Modified Butterfly Subdivision Research Articles

Construction of a modified butterfly subdivision scheme with [formula omitted]-smoothness and fourth-order accuracy

This article presents a modified butterfly subdivision scheme with improved smoothness over regular triangular meshes. The proposed technique is an approximating scheme with a tension parameter. It achieves fourth-order accuracy and generates C2 limit surfaces for a suitable range of the parameter while maintaining the same support of the original butterfly scheme. To validate the theoretical results, some numerical examples are provided.

Smooth fillet-end cutter tool path generation method on triangular-mesh surface based on Modified Butterfly subdivision

Triangular-mesh surfaces are widely applied in the design of aerospace, automobiles, ships, etc., because of their high computational efficiency and robustness compared with parametric surfaces. In order to improve the tool path continuity and reduce the cutter load fluctuation, a smooth tool path generation method on triangular-mesh surfaces based on Modified Butterfly subdivision is proposed. The machining model at the cutter-contact point is constructed based on the geometric analysis of the triangular-mesh surface and fillet-end cutter. The minimum cutter tilt angle and maximum machining stripe width are obtained to avoid the interference of cutter and workpiece. And then, the Modified Butterfly subdivision method is applied to the triangular-mesh surface to ensure the tool path interval within the strip width. The boundary of the surface is selected as the initial tool path based on which the subsequent tool paths are generated along the edge of triangular mesh, and the cutter location points are obtained according to the geometric model of the fillet-end cutter. The Discrete Domain of Feasible Orientation (DDFO) model is constructed to eliminate the singularity problem of the machining process, and Modified Butterfly subdivision is applied recurrently to guarantee the whole surface within the scallop height requirement. Finally, the spiral tool path with good continuity and smoothness is generated according to the specified cutter parameter and scallop height requirement, which can reduce the cutter load fluctuation.

Improving smoothness and accuracy of Modified Butterfly subdivision scheme

Motivated by the increasing request of surface representation techniques suitable for biomedical imaging applications, we construct a non-stationary subdivision scheme for regular 3-directional grids, which enjoys the following properties: (i) interpolation, (ii) affine invariance, (iii) C2 smoothness, (iv) approximation order 6 and (v) the capability of reproducing several trigonometric surfaces, especially ellipsoids. To study the smoothness properties of this new scheme via existing analysis tools, we also construct an auxiliary stationary subdivision scheme enjoying properties (i)–(iv). Taking into account that, when applied on regular 3-directional grids, the Modified Butterfly scheme is C1 and has approximation order 4, the subdivision schemes derived in this paper can be considered improved variants of the Modified Butterfly scheme.

Research on the Method of Geological Three-dimensional Modeling Based on Subdivision Virtual Drills

A concept and a realization principle of subdivision virtual drills which are introduced into modeling processing are put forward in this paper in order to solve the problems, which are difficult to ensure the topological consistency and continuity between adjacent stratums in detailed geological three-dimensional modeling. Firstly, spatial data structures used in the modeling processing are introduced. And then, stratigraphic information models of drills based on semantic description are designed. Also, a generating method of subdivision virtual drills based on the Adaptive Modified Butterfly Subdivision Algorithm is given. Finally, a detailed three-dimensional geological modeling method based on subdivision virtual drills technology is implemented. Simulation result shows that the three-dimensional geological models constructed by this method, not only conform to the law of geostatistics but also keep C1 geometric continuity.

