CURVE AND SURFACE INTERPOLATIONBY SUBDIVISION ALGORITHMS

Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference analysis, a systematic method to construct Cn (n≥ 0) interpolatorycurves by subdivision from given data is described and the mask (filter) of the algorithm is presentedexplicitly. This algorithm generates a Cn smooth curve which interpolates the initial control points.Control parameters are also provided so that the shape of the final curve can be adjusted according torequirements. An immediate generalisation of the method is the construction of smooth interpolatorysubdivision algorithms over uniform triangular networks (tensor product type data) in Rm. The mainresults of this algorithm for smooth interpolatory surface subdivision algorrthm are also included.AMS(MOS) : 65D05 , 65D15 , 65D17.

Neural Network Methods for NURBS Curve and Surface Interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS cuxve and surface interpolation. When all the knot spans are identical, the NURBS curve interpolation procedtire degenerates into that of uniform rational B-spline curves. If all the weights of data points are identical,then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.

SMOOTH SURFACE INTERPOLATION OVER ARBITRARY TRIANGULATIONS BY SUBDIVISION ALGORITHMS

A smooth interpolatory subdivision algorithm for the generation of surfaces over arbi-trary triangulations is introduced and its convergence properties over nonuniform triangulationsstudied. For uniform data, this method is a generalization of the analysis for univariatesubdivision algorithms and for nonuniform data, an extraordinary point analysis is introducedand the local subdivision matrix anaiysis presented. It is proved that the algorithm producessmooth surfaces over arbitrary triangular networks provided the shape parameters are kept with-in an appropriate range. Finally, two graphical examples of surface interpolation overnonuniform data are given to show the smoothing process of the algorithm.AMS (MOS): 65D05, 65D15,65D17.

Image Interpolation Through Surface Reconstruction

Reconstructing an HR （high-resolution） image which preserves the image intrinsic structures from its LR （low-resolution） counterpart is highly challenging. This paper proposes a new surface reconstruction algorithm applied to image interpolation. The interpolation surface for the whole image is generated by putting all the quadratic polynomial patches together. In order to eliminate the jaggies of the edge, a new weight fimction containing edge information is incorporated into the patch reconstruction procedure as a constraint. Extensive experimental results demonstrate that our method produces better results across a wide range of scenes in terms of both quantitative evaluation and subjective visual quality.

Feedback Mechanism-driven Mutation Reptile Search Algorithm for Optimizing Interpolation Developable Surfaces

Curvature lines are special and important curves on surfaces.It is of great significance to construct developable surface interpolated on curvature lines in engineering applications.In this paper,the shape optimization of generalized cubic ball developable surface interpolated on the curvature line is studied by using the improved reptile search algorithm.Firstly,based on the curvature line of generalized cubic ball curve with shape adjustable,this paper gives the construction method of SGC-Ball developable surface interpolated on the curve.Secondly,the feedback mechanism,adaptive parameters and mutation strategy are introduced into the reptile search algorithm,and the Feedback mechanism-driven improved reptile search algorithm effectively improves the solving precision.On IEEE congress on evolutionary computation 2014,2017,2019 and four engineering design problems,the feedback mechanism-driven improved reptile search algorithm is compared with other representative methods,and the result indicates that the solution performance of the feedback mechanism-driven improved reptile search algorithm is competitive.At last,taking the minimum energy as the evaluation index,the shape optimization model of SGC-Ball interpolation developable surface is established.The developable surface with the minimum energy is achieved with the help of the feedback mechanism-driven improved reptile search algorithm,and the comparison experiment verifies the superiority of the feedback mechanism-driven improved reptile search algorithm for the shape optimization problem.

Constrained Control of Interpolating Surfaces by Parameters

In order to meet the needs of practical design, an interpolation technique is employed to constrain the shape of surfaces. The method of preserving positivity on the interpolation surface and constraint on interpolating data is also developed. The advantage of this new method is that it can be used to constrain the shape of an interpolating surface only by selecting suitable parameters, and numerical examples are presented to show the performance of the method.

THE THEORY OF FRACTAL INTERPOLATED SURFACE AND ITS APPLICATIONS

In this paper, the principle of construction of a fractal surface is introduced, interpolation functions for a fractal interpolated surface are discussed, the theorem of the uniqueness of an iterated function system of fractal interpolated surface is proved, the theorem of fractal dimension of fractal interpolated surface is derived, and the case that practical data are used to interpolate fractal surface is studied.

Improved Butterfly Subdivision Scheme for Meshes with Arbitrary Topology

Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting vertex, 2n new vertices are inserted in the 2n triangle faces in each recursion, and the n old vertices are kept, special treatment is given to the boundary, achieving higher smoothness while using small stencils is realized. With the proposed scheme, the number of triangle faces increases only by a factor of 3 in each refinement step. Compared with the butterfly subdivision scheme and the modified butterfly subdivision scheme, the size of triangle faces changes more gradually, which allows one to have greater control over the resolution of a refined mesh.

Interpolating Quintic Polynomial with C^1 Continuity

A method constructinq C^1 Piecewise quintic polynomial over a triangular grid to interpo- late function values and partial derivatives at vertices is presented in this paper.The set of precise poly- nomials of this method is discussed.

C^2-(C^3-) Continuous Interpolation Spline Curve and Surface

Free-formed or sculptured surfaces in engineering products are frequently constructed from a set of measured 3D data points. C2- (C3-) continuity approach is important in this field. This paper presents a method of rectangular interpolation of given 3D data array which is regularly arranged. The interpolation surface which is constructed by tensor product has desirable properties (second-order or third-order continuity locality) and is implemented and adjusted easily. Higher order continuity methods are also briefly discussed.

Discrete Interpolation Surface

In this paper, a method to construct a surface with point interpolation and normal interpolation is presented. An algorithm to construct the discrete interpolation is also presented, which has the time com- plexity O (Nlog N), where N in the number of scattered points.

Generating Symbolic Interpolants for Scattered Data with Normal Vectors

Algorithms to generate a triangular or a quadrilateral interpolant with G^1-continuity are given in this paper for arbitrary scattered data with associated normal vectors over a prescribed triangular or quadrilateral decomposition. The interpolants are constructed with a general method to generate surfaces from moving Bezier curves under geometric constraints. With the algorithm, we may obtain interpolants in complete symbolic parametric forms, leading to a fast computation of the interpolant. A dynamic interpolation solid modelling software package DISM is implemented based on the algorithm which can be used to generate and manipulate solid objects in an interactive way.