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An Object-oriented design for hierarchical B-spline surfaces 1993

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AN OBJECT-ORIENTED DESIGN FOR HIERARCHICAL B-SPLINE SURFACES By Hailin Yan B.Sc University of Science and Technology of China A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES COMPUTER SCIENCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1993 © Hailin Yan, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature Department of  Computer Science The University of British Columbia Vancouver, Canada Date^March 15,1993 DE-6 (2/88) Abstract This thesis documents an object-oriented software system that supports free-form sur- face modelling based on the hierarchical overlay methodology of [Forsey88]. This work is motivated by the need to provide a space-efficient representation of tensor-product hierarchical spline surfaces, multiple offset method support, and general surface repre- sentation. This design uses a spatial data structure, the quadtree, to achieve this goal. The quadtree, itself a hierarchical data structure, is very suitable in this application be- cause of its ability to focus on the interesting subsets of the data. This focusing results in an efficient representation. The quadtree is also attractive because of its conceptual clarity and ease of maintenance. The package is implemented in C++ to provide: a) extensibility so that the new tools can be easily integrated into the existing package; b) reusability of code; and c) localization of code. Finally, this object-oriented hierarchical design keeps all of the original features of the hierarchical overlay method of [Forsey90]. ii Table of Contents Abstract^ ii List of Tables v List of Figures^ vi Acknowledgements viii 1 Introduction 1 1.1 The Problems and Motivations ^ 1 1.2 The Design Goals ^ 3 1.3 Overview^ 4 2 Background 6 2.1 The Hierarchical Surface Methodology ^ 6 2.2 Object-Oriented Programming and the Waterloo C++ Spline Classes 11 3 Design and Implementation 19 3.1 Analysis of Hierarchical Surface Representations ^ 19 3.2 Overview of Functional Relationships and Structures in the Design . 27 3.3 Functional Specification for Each Class ^ 35 3.3.1^Hierarchical Overlay Classes 35 3.3.2^Quadtree Classes ^ 43 3.3.3^The Support Classes 51 iii 3.4 Design Review^  58 4 Storage and Performance Analysis^ 59 4.1 Storage Benchmarks ^  59 4.1.1 Storage Analysis in the Single-overlay Quadtree Representation ^ 60 4.1.2 Storage Analysis in the Multiple-overlay Quadtree Representation 61 4.2 Performance Benchmarks^  65 4.2.1 CV Navigation  66 4.2.2 Traversals in a Hierarchical Surface ^  70 4.2.3 Evaluation  ^71 4.2.4 Refinement ^  72 4.2.5 Multi-level Editing  ^73 5 Conclusion and Future Work^ 75 5.1 Conclusion ^  75 5.2 Future Work  76 Appendices^ 79 A Mathematical Background on Tensor-product B-spline Surfaces^79 B The Definition of Quadtrees and Their Characteristics^84 Bibliography^ 87 iv List of Tables 3.1 Four kinds of quadtree data structures for hierarchical overlays ^ 24 4.2 Storage overheads for a fully refined surface in three kinds of data structures 62 4.3 Storage overheads for a sparsely refined surface in three kinds of structures 63 4.4 Execution time for getting a CV in the multiple-overlay quadtree ^ 67 4.5 Execution time for getting a CV in single-overlay quadtrees ^ 67 4.6 Execution time (in psec) for getting a CV and its neighbour in the multiple- overlay quadtree representation ^  68 4.7 Execution time (in psec) for getting a CV and its neighbour in the single- overlay quadtree representation ^  69 4.8 Execution time (in psec) for getting a CV and the CV's in its parent! child patch in the multiple-overlay quadtree representation ^ 69 v List of Figures 2.1 Surface S[k] and edited surface S [k+11  ^8 2.2 A 16-patch surface described by 49 control vertices  ^9 2.3 A 16-patch surface with refinement and overlay control vertices^ 10 2.4 Basis Class Hierarchy ^  14 2.5 The hierarchy for parametric tensor-product surfaces ^ 17 2.6 The basis spline surface object  ^18 3.7 A hierarchical surface and its multiple -overlay quadtree representation^25 3.8 A hierarchical surface and its single -overlay quadtree representation^26 3.9 A hierarchical bi-cubic B-spline surface  ^28 3.10 The initial hierarchical surface in a multiple-overlay quadtree  ^30 3.11 The hierarchical surface in a multiple -overlay quadtree after one refinement 31 3.12 The hierarchical surface in a multiple -overlay quadtree after the second refinement ^  32 3.13 The final hierarchical surface in a multiple-overlay quadtree ^ 33 3.14 The initial hierarchical surface in single -overlay quadtrees  ^34 3.15 The hierarchical surface in single -overlay quadtrees after one refinement^36 3.16 The hierarchical surface in single -overlay quadtrees after the second refine- ment ^  37 3.17 The final hierarchical surface in single-overlay quadtrees ^ 38 3.18 A hierarchical B-spline surface object in the multiple -overlay quadtree . ^ 38 vi 3.19 A hierarchical B-spline surface object in single -overlay quadtrees  ^39 3.20 HierOverlay Class Structure ^  39 3.21 QuadTree Class Hierarchy  44 3.22 The hierarchy for a control node definition ^  51 3.23 The Control Node Matrix Structure ^  57 4.24 The definition for the nth lowest level in a quadtree^  64 5.25 One non-uniform refinement case in the single -overlay quadtree structure^77 A.26 Regions for four equal patches ^  82 B.27 A region, its maximal blocks, and the corresponding quadtree. (a) Region. (b) Block decomposition of the region in (a). (c) Quadtree representation of the blocks in (b)  85 vii Acknowledgements I would like to thank Dr. David Forsey, my thesis supervisor, for his guidance and en- couragement throughout my work on this thesis. The discussion between us improved the content and presentation of this thesis significantly. I would also like to thank Dr. Jack Snoeyink for reading through the draft of this thesis, his comments and his help. Many thanks go to the graduate students of the Department of Computer Science, especially Christopher Healey, Karen Kuder, Gene Lee, Stanley Jang, Vishwa Ranjan and Tony Lau for their help and suggestions. I also wish to thank those people from the Waterloo Computer Graphics Laboratory for supplying the C++ Spline Software Package, which I made use of in my implementation. viii Chapter 1 Introduction This thesis is motivated by the need for an efficient representation of hierarchical B- spline surfaces. This chapter describes the problems inherent in constructing free-form hierarchical B-spline surfaces, briefly presents our design goals, and closes with a thesis overview. In this thesis, it is assumed that the reader has a basic knowledge of splines as used in the design of surfaces in computer graphics, at least to the level presented in Appendix A. For more information, refer to [Bartels87] or [Farin90]. 1.1 The Problems and Motivations In the traditional approach to B-spline surface modelling, a surface is defined by: an m x n matrix of control vertices, two knot vectors determining the location of each CV' in parametric space, and the appropriate basis functions (See Appendix A). The number of patches in the surface is increased from mn to (7n + 1)n or m(n + 1), using a process called knot insertion or refinement. A general, space-efficient approach for representing tensor-product spline surfaces was proposed in [Forsey90]. It offers greater editing flexi- bility than is normally found in systems using traditional representations ([Riesenfeld81], [Tiller83], [Maiorino85] and [Sederberg & Parry86]). Forsey introduced a data struc- turing technique which allows local refinement of tensor-product spline surfaces (such as 'Cy means control vertex. 1 Chapter 1. Introduction^ 2 B-splines and their rational counterparts) so that the number of patches in a given region can be increased without affecting the rest of the surface. Local refinement, coupled with a reference-plus-offset representation of the hierarchy, allows surface manipulation inde- pendent of refinement by exploring the local geometry of the surface. This "hierarchical form" is applicable to any spline with a refinement procedure and locally supported basis functions. These properties can be used in the task of creating a complex shape from a single, continuous, tensor-product spline surface, and are useful in free-form surface design in general. For details, refer to Section 2.1. Although the above hierarchical surface approach overcomes some of the shortcomings of traditional tensor-product spline surfaces, the current prototypical hierarchical B- spline editor exhibits the following cumbersome characteristics: • A flexible, but space-inefficient internal representation, in which each control vertex uses 6 pointers (4 pointers to its immediate neighbours in the CV matrix in the same overlay and 1 each to a corresponding CV in the two adjacent levels); • Only bi-cubic B-splines are available; • The process of adding new offset methods is not structured; • Multiple hierarchical surfaces cannot be manipulated within a single invocation of the editor. This thesis is the result of a re-evaluation of the software needs and requirements of a hierarchical spline editing system, motivated by: 1. Extensibility. The package must not be limited to one particular type of spline. It is essential that it is easy to develop new surface editors which can be used with a variety of spline formulations. Chapter 1. Introduction^ 3 2. Reusability of code. It is desirable to avoid repetition of code with similar func- tionality. Code should be reused rather than duplicated. 3. Localization of code. Code that deals with related matters, or structurally similar algorithms to achieve related goals, should be concentrated, as much as possible, in a single location. 4. More compact representation of the hierarchical surface itself. To provide a solid foundation, a C++ spline software system supporting interactive spline modelling was obtained from the University of Waterloo's Computer Graphics Laboratory ((Bartels91]). This system contains a reusable collection of support programs for splines, rendering, refinement, display and interaction. 1.2 The Design Goals The ultimate purpose of this research is to aid in the development of a highly interactive and intuitive modelling tool to replace the existing prototypical editor. This thesis details the implementation of C++ tools that both efficiently represent the hierarchical surface and provide a complete set of procedures to deal with the manipulation and instantiation of such a surface. A hierarchical surface is comprised of multiple levels of "spline overlay" with the parametric spacing of patches in each overlay being half of that in the preceding level. In this multi-resolution representation of a surface, level 0 has the largest parametric spacing (typically 1.0) and level n + 1 has a spacing that is half of that of level n. The criteria for choosing an efficient data structure to represent a hierarchical surface are: • fast access to an arbitrary control vertex in the hierarchy Chapter 1. Introduction^ 4 • fast access to the immediate neighbours of a given control vertex • fast access to level n ± 1 from level n • fast traversal of the hierarchy • minimized storage requirements An efficient data structure for a hierarchical surface representation is desired. This data structure must be easy to implement, minimize memory requirements and be flexi- ble enough to support future extensions. Since our design adopts the hierarchical overlay method, it is not surprising that a data structure has been chosen with a similar hierar- chical nature. That is, a hierarchical data structure, the quadtree, is used to represent a hierarchical spline surface. 1.3 Overview The rest of this thesis is organized as follows: Chapter 2 describes the hierarchical spline methodology, object-oriented programming and the Waterloo C++ spline classes. Chapter 3 gives a detailed description of our design of an object-oriented hierarchical B-spline surface. We capture the major characteristics of our design using an example, and then explore their advantages. Chapter 4 presents an assessment of the design with respect to the space- and time- efficiency of the main operations in our surface modeller. We then analyze the strengths and weaknesses of the design. Chapter 5 concludes the thesis with the results of our research and presents directions for future study. Appendix A gives B-spline and tensor-product B-spline surface definitions. Chapter 1. Introduction^ 5 Appendix B describes a hierarchical data structure, the quadtree, and its characteris- tics. Chapter 2 Background 2.1 The Hierarchical Surface Methodology Tensor-product B-splines are flexible surface representations, but they possess a de- ficiency when it comes to refinement. Refinement is usually advocated as a means of gaining finer control over a spline surface during editing. However, refinement may add more control vertices than required. One method of localizing the effect of refinement is the hierarchical surface representation presented in [Forsey90]. A degree (k, l) tensor-product B-spline surface has the form: n-1 rn-i S(U, V) = E E i.0 i=o The control vertices V j are arranged in an m x n rectangular array, called the control vertex mesh indexed by i, j. Each l/,, is a vector (Here, vector refers to the cardinal coordinates). Bi,k(u) and Bi,i(v) are the univariate B-spline basis functions. The variables k and 1 are the orders of the B-splines in the u and v parametric directions, respectively. Hierarchical surface representation is a data structuring technique imposed upon a tensor-product spline surface. This data structure relies on two properties of the B- spline basis function: local support and refinement. Local support means that the area on a surface affected by a single control vertex is bounded. The procedure of refinement produces an exact re-representation of the original surface but with a larger number of 'This section is referenced from [Forsey90]. 6 Chapter 2. Background^ 7 patches and control vertices. Local refinement exploits the local support property of the B-spline basis function to restrict the extent of refinement over the surface by increasing the number of patches in a restricted region. A level - k surface S[k](u, v) is defined over control nodes2 jk. 3 by the following equa- tion: s[k](u, v) E E fiiTB.1,2(u)BIy(v) The surface derived from the refinement of a restricted region of S[ 14 , defined by a subset of the V [k], is the level - k 1 surface s[k+i](u, v) E E .1,73k. +1, BIkk+1, 1 _ lu)-"3,1 j defined by control nodes Vi [ +1] and a different set of basis functions obtained through refinement of the restricted region. Each S [k] is called an overlay. The level - k surface is the parent of the child at level - k + 1 and the level - 0 surface is called the root-level. Only midpoint refinement is used. Therefore, each child overlay has uniform knot vectors with half of the spacing of its parent overlay. If one control node of 17,[ 1:+ 1] is moved (edited), the surface S[k+ 1] departs from its parent, S[k], in the local area influenced by that particular control node. If editing is restricted to those control nodes of V[ k+1] whose corresponding basis functions are zero at the boundary between the surfaces S[k] and S[k+ 1], discontinuities will not appear in the surface. Figure 2.1 shows the surface S [k] superimposed on the surface S[k+1] after the control vertex Qics+1] has been moved. The resulting composite surface retains the continuity properties of the underlying basis functions because the nature of the basic surface representation has not changed. 