The distance of a subdivision surface to its control polyhedron

Peters, Jörg

doi:10.1016/j.jat.2008.10.012

Cited by 9 publications

(6 citation statements)

References 20 publications

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“…Comparatively more subdivision steps are required in extraordinary regions in order to obtain good approximation polygons to the final limit surface for rendering. This would result in polygonal approximation lack of details in extraordinary region and require more subdivision steps to ensure that the subdivision surface is within a given tolerance of a control polyhedron [PW09]. [ADS09] tried to remove the rendering polar artifact through postprocessing algorithms namely adaptive subdivision and Bézier triangle approximation based on the work in [ADS06].…”

Section: Introductionmentioning

confidence: 99%

Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

2019

Computer Graphics Forum

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This paper presents subdivision schemes with subdivision stencils near an extraordinary vertex that are free from or with substantially reduced polar artifact in extraordinary regions while maintaining the best possible bounded curvature at extraordinary positions. The subdivision stencils are firstly constructed to meet tangent plane continuity with bounded curvature at extraordinary positions. They are further optimized towards curvature continuity at an extraordinary position with additional measures for removing or for minimizing the polar artifact in extraordinary regions. The polar artifact for subdivision stencils of lower valences is removed by applying an additional constraint to the subdominant eigenvalue to be the same as that of subdivision at regular vertices, while the polar artifact for subdivision stencils of higher valances is substantially reduced by introducing an additional thin‐plate energy function and a penalty function for maintaining the uniformity and regularity of the characteristic map. A new tuned subdivision scheme is introduced by replacing subdivision stencils of Catmull‐Clark subdivision with that from this paper for extraordinary vertices of valences up to nine. We also compare the refined meshes and limit surface quality of the resulting subdivision scheme with that of Catmull‐Clark subdivision and other tuned subdivision schemes. The results show that subdivision stencils from our method produce well behaved subdivision meshes with the least polar artifact while maintaining satisfactory limit surface quality.

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Section: Introductionmentioning

confidence: 99%

Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

2019

Computer Graphics Forum

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“…Peters and Wu [PW09] consider the problem for general subdivision surfaces, and use a reparameterization to show that the approximation error is proportional to for m subdivision steps. They also consider a posteriori estimates, based on measuring the error after subdivision, and recommend using a priori estimates only after one or two local subdivision steps.…”

Section: Analysis Of Subdivision Surfacesmentioning

confidence: 99%

Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface Methods

Cashman¹

2011

Computer Graphics Forum

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Subdivision surfaces allow smooth free-form surface modelling without topological constraints. They have become a fundamental representation for smooth geometry, particularly in the animation and entertainment industries. This survey summarizes research on subdivision surfaces over the last 15 years in three major strands: analysis, integration into existing systems and the development of new schemes. We also examine the reason for the low adoption of new schemes with theoretical advantages, explain why Catmull-Clark surfaces have become a de facto standard in geometric modelling, and conclude by identifying directions for future research.

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“…Initially, Nairn et al [16] and Karavelas et al [17] presented the bounds for the Bezier curve generated by control polygons. Peters and Wu [18], Rief [19], and Floater [20] computed the bounds on the subdivision surface, approximation of polynomials, and cubic spline interpolant by their control structure. Cai [21] introduced the error estimation technique for the four-point interpolating scheme.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

et al. 2023

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Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.

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The distance of a subdivision surface to its control polyhedron

Abstract: 2016-11-15T19:41:40

Cited by 9 publications

References 20 publications

Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface Methods

An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

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