Taylor Prediction for Mesh Geometry Compression

Abstract In this paper, we introduce a new formalism for mesh geometry prediction. We derive a class of smooth linear predictors from a simple approach based on the Taylor expansion of the mesh geometry function. We use this method as a generic way to compute weights for various linear predictors used for mesh compression and compare them with those of existing methods. We show that our scheme is actually equivalent to the Modified Butterfly subdivision scheme used for wavelet mesh compression. We also build new efficient predictors that can be used for connectivity‐driven compression in place of other schemes like Average/Dual Parallelogram Prediction and High Degree Polygon Prediction. The new predictors use the same neighbourhood, but do not make any assumption on mesh anisotropy. In the case of Average Parallelogram Prediction, our new weights improve compression rates from 3% to 18% on our test meshes. For Dual Parallelogram Prediction, our weights are equivalent to those of the previous Freelence approach, that outperforms traditional schemes by 16% on average. Our method effectively shows that these weights are optimal for the class of smooth meshes. Modifying existing schemes to make use of our method is free because only the prediction weights have to be modified in the code.

Adaptive surface-deformable model with shape-preserving spring

This paper presents a multi-resolutional surface deformable model with physical property adjustment scheme and shape-preserving springs to represent surface-deformable objects efficiently and robustly. In order to reduce the computational complexity while ensuring the same global volumetric behaviour for the deformable object, we introduce a multi-resolutional mass-spring model that is locally refined using the modified-butterfly subdivision scheme. For robust deformation, a shape-preserving spring, which helps to restore the model to the original shape, is proposed to reduce the animation instability. Volume and shape preservation is indirectly achieved by restoring the model to the original shape without computing the actual volume and associated forces at every iteration. Most existing methods concentrate on visual realism of multi-resolutional deformation and often neglect to maintain the dynamic behavioural integrity between detail levels. In order to preserve overall physical behaviour, we present a new scheme for adjusting physical properties between different levels of details. During the animation of deformable objects, the part of the object under external forces beyond a threshold or with large surface curvature variations is refined with a higher level of detail. The physical properties of nodes and springs in the locally refined area are adjusted in order to preserve the total mass and global behaviour of the object. The adequacy of the proposed scheme was analysed with tests using practical mesh examples. Experimental results demonstrate improved efficiency in object deformation and preservation of overall behaviour between different levels. Copyright © 2005 John Wiley & Sons, Ltd.

Modeling corneal topography with a subdivision scheme in keratoconus

Videokeratoscopy is often unable to obtain complete data sets in cases of irregular or asymmetric corneal topography. Subdivision schemes are very common in computer graphics for completion and smoothing of surfaces. Based on a network of triangular facets a smooth and complete surface in a standard coordinate system can be derived from topographic raw data. We examined 88 patients with keratoconus and 40 normal controls. Polygons and polyhedra were defined from videokeratoscopic height data, and a surface was modeled using a modified butterfly subdivision scheme for a nonuniform sampled grid. To assess the model quality topographic raw data were changed to missing values centrally (at the apex of the cone) and in four midperipheral quadrants. The target value was the root mean square error, comparing the remodeled value of the subdivision scheme to the raw data at the position of the missing values for each group. Due to the nonuniform mesh of our Placido-based topographer we used a dynamic adaptive model and the governing dynamic differential equation. With a single missing value, no difference was detected between normals and patients with keratoconus. For a missing area consisting of 13 raw data points, the remodeling error was significantly higher in patients with keratoconus than in normals. With the neighborhood of 13 missing data points, the apex of the cone was remodeled in both groups of patients with less precision than with the peripheral missing data. The locality of the equation systems to be solved and the easy calculation of explicit formulas for the normals may simplify ray-tracing techniques and make subdivision attractive for large datasets in corneal topography even with irregular patterns.

Postprocessing of Compressed 3D Graphic Data

This work presents a postprocessing technique applied to a 3D graphic model of a lower resolution to obtain a visually more pleasant representation. Our method is an improved version of the modified butterfly subdivision scheme developed by Zorin and his colleagues. Our main contribution is to exploit the flatness information of local areas of a 3D graphic model for adaptive refinement in relatively coarser areas. We can avoid unnecessary subdivision in regions which are smoother. Consequently, the new algorithm reduces the computational complexity and also saves the storage space for many models. Our algorithm can also remove overshoots around sharp edges and retain sharp features.