2Any control vertex in any level of a hierarchical surface is called a control node. In the remainder of this thesis, CV means control node in a hierarchical surface. Chapter 2. Background^ 8 Figure 2.1: Surface PI and edited surface S [k+1] Local refinement can be repeated on the interior of an overlay to create further overlays within overlays. The basic operation of creating an overlay consists of designating a control node fir[ksi on the surface at a particular level of refinement and executing a refinement step to re-represent the area influenced by the refined control node V this causes overlays to overlap each other in S[k+1], the overlapping overlays are made into a composite overlay by combining their respective control nodes into a single collection. Figure 2.2 shows a schematic plan of 7 x 7 matrix of control vertices (circled x's), along with the 16 bi-cubic B-spline patches and the surface that they define. This con- stitutes the minimal portion of the surface that would change due to any movement of the central control node V[s ] . Figure 2.3 illustrates the overlay that would result from local refinement. The black dots represent the control nodes in 0+ 1 ], and dashed boxes outline the smaller patches that the central control node V this central control node of 1-/[k+ 1] is moved, and all the others are held fixed, the patches given by the dashed boxes will remain an integral part of the surface and the boundary will remain continuous with the 12 surrounding patches defined by the control vertices - 47.[ ks + 1 ] . If - 1 1̀ .+ 1- 1 will influence. If only Chapter 2. Background^ 9 Figure 2.2: A 16-patch surface described by 49 control vertices in 17 [k] . The degree of continuity at the boundary depends upon the continuity of the splines themselves. For bi-cubic patches, it will be C 2 ; while for bi-quadric patches, it will be C 1 . For editing which involves the movement of several control vertices that modify a larger area of the surface, the new overlay must enclose the union of the individual single-vertex overlays. For editing that is to influence a smaller region, refinement will break each patch into several subpatches. The net effect of repeated local refinement is a surface composed of a collection of overlays at different levels of refinement. Two problems remain. Although editing the surface within an overlay cannot produce boundary anomalies (using the above definition of an overlay, the only control nodes that are allowed to move are those that will not affect the boundary), moving the surface immediately surrounding an overlay will cause tears to appear at the boundary between Chapter 2. Background^ 10 Figure 2.3: A 16-patch surface with refinement and overlay control vertices patches defined at two different levels. Also, when editing takes place at one level of surface definition, any overlays resting within the edited area are expected to remain embedded in that area. The embedded overlays will follow editing changes only if they can be dynamically influenced by changes in the shape of some ancestor overlay. This amounts to saying that their control vertices must move in accordance with the movement of the section of the surface being edited. These properties are achieved by a method called offset-referencing. The positions of the control nodes of any overlay are defined relative to a point on the parent surface S [k] , rather than relative to a single fixed frame of reference defined by some external coordinate system. In this reference-plus-offset formulation, each control node Vi [3k. +1] of any part of the surface, whether the root-level parent surface or an overlay at any level of refinement, is written in the form T,[k+1] - 6[k+1] Er, ri[+1] ij Chapter 2. Background^ 11 where illi:j+11 is the reference position, and 61:7 11 is the offset vector. Both are specified separately for each control node Ojk+11 in each overlay. The position of Viijk+11 immediately after creating a new overlay using refinement is equal to A kat il and, by definition, the value of CR:7 11 is zero. Any change to the position of a control node Vr il, is stored in the offset vector 617 11 as a relative change in position from the reference pointThe, 73 operator El) specifies how fiV:7 11 is combined with OlV 1 and is called the offset method. This operator can be defined as one of several kinds of operations, such as vector addition, tangent plane, skeletal frame and dynamic function methods in our implementation. Changes to the surface at any level of refinement close to the root level appear as global changes to the overlay surface at the current level of refinement. The current level of the surface has its control vertices modified in accordance with changes in the reference position information, and this in turn affects the reference information for finer levels of the surface. Likewise, adjustments to the offset affect the surface at the current level of refinement and all finer levels. Rendering proceeds for any point on the surface using reference and offset information at the lowest level of the tree containing that point. 2.2 Object-Oriented Programming and the Waterloo C++ Spline Classes This section briefly introduces the main concepts of object-oriented programming (en- capsulation, inheritance, polymorphism, overloading and genericity) using examples from the Waterloo C++ Spline Software Package (WaSP). The promise of object-oriented languages is that code written in an object-oriented fashion is highly modular, maintainable, flexible and reusable. These advantages arise from the virtues of encapsulation, inheritance and polymorphism manifested in various object-oriented languages in the form of objects defined by classes. A good introduction Chapter 2. Background^ 12 to the basics of object-oriented programming may be found in [VIeyer88]. For more details on the specific features of C++, consult [Ellis90, 14opman90, or Stroustrup9l]. An informal explanation of these concepts in the context of the design philosophy for the WaSP spline classes will be presented. Encapsulation is a fundamental aspect of object-oriented programming. Code and data are encapsulated when related data and algorithms are tightly bound together and can be accessed only through a well defined interface. This interface should be indepen- dent of the algorithms, data storage, and data management within the code. Encapsu- lated code provides services, but does not reveal the details of how those services are carried out. Code and data that have been encapsulated are often referred to as an abstract data type. An abstract data type definition is called a class in C++. An instance of a class is called an object. A class definition provides the template of an abstract data type, which includes data, construction, destruction, access, management, and service. The following code segment shows part of the class definition for the class NUBBasis from WaSP: // NUBBasis represents a non-uniform B-spline basis class NUBBasis: public BBasis { private: // Private members can be accessed only by members // in this class or its friend classes/functions. NumberSequence knts; // knots. public:^// Public members can be accessed by all the // instances of this class. // Constructor, set knts to [0,1,-3. NUBBasis( int dimension, int order ); // Destructor "NUBBasis(); Chapter 2. Background^ 13 // Make knts a copy of ns. void setKnots( const NumberSequencek ns ); // Add the value of a knot. void addKnot( double knt, int mult=1 ); // Delete the value of a knot. void deleteKnot( int i ); Management and service are incorporated using code that defines the public interface in the form of member functions. The functions setKnots, addKnot, and deleteKnot are examples of member functions that provide various services to users of a NUBBasis object. Member data is also encapsulated in the class definition. Data can be declared to be either public or private. Public data can be accessed and modified directly by users of a NUBBasis object. Private data can only be accessed by the member functions of the given object. The variable knts is an example of a private data variable. Users of a NUBBasis object must "request service" through a call to a member function in order to access or modify private member data. Every new instantiation of an object gets its own unique copy of the member data variables in the class definition. Modifying a given NUBBasis object's member data has no effect on the data values of other NUBBasis objects. This method of access control enforces encapsulation by making it impossible for a program using NUBBasis objects to depend on anything other than the public interface. In particular, the implementation of member functions can be changed freely, as long as the original public interface is maintained. Initialization, allocation, and deallocation of objects and their corresponding member data are performed automatically through the use of special constructor and destructor member functions. The semantics of C++ provide rules that specify when constructors and destructors are to be invoked. The function NUBBasis(int dimension, int order) is an example of a constructor for the NUBBasis class. The C++ statement Chapter 2. Background^ 14 FuncBasis BBasis I^I BezBasis NUBBasis Figure 2.4: Basis Class Hierarchy NUBBasis nub_obj(8,4); defines a new NUBBasis object called nub_obj, and automatically invokes the constructor function, which initializes the dimension and order of the object to 8 and 4 respectively. The destructor rNUBBasisO is called whenever a NUBBasis object is destroyed either explicitly by an invocation of "delete" or implicitly upon the return of a procedure in which the object appears as a local variable. The destructor "cleans up" by freeing memory allocated to the object before it disappears. Another fundamental concept in object-oriented programming is inheritance. Classes can be grouped into type hierarchies using inheritance. This feature is used to define a new class, called a derived class, in terms of an existing class, which is referred to as a base class. The derived class "inherits" the base class' member functions and data. In addition to defining its own functions and data, the new class can replace "inher- ited" information, providing a natural supporting mechanism for incremental software development. Inheritance also provides a logical way of organizing services into several layers of complexity, represented by classes in several layers of a hierarchy.' Through the use of inheritance, the C++ hierarchy for spline basis functions is organized in several layers (Figure 2.4). 'Reference from C++ Prime r[Lippman91]. Chapter 2. Background^ 15 Class derivation permits code to be extended with a minimum of effort and disruption. For example, the member function evaluate is defined in the class BBasis, and thus it does not need to be redefined as a member function in the class NUBBasis because NUBBasis is a derived class of BBasis. Inheritance and class derivation provide the ability to modify services as well as to augment them. For example, there are no member functions order() and knots() in the class FuncBasis, but they do exist in the class BBasis which is derived from FuncBasis. BBasis augments the services of FuncBasis. On the other hand, both BBasis and FuncBasis have the member function evaluate(double u), however these two functions have different definitions. BBasis modifies the evaluate function service inherited from FuncBasis. Objects that come from the same class hierarchy are permitted to act as members of a common family, whenever this is appropriate. However, C++ requires that the declaration of a member function be virtual, if it will be redefined further along in a hierarchy. This redefinition of functions is called polymorphism. Sometimes it is desirable to omit a definition for a virtual function in a base class. This can be done by putting =0 after the member's declaration in place of code. Such a function is called a pure virtual function. If a class contains a pure virtual function, it is known as a virtual class. No objects of this class type may be declared since the class contains a gap in its definition. A derived class may provide only some of the definitions for the pure virtual functions that it inherits. It might omit others which would be inappropriate for it to provide. Only when all pure virtual functions have been defined through a chain of class derivations does it become possible to instantiate objects of a particular class. Overloading is another basic concept of object-oriented programming. It provides efficiency and compactness of syntax by allowing function names to be reused within a class. The various commonly named functions can be distinguished by the type and Chapter 2. Background^ 16 number of parameters passed to them. There is no need to invent different names to apply to variants of a single concept. In C++, both functions and standard operators can be given overloaded definitions, provided that any ambiguity can be resolved in the context of usage. For example, evaluate in the class FuncBasis has overloaded definitions which are dependent upon the type of operations. Recently, genericity has become a popular term in object-oriented programming. This term was coined in [Meyer88] to describe a controlled form of macro which is useful in defining collections of similar classes or functions. Because C++ requires all variables and constants to be declared with a type, it can be inconvenient to construct classes of a generic nature. Templates provide a way of getting around this inconvenience. For example, a class can be defined as an array of integer numbers or an array of real numbers. A class or function definition can be headed by a template list of formal tokens, which are then used as type declarations in the definition. Thus a generic class for such array can be created by: template<class DATATYPE> class array { DATATYPE data; } If an instance array<int> x is declared, x will be an array of integers; if an instance array< double> x is declared, x will be an array of double precision floating