曲面上葉狀結(jié)構(gòu)的研究與多元Birkhoff插值
發(fā)布時間：2018-09-10 06:13
【摘要】：計算共形幾何是計算機(jī)科學(xué)和純粹數(shù)學(xué)之間的交叉學(xué)科,其目的是將現(xiàn)代幾何、經(jīng)典幾何的概念和定理轉(zhuǎn)化為計算機(jī)算法,為工程實踐服務(wù).計算機(jī)算法在某種程度上促進(jìn)和傳播了數(shù)學(xué),同時也拉近了數(shù)學(xué)和工程應(yīng)用之間的距離.全純二次微分作為共形幾何中的重要概念.在Teichmuller空間的研究中具有重要的意義,它和曲面上的葉狀結(jié)構(gòu)有著直接的關(guān)聯(lián),全純二次微分的水平軌線和豎直軌線都構(gòu)成了曲面上的葉狀結(jié)構(gòu).但由于全純二次微分的艱深晦澀,少有人能領(lǐng)略其美感,更無法想象其在工程領(lǐng)域能有什么樣的應(yīng)用.本文首次實現(xiàn)了高虧格黎曼曲面到圖(graph)的廣義調(diào)和映照算法.得到曲面上的葉狀結(jié)構(gòu),并進(jìn)行了可視化,同時,由于黎曼曲面到圖(graph)的廣義調(diào)和映照誘導(dǎo)的Hopf微分就是全純二次微分(Strebel微分),于是算法可以進(jìn)一步計算全純二次微分.算法通過輸入不同的曲面上的環(huán)路和高度參數(shù),可以控制葉狀結(jié)構(gòu)的拓?fù)渫瑐愵惡蛶缀螌傩?因而得到的全純二次微分也是可控的.本文首次對高虧格曲面上的葉狀結(jié)構(gòu)和全純二次微分進(jìn)行了全面豐富的可視化.使得我們能夠更加直觀理解這些概念,進(jìn)而將之應(yīng)用到工程領(lǐng)域.在六面體網(wǎng)格生成中.本文首次闡述了可染色四邊形網(wǎng)格、有限可測葉狀結(jié)構(gòu)和Strebel微分三個概念之間的等價關(guān)系,為六面體網(wǎng)格生成算法提供了堅實的理論基礎(chǔ);并且基于曲面上的葉狀結(jié)構(gòu)和Strebel微分,提出并實現(xiàn)了全自動的六面體網(wǎng)格生成算法,對六面體網(wǎng)格自動生成這一所謂的“圣杯”問題實現(xiàn)了重大突破.此外,基于曲面上的葉狀結(jié)構(gòu),我們提出一種新的曲面配準(zhǔn)算法,可以進(jìn)行高虧格、大形變非等距變換下的曲面配準(zhǔn).另外,本文還介紹了Birkhoff插值中,在任意單項序下具有相同極小單項基的一類問題,提出了這類問題的判別標(biāo)準(zhǔn),使得我們可以使用字典序下的極小單項基快速算法來計算其他單項序下的問題.曲面上的葉狀結(jié)構(gòu)和Strebel微分.所謂葉狀結(jié)構(gòu)(foliation),就是將n維流形分解成(n-1)維子流形,其分解方式局部上具有直積結(jié)構(gòu),如圖1中(a)所示,我們將虧格為2的曲面分解成一族曲線，每條曲線被稱為一片葉子.曲面上三條葉子交匯的點被稱為奇異點.圖1 虧格為2的曲面上的葉狀結(jié)構(gòu)和Strebel微分.為得到曲面上的葉狀結(jié)構(gòu)，我們考慮曲面到圖(graph)的廣義調(diào)和映照.如圖1(a)中三條橙色曲線所示，我們在曲面上指定了一組環(huán)路(封閉曲線){γ1，γ2，γ3}，并相應(yīng)配以高度參數(shù)這組環(huán)路將曲面分割成2條褲子(帶有三條邊界的虧格為零的曲面).這種分解方式可以定義出一個圖(graph),如圖1(b)，每條褲子對應(yīng)一個節(jié)點，每條環(huán)路對應(yīng)一條邊，邊的長度即為對應(yīng)的高度參數(shù)；如果環(huán)路連接兩條褲子(可以相同)，，那么此環(huán)路對應(yīng)的邊連接這兩條褲子對應(yīng)的節(jié)點.Gromov和Schoen[20]定義了圖(graph)的雙曲性質(zhì)，證明了從曲面到圖的廣義調(diào)和映照的存在性和唯一性.我們的算法使用非線性熱流方法求解曲面到圖的廣義調(diào)和映照，并得到曲面上的葉狀結(jié)構(gòu).如圖1所示，(b)中圖(graph)的節(jié)點的原像是(a)中標(biāo)識紅色的曲線；(b)中圖(graph)中邊上的點的原像是(a)中曲面上的葉子.曲面上葉狀結(jié)構(gòu)的葉子可以是環(huán)路(封閉曲線)，也可以是無限延伸的螺旋線，所有葉子均是環(huán)路的葉狀結(jié)構(gòu)稱為有限可測葉狀結(jié)構(gòu).我們算法通過廣義調(diào)和映照算得的葉狀結(jié)構(gòu)是有限可測葉狀結(jié)構(gòu).經(jīng)典的Hubbard-Masure理論[25]證明了可測葉狀結(jié)構(gòu)和全純二次微分之間的等價關(guān)系,即任給一個可測葉狀結(jié)構(gòu),如圖1(a),則存在一個全純二次微分如圖1(c),其水平軌道誘導(dǎo)的葉狀結(jié)構(gòu)恰好是所給的葉狀結(jié)構(gòu).具有有限可測葉狀結(jié)構(gòu)的全純二次微分叫做Strebel微分.我們在計算出葉狀結(jié)構(gòu)后,對其進(jìn)行Hodge星算子操作,就可以計算Strebel微分.六面體網(wǎng)格生成.本文首次闡述和證明了下面這三個概念的等價關(guān)系:{可染色四邊形網(wǎng)格}(?){有限可測葉狀結(jié)構(gòu)}(?){Strebel微分}定理0.0.1(三位一體)設(shè)S是虧格大于1的封閉黎曼曲面.給定曲面上可染色四邊形網(wǎng)格Q,則存在Q誘導(dǎo)的有限可測葉狀結(jié)構(gòu)(FQ,μQ),同時存在唯一的Strebel微分Φ.使得Φ誘導(dǎo)的水平有限可測葉狀結(jié)構(gòu)(FΦ,μΦ)恰好等于(FQ,μQ).相反地.給定Strebel微分Φ,可構(gòu)造有限可測葉狀結(jié)構(gòu)(FΦ,μΦ).誘導(dǎo)可染色四邊形網(wǎng)格Q.此三位一體理論框架使得我們可以使用Strebel微分來構(gòu)造可染色四邊形網(wǎng)格,然后拓展生成曲面內(nèi)部實體的六面體網(wǎng)格.算法流程為:輸入虧格為g1的封閉曲面,曲面的內(nèi)部空間構(gòu)成高虧格的實體.(1),用戶在輸入曲面上設(shè)定一組不相交的簡單環(huán)路,并對每個環(huán)路指定一個高度參數(shù);(2),計算出唯一的Strebel微分;(3),Strebel微分將輸入曲面分割成圓柱面,并生成可染色四邊形網(wǎng)格;(4),根據(jù)曲面的圓柱面分割,向曲面內(nèi)部空間延伸,生成體的圓柱體分割;(5),計算分割出的每個拓?fù)鋱A柱體到標(biāo)準(zhǔn)圓柱體的微分同胚映射;(6),將標(biāo)準(zhǔn)圓柱體上的六面體網(wǎng)格拉回映射到原拓?fù)鋱A柱體上,即完成了各個分割圓柱體上的六面體網(wǎng)格生成,最后將其拼接,得到整體的六面體網(wǎng)格.高虧格曲面配準(zhǔn).基于曲面上的葉狀結(jié)構(gòu),我們提出一種新的曲面配準(zhǔn)方法.葉狀結(jié)構(gòu)將曲面分解為一組封閉曲線,這種分解具有局部張量積結(jié)構(gòu).并且對應(yīng)一個圖(graph).對于兩個具有相同拓?fù)涞娜~狀結(jié)構(gòu)的同胚曲面.我們首先對它們的圖進(jìn)行配準(zhǔn),然后配準(zhǔn)對應(yīng)的葉子.圖2展示了我們曲面配準(zhǔn)算法的流程.給定兩個虧格為g= 4的曲面,我們自動計算出能將曲面分解為2g-2條褲子的3g-3條環(huán)路,進(jìn)而誘導(dǎo)褲子分解圖.兩個曲面上拓?fù)湎嗤难澴臃纸?誘導(dǎo)相同的褲子分解圖.我們?yōu)閳D(graph)中的邊賦予長度.計算曲面到圖的廣義調(diào)和映照,進(jìn)而得到曲面的葉狀結(jié)構(gòu).圖上的一個點對應(yīng)著源曲面上的一片葉子,也對應(yīng)著目標(biāo)曲面上的一片葉子.這給出了葉子之間的對應(yīng)關(guān)系,進(jìn)而保證了圓柱面和奇異軌線之間的對應(yīng)關(guān)系.如圖2所示,兩個曲面上對應(yīng)的圓柱面用相同的顏色進(jìn)行渲染.最后調(diào)整相應(yīng)葉子之間的映射,得到曲面整體之間的微分同胚映射.圖2 虧格為4的曲面配準(zhǔn).Birkhoff插值的極小單項基問題.本文另外研究了多元Birkhoff插值中，在任意單項序下具有相同極小單項基的一類問題，得出如下結(jié)論：若在所有消去序下，某—Birkhoff插值問題存在唯一的極小插值單項基B，則該插值問題在任意單項序下的極小插值單項基都是B.我們結(jié)合任意非插值基中的單項對應(yīng)的向量都與嚴(yán)格小于該單項的單項對應(yīng)向量線性相關(guān)這一結(jié)論，利用歸納法.證明了該定理的正確性.這一定理使得我們可以使用字典序下的極小單項基快速算法來計算其他單項序下的問題.
