本文選題：基函數(shù) + 金字塔算法��； 參考：《華北理工大學》2017年碩士論文

【摘要】：金字塔算法(Pyramid Algorithms)是由美國數(shù)學家Ron Goldman首先提出來的,是一種動態(tài)編程算法,因其形似金字塔,結構清晰簡單,表現(xiàn)算法全局能力強,因此在多項式插值和逼近理論中得到廣泛應用。在復雜的自由曲線曲面造型中,不可避免地要對曲線曲面的切向、曲率及包絡等變量進行求解運算,而該類問題一般都可轉化為基函數(shù)和導函數(shù)的求解。所以,研究樣條基函數(shù)快速且通用的構造方法具有重要意義�；趥鹘y(tǒng)樣條插值逼近理論,系統(tǒng)地研究了金字塔算法在基函數(shù)構建過程中的運用。首先針對Lagrange,Newton及Hermite三類插值基函數(shù),通過線性插值來構建算法金字塔,綜合Neville和Aitken算法,分析了路徑路標仿射組合系數(shù)的性質,得到了節(jié)點下標的可交換性,有效減少了計算復雜度。通過數(shù)值算例,分析插值曲線拼接處的光滑性,驗證了算法的優(yōu)越性。其次結合開花(Blossom)理論,通過利用對稱性及多仿射性將舊的開花值遞推得到新的開花值,來推導一元B樣條基函數(shù)的算法金字塔�；诼窂铰窐说膶ΨQ平行性質,通過倒轉金字塔來減少計算復雜度,得到基函數(shù)的向下遞推算法。并且對相鄰路徑進行交叉重疊,實現(xiàn)了拼接節(jié)點處光滑性的簡單證明。進一步將算法在x,y兩個方向上進行雙線性插值,得到矩形張量積基函數(shù)的金字塔算法,并分析了計算復雜度。針對矩形張量積節(jié)點處計算復雜度較高的問題,將節(jié)點定義在三角形網(wǎng)格上,利用重心坐標的仿射不變性,推廣到局部三角形B樣條曲面。通過對基函數(shù)構造理論的分析和研究,設計了基于動態(tài)編程的一般插值多項式及樣條曲線曲面生成的金字塔算法,為復雜曲線曲面和實體造型問題提供了新的思路與方法,特別是對CAGD中需要對幾何變量進行編程求解的問題具有重要的應用價值。
[Abstract]:Pyramid algorithm was first put forward by American mathematician Ron Goldman. It is a kind of dynamic programming algorithm. So it is widely used in polynomial interpolation and approximation theory. In the complex modeling of free curve and surface, it is inevitable to solve the tangential, curvature and envelope variables of curve and surface, but this kind of problem can be transformed into the solution of basis function and derivative function. Therefore, it is of great significance to study the fast and general construction method of spline basis function. Based on the traditional spline interpolation approximation theory, the application of pyramid algorithm in the construction of basis functions is systematically studied. Firstly, the algorithm pyramid is constructed by linear interpolation for Lagrange Newton and Hermite interpolation basis functions. The properties of affine combination coefficients of path signs are analyzed by synthesizing Neville and Aitken algorithms, and the commutativity of node subscript is obtained. The computational complexity is reduced effectively. By numerical example, the smoothness of interpolation curve splicing is analyzed, and the superiority of the algorithm is verified. Secondly, based on the Blossom-blooming theory, the new flowering value is obtained by using symmetry and multi-affine to derive the algorithm pyramid of B-spline basis function. Based on the symmetry and parallelism of path signs, the computational complexity is reduced by turning the pyramids upside down, and the downward recursive algorithm of the basis function is obtained. And the adjacent paths are crossed and overlapped, and a simple proof of smoothness of the spliced nodes is realized. The algorithm is further interpolated bilinear in XY direction to obtain the pyramid algorithm of rectangular tensor product basis function, and the computational complexity is analyzed. In order to solve the problem of high computational complexity at rectangular tensor product nodes, the nodes are defined on triangular meshes and generalized to local triangular B-spline surfaces by using affine invariance of barycentric coordinates. Based on the analysis and research of basis function construction theory, a pyramid algorithm for generating general interpolation polynomial and spline curve and surface based on dynamic programming is designed, which provides a new way of thinking and method for complex curve and surface and solid modeling problem. Especially, it has important application value to solve the problem of geometric variables in CAGD.
【學位授予單位】：華北理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.3

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本文編號：1977845