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附錄：外文文獻(xiàn)與中文翻譯
外文文獻(xiàn)A1
Variational Surface modeling
We present a new approach to interactive modeling of free-from surfaces. Instead of a fixed mesh of control points, the model presented to the user is that of an infinitely malleable surface, with no fixed controls. The user is free to apply control points and curves which are then available as handles for direct manipulation. The complexity of the surface’s shape may be increased by adding more control points and curves, without apparent limit. Within the constraints imposed by the controls, the shape of the surface is fully determined by one or more simple criteria, such as smoothness. Our method for solving the resulting constrained variational optimization problem rests on surface representation scheme allowing nonuniform subdivision of B-spline surfaces. Automatic subdivision is used to ensure that constraints are met, and to enforce error bounds. Efficient numerical solutions are obtained by exploiting linearities in the problem formulation and the representation.
The most basic goal for interactive free-form surface design is to make it easy for the user to control the shape of the surface. Traditionally, the pursuit of this goal has taken the form of a search for the “right” surface representation, one whose degrees of freedom suffice as controls for direct manipulation by the user. The dominant approach to surface modeling, using a control mesh to manipulate a B-spline or other tensor product surface, clearly reflects this outlook.
The control mesh approach is appealing in large measure because the surface’s response to control point displacements is intuitive: pulling or pushing a control point makes a local bump or dent whose shape is quite easily controlled by fine interactive positioning. Unfortunately, local bumps and dents are not the only features one wants to create. For example, almost anyone who has used a control mesh interface has had the frustrating experience of trying to make a conceptually simple change, but being forced in the end to precisely reposition many—even all—the control points to achieve the desired effect.
This sort of problem is bound to arise whenever the controls provided to the user are closely tied to the representation’s degrees of freedom, since no fixed set of controls can be expected to anticipate all of the users’ needs.
The work we will describe in this paper represents an effort to escape this kind of inflexibility by severing the tie between the controls and the representation. The model we envision presenting to the user is that of an infinitely malleable piecewise smooth surface, with no fixed controls or structure of its own, and with no prior limit on its complexity or ability to resolve detail. To this surface, the user may freely attach a variety of features, such as points and flexible curves, which then serve as handles for direct interactive manipulation of the surface.
Within the constrains imposed by these controls, surface behavior is governed not by the vagaries of the representation, but by one or more simply expressed criteria—that the surface should be as smooth as possible, should conform as closely as possible to a prototype shape, etc.
Our choice of this formulation is motivated by the desire to present a simple representation-independent fa?ade to the user, however, maintaining the fa?ade is anything but simple. Formally, our approach entails the specification of surface as solutions to constrained variational optimization problems, i.e. surfaces that extremize integrals subject to constraints. To realize our goal of forming and solving these problems quickly enough to achieve interactivity, yet accurately enough to provide useful surface models, we must address these key issues:
We require a surface representation that is concise, yet capable of resolving varying degrees of detail with no inherent limit to surface complexity; that is capable of representing surfaces (in practice we are usually content with continuity) and that supports efficient solution of the constrained optimization problems we wish to solve. On the other hand, since the representation is to be hidden from the user, we do not require the surface to respond in an intuitive or natural way to direct control-point manipulation.
We must be able to accurately and efficiently impose and maintain a variety of constraints on the surface, including those requiring the surface to contain a curve, or requiring two surfaces to join along a specified trim curve. Such constrains raise special problems because the constraint equation involves an integral which must be extremized. Subject to the constraints, we must be able to extremize any of a variety of surface integrals—to create fair surfaces, minimize deviation from a specified rest shape, etc.
To create surfaces that reflect the variational solution, without letting the limitations of the representation show through, the resolution of the surface representation must be automatically controlled. Ideally, subdivision should be driven by a measure of the error due to the surface approximation. As constraints are added, additional degrees of freedom must be provided to allow all constraints to be satisfied simultaneously without ill conditioning. Unlike point constraints, which can be met exactly, integral constraints require subdivision to bring their approximation error within a specified tolerance. Additional subdivision should be driven by estimates of the error with which the constrained variational minimum is approximated.