[Abstract]:Computational conformal geometry is an interdisciplinary subject between computer science and pure mathematics. Its purpose is to transform the concepts and theorems of modern geometry and classical geometry into computer algorithms to serve engineering practice. Quadratic differentiation is an important concept in conformal geometry. It is of great significance in the study of Teichmuller space. It is directly related to the leaf structure on a surface. Horizontal and vertical trajectories of holomorphic quadratic differentiation form the leaf structure on a surface. However, due to the obscurity of holomorphic quadratic differentiation, few people can master it. In this paper, a generalized harmonic mapping algorithm for high genus Riemannian surfaces to graphs is implemented for the first time. The leaf structure on the surface is obtained and visualized. Meanwhile, the Hopf differential induced by the generalized harmonic mapping from Riemannian surfaces to graphs is holomorphic. Quadratic Differentiation (Strebel Differentiation) allows the algorithm to further compute holomorphic quadratic differentials. By inputting loop and height parameters on different surfaces, the algorithm can control topological homotopy classes and geometric properties of leaf structures, and thus the obtained holomorphic quadratic differentials are controllable. Holomorphic quadratic differentials have been visualized extensively, which enables us to understand these concepts more intuitively and then apply them to engineering fields. In the generation of hexahedron meshes, the equivalence relations among the three concepts of dyeable quadrangular meshes, finite measurable leaf structures and Strebel differentials are expounded for the first time in this paper. Lattice generation algorithm provides a solid theoretical foundation; and based on the leaf structure on the surface and Strebel differentiation, a fully automatic hexahedron mesh generation algorithm is proposed and implemented, which makes a significant breakthrough in the so-called "holy grail" problem of hexahedron mesh generation. In addition, this paper also introduces a class of problems in Birkhoff interpolation which have the same minimal monomial basis under any monomial order, and puts forward the criteria for this kind of problems, so that we can use the minimal monomial basis under dictionary order to calculate quickly. The so-called foliation is to decompose a n n-dimensional manifold into (n-1) dimensional submanifolds. The decomposition method has a direct product structure locally. As shown in Figure 1 (a), we decompose a surface with genus 2 into a family of curves. Each curve is called a piece. Leaves. Points where three leaves intersect on a surface are called singular points. The leaf structure and Strebel differentiation on a surface with genus 2 in Fig. 1. To obtain the leaf structure on a surface, we consider a generalized harmonic mapping from a surface to a graph. As shown in the three orange curves in Fig. 1 (a), we specify a set of loops (closed curves) {gamma} on the surface. 