Optimization methods for scattered data approximation with subdivision surfaces

Marinov, Martin; Kobbelt, Leif

doi:10.1016/j.gmod.2005.01.003

Cited by 51 publications

(87 citation statements)

References 15 publications

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“…Several families of functions have been applied to this problem. They include subdivision surfaces [6,7], function reconstruction [8], radial basis functions [9], implicit surfaces [10], hierarchical splines [11], algebraic surfaces [12], polynomial metamodels [13], and many others. However, the most popular choice is given by the free-form parametric basis functions because they are very flexible and very well suited to represent any smooth shape with only a few parameters.…”

Section: Surface Approximation In Reverse Engineeringmentioning

confidence: 99%

“…All these methods are not commonly supported within current modeling software systems. More popular choices include triangular meshes and Loop subdivision surfaces [7]. However, structures with quadrilateral sets of poles, such as Catmull-Clark subdivision surfaces and free-form parametric surfaces are de facto industry standard in CAD/CAM and other fields [1,6,22,23].…”

Section: Previous Workmentioning

confidence: 99%

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Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering

et al. 2018

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This paper concerns several important topics of the Symmetry journal, namely, computer-aided design, computational geometry, computer graphics, visualization, and pattern recognition. We also take advantage of the symmetric structure of the tensor-product surfaces, where the parametric variables u and v play a symmetric role in shape reconstruction. In this paper we address the general problem of global-support parametric surface approximation from clouds of data points for reverse engineering applications. Given a set of measured data points, the approximation is formulated as a nonlinear continuous least-squares optimization problem. Then, a recent metaheuristics called Cuckoo Search Algorithm (CSA) is applied to compute all relevant free variables of this minimization problem (namely, the data parameters and the surface poles). The method includes the iterative generation of new solutions by using the Lévy flights to promote the diversity of solutions and prevent stagnation. A critical advantage of this method is its simplicity: the CSA requires only two parameters, many fewer than any other metaheuristic approach, so the parameter tuning becomes a very easy task. The method is also simple to understand and easy to implement. Our approach has been applied to a benchmark of three illustrative sets of noisy data points corresponding to surfaces exhibiting several challenging features. Our experimental results show that the method performs very well even for the cases of noisy and unorganized data points. Therefore, the method can be directly used for real-world applications for reverse engineering without further pre/post-processing. Comparative work with the most classical mathematical techniques for this problem as well as a recent modification of the CSA called Improved CSA (ICSA) is also reported. Two nonparametric statistical tests show that our method outperforms the classical mathematical techniques and provides equivalent results to ICSA for all instances in our benchmark.

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Section: Surface Approximation In Reverse Engineeringmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering

et al. 2018

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“…Subdivision surfaces [15][16][17][18][19] remove the need for explicitly maintaining continuity, but while regular areas of the subdivision mesh can be approximated by splines, the surface is not everywhere analytic.…”

Section: Previous Workmentioning

confidence: 99%

Adaptive smooth surface fitting with manifolds

Grimm

Phan

et al. 2009

Vis Comput

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We present a smooth, every-where C k , analytic surface representation for closed surfaces of arbitrary topology. We demonstrate fitting this representation to meshes of varying resolutions and sampling quality. The fitting process is adaptive and provides controls for both the average and the maximum allowable error. The representation is suitable for applications which require consistent parameterizations across different surfaces.Keywords manifolds · surface fitting · analytic surface MotivationWe present a smooth, every-where C k , analytic surface representation for closed surfaces of arbitrary topology. In order for a representation to be useful we need to be able to build surfaces from data. In this paper, we show how to fit our representation to an input mesh of the same topology, taking advantage of the adaptive nature of the representation to better control both average and maximum fit levels. Most fitting approaches typically minimize just the average error; this can lead to areas which are inadequately represented, even though the average is still low. We address this by allowing the user to specify both a desired average and maximum error. OverviewIn our representation the surface is defined as an embedding of a global domain of the desired topology, such as a sphere or torus. The embedding is created by locally describing what the surface should look like and blending the results together. Unlike splines, the continuity of the surface is maintained by the blending process, not constraints between the local descriptions. This means these local surface pieces can be added, changed, or deleted without needing to re-establish continuity constraints.More specifically, we define a general method for defining local, planar parameterizations on each of the basic global domain topologies (sphere, torus, or n-holed torus). This parameterization is a C ∞ , invertible map from part of the global domain to a disk in the plane. The embedded surface is created by writing an embedding function for each of these local parameterizations, then blending them together based on how they overlap in the global domain.Our surface representation naturally supports additional data sets with different resolution requirements by simply defining new local data descriptions on the global domain. The correspondence between the additional data and the geometry is maintained through the global domain. Fig. 1 A) Defining a surface using our representation. Surface is an embedding of a global domain (in this case a circle). The embedding is defined by several local embeddings (two shown) which are blended together.To build our surface representation from a data set we use a modified leastsquares based approach. The input data is a manifold mesh of the desired topology. The fitting process has two steps: 1) Build a mapping between the global domain and the mesh. 2) Adaptively cover the domain (and mesh) with local parameterizations, each of which is fit to the corresponding part of the input mesh. This fitting process is enti...

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“…So, starting from the reference original subdivision surface, a control mesh synchronization moves iteratively its control points in order to match it with the suspect smooth surface. For a given target smooth surface (attacked subdivided watermarked surface) (see Figure 3.a) and a given reference subdivision control mesh (see Figure 3.b), our process displaces control points by minimizing a global error between the corresponding limit surface and the target one, based on the quadratic distance approximants defined by Pottmann and Leopoldseder [14]; This algorithm, used for subdivision surface approximation by Lavoué et al [11] and Marinov and Kobbelt [10] allows a quite accurate and rapid convergence. and 0.03 × 10 −3 after 2 and 5 iterations (surfaces were normalized in a cubic bounding box of length equal to 1).…”

Section: Control Mesh Synchronizationmentioning

confidence: 99%

“…Figure 1 shows an example of subdivision surface (Catmull-Clark rules [9]), at each iteration, the base mesh is linearly subdivided and smoothed. A lot of algorithms exist to convert a 3D mesh into a subdivision surface [10,11] because this model is much more compact, in term of amount of data, than a dense polygonal mesh.…”

Section: Introductionmentioning

confidence: 99%

A Watermarking Framework for Subdivision Surfaces

Lavoué¹,

Denis²,

Dupont³

et al. 2006

Multimedia Content Representation, Classification and Security

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Abstract. This paper presents a robust watermarking scheme for 3D subdivision surfaces. Our proposal is based on a frequency domain decomposition of the subdivision control mesh and on spectral coefficients modulation. The compactness of the cover object (the coarse control mesh) has led us to optimize the trade-off between watermarking redundancy (which insures robustness) and imperceptibility by introducing two contributions: (1) Spectral coefficients are perturbed according to a new modulation scheme analyzing the spectrum shape and (2) the redundancy is optimized by using error correcting codes. Since the watermarked surface can be attacked in a subdivided version, we have introduced a so-called synchronization algorithm to retrieve the control polyhedron, starting from a subdivided, attacked version. Through the experiments, we have demonstrated the high robustness of our scheme against both geometry and connectivity alterations.

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Optimization methods for scattered data approximation with subdivision surfaces

Cited by 51 publications

References 15 publications

Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering

Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering

Adaptive smooth surface fitting with manifolds

A Watermarking Framework for Subdivision Surfaces

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