In this paper we report on our progress to date in pursuing the substantial research agenda that these requirements define. Following a discussion of background and related work, we will address each of the issues outlined above. First, the need to compactly represent arbitrarily detailed surfaces leads us to consider schemes for locally refinable representations. Although many have been developed, none meets all of our requirements. We describe a surface representation based on sums of tensor-product B-splines at varying levels of detail. Next we consider the constrained optimization problem itself. We give formulations for several quadratic objective functions, and discuss linear constraints for controlling arbitrary points and curves on the surface. We then turn to the problem of automatic surface refinement based on two kinds of approximation error: objective function error, and constraint error. Finally, we describe a preliminary implementation and present results.
The limitations of control meshes as interactive handles have been noted before. To address them, Fowler and Bartels present techniques that allow the user to directly manipulate arbitrary points on linear blend curves and surfaces: the curve/surface is constrained to interpolate the grabbed point. As the point is moved interactively, the change to control points is minimized subject to the interpolation constraint. Parametric derivatives are also presented to the user for direct manipulation, to control surface orientation and curvature at a point. Moving beyond point constraints, Celniker and Welch presented a technique for freezing the shape along an embedded curve, although the issues involved in having the surface track a moving control-curve were not addressed.
One of our key requirement is the ability to represent smooth surfaces with no a priori limit on the detail that can be resolved. Although a number of nonuniform refinement schemes have been developed, no existing one meets all of our needs. Most of these fail to provide continuity we require. In computer graphics, Bezier patches have been widely used for nonuniform refinement. In general, however, higher-order continuity between Bezier patches is not preserved if they are manipulated after subdivision, though formulates adaptive Bezier patch refinement with continuity. Triangular patch, which support topologically irregular meshes, are widely used in finite element analysis, but have been restricted to first-order continuity. Recent developments point to triangular B-spline patches as a way of constructing a surface with high-order continuity across a triangular mesh, although a computationally efficient refinement scheme for such a representation has not yet been presented.
Forsey presents a refinement scheme that uses a hierarchy of rectangular B-spline overlays to produce surfaces. Overlays can be added manually to add detail to the surface, and large- or small-scale changes to the surface shape can be made by manipulating control points at different levels. The hierarchic offset scheme may be well-suited to direct user manipulation of the control points, but it does not meet our need for a refinable substrate for constrained variational optimization. One of the fundamental advantages of conventional tensor product surface is linearity: surface points and derivatives are linear functions of the control points. Under Forsey’s formulation linearity is lost because unit normals are used to compute offsets. We depend heavily on linearity in later sections; use of the hierarchic offset representation would have a devastating impact on performance.
Variational constrained optimization plays a central role in the formulation of so-called natural splines, piecewise cubic plane curves that interpolate their control points. The proof that natural splines minimize the integral of second derivative squared subject to the interpolation constraints frequently appears as a demonstration problem in the calculus of variations.
Surface models based on variational principals have been widely used in computer vision to solve surface reconstruction problem, in which a surface is fit to stereo measurements, noisy position date, surface orientations, shading information etc. Similar formulations have been employed in computer graphics for physically based modeling of deformable surfaces. All of these are based on regular finite difference grids of fixed resolution.
Constrained optimization based on second-derivative norms has been used in fairing B-spline surfaces. Moreton minimizes variation of curvature to generate surfaces which skin networks of curves while seeking circular or straight-line cross-sections. Such schemes can give rise to very fail surfaces, but the nonlinearity of their fairness metrics prevents them from being used for interactive surface design.
Celniker proposed a physically-based model for interactive free-form surface design, in which the surface is modeled using a mesh of triangular patches, and position and normal may be controlled along patch boundaries. Interactivity is possible because the surface fairing problem is formulated as a minimization of a quadratic functional subject to linear constraints. Our approach is closely related in this respect, although we consider more general formulations for both surface functionals and shape control constraints.
We require a representation for smoothly deformable surfaces, which has no a priori limit on the detail that can be resolved. Further, we require that points on such a surface be linear functions of its shape control parameters, yielding a more tractable control problem.