1, gamma 2, gamma 3} and the corresponding set of loops with height parameters divide the curved surface into two pairs of pants (a surface with three edges of genus zero). This decomposition can be defined as a graph, as shown in Figure 1 (b), each pair of pants corresponds to a node, each loop corresponds to an edge, and the length of the edge corresponds to a height parameter; if the loops have three edges, each pair corresponds to a node. Gromov and Schoen [20] define the hyperbolic properties of a graph and prove the existence and uniqueness of generalized harmonic mappings from a surface to a graph. Our algorithm uses the nonlinear heat flux method to solve the generalized harmonic mappings from a surface to a graph. As shown in Figure 1, (b) the original image of the node in the middle graph is a red curve; (b) the original image of the point on the edge of the middle graph is a leaf on the surface in (a). The leaf on the surface of the leaf structure can be a loop (closed curve) or an infinite extension of the helix, all the leaves are equal. The foliate structure of a loop is called a finite measurable foliate structure. The foliate structure computed by our algorithm through generalized harmonic mapping is a finite measurable foliate structure. A holomorphic quadratic derivative is shown in Fig. 1 (c). The foliate structure induced by the horizontal orbit is exactly the given foliate structure. The holomorphic quadratic differential with finite measurable foliate structure is called Strebel differential. The equivalence relations of the following three concepts are first expounded and proved: {dyeable quadrilateral mesh} (?) {finite measurable foliate structure} (?) {Strebel differential} theorem 0.0.1 (trinity) Let S be a closed Riemannian surface with genus greater than 1. Given a dyeable quadrilateral mesh Q on a surface, there exists a Q-induced finite measurable foliate structure (FQ, muQ), the same as There exists a unique Strebel differential_. So that_-induced horizontal finite measurable foliate structures (F, mu_) are exactly equal to (FQ, mu Q). Conversely, given Strebel differential_, finite measurable foliate structures (F, mu_) can be constructed. Induced dyeable quadrilateral grid Q. This Trinity theoretical framework allows us to construct dyeable foliate structures using Strebel differential. Color quadrilateral meshes are then extended to generate hexahedral meshes of the interior entities of the surface. The algorithm flow is as follows: the input genus is g 1 closed surface, the interior space of the surface constitutes a high genus entity. (1) The user sets a set of disjoint simple loops on the input surface, and assigns a height parameter to each loop; (2) calculates the unique one. Strebel Differential; (3) Strebel Differential divides the input surface into a cylindrical surface and generates a dyeable quadrilateral mesh; (4) According to the cylindrical division of the surface, extending to the inner space of the surface, the cylindrical division of the generated body; (5) Calculate the differential homeomorphism mapping of each separated topological cylinder to the standard cylinder; (6) On the standard cylinder. The hexahedron mesh is pulled back to the original topological cylinder, that is, the hexahedron mesh is generated on each partitioned cylinder. Finally, the hexahedron mesh is obtained by splicing the hexahedron mesh. High genus surface registration. Based on