point numbers. WaSP includes a rich collection of spline types in a well-organized collection of hi- erarchies. Spline bases are defined separately from classes that assemble splines from a basis and coefficients. For extensibility, the C++ spline classes were designed to provide several levels of inheritance so that new classes can be inserted at an appropriate level, and so that programs can use as general a class as possible. Chapter 2. Background^ 17 TPSurf^(a container class for parametric tensor-product surfaces) LCSurf^(a base class for linear combination tensor-product surfaces) BSurf^(a container class for basis spline tensor-product surfaces ) BezSurf - - -^NUBSurf - - -^NURBSurf Figure 2.5: The hierarchy for parametric tensor-product surfaces The goal of reusability led to the separation of the class for a basis from the class for parametric splines. The particular basis used to build spline functions, curves and surfaces is separated from the algorithms and data that a basis needs. The class FuncBasis is the general base class for any function basis. It would be appropriate to derive a class for trigonometric functions from FuncBasis. The BBasis class is the base class for any basis spline. A basis spline is defined by its degree, its knot vector, and its evaluation algorithm. The evaluation algorithm may use subsidiary information (e.g., a control vertex mesh). A version of the parametric tensor-product surface hierarchy from WaSP is shown in Figure 2.5. This set of classes is used in our implementation. The class TPSurf is a container class for parametric tensor-product surfaces, from which any other special parametric tensor-product surface can be derived. The class LCSurf is a base class for linear combination tensor-product surfaces, derived from the class TPSurf. The class BSurf is a container class for basis spline surfaces, derived from the class LCSurf. It contains a control vertex mesh, TupleMatrix, and two linear combination bases in order to describe a completed basis spline surface. A pictorial Chapter 2. Background^ 18 representation of a BSurf object is shown in Figure 2.6. The class TupleMatrix is a matrix of vectors, which is used to describe the control vertex mesh in a tensor-product B-spline surface. The TupleMatrix class is also defined in WaSP. More information about the above is found in [Bartels9l]. Chapter 3 Design and Implementation This chapter presents the design and implementation for two different quadtree data structures. Both data structures are specifically devoted to representing hierarchical B-spline surfaces. 3.1 Analysis of Hierarchical Surface Representations This section describes the functional requirements of any implementation of a hierarchical surface, and proposes appropriate data structures for its representation. The prototypical hierarchical surface editor of [Forsey88] is limited in several ways (see Chapter 1). For example, only a bi-cubic B-spline hierarchical surface can be instantiated and the process of adding new offset methods is not structured. This implementation, however, has served as the model for the functional requirements of a hierarchical sur- face. The operations can be divided into three groups: surface definition dealing with instantiation, creation and evaluation; surface navigation dealing with accessing CV's; and control node operation dealing with CV attributes and storage. 1. Surface definition: • create a new hierarchical surface with a given dimension, order, and CV mesh. • evaluate (position/derivatives) the surface at a parametric point (u,v). • traverse one level overlay with a given function. 19 Chapter 3. Design and Implementation^ 20 • traverse an entire hierarchy with a given function. • generate an overlay (i.e., refine a surface locally around a given CV). • increase/decrease the dimension (i.e., the number of rows/columns) of a sur- face 2. Surface navigation: • get the CV at the position (i,j,level) 1 . (this corresponds to the Vi ,[lievell in Section 2.1) • get the north/south/east/west neighbour of a given CV. • return neighbours of a given CV within a certain range. • get the CV at the corresponding parametric location on a different level overlay for a given CV. 3. Control nodes: • query/set attribute —offset vector qj —reference vector 8,3 —offset method/function —visibility —colour —movability (i.e., Can the position of the control vertex be altered without causing discontinuities in the surface?) • get final position (for a given offset method), i.e., 17, i !.,k). 1 (i,j,level) identifies a CV position in the hierarchy. level represents which level overlay this CV lies in; (i,j) represents the indexes in the m x n matrix of control vertices at that level. Chapter 3. Design and Implementation^ 21 A hierarchical surface is considerably more complicated than a single tensor-product surface. It has: • multiple levels, each procedurally connected through refinement, offset vector and offset function. • levels of overlays, each containing one or more continuous spline surfaces. • an m x n matrix of CV's for each fully-refined level. If the its, ^surface has an mi x ni matrix of CV's, the (i 1) 3t overlay will have a (2m i — u_order + 1) x (2n i — v_order + 1) matrix of CV's. Here, u_order and v_order are the orders in the u and v parametric directions, respectively. However, depending upon the pattern of refinement, this matrix can be — empty, — partially filled with an arbitrary clustering of CV's (the minimum size of each cluster is dependent upon the order and type of spline), — or full. Given the wide range of possible distribution of CV's in an overlay, the data structure chosen to represent a hierarchical surface is the quadtree since it has proven very effective in representing rectangular clustering of two dimensional information. Briefly, a quadtree is a tree whose nodes are either leaves or have at most four children (ordered 1, 2, 3, and 4). If a node in the quadtree is not a leaf, it is called an internal node; otherwise, it is called a leaf node. A quadtree partitions R 2 into rectangular quad- rants recursively according to the similitude of the components within the domain being represented. A general description of quadtrees is provided in Appendix B. Quadtrees have been used to represent images, point data, areas, curves, and surfaces. Chapter 3. Design and Implementation^ 22 DeFloriani et al. ([DeFloriani82]) discuss a data structure for multilevel surface represen- tation consisting of a nested, triangulated, irregular network ([Lee and Schachter80]) that is used for surface interpolation and can also serve as a mechanism for data compression. Gomez and Guzman ([Gomez79]) use a data structure related to the quadtree that is based upon a recursive subdivision of the surface into four triangles of unequal size. This data structure uses a process that stops when a triangle matches the surface to within a prespecified error. Carlson ([Carlson82]) describes a quadtree-based data structure used to model three-dimensional objects in computer graphics, which allows for the in- tersection of polygonal objects or other primitive shapes to create more complex objects. Wilhelms ([Wilhelms90]) utilizes octrees, the 3D equivalent of the quadtree, for faster isosurface generation by storing summary information to prevent useless explorations of regions of little or no interest within a volume. The various kinds of quadtree differ in: • the type of data represented, • the decomposition process, • the resolution (variable or constant). The decomposition of the domain proceeds either by partitioning into four equal parts (regular decomposition) or into unequal parts governed by the input (irregular decomposition). For a hierarchical surface, regular decomposition is adopted because of the regular subdivision of the surface by midpoint refinement. Irregular decomposition may prove useful in a hierarchical surface with non-uniform surface refinement, but this is left as a subject for future work. Recursive local refinement in hierarchical surfaces results in a surface composed of spline patches with different parametric spacing. In a hierarchical surface, each overlay may be only a subset of the full m x n array of CV's possible for that particular overlay Chapter 3. Design and Implementation^ 23 (i.e., some entries in the array will be null or undefined). A homogeneous region in the array is one where either none or all of the CV's are defined and it is used for determining the regions in a quadtree decomposition. We consider four different approaches for using quadtree data structures to represent a hierarchical surface. Each approach is characterized according to the type of data stored in each node or decomposition principle (patch definition versus CV definition), and the mechanism used to represent multiple level overlays (one quadtree per overlay versus multiple overlays per quadtree). If quadtree decomposition is governed by the distribution of patches in each hierar- chical overlay, each node in the quadtree represents one or more tensor-product B-spline patch(es). Specifically, each leaf node in the quadtree contains one s x t control vertex matrix2 defining one or more B-spline patch(es). Thus, adjacent nodes in the quadtree will have multiple references to any common CV's. If quadtree decomposition is based upon a partition of control vertices in a hierarchical surface, each node in a quadtree stores an s x t control vertex mesh defining part or all of an overlay. This method's advantage is that no control vertex is shared. However, it may be necessary to access several nodes in a quadtree to get one patch definition in an overlay. To represent the multiple overlays in a hierarchical surface, each level overlay can be represented by a separate quadtree, or the entire hierarchical surface can be stored in a single quadtree, that is, each overlay can be stored in nodes at the corresponding depth in the quadtree. A quadtree representing only one level overlay is called a single-overlay quadtree, and one representing the entire hierarchical surface is called a multiple-overlay quadtree. In a multiple-overlay quadtree, nodes at a given level represent the corresponding level 2 Here s,t >= 4 in the bi-cubic B-spline surface case. Chapter 3. Design and Implementation^ 24 omposition principlemethod of representation control vertex domain patch domain single-overlay quadtree implemented * multiple-overlay quadtree * implemented Table 3.1: Four kinds of quadtree data structures for hierarchical overlays overlay in a hierarchical surface. Every node, represented by a control node matrix and five pointer fields, is associated with a certain part of the surface. The root node is associated with the root level overlay of the hierarchical surface. In a single-overlay quadtree, each overlay is represented by an independent quadtree that describes the subset of the overlay that has so far been defined by refinement of the parent overlay. Each leaf node in a quadtree representing a given level overlay describes the subset of the 224evel /22*depth possible patches in that level. Here, depth is defined to be the depth of the leaf node in the quadtree. All internal nodes are phantom nodes. The entire hierarchical surface structure requires as many quadtrees as there are overlay levels. In total, there are four kinds of quadtree data structures (Table 3.1) for representing hierarchical overlays, but only two of them have been implemented. The remainder of this chapter describes how these two approaches are used to rep- resent the same hierarchical surface. Figure 3.7 illustrates the multiple-overlay quadtree representation and Figure 3.8 shows the single-overlay quadtree representation. In the following, the detailed operations for forming these two quadtree structures are explored. Chapter 3. Design and Implementation^ 25 Figure 3.7: A hierarchical surface and its multiple-overlay quadtree representation Chapter 3. Design and Implementation^ 26 Figure 3.8: A hierarchical surface and its single-overlay quadtree representation Chapter 3. Design and Implementation^ 27 3.2 Overview of Functional Relationships and Structures in the Design Software support for hierarchical uniform B-spline patches with local refinement re- stricted to midpoint subdivision has been implemented. The system is written in C++ and takes advantage of the object-oriented nature of the language. As described in the previous chapter, C++ allows for the creation of structures that inherit functionality from other structures. The first task in our object-oriented design was to decompose the problem domain into a set of classes. The criterion of extensibility shaped the various class hierarchies to provide several levels of inheritance. This is arranged to enable new classes to be inserted at an appro- priate level (for example, the creation of the class BSurfQuadTree derived from the class Quad Tree) such that programs can use as general a class as possible. The criterion of reusability led to the decision to separate the class for surface repre- sentations (quadtrees) from the class for hierarchical overlays. Because one might want to add new surface representation methods into our hierarchical surface modeller, this design allows for a maximal amount of code reuse. The criterion of localization shaped decisions about where in the hierarchy the classes should be included and what mechanisms in C++ should be used to achieve certain goals. The following section examines functional relationships between classes in our design and data structures, and provides one simple example to explain how they fit together. The organization of our hierarchical overlays provides a mechanism for the creation and manipulation of hierarchical surfaces using Bezier splines, uniform non-rational B- splines and uniform rational B-splines. The following example describes how to construct and represent the particular hierarchical bi-cubic B-spline surface shown in Figure 3.9. This surface has 3 overlay levels obtained in 3 refinement steps around Vi[T, VA} and VP). Multiple-overlay quadtree representation Chapter 3. Design and Implementation^ 28 Figure 3.9: A hierarchical bi-cubic B-spline surface Chapter 3. Design and Implementation^ 29 A multiple-overlay quadtree hierarchical surface (an instance of the class HierBOver- lay) begins with one overlay level representing a bi-cubic B-spline patch (surface) defined by a 4 x 4 control vertex mesh, and is represented by an instance of multiple-overlay quadtree (an instance of the class BSurfQuadTree). Each overlay at a given level of the surface hierarchy is represented by the nodes of the quadtree at the corresponding depth. Initially, for a new surface (consisting of a single patch), this multiple-overlay quadtree has just one root node storing the original patch definition, that is, one 4 x 4 control node matrix and its corresponding B-spline surface definition. Of course, each node in the B-spline surface quadtree also contains general information found in any quadtree, such as its parent pointer, its four child pointers, the index in its parent, and so on. Before refinement, the original bi-cubic B-spline patch is represented internally in the multiple-overlay quadtree, as shown in Figure 3.10. To get the hierarchical surface representation shown in Figure 3.9, the region around control node v) is refined to create the first overlay consisting of 4 spline patches. The modified data structure in the multiple-overlay quadtree is displayed in Figure 3.11 (augmenting the data appearing in Figure 3.10). In a similar way, the control node 1/1.