Tensor-product B-splines conveniently represent piecewise polynomial surfaces as control-point weighted sums of nonlinear shape functions, and they form the basis of our representation scheme. Unfortunately, the standard tensor-product construction does not allow detail to be nonuniformly added to the surface through local refinement. We instead represent such a locally refined region as a sum of the original surface and smaller, more finely parameterized surfaces. Surface patches at various levels are evaluated and summed to compute the nonuniform surface's value. This is related to Forsey's overlay scheme for B-spline surface refinement [10], but the formulation is much simpler because there is no notion of hierarchic offsets for overlays. The nonuniform surface is a simple sum of sparse, uniform surface layers, which may overlap in arbitrary ways. Further, the resulting surface shape remains a linear function of the control-points, leading to a tractable surface control problem.
譯文B1
變動的曲面造型
我們提出了一種新的手段，能使自由行態(tài)的曲面造型相互影響。這種造型方法提供給用戶們的是一種無限的，柔順的，沒有固定控制的曲面，從而取代了那種固定的網(wǎng)狀控制點。用戶們自由地實施那些經(jīng)過處理的適合操作指令的控制點和曲線。這些復(fù)雜的曲面形狀也許會因為增加更多的控制點和曲面而變得沒有明顯的界限。在利用那些控制的約束，這些曲面的形狀會在一種或多種的簡單的標(biāo)準(zhǔn)下而變得十分確定，就比如光滑度。我們解決導(dǎo)致強迫變形的最優(yōu)化問題的方法停留在一個允許不一致的B型活動曲線規(guī)曲面細(xì)分曲面描寫上。自動細(xì)分是用來確保那些約束是滿足要求，而不去執(zhí)行錯誤的領(lǐng)域。高效的數(shù)字化表示會在公式和描述問題上的線性開發(fā)中獲得。
相互影響的自由形態(tài)曲面設(shè)計的最基本目標(biāo)是能使用戶能簡單的控制曲面的形狀。一般來說，這個目標(biāo)的追尋已經(jīng)由一種尋找“正確”的曲面描述所構(gòu)成，對于用戶來說，他們的自由程度是足以控制指揮操作的。處理曲面造型的要素，是用控制操作B型活動曲線規(guī)的嚙合或其他曲面制作的張力，清楚得地反映這種看法。
這種控制嚙合處理出現(xiàn)在大型的測量上，因為曲面控制點轉(zhuǎn)移的響應(yīng)是直觀的：拉或推一個控制點會造成那些本來能輕易地通過良好的相互影響位置的確定來控制的形狀，發(fā)生一個局部撞擊或凹陷。不幸的是，那些局部撞擊或凹陷不會只對想創(chuàng)作的人起重要作用。舉例來說，盡管幾乎任何用控制嚙合面方法的人都有試著去做一個概念化的簡單變化的失敗經(jīng)驗，但是最后他們強迫去精確地復(fù)位許多甚至是全部圖形，通過控制點去實現(xiàn)所希望的外形。