the leaf structure on the surface, a new surface registration method is proposed. For two homeomorphic surfaces with the same topology, we first register their graphs and then register the corresponding leaves. Fig. 2 shows the flow of our surface registration algorithm. Given two surfaces with genus g = 4, we give a graph. The curved surface can be decomposed into 3g-3 loops of 2g-2 trousers automatically, and then the trousers decomposition graph can be induced. The same trousers decomposition graph can be induced by decomposition of the same topology on two curved surfaces. Points correspond to a leaf on the source surface and a leaf on the target surface. This gives the correspondence between the leaves and ensures the correspondence between the cylindrical surface and the singular trajectory. As shown in Figure 2, the corresponding cylindrical surface is rendered with the same color on the two surfaces. Finally, the mapping between the corresponding leaves is adjusted. The problem of minimal monomial basis of Birkhoff interpolation is also studied in this paper. In addition, a class of multivariate Birkhoff interpolation problems with the same minimal monomial basis under any monomial order are studied. The following conclusions are obtained: If all elimination orders are given, a certain-Birkhoff interpolation problem is solved. If there exists a unique minimal interpolation monomial basis B, then the minimal interpolation monomial basis of the interpolation problem under any monomial order is B. We prove the correctness of the theorem by induction, combining the conclusion that the vector corresponding to a single term in any non-interpolation basis is linearly related to the vector corresponding to a single term strictly smaller than the single term. We can use the fast algorithm of minimal monomial basis under dictionary order to solve other problems under monomial order.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O18
本文編號：2233612
[Abstract]:Computational conformal geometry is an interdisciplinary subject between computer science and pure mathematics. Its purpose is to transform the concepts and theorems of modern geometry and classical geometry into computer algorithms to serve engineering practice. Quadratic differentiation is an important concept in conformal geometry. It is of great significance in the study of Teichmuller space. It is directly related to the leaf structure on a surface. Horizontal and vertical trajectories of holomorphic quadratic differentiation form the leaf structure on a surface. However, due to the obscurity of holomorphic quadratic differentiation, few people can master it. In this paper, a generalized harmonic mapping algorithm for high genus Riemannian surfaces to graphs is implemented for the first time. The leaf structure on the surface is obtained and visualized. Meanwhile, the Hopf differential induced by the generalized harmonic mapping from Riemannian surfaces to graphs is holomorphic. Quadratic Differentiation (Strebel Differentiation) allows the algorithm to further compute holomorphic quadratic differentials. By inputting loop and height parameters on different surfaces, the algorithm can control topological homotopy classes and geometric properties of leaf structures, and thus the obtained holomorphic quadratic differentials are controllable. Holomorphic quadratic differentials have been visualized extensively, which enables us to understand these concepts more intuitively and then apply them to engineering