[1] is chosen and refined to get the second level overlay which also has 4 patches. The corresponding multiple-overlay quadtree represent- ing this hierarchical surface is shown in Figure 3.12. The final step in obtaining the hierarchical surface as shown in Figure 3.9, is local re- finement around the control node V2i2j. This refinement produces the desired hierarchical surface represented by the multiple-overlay quadtree (Figure 3.13). Single-overlay quadtree representation In the single-overlay quadtree representation for the hierarchical surface (Figure 3.9), a separate quadtree is required for each overlay level. A hierarchical surface (an instance of the class HierBOverlay) using the single-overlay Chapter 3. Design and Implementation^ 30 Figure 3.10: The initial hierarchical surface in a multiple-overlay quadtree Chapter 3. Design and Implementation^ 31 Figure 3.11: The hierarchical surface in a multiple-overlay quadtree after one refinement Chapter 3. Design and Implementation^ 32 Figure 3.12: The hierarchical surface in a multiple-overlay quadtree after the second refinement quadtree data structure is constructed. This surface (containing only one level of overlay) is stored in one single-overlay quadtree list (an instance of the class BSurfQuadTreeList) to represent the entire hierarchical surface. For an initial surface consisting of one level overlay, the single-overlay quadtree list has only one element, a single-overlay quadtree that stores the original surface definition (one control node matrix and its corresponding B-spline surface definition). The single-overlay quadtree and multiple-overlay quadtree have the same quadtree class definition, except that each node in the multiple-overlay quadtree contains one u_order x v_order control node matrix, whereas a node in the single- overlay quadtree could contain up to a (2' + u_order — 1) x (2n + v _order — 1) control node matrix. The remaining objects have the same definition and similar functional relationships as in the multiple-overlay quadtree. Figure 3.14 shows the data structure for a single level single patch surface. After refining the patches around the control node VP) to create the first level overlay, Chapter 3. Design and Implementation^ 33 Figure 3.13: The final hierarchical surface in a multiple-overlay quadtree Chapter 3. Design and Implementation^ 34 Figure 3.14: The initial hierarchical surface in single-overlay quadtrees Chapter 3. Design and Implementation^ 35 a second quadtree is added to the single-overlay quadtree list (Figure 3.15). Refinement around the control node 171.[ results in the single-overlay quadtree repre- sentation shown in Figure 3.16. 2]Finally, refinement around the control node V2[,2 results in the hierarchical surface shown in Figure 3.9 and has the corresponding quadtree representation as shown in Figure 3.17 (augmenting the data appearing in Figure 3.16). In our design, an object from the class for hierarchical overlays holds an object from the class for quadtree/quadtree list as member data for surface representation. The quadtree list is represented by a doubly-linked list of quadtrees. In the quadtree data structure itself, there is a pointer pointing to the root node in the tree. Each node in the quadtree has a pointer to its parent and four pointers to its children. Each node contains an object of class CntrlNodeMatrix as member data used to define the B-spline patch(es) represented by the node. A pictorial representation of a HierBOverlay object in the multiple-overlay/single-overlay quadtree structure is shown in Figure 3.18/3.19 . 3.3 Functional Specification for Each Class This section will explore all the classes related to our hierarchical B-spline surface design and their functional specifications. 3.3.1 Hierarchical Overlay Classes A simplified version of the hierarchical structure for the class HierOverlay (Figure 3.20) shows HierOverlay, a general base class for any hierarchical overlay, HierBOverlay, a base class for any hierarchical basis spline tensor-product surface, and three leaf classes, HierNUBOverlay, HierNURBOverlay and HierBezOverlay, for hierarchical non-rational B-spline surfaces, hierarchical rational B-spline surfaces and hierarchical Bezier spline Chapter 3. Design and Implementation^ 36 Figure 3.15: The hierarchical surface in single-overlay quadtrees after one refinement Chapter 3. Design and Implementation^ 37 Figure 3.16: The hierarchical surface in single-overlay quadtrees after the second refine- ment Chapter 3. Design and Implementation^ 38 Figure 3.17: The final hierarchical surface in single-overlay quadtrees Figure 3.18: A hierarchical B-spline surface object in the multiple - overlay quadtree Chapter 3. Design and Implementation^ 39 Figure 3.19: A hierarchical B-spline surface object in single-overlay quadtrees Chapter 3. Design and Implementation^ 40 surfaces, respectively. The HierOverlay Class The HierOverlay class describes any type of hierarchical overlay. The following code fragment shows key members of this class: class HierOverlay { protected: int name; // Identify this hierarchical overlay -- a unique name. public: virtual short totalLevel()=0; // Return the total number of levels of overlays in this hierarchy. virtual void traversal( HierTravFn function )=0; // Traverse this hierarchy with a given function. inline int getName() const; inline void setName( int i ); // Get or set this hierarchical overlay's identification. }; All the hierarchical overlay structures have some common operations, such as retriev- ing the total number of overlays in the hierarchy, or traversing the hierarchy with a given function, but employ different methods to perform these operations. Therefore, we make these methods pure virtual functions in this abstract class. This ensures a definition for each method will be provided by one of HierOverlay's derived classes. The HierBOverlay Class HierBOverlay is a base class for any hierarchical basis spline surface. In this and all subse- quent classes, only a selected subset of the member functions will be shown. Destructors and some variants of other member functions designed for special cases or efficiency do not appear for the sake of brevity. Chapter 3. Design and Implementation^ 41 template <class SurfaceRepresentation> class HierBOverlay : public HierOverlay { protected: SurfaceRepresentation *tree; // Pointer to the whole hierarchical data structure. RWBoolean symmetry; // Indicate whether all the operations are handled symmetrically. public: HierBOverlay( BSurfk ); HierBOverlay( BSurf * ); HierBOverlay( const HierBOverlayk ); // Copy constructor creates an instance pointing to the same surface. short totalLevel(); // Return the total number of all the overlays in this hierarchy. void updateCV( short level, int i, int j, DoubleVec vector ); // Edit a CV at position (level,i,j) in the hierarchical surface. // Update its offset vector and all affected control nodes in finer overlays. void traversal( TravFn fn ); // Traverse the hierarchy with a given function. void leavesTraversal( TravFn fn ); // Traverse all leaves of hierarchy applying fn to each leaf node of. // the quadtree. fn returns a boolean. If the boolean is TRUE, the // traversal continues; if it is false, the traversal terminates. void nodesTraversallnLevel( short i, TravFn fn ); // Traverse all nodes in the level "i" applying fn to each node. void refinement( short level, int i, int j ); // Only refine where needed according to the specified CV. CntrlNodeList* CVInDownLevel( short level, int i, int j ); CntrlNodeList* CVInUpLevel( short level, int i, int j ); // Get a CV in the down/up level from a given CV(same u,v location). ControlNode* CVInDownPatch( short level, int i, int j ); ControlNode* CVInUpPatch( short level, int i, int j ); // Get a CV in the down/up patch from a given CV(same patch definition). ControlNode* CVNeighbour( short level, int i, int j, int flag ); // Return north/south/east/west neighbour of a given CV // according to flag = NORTH/SOUTH/EAST/WEST. Cntr1NodeMatrix* CVNeighbours( short level, int i, int j, Chapter 3. Design and Implementation^ 42 int low_i, int high_i, int low_j, int high_j ); // Return given range neighbours of a given CV. ControlNode* CVNavigate( short level, int i, int j ); // Return the control node at (level,i,j). DoubleVec getUVParameters( short level, int i, int j ); // Get the corresponding (u,v) values in hierarchy. virtual void evaluate( double u, double v,Triple * ); virtual void evaluate( double u, double v,int du,int dv,Triple * ); virtual void normal( double u, double v, Triple * ); virtual void curvature( double u, double v, double * ); virtual void curvMaxMin( double u, double v, Triple *, Triple * ); // Evaluate the surface at a given point in parametric space. inline void setSymmetry( RWBoolean value ); inline RWBoolean getSymmetry(); // Get or set the symmetry attribute in this hierarchy. inline BSuriQuadTree *refQuadTree() const; 1; The most important member function in the HierBOverlay class is refinement. Only via refinement can we get multi-level overlays in the hierarchical surface. Refinement occurs only where it is necessary to gain finer control over the B-spline surface during editing. Another important member function is updateCV, which allows us to edit the final position of a CV in the hierarchical surface. The class HierBOverlay also provides a number of basic functions including various kinds of traversal of the hierarchy, getting the control node via its position (level,i,j), finding the neighbour(s) of a given CV, and evaluating a point in the hierarchical surface corresponding to the point (u,v) in its parametric space. The initially input B-spline surface is defined by its degree/order, knots and control vertex mesh. An object tree of the template class SurfaceRepresentation provides the entire hi- erarchical surface representation. In the multiple-overlay quadtree representation, the template class SurfaceRepresentation is specified as the class BSurfQuadTree. In the Chapter 3. Design and Implementation^ 43 single-overlay quadtree representation, the template class SurfaceRepresentation is spec- ified as the class BSurfQuadTreeList. For example, if we want to declare an object x as a hierarchical surface using the multiple-overlay quadtree structure, we could specify HierBOverlay<BSurfQuadTree> x. Similarly, if we want to declare an object x as a hierarchical surface using the single-overlay quadtree structure, we need to specify HierBOverlay<BSurfQuadTreeList> x. All of these quadtree-related classes are de- fined in the Quad Tree class library. The class BSurf provides a general B-spline surface description. It is defined in WaSP. The class ControlNode and the class CntrlNodeList provide the control vertex and the list of control vertices representations, respectively. They are defined in the Support Classes library. 3.3.2 Quadtree Classes A version of the hierarchy for the class Quad Tree is shown in Figure 3.21. The figure shows SNTree, a general base class for an N-ary tree, and BSurfQuadTree, a class for the quadtree representing a hierarchical B-spline surface in the multiple-overlay quadtree representation, or representing one overlay of a hierarchical B-spline surface in the single-overlay quadtree representation. Correspondingly, each BSurfQuad Tree has its own node class definition. The node classes themselves have the same hierarchical structure. The SNTree Class The SNTree class from WaSP describes an N-ary tree data structure, each node of which has one parent node and any number of child nodes. The key members of this class are: class SNTree { protected: SNTreeNode *root; Chapter 3. Design and Implementation^ 44 Figure 3.21: QuadTree Class Hierarchy Chapter 3. Design and Implementation^ 45 // Pointer to the root of the tree. public: void insertRoot (SNTreeNodek newRoot); // Make newRoot the root of this tree. RWBoolean insert(SNTreeNodek parent,int childPos,SNTreeNodek element); /* Insert an element as the childPos child of parent. All current children of element are discarded. Insert returns TRUE if successful, FALSE if something went wrong. */ RWBoolean removeSubTree (SNTreeNodek node, SNTreeTravFn fn = NULL); /* Remove node from the tree. All of node's children are removed too. returns TRUE if successful, FALSE otherwise Each node in the subtree is removed from the tree recursively by doing a prefix traversal on the subtree and calling fn on each node. If no function is supplied a default routine is used. */ SNTreeNode *removeSubTree (SNTreeNodek parent, int childPos, SNTreeTravFn fn = NULL); /* Remove the childPos child of parent. All of the removed node's children are removed as well. Returns a pointer to the removed root of the subtree or NULL if something went wrong Each node in the subtree is removed from the tree recursively by doing a prefix traversal on the subtree and calling fn on each node. If no function is supplied a default routine is used. */ SNTreeNode *remove (SNTreeNodek parent, int childPos); /* Remove the childPos child of parent. All of the removed node's children remain in the tree and become children of node's parent. Returns a pointer to the removed node/NULL if something went wrong. */ RWBoolean remove (SNTreeNodek node); /* Remove node from the tree. All of node's children remain in the tree and become children of node's parent. Returns TRUE if successful, FALSE otherwise. */ RWBoolean moveSubTree(SNTreeNodek node,SNTreeNodek destParent,int posy; /* Move the subtree rooted at node to be the pos-th child of parent. For the operation to be successful, parent must not already have a child in that position. Returns TRUE if successful, FALSE otherwise. */ RWBoolean isEmpty (); // Return TRUE if the tree is empty. RWBoolean isNodeInTree (SNTreeNode& node); // Check to see if the node is really in the tree. int whichChild (SNTreeNodek parent, SNTreeNodek node); /* Search for the child number of node in parent. If node isn't a child of parent then return -1. */ Chapter 3. Design and Implementation^ 46 void prefixTraversal (SNTreeTravFn fn); void postfixTraversal (SNTreeTravFn fn); /* Traverse the tree in a pre or postfix order applying fn to each element of the tree. fn returns a