這種問題的性質(zhì)是有限制的。在沒有設(shè)置固定的控制就有希望達(dá)到用戶要求的預(yù)期之前，提升任何時候這種為用戶準(zhǔn)備的控制是與自由度描述精密結(jié)合的能力是有限制的。
這種我們將在紙上描述的工作表明了一個通過切斷控制與描述之間聯(lián)系來避開不可彎曲性的能力。我們想象著提供給用戶的造型是一塊無限的柔性片狀光滑曲面它本身沒有固定的控制或構(gòu)造，按它的復(fù)雜性和能力性決定細(xì)節(jié)方面也沒有前端限制。對這塊曲面來說，用戶也許能很自由地附加一種特征變化，就像那些為了處理知道相互影響的曲面操作而年切斷的點和彎曲曲線。
約束在這些控制的利用下，曲面形態(tài)不是被那些描述的奇特行為所左右，而是被一種或多種簡單直接的標(biāo)準(zhǔn)所決定，就不如說曲面應(yīng)該越光滑越好，與原型形狀越一致越緊密越好，如此等等。
我們這種陳述的選擇是被為了提供給用戶一種簡單的獨立描述的外觀所激發(fā)的；但是，維持這種外觀確實非常困難的。正式地說，我們的方法是使曲面的詳述承擔(dān)對約束的變化性和最優(yōu)化問題的解釋，換言之，是在極端完整的條件下進行約束的曲面。為了認(rèn)識到我們不僅要盡快形成和解決滿足相互影響的目的，也要足夠精確地準(zhǔn)備有用的曲面造型，我們必須要做到以下關(guān)鍵問題：
我們需要的一個曲面是簡明的，是有能力在對曲面復(fù)雜性沒有固有限制的情況下決定詳細(xì)程度的改變；是有能力描述的曲面（在練習(xí)中，我們經(jīng)常滿足連續(xù)）和提供我們所希望的有效率的約束最優(yōu)化問題的解決方法。從另外的方面來說，在描述被向用戶隱藏之前，我們不需要曲面負(fù)責(zé)一種直觀的或自然的方法去控制點的操作。
我們必須能夠精確的，高效的利用和維持約束在曲面上的變化，包括那些需要曲面包含一條曲線，或者需要兩個曲面用一條被詳細(xì)說明的整齊的曲線所連接。這樣的約束產(chǎn)生了特殊的問題，因為這種約束平均含有一個必須極端化的整體。依照這種約束下，我們必須能夠極端化任何一種曲面整體的變化，去產(chǎn)生清楚的曲面，使與詳細(xì)的安置形狀之間的偏差最小化，如此等等。
產(chǎn)生沒有制定描述顯示完全界限而能反映變化的解決方法的曲面，曲面描述的決定必須被自動化控制。理想地來說，細(xì)分應(yīng)該被一種應(yīng)歸于曲面近似值錯誤的測量驅(qū)動的。隨著約束的增加，額外的自由度必須被準(zhǔn)備去容許所有約束在沒有錯誤的調(diào)節(jié)下同時被滿足。不像點約束那樣需要被精確的滿足，整體的約束需要對帶給它們有詳細(xì)公差在內(nèi)的近似值誤差。額外的細(xì)分部分應(yīng)該被誤差的估計所驅(qū)動的，而這些誤差是那種約束變化最小化是被近似的誤差。
在這篇文章中，我們報道了我們在追蹤那些需要詳細(xì)說明的實質(zhì)研究事項上的進步。根據(jù)工作的背景和聯(lián)系的討論，我們將在每個產(chǎn)生的外形上標(biāo)明地址。首先，簡潔的描述能任意詳述曲面的要求使我們?nèi)ニ伎季植烤?xì)描述的方案。經(jīng)管很多方面已經(jīng)得到發(fā)展，但是不能滿足我們描述的所有要求。我們描述一個曲面是基于B型活動曲線規(guī)在不同的詳述水平上的制造張量的總計上。其次，我們考慮約束本身的最優(yōu)化問題。我們給出一些客觀的方程式函數(shù)，討論為了控制在曲面上的任意點和曲線而做的線性約束。然后我們就把問題轉(zhuǎn)到自動化曲面磨光基于兩種近似值誤差上：客觀函數(shù)誤差和約束誤差。最后，我們描繪初步的實施方法和提供結(jié)果。
控制相互作用的網(wǎng)孔局限性的操作在以前就已經(jīng)很著名了。對它們的解說，F(xiàn)owler和Bartel提出允許用戶熟練操作任意線性曲線和曲面上的點的方法：曲線/表面被強迫竄改被抓取的點。當(dāng)點被相互作用地移動，控制點的修改是使限制的修改服從最小化。參數(shù)的導(dǎo)數(shù)也為直接的處理被呈現(xiàn)給使用者,用點去控制曲面的方位和曲率。通過超越點的約束，Celnike和Welch提出了一種凍結(jié)內(nèi)含式曲線形狀的技術(shù)。盡管有關(guān)曲面沿著一條沿著控制曲線移動的論點還沒有被提出來。