fields. In the generation of hexahedron meshes, the equivalence relations among the three concepts of dyeable quadrangular meshes, finite measurable leaf structures and Strebel differentials are expounded for the first time in this paper. Lattice generation algorithm provides a solid theoretical foundation; and based on the leaf structure on the surface and Strebel differentiation, a fully automatic hexahedron mesh generation algorithm is proposed and implemented, which makes a significant breakthrough in the so-called "holy grail" problem of hexahedron mesh generation. In addition, this paper also introduces a class of problems in Birkhoff interpolation which have the same minimal monomial basis under any monomial order, and puts forward the criteria for this kind of problems, so that we can use the minimal monomial basis under dictionary order to calculate quickly. The so-called foliation is to decompose a n n-dimensional manifold into (n-1) dimensional submanifolds. The decomposition method has a direct product structure locally. As shown in Figure 1 (a), we decompose a surface with genus 2 into a family of curves. Each curve is called a piece. Leaves. Points where three leaves intersect on a surface are called singular points. The leaf structure and Strebel differentiation on a surface with genus 2 in Fig. 1. To obtain the leaf structure on a surface, we consider a generalized harmonic mapping from a surface to a graph. As shown in the three orange curves in Fig. 1 (a), we specify a set of loops (closed curves) {gamma} on the surface. 1, gamma 2, gamma 3} and the corresponding set of loops with height parameters divide the curved surface into two pairs of pants (a surface with three edges of genus zero). This decomposition can be defined as a graph, as shown in Figure 1 (b), each pair of pants corresponds to a node, each loop corresponds to an edge, and the length of the edge corresponds to a height parameter; if the loops have three edges, each pair corresponds to a node. Gromov and Schoen [20] define the hyperbolic properties of a graph and prove the existence and uniqueness of generalized harmonic mappings from a surface to a graph. Our algorithm uses the nonlinear heat flux method to solve the generalized harmonic mappings from a surface to a graph. As shown in Figure 1, (b) the original image of the node in the middle graph is a red curve; (b) the original image of the point on the edge of the middle graph is a leaf on the surface in (a). The leaf on the surface of the leaf structure can be a loop (closed curve) or an infinite extension of the helix, all the leaves are equal. The foliate structure of a loop is called a finite measurable foliate structure. The foliate structure computed by our algorithm through generalized harmonic mapping is a finite measurable foliate structure. A holomorphic quadratic derivative is shown in Fig. 1 (c). The foliate structure induced by the horizontal orbit is exactly the given foliate structure. The holomorphic quadratic differential with finite measurable foliate structure is called Strebel differential. The equivalence relations of the following three concepts are first expounded and proved: {dyeable quadrilateral mesh} (?) {finite measurable foliate structure} (?) {Strebel differential} theorem 0.0.1 (trinity) Let S be a closed Riemannian surface with genus greater than 1. Given a dyeable quadrilateral mesh Q on a surface, there exists a Q-induced finite measurable foliate structure (FQ, muQ), the same as There exists