boolean. If the boolean is TRUE, traversal continues, if it is FALSE, traversal terminates. */ 1 ; SNTree provides many basic operations for a tree data structure, such as removal of a node, insertion of a node, subtree movement, traversal in pre/postfix order, and so on. Since these operations are needed in all the tree data structures, we could reuse a large amount of code if we derived the Quad Tree and Octree classes from SNTree. The SNTreeNode class from WaSP defines a node in the N-ary tree class SNTree. It can be inserted into trees of the type SNTree. The key members of the class SNTreeNode are: class SNTreeNode { protected: int numChildren; SNTreeNode **children; // Pointers to the children of this node. SNTreeNode *parent; // Pointer to the parent of this node. public: SNTreeNode(); SNTreeNode(const SNTreeNode& n); SNTreeNode(int numchildren); // The third constructor constructs a node with the specified number of children. RWBoolean hasChildren(void) const; // Return true if the node has any children, false otherwise. inline SNTreeNode *getParent(void) const; // Return the parent of this node. virtual SNTreeNode *getChild(int i) const; // Return the ith child of this node. inline int getNumChildren(void) const; // Return the number of children this node can accommodate. Chapter 3. Design and Implementation^ 47 int growNumChildren(int newNumChildren); // Increase the number of children this node has to newNumChildren. // newNumChildren must be larger than the position of the last // non-NULL child. Returns old numChildren. ; The class SNTreeNode holds basic information needed for a node in a tree, such as the number of children, its parent pointer, and its child pointers. It also provides basic operations for the node, like getting its parent, fetching one of its children, obtaining the number of children, and judging whether or not it has children. It would be an appropriate base class for other tree node classes. The QuadTree Class The class Quad Tree is used to implement a general class of quadtree data structures which are based on the principle of recursive decomposition of space into four partitions. The key members of the class QuadTree are: class QuadTree : public SNTree { public: QuadTree(); // Constructor for the empty tree. QuadTree( const QuadNodek ); QuadTree( QuadNode *); QuadTree( const QuadTreek ); // For copying. inline QuadNode *rootNode() const; // Get the root node in the current tree. void leavesTraversal( QuadTreeTravFn fn ); /* Traverse all leaf nodes in the tree applying fn to each leaf node. fn returns a boolean. If the boolean=TRUE, the traversal continues; if it is false the traversal terminates. */ void nodesTraversallnLevel( short level, QuadTreeTravFn fn ); /* Traverse all nodes in the depth "level" of the tree applying fn to each node. fn returns a boolean. If the boolean is TRUE, the traversal continues; if FALSE, the traversal terminates. */ ; Chapter 3. Design and Implementation^ 48 Since Quad Tree inherits all the operations from the class SNTree, we do not need to provide additional operations specifically for manipulating quadtrees. In order to make it easy for reading and using, we list prefixTraversal and postfixTraversal here again with two other traversal operations, leaves Traversal and nodesTraversallnLevel. Note these methods are simply calling the corresponding functions in the base class SNTree. SNTree also provides another basic function rootNode to let us access the root node of a quadtree. The key members of the class QuadNode are: class QuadNode : public SNTreeNode { protected: int indexInParent; // Identify which child of its parent this node belongs to; // If it has no parent, indexInParent = -1. public: inline void setParent( QuadNode * ); void putNewChild( QuadNode *, int ); // Put the given node as the indexed child. Return this child. RWBoolean isLeaf() const; // Return true if the node is a leaf. inline int index() const; // Return which child of its parent this node belongs to. inline int depth() const; // Return this node's depth in the tree. }; Common function members which are often used in the quadtree are listed in the class QuadNode. This ensures they are localized and easy to read and use. The BSurfQuadTree Class This class implements the multiple-overlay quadtree approach for hierarchical overlays. In the hierarchical surface design, the class BSurfQuadTree is derived from the general Chapter 3. Design and Implementation^ 49 class QuadTree. The data stored in each node represents some of the B-spline patches. Decomposition is based on subdividing the parametric space, which is a reflected in the original surface. Resolution depends on the refinement of a surface in the multiple -overlay quadtree representation. The key members of the class BSurfQuadTree are: class BSurfQuadTree : public QuadTree { protected: ** short virtualHeight; // Indicate which level contains the actual surface node. // In the single-overlay quadtree, we don't need this member. short totalHeight; // Indicate the total height in this quad tree. ** int patchNoInU, patchNoInV; // Indicate how many patches are contained in the root level. // In the single-overlay quadtree, we don't need these two members. public: BSurfQuadTree( BSurf& ); BSurfQuadTree( BSurf * ); // Set the root node's surface to the given one, constructor BSurfQuadTree( const BSurfQuadNode&, BSurfQuadTree *tree ); // Set the given node to the root node, copy its subtree ** short actualHeight(); // Return the actual height of the tree which represents the // different refined overlays. // In the single-overlay quadtree, we don't have this function. short height(); // Return the height of the tree. BSurfQuadNode *getQuadNode( **short level, int i, int j ); // Get the quadnode which patch is identified by CV (level,i,j) // In the single-overlay quadtree, we don't have the parameter - level. ControlNode* CVNavigate(**short level,int i,int j,HierBOverlay *x); // According to this control vertices's index (i,j), return its // corresponding control node. // In the single-overlay quadtree, we don't have the parameter - level. void updateCV( **short level, int i, int j, DoubleVec ); // Set the double vector to a control node offset, given // index (i,j). We also need to modify all the affected // reference positions in the finer level(s). Chapter 3. Design and Implementation^ 50 // In the single-overlay quadtree, we don't have the parameter - level. CntrlNodeMatrix* CVNeighbours( **short level, int i, int j, int low_i, int high_i, int low_j, int high_] ); // According to this control node index (i,j) , return the // corresponding control node matrix , which index range is from // (i+low_i, j+low_j) to (i+high_i, j+high_j) // In the single-overlay quadtree, we don't have the parameter - level. CntrlNodeList* // Return a CV CntrlNodeList* // Return a CV CVInDownLevel(short level,int i,int j,HierBOverlay *x); in the down level from a given CV (same u, v location). CVInUpLevel(short level,int i,int j,HierBOverlay *x); in the up level from a given CV (same u, v location). The class BSurfQuadTree uses the member virtualHeight from the multiple-overlay quadtree structure to handle an input initial surface which is not regular (if the number of patches in the u or v parametric direction is not 2n, phantom internal quadtree nodes are needed to obtain the desired regular decomposition). In the single-overlay quadtree structure, this member is not needed since it is identified in the class BSurfQuadNode in order to merge different patches during refinement. The members virtualHeight, patch- NoInU and patchNolnV are used to identify phantom internal nodes in the multiple- overlay quadtree representation. All member functions in this class support the corresponding operations in the class HierBOverlay, no matter what kind of quadtree representation is adopted. The key members of the class BSurfQuadNode are: class BSurfQuadNode : public QuadNode { protected: CntrlNodeMatrix *CVnodes; // Store the control nodes' information related to this surface patch. public: BasicControlNode *getInternalControlNode( int i, int j ); // Get the corresponding internal control node in the quadtree. .1 Chapter 3. Design and Implementation^ 51 CntrlNodeMatrix* getCntr1NodeMatrix(); // Return the related control node matrix. TupleMatrix getCpFinalMat(); // Get the current CV matrix which patch is defined in this quad node. 1; Each node in the quadtree for representing hierarchical B-spline surfaces describes a B-spline patch(es). In other words, the class BSurfQuadNode contains the member CVnodes, an object of the class CntrlNodeMatrix. The other function members provide operations for surface representation and offset-referencing method. In the single-overlay quadtree, each node holds two members patchNoInU and patchNoInV to store the actual number of patches in the u or v parametric direction, since they might not be 2". 3.3.3 The Support Classes To take advantage of the offset-referencing mechanism and gain better offset structure in our hierarchical surface design, we define classes suitable for describing a control node in a hierarchical surface. This definition supports various kinds of offset methods, such as vector addition, tangent plane, skeletal frame and dynamic function. A version of the hierarchy for a control node class is shown in Figure 3.22. The key members of the base class BasicControlNode are: class BasicControlNode { private: Chapter 3. Design and Implementation^ 52 void setCpDerived( DoubleVec vector ); // Set the control node's derived position. protected: DoubleVec CpDerived; // Obtained by subdivision from parent. DoubleVec *CpOffset; // Set by editing of this control node. RWBoolean moveable; RWBoolean visible; long color; // Indicate the control node's attributes. public: BasicControlNode( unsigned dimension=3 ); BasicControlNode( DoubleVec vector ); // If we are given the derived vector. BasicControlNode( const BasicControlNode& ); // Constructor. inline DoubleVec getCpDerived() const; // Get the control node's derived position. inline DoubleVec getCpOffset(); void setCpOffset( DoubleVec vec ); // Get or set the control node's offset vector. BasicControlNode& operator=( const BasicControlNodet x ); // Set "x" to the current BasicControlNode. BasicControlNode copy() const; // Make a copy for the current control node. RWBoolean getBooleanFlag( CntrlNodeFlag flag ); void setBooleanFlag( CntrlNodeFlag flag, RWBoolean value ); // Note the definition : enum CntrlNodeFlag {VISIBLE, MOVEABLE}. long getAttribute( CntrlNodeAttr attr ); void setAttribute( CntrlNodeAttr attr, long value ); // Note the definition : enum CntrlNodeAttr {COLOR}. virtual DoubleVec getCpFinal(); // Use the offset method to calculate the final CV position. ; The most important member function in the class BasicControlNode is getCpFinal. This function calculates the control node coordinates from A j,^and ®. Chapter 3. Design and Implementation^ 53 The class BasicControlNode is the type for the private control node. It contains the following information: • reference position (obtained by subdivision from the parent patch), • offset vectca. pointer, • how to get a reference position, • how to get/reset an offset vector, • how to get/set an attribute (attr_name,value), • how to calculate a final position (apply offset method to offset). The class ControlNode is used as the type for the public control node in the application, and contains more information that is not in a BasicControlNode: • a pointer to the hierarchical surface that contains the node, • which level overlay of the hierarchy contains the node, • the node's position (i,j) in the current level, • an internal/private control node object. The key members of the class ControlNode are: class ControlNode { private: int ith, jth; // Represent the control node's position in one level surface overlay. short level; // Represent which level of the hierarchy the control node lies in. HierBOverlay *surface; // Identify which hierarchic surface this control node lies in. BasicControlNode *node; // The basic control node information responds to this Control Node. Chapter 3. Design and Implementation^ 54 public: ControlNode copy() const; // Copy the current control node. ControlNode operator=( const ControlNode& a ); // Should point to the same node as BasicControlNode. inline HierBOverlay *getSurface() const; // Get which hierarchical surface this control node lies in. IntVec getPosInHier(); // Get the control node's position (i,j,level) in the hierarchy. inline short whichLevel() const; // Get which level of the hierarchy this control node lies in. inline BasicControlNode *getControlNode() const; // Get the internal control node. DoubleVec getCpFinal(); // Get this control node's final position in world coordinate frame. DoubleVec getCpDerived(); // Get this control node's reference position in world coordinates. DoubleVec getCpOffset(); // Get this control node's offset vector in world coordinates. RWBoolean getBooleanFlag( CntrlNodeFlag flag ); void setBooleanFlag( CntrlNodeFlag flag, RWBoolean value ); long getAttribute( CntrlNodeAttr attr ); void setAttribute( CntrlNodeAttr attr, long value ); I; The key members of the class AdditionCntrlNode for the addition offset method (de- rived from the class BasicControlNode) are: class AdditionCntrlNode : public BasicControlNode { public: DoubleVec getCpFinal(); // Calculate a control node's final position via the addition offset method. }; The addition offset method applies vector addition to calculate the final position of the control node rather than the virtual calculation getCpFinal. Chapter 3. Design and Implementation^ 55 The key members of the class Tangent0117-11VodefOrthetangent plane offset method (derived from the class BasicControlNode) are: class TangentCntrlNode : public BasicControlNode { private: DoubleVec Uij, Vij, Nij; // Tangent plane is defined by the two partial derivative: Uij, Vij; // Nij is the normal to this tangent plane, i.e. Uij X Vij. void setTangentPlane( DoubleVec uij, DoubleVec vij ); // Set the tangent plane for this offset method. public: TangentCntrlNode( DoubleVec vector, DoubleVec du, DoubleVec dv ); TangentCntrlNode( const TangentCntrlNodek ); // Constructor TupleVec getTangentPlane(); // Get the tangent plane for this offset method. DoubleVec getCpFinal(); // Get a control node's final position in the world coordinate frame. } ; This class calculates the position of a CV using a local coordinate frame taken from the parent overlay and an offset interpreted as a position in that local frame. The key members of the class FrameOffset for the skeletal frame offset method (de- rived from the class BasicControlNode) are class FrameOffset : public BasicControlNode { private: STransformLinHandle *frame; // Store the frame. public: FrameOffset( int dim = 3 ); FrameOffset( DoubleVec vector ); // Default constructor containing an identity transformation matrix // of the supplied dimension. FrameOffset( const DoubleGenMatk basis, DoubleVec vector ); Chapter 3. Design and Implementation^ 56 FrameOffset( const DoubleGenMat& basis, const DoubleVeck origin, DoubleVec vector ); FrameOffset( const FrameOffset& v ); FrameOffset( STransformLinHandle *x, DoubleVec vector ); FrameOffset( const STransformLinHandle& x, DoubleVec vector ); // Constructor void setBasis( const DoubleGenMatk newBasis ); // Use this new matrix as the basis matrix for this frame. // Here the basis matrix is for a linear transformation, not a spline basis. void setOrigin (const DoubleVeck newOrigin); // Use this new DoubleVec as the origin for this frame. DoubleGenMat getBasis (void); DoubleVec^getOrigin(void); // Return the basis vectors and the origin. inline STransformLinHandle* getFrame(); // Get the frame for this control node. DoubleVec getCpFinal(); // Apply the linear transformation to basic control node and // get the control node's final position. ; This class provides the coordinate frame and the method for getting the control node final position in a global coordinate frame. The key members of the class GeneralOffset for the user-defined dynamic offset method (also derived from the class BasicContmlArode) are: class GenericOffset : public BasicControlNode { private: FuncFromFunction *function; public: GenericOffset( FuncFromFunction *f, int dimension=3 ); GenericOffset( FuncFromFunction *f, DoubleVec vector ); DoubleVec getCpFinal(); // Apply the generic function to the basic control node and // get the control node's final position. 