我們的一項主要需求是在能被決定的細(xì)節(jié)上沒用先驗的限制表現(xiàn)平滑的表面能力。雖然一些不均勻細(xì)化方案已經(jīng)被發(fā)展了，但是還沒有一種現(xiàn)有的符合我們的全部需要的方案。它們中的大多數(shù)不能提供我們所需要的連續(xù)性。在計算機圖形方面，貝塞爾曲線片已經(jīng)廣泛地用來做不均勻細(xì)化。但是一般來說，如果在細(xì)分之后被操作，貝塞爾曲線碎片之間的高次序連續(xù)性是不被保護的，雖然用連續(xù)性闡明貝塞爾曲線碎片的細(xì)化。雖然支持拓?fù)錈o規(guī)律網(wǎng)孔的三角形片被廣泛地應(yīng)用于有限元分析，但是已經(jīng)被限制在第一次序的連續(xù)性上。最經(jīng)發(fā)展的指向三角形B型活動曲線規(guī)碎片作為一種構(gòu)造一個橫跨三角形網(wǎng)孔的高次序連續(xù)性曲面的方法，盡管對于一個如此表現(xiàn)還沒有出現(xiàn)一個有效率計算細(xì)分的方案。
Forsey提出一種用一種矩形層的B型活動曲線規(guī)覆蓋來創(chuàng)建曲面精確方案。覆蓋能手動對曲面增加細(xì)節(jié)，已經(jīng)大規(guī)模和小規(guī)模改變曲面形狀能通過操作不同高度控制點來實現(xiàn)。雖然分層抵消也許能適當(dāng)指導(dǎo)使用者控制點的操作，但是這并不能滿足我們對于一種用于約束變化最優(yōu)化的精制基礎(chǔ)的要求。一種常規(guī)張量積曲面的基本優(yōu)勢是線性的：曲面的點和派生物是控制點的一次函數(shù)。因為單位法線用于計算抵消，所以在Forsey的線性公式形成下被丟失。我們在較后的區(qū)域倚重線性；主要抵消表示法的使用有可能對性能有破壞性的影響。
約束變化最優(yōu)化對所謂的自然樣條的闡述起著非常重要的作用，把篡改控制點的立方的平面曲線分段。自然樣條把第二派生的正方形的整體最小化的試驗使之遭遇頻繁地添加約束作為一個變化的微積分示范問題。
首要是變化為基礎(chǔ)的曲面造型已經(jīng)廣泛的用于計算機現(xiàn)象去解決曲面重建問題，在一個曲面上適合立體地測量，日期的嘈雜定位，表面定方向，投影等等。類似的闡述已經(jīng)被物理地基于可變表面的造型的計算機圖像所使用。這些全部以有規(guī)則的，有限的，有規(guī)則的確定解釋的格子為基礎(chǔ)。
基于第二派生物規(guī)則的約束最優(yōu)化已經(jīng)被用于平的B行活動曲線規(guī)的曲面。當(dāng)在尋找彎的或直的橫截面線時，Moreton把發(fā)生在表面是曲線網(wǎng)格的曲面上的曲率變化最小化。雖然這樣的方法會造成非常失敗的曲面，但是他們的平順性的非線性阻止它們被用于交互式曲面設(shè)計。
Celniker提議一種為了交互式自由形態(tài)的曲面設(shè)計，以身體為基礎(chǔ)的造型，那種表面用一種三角形片的網(wǎng)孔，而且位置和常態(tài)可能沿著片邊界被控制。相互影響是可能的，因為曲面平整問題被闡述成一個二次函數(shù)最小化服從線性約束。我們的方法是近似地講述這方面的相互關(guān)系。
我們需要一種平滑可變曲面的表示方法，使之在可以決定的細(xì)節(jié)上沒有先前的限制。更進一步，我們需要這樣一個曲面上的點是形狀控制參數(shù)的線性函數(shù)，屈從一個更容易的控制問題。
B型活動曲線規(guī)的張量積方便地表示分段多項式曲面作為控制點集合非線性形狀功能的總數(shù)，而且他們形成我們表示方案的基礎(chǔ)。不幸的是，標(biāo)準(zhǔn)的張緊積結(jié)構(gòu)不允許細(xì)節(jié)通過局部改進被不均勻地添加添加在曲面上。我們替換如局部改進的區(qū)域作為曲面的總和，更加細(xì)微化地參數(shù)化曲面。不同水平的表面片被評價和總計去計算曲面值的不均勻。雖然這是涉及到對B型活動曲線規(guī)的Forsey的覆蓋方案，但是因為為覆蓋沒有分層抵銷的觀念，形成非常簡單。不均勻表面是簡單的稀疏的，統(tǒng)一的分層堆積總和，可能以任意方式重疊。更近一步來說，產(chǎn)生的曲面形狀保持著一個控制點的一次函數(shù)，引導(dǎo)一個易于控制的曲面控制問題。