a unique Strebel differential_. So that_-induced horizontal finite measurable foliate structures (F, mu_) are exactly equal to (FQ, mu Q). Conversely, given Strebel differential_, finite measurable foliate structures (F, mu_) can be constructed. Induced dyeable quadrilateral grid Q. This Trinity theoretical framework allows us to construct dyeable foliate structures using Strebel differential. Color quadrilateral meshes are then extended to generate hexahedral meshes of the interior entities of the surface. The algorithm flow is as follows: the input genus is g 1 closed surface, the interior space of the surface constitutes a high genus entity. (1) The user sets a set of disjoint simple loops on the input surface, and assigns a height parameter to each loop; (2) calculates the unique one. Strebel Differential; (3) Strebel Differential divides the input surface into a cylindrical surface and generates a dyeable quadrilateral mesh; (4) According to the cylindrical division of the surface, extending to the inner space of the surface, the cylindrical division of the generated body; (5) Calculate the differential homeomorphism mapping of each separated topological cylinder to the standard cylinder; (6) On the standard cylinder. The hexahedron mesh is pulled back to the original topological cylinder, that is, the hexahedron mesh is generated on each partitioned cylinder. Finally, the hexahedron mesh is obtained by splicing the hexahedron mesh. High genus surface registration. Based on the leaf structure on the surface, a new surface registration method is proposed. For two homeomorphic surfaces with the same topology, we first register their graphs and then register the corresponding leaves. Fig. 2 shows the flow of our surface registration algorithm. Given two surfaces with genus g = 4, we give a graph. The curved surface can be decomposed into 3g-3 loops of 2g-2 trousers automatically, and then the trousers decomposition graph can be induced. The same trousers decomposition graph can be induced by decomposition of the same topology on two curved surfaces. Points correspond to a leaf on the source surface and a leaf on the target surface. This gives the correspondence between the leaves and ensures the correspondence between the cylindrical surface and the singular trajectory. As shown in Figure 2, the corresponding cylindrical surface is rendered with the same color on the two surfaces. Finally, the mapping between the corresponding leaves is adjusted. The problem of minimal monomial basis of Birkhoff interpolation is also studied in this paper. In addition, a class of multivariate Birkhoff interpolation problems with the same minimal monomial basis under any monomial order are studied. The following conclusions are obtained: If all elimination orders are given, a certain-Birkhoff interpolation problem is solved. If there exists a unique minimal interpolation monomial basis B, then the minimal interpolation monomial basis of the interpolation problem under any monomial order is B. We prove the correctness of the theorem by induction, combining the conclusion that the vector corresponding to a single term in any non-interpolation basis is linearly related to the vector corresponding to a single term strictly smaller than the single term. We can use the fast algorithm of minimal monomial basis under dictionary order to solve other problems under monomial order.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O18
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