1; The class FuncFromFunction is a user-defined function based on a C++ function. Chapter 3. Design and Implementation^ 57 BasicControlNode *^BasicControlNode* BasicControlNode *^BasicControlNode*  BasicControlNode * BasicControlNode * • •••• BasicControlNode *^BasicControlNode*^BasicControlNode * Figure 3.23: The Control Node Matrix Structure The class CntrlNodeMatrix represents the control node mesh of a B-spline surface. In the class CntrlNodeMatrix, each element is a control node pointer (Figure 3.23). This class is used for space- and time-efficiency, especially when updating the offset vector or reference position for a control node. All the references to this control node object share the same data, thus it is not necessary to update the different control node objects representing the same control node. Operations on a control node include obtaining its CpDerived reference vector (M), Cp0ffset offset vector (On, CpFinal coordinates (0,3k.), and its attributes (such as moveable, visible and colour). CntrlNodeMatrix also provides basic operations on the matrix, such as: • how to set one element in the control node matrix, • how to update its offset vector in an element, • how to get its subMatrix, • how to insert/delete the new rows or columns in a control node matrix, • how to get an element, row number, and column number. The above are primitive classes which are used in the hierarchical surface design. The next chapter will examine the space- and time-efficiency for basic operations provided by each of those classes. • 111 • • • • • • Chapter 3. Design and Implementation^ 58 3.4 Design Review In the above, two kinds of quadtree data structures have been proposed as representations for hierarchical B-splines surfaces. Both of these maintain the nature of the surface hierarchy. It is easy to understand this design and implement hierarchical B-splines surfaces via such quadtree data structures. The C++ classes for hierarchical splines in our design have the following characteristics: • multiple surface support • better offset structure • rational/non-rational uniform refinement (midpoint subdivision) • compatible with the spline formulation for curves, surfaces and volumes This system can be used to create a new hierarchical surface given the size, degree, and CV mesh. That is, it supports arbitrary order B-splines, and has the ability to increase/decrease the number of rows/columns in the root-level surface. Our design and implementation is still incomplete at present. For example, only midpoint subdivision is used. Whenever multi-level editing3 is executed, all affected control nodes in finer overlays are updated immediately. From the time-efficiency point of view, we could buffer edited control nodes at a certain level, then update their offsets at that level and all affected control nodes in finer overlays at the same time when necessary. This would save a lot of time when doing multi-level editing in a hierarchical surface. In the next chapter, this point will be further discussed during the examination of performance. 3 When editing takes place at one level overlay of surface definition, any finer overlays resting within the edited area will follow editing changes, which amounts to saying that their control vertices will move in accord with the movement of the edited CV. Also refer back in Section 2.1. Chapter 4 Storage and Performance Analysis One of the primary motivations for using a quadtree data structure to represent hierar- chical B-spline surfaces is to reduce the storage requirement via the use of aggregation of homogeneous quadtree nodes. This chapter assesses the space- and time-efficiency of this design. 4.1 Storage Benchmarks This section examines storage costs of two quadtree representations for a hierarchical surface and compares them in the best/worst case, respectively. The "average" storage requirement for representing a hierarchical surface will not be fully explored because of the difficulty of determining what the "average" case is. To evaluate the quadtree representation, the two approaches will be compared with each other, and with the rep- resentation used in the prototype hierarchical surface editor. Bi-cubic B-spline surfaces are taken as an example to assess storage costs for these representations. In the prototype hierarchical surface editor, it is necessary to represent six neighbours of a control node. If we define that each pointer needs Mpointer bytes of memory, this costs— a fixed 6Mpointer- i ter bytes of memory (6 pointers to north, south, east, west, up and down neighbour, respectively). Suppose that Ncv is the number of control nodes in a hierar- chical surface, in total, the storage overhead for a hierarchical surface is 6MpoinierNCV bytes. 59 Chapter 4. Storage and Performance Analysis^ 60 4.1.1 Storage Analysis in the Single-overlay Quadtree Representation The single-overlay quadtree approach uses a separate quadtree to represent each level of overlay in the hierarchical surface. The number of nodes required for each quadtree (i.e., for each level of overlay) depends upon the extent that the level has been defined. When a new level is initiated, the quadtree nodes at different levels are generated. As further refinement occurs at this level, every four sibling leaf nodes in the quadtree that contain only non-null references to CV's are coalesced into a parent node that contains a CV matrix containing all of the pointers to CV's in its children. This parent node is converted into a leaf node and its children are deleted. Such processing proceeds until no four homogeneous sibling leaf nodes exist in the quadtree. When this level is fully refined (i.e., all possible CV's are generated), the quadtree evolves to one that has only a single root node containing a CV matrix with all the CV's at that level. Because each node in the quadtree contains five pointers (4 pointers to its children and 1 pointer to its parent), it needs 5Mpoinier bytes of memory in total. Here, an internal node in the quadtree does not contain a CV matrix but simply contains 5 pointers. Each leaf node in the quadtree holds a CV matrix in addition to 5 pointers. The size of this CV matrix is initially 1 x 1 but as nodes are coalesced (described above), this matrix can grow to a size that encompasses all the CV's at this level. If there are N node quadtree nodes for the entire hierarchical surface, the storage overhead for quadtrees alone is 5Mpointer N node bytes. In the single-overlay quadtree representation, the storage requirement is minimal when a hierarchical surface is fully refined. In that case, if we define that level is the number of overlays in a hierarchical surface, the storage requirement is NcvMpointer + 5Mpoinier *level bytes because each single-overlay quadtree contains only one node. If the hierarchical surface is not fully refined, it is necessary to add some internal nodes in order to reach all Chapter 4. Storage and Performance Analysis^ 61 leaf nodes from the root node. Each internal node needs 5 M.— pointer bytes of space. This storage cost depends upon refinements of the hierarchical surface. Suppose that Nu is the number of CV's in the u parametric direction at the root level, N„ is the the number of CV's in the v parametric direction at the root level. In the worst case, when every lowest internal node has three leaf nodes , the storage overhead is 5Mposnt er "Y. 12•^,_ 12 A level +ilogmax(Nu bytes. In the single-overlay quadtree representation, if an initial input surface does not con- tain the 2" * r CV mesh required by a regular decomposition, the size of the CV matrix needs to be stored in every node of the quadtree so that during a refinement operation, it is possible to tell when a leaf node becomes a candidate for merging with its sibling nodes. Though such information can be calculated, two extra fields in each node will save a lot of execution time. Suppose that M— int eg er represents the size of an integer, this requires 2M .— integer Nn ode bytes extra space for single-overlay quadtrees representing a hierarchical surface if each field needs M— integer bytes of memory. 4.1.2 Storage Analysis in the Multiple -overlay Quadtree Representation In the multiple-overlay quadtree approach, a single quadtree stores all the CV's for the entire hierarchical surface. The storage requirement for a multiple-overlay quadtree is 5Mpo inter Nn ode bytes if there are Nuode quadtree nodes in this multiple-overlay quadtree. The number of nodes required for such a quadtree depends upon how many patches are in all levels of a hierarchical surface. When a new level initiated, a number' of new quadtree nodes are generated. Every refinement at this level generates more nodes. Each leaf node in the quadtree holds one 4 x 4 CV matrix defining one bi-cubic B-spline patch. Thus, it will require 16Mpointeri ter + 5Mpointer bytes of memory for each leaf node. To speed up processing and evaluation in the "leaf" node, the CV matrix in each internal 1 4, 8, or 16, depending upon where refinement occurs Chapter 4. Storage and Performance Analysis^ 62 level No. of control nodes No. of overhead pointers in original editor single quadtree multiple quadtree 0 16 96 23 21 1 25 150 32 84 2 49 294 56 336 3 121 726 128 1344 4 361 2166 368 5376 5 1225 7350 1232 21504 6 4489 26934 4496 86016 7 17161 102966 17168 344064 8 67801 406806 67808 1376256 9 265225 1591350 265232 5505024 10 1054729 6328374 1054736 22020096 Table 4.2: Storage overheads for a fully refined surface in three kinds of data structures node is retained (defining a patch at other than the lowest level). A storage overhead of 21 Mpointer Nnode bytes exists just for the multiple-overlay quadtree. In such a data structure, the worst case for storage overhead occurs when a multiple-overlay quadtree is full, i.e., each level in the hierarchical surface is fully refined. Each control node requires nearly 21Mpointert r bytes of extra space because each control node affects almost 16 patches of a hierarchical surface and Nnode almost equals Arcv. In the multiple-overlay quadtree representation, if an initial input surface does not contain 2n * 2n patches required by a regular decomposition, it is necessary to add some virtual patches' (implicitly) in the initial input surface to make regular decomposition possible in the patch domain. That is to say, some phantom internal nodes' are created to reach the root overlay. The memory cost for this is low because it does not add any extra fields into the nodes in the quadtree. 2Virtual patch means it is undefined in the initial surface. 3A phantom internal node is an internal node which does not contain a CV matrix. Chapter 4. Storage and Performance Analysis^ 63 level No. of CV representative case No. of overhead pointers in each level prototype editor single quadtree multiple quadtree 0 16 M 96 23 (1.44 / CV) 21 (1.31 /CV) 1 25 ME MO 150 32 (1.28 / CV) 84 (3.36 / CV) 2 25 150 133 (5.32 / CV) 84 (3.36 / CV) 11111 MO 3 25 150 140 (5.6 / CV) 84 (3.36 / CV) 4 25 150 147 (5.88 / CV) 84 (3.36 / CV) 5 25 150 154 (6.16 / CV) 84 (3.36 / CV) 6 25 150 161 (6.44 / CV) 84 (3.36 / CV) 7 25 150 168 (6.72 / CV) 84 (3.36 / CV) 8 25 150 175 (7 /CV) 84 (3.36 /CV) 9 25 150 182 (7.28 / CV) 84 (3.36 / CV) 10 25 150 189 (7.56 / CV) 84 (3.36 / CV) Table 4.3: Storage overheads for a sparsely refined surface in three kinds of structures Chapter 4. Storage and Performance Analysis^ 64 the nth lowest level (containing 2 "lx 2 '1 CV matrix)  the 3rd lowest level (containing 4 x 4 CV matrix) the 2nd lowest level (containing 2 x 2 CV matrix) the 1st lowest level (containing 1 x 1 CV matrix) Figure 4.24: The definition for the nth lowest level in a quadtree. Table 4.2 shows the storage overhead for a fully-refined hierarchical surface using the three different data structures. Table 4.3 shows the storage overhead for a sparsely refined hierarchical surface' using the three different data structures. Comparing the storage requirements of the two quadtree representations, there is no doubt that a single-overlay quadtree surface representation is the best when a hierarchical surface is fully refined or when the patches of a hierarchical surface are clustered. In this case, the multiple-overlay quadtree representation requires a lot more storage than the single-overlay quadtree representation. On the other hand, the best case for the multiple- overlay quadtree representation (when all patches in a hierarchical surface are scattered in different areas) is the worst case for the single-overlay quadtree representation, since it will contain a large number of phantom internal nodes. As shown in Table 4.3, when the patches of a hierarchical surface are extremely sparse or scattered, the single-overlay quadtree representation has the highest storage overhead. To avoid the worst case of storage overhead in the single-overlay quadtree represen- tation, the following strategy could be adopted: whenever a node lies in the nth lowest 4This is almost the worst case for single-overlay guadtrees. Chapter 4. Storage and Performance Analysis^ 65 depth of a quadtree (as shown in Figure 4.24) and satisfies the relationship: 2n-1 * 2n-1 — #Ncv < 7 * #Nnodel 5 (i.e., the cost of the quadtree nodes exceeds the number of null entries in all the children) all the subtrees are coalesced into one 2n-1 x 2n-1 CV matrix. This