surfaces by adjusting the control mesh iteratively [Chen et al. (2008), Cheng et al. (2009), Deng and Ma (2012)]. ...
Volumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the Catmull–Clark subdivision scheme interpolates vertices of the original hexahedral mesh. The new control lattice is derived by the local push-back operation from one Catmull–Clark subdivision step with modified geometric rules. This interpolating method is simple and efficient, and several shape parameters are involved in adjusting the shape of the limit volume. The second method is based on progressive-iterative approximation using limit point formula. At each iteration step, we progressively modify vertices of an original hexahedral mesh to generate a new control lattice whose limit volume interpolates all vertices in the original hexahedral mesh. The convergence proof of the iterative process is also given. The interpolatory subdivision volume has C2-smoothness at the regular region except around extraordinary vertices and edges. Furthermore, the proposed interpolatory volumetric subdivision methods can be used not only for geometry interpolation, but also for material attribute interpolation in the field of volumetric material modeling. The application of the proposed volumetric subdivision approaches on isogeometric analysis is also given with several examples.
... Surface Parameterization. Surface parameterization was first introduced to computer graphics as a method for mapping texture onto surfaces [16,17] and has gradually become a useful tool for many geometry processing applications, such as detail mapping, synthesis and transfer, mesh editing and compression, remeshing, fitting, morphing, and so on [18][19][20][21][22][23][24] . Surface conformal mapping as the most popular surface parameterization method is angle preserving and has been widely used for various shape analysis applications [25] . ...
This paper presents a novel method to compute the quasiconformal rectilinear map for a 2D polygonal subdivision, which keeps the original topology and best preserves the shape of the original input in the elasticity sense by the curve-driven quasiconformal mapping. We first automatically compute the straightening styles of the subdivision curves, which are then utilized to generate the optimal rectilinear map with the minimal harmonic energy. The quasiconformal rectilinear map is aesthetically pleasing for human perception, and hierarchically and progressively flexible for the design of spatial data structures. Based on the quasiconformal rectilinear map, an approach is given to compute the polyomino map. We evaluate the proposed structural maps in the data visualization application.
... Due to (6), primal interpolatory schemes have the so called stepwise interpolation property, i.e. at each subdivision step they maintain the data of the previous one. Beside the use of primal interpolatory schemes, there exist other approaches to interpolate points via subdivision, such as those that apply an approximating scheme after suitably preprocessing the data to be interpolated (see, e.g., [14,30,32]). However, before [29], none of the existing approaches took into account the possibility of constructing a native dual interpolatory scheme which does not have the property of retaining the initial data at each iteration, but achieves the interpolation in the sense that the initial data are obtained again in the limit function, see Figure 1. ...
... At this point we are ready to apply Poisson summation formula to both sides of (14) obtaining, for the left-hand-side, ...
... Combining (14) with (15) and (16) we obtain ...
A new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts.
... To create smoother point cloud models, hole-filling is employed using the Catmull-Clark [31] subdivision method, which employs surface-interpolating corner vertices and boundary curves to fill holes. In the proposed multi-camera system, the holograms are generated from synthesized real object point clouds. ...
A multiple-camera holographic system using non-uniformly sampled 2D images and compressed point cloud gridding (C-PCG) is suggested. High-quality, digital single-lens reflex cameras are used to acquire the depth and color information from real scenes; these are then virtually reconstructed by the uniform point cloud using a non-uniform sampling method. The C-PCG method is proposed to generate efficient depth grids by classifying groups of object points with the same depth values in the red, green, and blue channels. Holograms are obtained by applying fast Fourier transform diffraction calculations to the grids. Compared to wave-front recording plane methods, the quality of the reconstructed images is substantially better, and the computational complexity is dramatically reduced. The feasibility of our method is confirmed both numerically and optically.
... Due to (6), primal interpolatory schemes have the so called stepwise interpolation property, i.e. at each subdivision step they maintain the data of the previous one. Beside the use of primal interpolatory schemes, there exist other approaches to interpolate points via subdivision, such as those that apply an approximating scheme after suitably preprocessing the data to be interpolated (see, e.g., [14,30,32]). However, before [29], none of the existing approaches took into account the possibility of constructing a native dual interpolatory scheme which does not have the property of retaining the initial data at each iteration, but achieves the interpolation in the sense that the initial data are obtained again in the limit function, see Figure 1. ...
... At this point we are ready to apply Poisson summation formula to both sides of (14) obtaining, for the left-hand-side, ...
... Combining (14) with (15) and (16) we obtain ...
A new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts.
... Since the method has good adaptive ability and convergent stability, it can be used to interpolate scattered data points. Thus, it has been used in approximating different types of subdivision surfaces such as Doo-Sabin, Loop, and Catmull-Clark subdivision surfaces [8][9][10][11]. Several methods were proposed for improving the convergence rate and the flexibility of the method lately. For example, local geometric iterative method [12] characteristically selects a subset of the data points as the approximate points to make this method more flexible; weighted geometric iterative method was proposed by Lu [13] to improve the convergence rate. ...
In this paper, a novel geometric iterative fitting method is presented which has a set of mutually different weights. It possesses the advantages of least square progressive iterative approximation method (abbr. LSPIA method) which can handle point sets of large sizes and adjust the number of control points and knot vector flexibly. The presenting method degrades into LSPIA method with appropriate choices of weights, and it illustrates better effects for the previous iteration steps comparing with the LSPIA method. Also, this method is further applied to generalized B-splines which have changing core functions (The mentioned generalized B-spline is a special generalization of classical B-spline with linear core function). Combining the advantages of generalized B-splines and choice of different weights, it can handle much more complicated practical problems. Detailed discussion about the choosing of core functions and weights is also given. Plentiful numerical examples are also presented to show the effectiveness of the method.
... Since the method has good adaptive ability and convergent stability, it can be used to interpolate scattered data points. Thus it has been used in approximating different types of subdivision surfaces such as Doo-Sabin, Loop, Catmull-Clark subdivision surfaces [10][11][12][13][14]. Furthermore, Lin [15] also developed a constrained volume iterative fitting algorithm to fill a given triangular mesh model with an all-hex volume mesh. ...
Generalized B-spline bases are generated by monotone increasing and continuous “core” functions; thus generalized B-spline curves and surfaces not only hold almost the same perfect properties which classical B-splines hold but also show more flexibility in practical applications. Geometric iterative method (also known as progressive iterative approximation method) has good adaptability and stability and is popular due to its straight geometric meaning. However, in classical geometric iterative method, the number of control points is the same as that of data points. It is not suitable when large numbers of data points need to be fitted. In order to combine the advantages of generalized B-splines with those of geometric iterative method, a fresh least square geometric iterative fitting method for generalized B-splines is given, and two different kinds of weights are also introduced. The fitting method develops a series of fitting curves by adjusting control points iteratively, and the limit curve is weighted least square fitting result to the given large data points. Detailed discussion about choosing of core functions and two kinds of weights are also given. Plentiful numerical examples are also presented to show the effectiveness of the method.