matrix may contain some null entries corresponding to undefined CV's, but the number of null entries will be less than the total storage required for all of the descendant nodes generated according to homogeneousness. Another way to avoid the worst case would be to utilize both the single-overlay quadtree and multiple-overlay quadtree representations, i.e., when a hier- archical surface is sparsely refined, adopt the multiple-overlay quadtree representation; otherwise, use the single-overlay quadtree representation. These methods have not been implemented, and are the subject of future work. 4.2 Performance Benchmarks This section describes and analyzes the performance of representative operations on a hierarchical surface stored in two kinds of quadtree data structures. To get an intuitive feeling of their performance, primitive operations in a hierarchical surface were tested to give their execution times when using one of the two different quadtree data structures. At the same time, we analyzed where the execution time was spent in both quadtree representations. The following primitive operations on a hierarchical surface were examined: • CV navigation. — getting the CV at a given position, — getting the north/south/east/west neighbour of a given CV, 5 Here, #Ncv is the number of generated CV's in this node's CV matrix; #N„ ode is the number of homogeneous nodes for this node's descendants. Chapter 4. Storage and Performance Analysis^ 66 —getting neighbours of a given CV within a certain range, —getting the CV at the corresponding parametric location on a different level overlay for a given CV. • Various traversals — traverse one level overlay with a given function. — traverse an entire hierarchy with a given function. • Evaluate the surface at a given point • Refinement • Multi-level editing 4.2.1 CV Navigation Getting the control node with given indexes (level, i,j) from a hierarchical surface is a basic operation, and its execution time 6 in a quadtree data structure is compared with the execution time of retrieving a control vertex with given indexes (i,j) from a TupleMatrix (a standard two dimensional array where each element is a vector). • Execution time for getting a control node from a TupleMatrix is 7 microseconds; • Execution time (in microseconds) for getting a control node from a hierarchical surface in the multiple-overlay quadtree representation is shown in Table 4.4. • Execution time? (in microseconds) for getting a control node from a hierarchical surface in the single-overlay quadtree representation is shown in Table 4.5. 6The execution time was measured on a Silicon Graphics IRIS 4D crimson workstation. 7This result comes from a sparsely refined hierarchical surface (the worst case). Chapter 4. Storage and Performance Analysis^ 67 Overlay level 1 2 3 4 5 6 7 8 9 10 Execution time (psec) 34 39 40 60 72 82 90 101 112 122 Table 4.4: Execution time for getting a CV in the multiple-overlay quadtree Overlay level 1 2 3 4 5 6 7 8 9 10 Execution time (fisec) 43 44 47 70 78 87 101 112 122 131 Table 4.5: Execution time for getting a CV in single-overlay quadtrees It will take more time to get one control node from a hierarchical surface than from a TupleMatrix even when the control node lies in the root level overlay, because there are some extra function calls and pointer operations (e.g., getting the quadtree from the hierarchy, getting the root node from the quadtree, getting the control node matrix from the root node, and getting the control node from the matrix). Table 4.4 and Table 4.5 show that when going to a finer level to search for the control node, an extra 10 microseconds might be required. Most of the extra time is used by bit-shift operations to get the proper node in a quadtree for the given indexes (level,i,j). In a multiple-overlay quadtree, execution time has a linear relationship with level of the control node. The upper bound on execution time for getting a control node is O(level) in the multiple-overlay quadtree representation. If we define that Ti„ei is execution time for getting the proper quadtree representing a certain level overlay; depth is the level of the quadtree that the node, where the CV at (level, i , j) is contained, lies in, then in the single-overlay quadtree representation, the time required to get the control node (level,i,j) is bounded by O(depth) -I-Ti„ei. It is necessary to first go to the quadtree representing the level (which costs Ti„, / execution time); and then in the corresponding quadtree, it is necessary to go to the node that Chapter 4. Storage and Performance Analysis^ 68 Level CV North CV South CV West CV East CV 1st 34 40 27 36 33 2nd 39 40 30 35 37 3rd 40 35 29 35 31 4th 60 64 28 43 51 Table 4.6: Execution time (in psec) for getting a CV and its neighbour in the multi- ple-overlay quadtree representation contains the required control node. This takes O(depth) execution time. In total, its execution time is bounded by O(depth)8 . For comparison, if a single TupleMatrix is used to represent each level of a hierarchical surface, approximately 3 psec is required to obtain a proper TupleMatrix (i.e., a proper level) and 7 psec is required to retrieve a CV from the TupleMatrix. In total, it takes about 10 psec. Getting the north/south/west/east neighbour of a control node with given indexes (level,i,j) is another primitive operation in a hierarchical surface. This takes almost the same execution time as getting the control node with given indexes (level,i,j) except that it has one extra "add" operation (i + 1 or i — 1 or j -I- 1 or j — 1). For both quadtree representations, this point is the same. Table 4.6 shows the execution time for getting a control node and its north/south/west/east neighbour in the multiple-overlay quadtree representation of a hierarchical surface. Table 4.7 shows the execution time for getting a control node and its north/south/west/east neighbour in the single-overlay quadtree representation of a hierarchical surface. Getting the control node(s) at the corresponding parametric location of a different level overlay from a given CV (level,i,j) is an elementary operation. No matter what kind of quadtree representation is adopted, this has the same upper-bound execution time as 8Tlevel is far less than 0(depth) and ignored. Chapter 4. Storage and Performance Analysis^ 69 Level CV North CV South CV West CV East CV 1st 43 45 47 47 48 2nd 44 49 48 46 50 3rd 47 51 52 51 53 4th 70 78 74 76 75 Table 4.7: Execution time (in psec) for getting a CV and its neighbour in the sin- gle-overlay quadtree representation Level position CV CV in the parent patch CV in the child patch 1st (2,1) 34 - 60 2nd (1,1) 39 70 65 3rd (2,2) 40 364 60 4th (2,2) 60 366 - Table 4.8: Execution time (in psec) for getting a CV and the CV's in its parent' child patch in the multiple-overlay quadtree representation getting the control node with given indexes (level,i,j) except that some extra calculation is required to get the corresponding control node index(es) (develn , in , j„) in the parent or child patch of a given CV. Table 4.8 shows the execution time for getting a control node and the corresponding CV(s) in the parent and child patches of a hierarchical surface using the multiple-overlay quadtree representation.' Another often-used operation is to get neighbours of a given control node (level,i,j) within the index range [(inrin)imin), (imax)imax)]. Suppose the execution time for getting one control node with given indexes (level,i,j) is Tcv. Then, its upper bound execution time will not be more than 0((imax — imin)(imar min)T CV) using either of the two quadtree data structures. 9 Table 4.8 has two testing cases which take 364 and 366 psec execution time, because the control node lists are obtained in these two testing cases instead of one control node. The control nodes in the control node list form a convex hull containing the given CV. Chapter 4. Storage and Performance Analysis^ 70 All of the above operations involve a CV navigation. 4.2.2 Traversals in a Hierarchical Surface Next, we describe traversal, another kind of elementary operation in a hierarchical sur- face. In our design, we can traverse an entire hierarchy or one level overlay of a hierarchy with a given function. In the multiple-overlay quadtree representation, execution time for traversing an entire hierarchy has a linear relationship with the number of nodes in the quadtree because every node needs to be visited and has the given function applied to it. Execution time for traversing a certain level of a hierarchy represented by a multiple-overlay quadtree includes visiting all the nodes above this level (since it is necessary to first visit nodes above the level in order to reach nodes in the desired level) and applying the given function to all the nodes at this level. Assume that Nnode is the number of nodes in a multiple-overlay quadtree, NnodeAboveLevel is the number of nodes above the specific level of a hierarchy in a multiple-overlay quadtree, NnodeInLevel is the number of nodes at the specific level of a hierarchy in a multiple-overlay quadtree, and Tfun, is execution time for applying the given function to each node. Then execution times for traversing an entire hierarchy or one level of a hierarchy are bounded by O(NnodeTfunc) and 0 (NnodeAboveLevel NnodeInLevelTfunc), respectively. In the single-overlay quadtree representation, the execution time for traversing an entire hierarchy is related to the number of nodes and the number of leaf nodes in the quadtrees because it is necessary to visit every node and apply the given function to every leaf node. Assume that Nnode is the number of nodes in all single-overlay quadtrees, &ay e, is the number of leaf nodes in all single-overlay quadtrees, NleavesInOneTree and NnodeInOneTree is the number of leaf nodes and the number of all nodes in one single quadtree representing a certain level overlay of a hierarchical surface in the single-overlay Chapter 4. Storage and Performance Analysis^ 71 quadtree representation. Then execution time for traversal in a specific level overlay of a hierarchy represented by a single-overlay quadtree data structure includes the execution time to reach the proper single-overlay quadtree describing that level overlay and to traverse this single-overlay quadtree. Execution times for traversal in an entire hierarchy or one level overlay of a hierarchy are bounded by 0(Nnode + Niea„„Tfitnc ) and O(level +NleavesInOneTree Tfunc + NnodelnOneTree), respectively. 4.2.3 Evaluation Evaluating a point in a hierarchical surface is also a primitive operation (including normal vector and curvature evaluation). Because extra storage has been used to retain the R[k] at each level, in the multiple-overlay quadtree representation, execution time for evaluating a point is equal to the execution time required to reach the leaf node that contains this point plus the execution time required to evaluate a point in a patch (which is defined by the leaf node). In the single-overlay quadtree representation, it is necessary to first get a CV matrix which defines a patch containing the point. Such an operation may involve several nodes of a quadtree, thus having a longer execution time than when using the multiple-overlay quadtree representation. In total, when using the single-overlay quadtree representation, the execution time for evaluating a point is equal to the execution time for getting the required CV matrix plus the execution time for evaluating the point in the patch defined by the CV matrix. For this kind of evaluation operation, the multiple-overlay quadtree representation can be better than the single-overlay quadtree representation. Chapter 4. Storage and Performance Analysis^ 72 4.2.4 Refinement Refinement is a crucial operation to hierarchical surfaces. The execution time for refine- ment of a hierarchical surface in two different quadtree data structures will be analyzed. First, it takes some time to judge whether a given CV (level,i,j) can be refined in the current level overlay. If not, it is necessary to go to the parent level to get the corresponding point, repeating this procedure until a proper refinable point is found. Such judgement involves deciding whether at most (2 Ru_order/21/2-1 -I- u_order mod 2)(2 r[v_order/2]/21 + v_order mod 2) patches exist at the current level. Here, u_order and v_order are the B-spline orders in the u and v direction of parametric space, respectively. After the proper refinable point in a hierarchical surface is obtained, we need to refine a surface that contains at most (2 i[u_order/2]/2] + u_order mod 2)(2 r[v_order/1]/2 -1 + v_order mod 2) patches. During this refinement, space has to be allocated for every recently created control node in the finer level if it does not already exist in the finer level overlay. In the worst case, it is necessary to allocate space for (2(2 i[u_order/ 2]/21 + u_order mod 2) + u_order — 1)(2(2 [[v_order/2]/21 + v_order mod 2) + v_order — 1) control nodes, and set pointers to those objects appropriately. All of the above steps in both quadtree surface representations have to be executed, and thus both representations have almost the same execution time for these operations. To decide whether to-be-created control nodes have existed in the lower level overlay takes some time in the multiple-overlay quadtree representation because each control node might lie in u_order * v_order patches, i.e., u_order * v_order nodes of a multiple- overlay quadtree. Average judgement involves (u_order * v_order)/4 nodes of a multiple- overlay quadtree. However, this execution time still has a constant upper bound 0(1), Chapter 4. Storage and Performance Analysis^ 73 although this constant is a little larger. In the multiple-overlay quadtree representation, one refinement operation takes between 0.5 and 1 seconds'. In the single-overlay quadtree representation, one additional step for refinement is still required, i.e., merging every cluster of four nodes into one larger node in the quadtree. In this step, each newly-created node has to check whether it can be merged with three neighbours. If it can be merged, it is necessary to change its phantom parent node to a leaf node, set the corresponding control node pointer matrix properly, and delete the original four leaf nodes in the quadtree. This procedure is repeated for the newly-created leaf node until it can not be merged. This recursive step will take considerable execution time. For a refinement operation, execution time in the multiple-overlay quadtree represen- tation is less than in the single-overlay quadtree representation. 4.2.5 Multi-level Editing Editing a hierarchical surface interactively is an important task in our design. Hierar- chical B-spline refinement and offset-referencing provide a mechanism that allows ma- nipulation of the surface regardless of any previous refinement of the surface. A change to the surface at any level of refinement closer to the root (level 0 through level k — 1) changes _Mk] altering the shape of the surface. This, in turn, changes the reference infor- mation for any recursively defined overlay SR+ 11 , percolating the effect down through the hierarchy. Similarly, changes to S[k], effected through the offsets at that level, influence the shape of the surface S[k] and all finer levels of overlay within the affected region. In our implementation, this is a non-trivial operation because each affected control node (for which d [i] or RN has been changed) lying in a non-leaf node of the quadtree will affect (u_order +1)* (v_order +1) control nodes in the next finer level. If a control node 10 Run on a Silicon Graphics IRIS 4D crimson workstation. Chapter 4. Storage and Performance Analysis^ 74 close to the root level is edited and a hierarchy contains a lot of overlays which might be affected, it will take considerable time to update an entire hierarchical surface. This operation in the two kinds of quadtree surface representations has the same execution time bound because each edited CV will affect the same number of CV's in the finer level overlays. It is better to adopt the "lazy evaluation" method to deal with this situation in order to speed up execution. That is to say, some flag could be set to indicate which control nodes need updating in the future, but those control nodes are not actually updated until they are needed. Up until now, we have roughly examined the space/performance efficiency in our object-oriented design for hierarchical B-spline surfaces in two different quadtree data structures. The next chapter will look at what can be improved in the future. Chapter 5 Conclusion and Future Work This chapter briefly summarizes the design for hierarchical surface representations and discusses what can be further studied in the future. 5.1 Conclusion Surface representations are important in the domains of computer graphics, geometric modelling, image processing, geographic information systems and robotics. The hierarchi- cal overlay surface is proposed as a general, flexible, space-efficient surface representation. It is coupled with a hierarchical data structure, the quadtree, in order to further reduce memory requirements and to keep its time-efficiency via the nature of homogeneous hi- erarchies. In this thesis, two object-oriented schemes based on quadtree data structures for the representation of hierarchical B-spline surfaces have been presented. The schemes have the following advantages: • They keep all the characteristics of hierarchical surfaces, such as the reference- plus-offset feature. This represents regions as a series of overlays with different knot spacing allowing the shape of a hierarchical surface to be modified at multiple levels of overlay. It circumvents the effect of knot insertion for the traditional tensor-product surface upon shape manipulation. • They support multiple surface representations. 75 Chapter 5. Conclusion and Future Work^ 76 • They provide different offset methods. • They have an ability to focus on the interesting subsets of the data, which results in an efficient representation and improved execution time. • It is easy to experiment with new types of hierarchical spline surfaces. The main algorithms for the hierarchical B-spline surface operations and its corre- sponding quadtree operations have been described in the previous chapters, and their complexity and memory requirements have also been discussed. Finally, we took advantage of the object-oriented nature in our implementation be- cause we aimed for flexibility, extensibility, portability and re-use in our design and coding. C++ exhibits those properties and enables larger programs to be structured in a rational way. Therefore, our modelling tools were written in the C++ programming language on SGI workstations at the GraFiC/Imager lab. 5.2 Future Work It is clear that many questions remain. Some of them, by themselves, are topics deserving special attention. Others have known solutions but implementing them can nevertheless be difficult. Thus, it is not surprising that various portions of our modelling system are incomplete. This section briefly describes some of the problems left unsolved and extensions that would be considered useful capabilities within our modelling tools. One extension is to integrate the non-uniform refinement with the current uniform midpoint subdivision. Thus, the more powerful NURBS could be taken as a mathemati- cal form for representing and designing both standard analytic shapes (conics, quadrics, surfaces of revolution, etc.) and free-form shapes precisely since NURBS are genuine generalizations of non-rational B-spline forms as well as rational and non-rational Bezier CVI area 1 CV' area 2 ‘.. Chapter 5. Conclusion and Future Work^ 77 aria 3 _.-._!_ -^1 ' 1 • / - - -11-Cif -', : .‘^-..+4 -i - -43  .•. j _.ar._ _._.1 1 -. 01 1 1 K1 1 1 Figure 5.25: One non-uniform refinement case in the single-overlay quadtree structure surfaces. In the multiple-overlay quadtree representation, this extension is easy to imple- ment to a certain degree since we only need to add two extra fields to each node in the quadtree in order to present non-uniform knots for each patch. However, a lot of diffi- culty still exists. For example, when there is a multiple-overlay quadtree representing a hierarchical surface (Figure 5.25), we cannot further refine this surface at the boundaries between areas 1, 2 and 3. In the single-overlay quadtree representation, such extension is more difficult. For example, it is hard to deal with the case shown in Figure 5.25 during the aggregation of homogeneous nodes of a quadtree, even if they lie in the finest level overlay. Suppose we want to do refinements at CV1 , CV2 and CV3 according to the area 1, 2 and 3 patterns, respectively. This design is constructed to easily extend to hierarchical volumes using octrees. This is an interesting area worth studying further. Besides these, there are still a number of interesting and attractive avenues for future work. For example, an alternative to the quadtree representation is to use a decompo- sition method that is not regular (i.e., rectangles of arbitrary size rather than squares). This alternative has the potential of requiring less space via coalescence in a quadtree. Another interesting topic is the extension of our design to Box-splines, which have Chapter 5. Conclusion and Future Work^ 78 a triangular grid in parametric space. The type of quadtree used often depends on the grid formed by the image sampling process or surface representation. Square quadtrees are appropriate for square grids and triangular quadtrees are appropriate for triangular grids (relating to Box-spline basis functions). Thus, there is a lot of interesting work which can still be done in surface representation with hierarchical methods. Appendix A Mathematical Background on Tensor-product B-spline Surfaces A parametric spline is defined analytically as a set of polynomials over a knot vector. A knot vector is a vector of real numbers, called knots, in nondecreasing order; i.e. u = [uo , u i ,^uq] such that ui._ 1 < ui, i = 1, ..., q. A spline of order k is a Ck-2 continuous polynomial of degree at most k — 1 on each interval [ui_ 1 , ui). The ith B-spline basis function of order k (degree k —1) for the knot vector [ui,^uj+k] is denoted Bi,k(Ui,^Ui+k; u) and can be expressed as the following recurrence relation: u - ui tii±k;u) = ^Bi,k-1(Ui, • • • 7 Ui+k-1; U) Ui+k-1 ui Ui+k - ^ .101:+1,k-1(Ui-1-1, • .. tti+k; u) Ui+k Ui+1 Vui < u < ui+i and Bi, i (ui, ui+1 ; u) = ^1 ui < U < 2.141 0 otherwise The above equation means that the B-spline of order k in the ith span is the weighted average of the B-splines of order k — 1 on the i th and (i 1)st spans, each weight being the ratio of the distance between the parameter and the end knot to the length of the k — 1 spans. The computation of B i,k(ui, ui+k; u) involves all the knots from ui to Ui+k but no others, since the width of support is k spans. The restrictions on the specification of a knot vector are that the same value cannot appear more than k times and that the knots must be in nondecreasing order. If the same 79 Appendix A. Mathematical Background on Tensor-product B-spline Surfaces^80 knot value ui occurs t times (i.e., ui = u i+1 = ••• =^where t < k, the continuity at this knot is reduced by t — 1. Moving through the knot vector, each basis function is nonzero over a successive set of k + 1 knots. So, k m 1 knots define M + 1 basis functions that correspond to the m + 1 control vertices. A degree (k, I) tensor-product B-spline surface has the form: n m S(u, v) = E E i=0 j=0 The control vertices 14,; are arranged in a topologically rectangular array called a control vertex mesh. Bi,k(u) and Bi,i(v) are the univariate B-spline basis functions. The rational B-spline surface representation has one more degree of freedom, weight. The degree (k,l) rational B-spline surface (in 3D) is defined as the map of a tensor-product B-spline surface in 4D: n m S(u, v) = E E i=0 j=0 EL() E 7210 Bi,k( U)Bi$ 1 (V )WijVij E i1=0 E71=O Bi,k(U)Bjj(V)Wij n m = E E Ri,k;j,l(t i=0 j=0 where 14,.; are the 3D control points and k(u)BB /(v)wijVii Ri,k ;jj(u, v) = n Er=o ET-o BrA(u)B,,i(v)wr,s The Ri,k ;i,i(u, v) functions are the bivariate rational basis functions. For some applications, subdivision of a B-spline surface is desirable. Subdivision means that a surface constructed V0,0 •^VO,n • from one net of control vertices, Vm ,0 • • • Vm,n Appendix A. Mathematical Background on Tensor-product B-spline Surfaces^81 • weighted by two sets of B-splines, Bi,k and B;,1 • and defined on two sequences of knots, {iii}gil-k and {iiii }70 “, respectively can be represented in terms of W0,0^• • •^WO,n+nl • a large net of control vertices, Wm+mi . . . Wm+m i • weighted by two refined sets of B-splines, BP] and B.111 • and defined on two finer sequences of knots, {i/ i }rm i +k and {thi}rn i +I . After refinement, the surface becomes m+m1 n+n1 S(U [11 , v [1]) = EE Bl iku[ii)Blikv[1])wij i.0 J=0 which is defined by a finer control vertex mesh In our implementation, we use the Olso algorithm ([Cohen80, Riesenfeld8l, Prautzsch84, Prautzsch85, and Lee85]) for our midpoint subdivision. The re-representation of the sur- face defined by (m+1) x (n+1) control vertices [V] as a surface of (m+m 1 +1) x (n+n 1 +1) new control vertices [W] is accomplished by two matrices composed of the a coefficients, [a teit] and [ar ight ] [HI= [aleft][v] [ar ight]T where 14—k+1,v-1+1 • • • V5—k+1,v [V] = V5,v-1+1^•^V6,1) WA— k -I- 1,A- 1-1-1 • • • WA— k -1-1,A = 141;4,A-1+1 Appendix A. Mathematical Background on Tensor-product B-spline Surfaces^82 0 I^II Iv 0  u [ale f t] Figure A.26: Regions for four equal patches as-k+1,k(1t — k 1) ...^— k +1) (15-k+1,k(11 )^as,k(p) a.),_/+1 ,/(A — / + 1) ... a..,„/(A — /^1) [aright] a-y- 1+1,1 (A ) The simplest example is derived from the uniform cubic case where each parametric range in u and v is broken at its midpoint. This converts each patch determined by the control vertices [V] into four equal patches according to the diagram (Figure A.26) [aleft] [aright] = [A1] for regions I and III [A2] for regions II and IV [A1] for regions I and II [A2] for regions III and IV Appendix A. Mathematical Background on Tensor-product B-spline Surfaces^83 The matrices [A 1] and [A 2] are given by 0 [A1] = 1 1 0 0 i 2 1 3 1 0 8 4 8 1^1 2 2 0 0 1 3 18 4 8 [A2] = 1 3 1 0 g 4 8 0 1 1 0 2 2 0 1 3 1 8 4 8 and 1^1 _ 2 2 0 0 _ A more detailed treatment of this material, and computational algorithms for B- spline, Bezier, and NURB surfaces can be found in [Bartels87]. Appendix B The Definition of Quadtrees and Their Characteristics A quadtree is a data structure, originally used for image representation, that is based on the successive subdivision of the image into rectangular quadrants. It is represented by a tree of outdegree 4 1 , where the root corresponds to the image itself and its four children correspond to the northwest, northeast, southwest and southeast quadrants respectively. The root node has no parent and leaf nodes have no children. Each node in the quadtree contains six pieces of information. The first five items are pointers to the node's parent and to its four children. The sixth piece of information describes the node itself (such as colour, etc). Figure 3b is a block decomposition of the region in Figure 3a. Figure 3c is the corresponding quadtree, which encoded image array (Figure 3b). It exploits two-dimensional coherence by recursively decomposing such an image into square areas of identical colour. This decomposition begins with a tree structure consisting of a single root node corresponding to the whole image. Unless the image is homogeneous (i.e., all the same colour), it is subdivided into four quadrants. This process is repeated for any non-homogeneous sub-region. Each of the leaf nodes in the resulting quadtree represent a region having the same colour. In our design, if the quadtree decomposition is governed by control vertices, each rn i x n i CV matrix for one level i when fully-refined in a hierarchical surface is taken as lA tree of outdegree n means that each node in the tree has at most n children. 84 Appendix B. The Definition of Quadtrees and Their Characteristics^85 Figure B.27: A region, its maximal blocks, and the corresponding quadtree. (a) Region. (b) Block decomposition of the region in (a). (c) Quadtree representation of the blocks in (b). Appendix B. The Definition of Quadtrees and Their Characteristics^86 an image array; each CV is an element in such an image array. In a hierarchical surface, each overlay may be only a subset of the full m i x ni array of CV's. A homogeneous region in the array is one where either none or all of the CV's are defined. That is to say, the CV matrix is recursively decomposed. If the quadtree decomposition is governed by patches, each si x ti patch matrix for one level i when fully-refined in a hierarchical surface is taken as an image array; each patch is an element in such an image array and defined by one CV matrix. In a hier- archical surface, each overlay may be only a subset of the full si x ti array of patches. A homogeneous region in the array is one where either none or all of the patches are defined. That is to say, patches are recursively decomposed. There are numerous ways to represent quadtrees. These range from fully pointered quadtrees in which each pointer from parent to child is stored explicitly, to pointerless quadtrees in which no pointers are stored. Mark and Lauzon [Mark85] compare the merits and space-efficiency of several quadtree data structures. Fully pointered quadtrees offer maximum flexibility because they can be traversed in any order, but more storage space is required for the pointers. 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