~ Computer Graphics, Volume 23, Number 3, July 1989
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~ Computer Graphics, Volume 23, Number 3, July 1989 Combinatorial Analysis of Ramified Patterns and Computer Imagery of Trees Xavier Gdrard Viennot 1, Georges Eyrolles 1, Nicolas Janey 2, Didier Arquds 2 1. LABRI, D6partement ffInforrnatique, Universit6 de Bordeaux I, FRANCE (U.A. CNRS 0726) 2. LIB, D~partement d'Informatique, Universit~ de Franche-Comt~, FRANCE (U.A CNRS 0822) Abstract Hodges, Naylor [7], Oppenheimer [31], Prusinkiewicz [35], Herein is presented a new procedural method for generating Beyer et Friedel [3], Prusinkiewicz, Lindertmayer et Hanan images of trees. Many other algorithms have already been [36], De Reffye, Edelin, Franqon, Jaeger, Puech [8]. proposed in the last few years focusing on particle systems, In these works, tree generation was mainly made at two fractals, graftals and L-systems or realistic botanical models. levels. The first one is the generation of the topological Usually the final visual aspect of the tree depends on the (combinatorial) tree underlying the real tree. In general, this development process leading to this form. Our approach differs topological tree is binary or ternary. The second level is the from all the previous ones, We begin by defining a certain generation of the geometrical tree. Each method applies a more "measure" of the form of a tree or a branching pattern. This is or less sophisticated geometrical model to the topological done by introducing the new concept of ramification matrix of tree. The minimum geometry consists in a 2D-drawing of trees, a tree. Then we give an algorithm for generating a random tree with width and length choices for each branch and angle having as ramification matrix a given arbitrary stochastic choices for each branching node. A more sophisticated triangular matrix. The geometry of the tree is defined from the geometry consists in a 3D visualization to which are added combinatorial parameters implied in the analysis of the forms vegetal elements (leaves, flowers), bark texture on branches of trees. We obtain a method with powerful control of the final and outside constraints such as light, wind or gravity. form, simple enough to produce quick designs of trees without In all previous mentioned works, these two topological and loosing in the variety and rendering of the images. We also geometrical levels are more or less separated during the gene- introduce a new rapid drawing of the leaves. The underlying ration. These methods can be roughly classified as follows : combinatorics constitute a refinment of some work introduced - Fixed topology methods, like the ones of Kawaguchi [16], in hydrogeology in the morphological study of river net- Aono & Kunii [1] which only generate perfect trees. Due to the works. The concept of ramification matrix has been used very lack of variation of topology, geometry is of great importance recently in physics in the study of fractal ramified patterns, in order to create a large diversity of forms. CR Categories and Subject Descriptors : 1.3.5 - Generation methods by development models which include [Computer Graphics]: Computational Geometry and Object a real growth strategy of trees. For example : generation by Modeling. L3.7.[Computer Graphics]: Three-Dimensional fractals of Mandelbrot, Oppenheimer [31], or by stochastic Graphics and Realisms. J.3 [Life and Medical Sciences]: and recursive growth of branching nodes, Niklas [30]; Biology. J.5 [Arts and Humanities]: Arts, fine and performing. generation by rewriting rules using L-systems theory General terms: Trees, plants, algorithms, realistic image developed by Lindenmayer, Smith [39], Prusinkiewicz [35], synthesis, figurative image synthesis Prusinkiewicz, Lindenmayer, Hanan [36]; generation by a Additional keywords and phrases: branching botanical development model De Reffye, Edelin, Fran~on, patterns in physics, stochastic modeling, analysis of form, Jaeger, Pueeh [8], fractals, self-similarity, combinatorics, ramification matrix Papers [16],[I],[37],[4], [31] essentially focus on geometry, 1. Introduction while papers [3], [8], [35], [36], [39] are mainly interested in Computer Image Synthesis generation of trees and plants has the development of topology. been the subject of many papers in the past few years. Let us These various methods use geometry in order to obtain the mention for example : Marshall, Wilson, Carlson [24], final shape (or form) of the tree. The aim is to realize realistic Kawaguchi [16], Reeves, Blau [37], Gardner [11], Aono, Kunii drawings of plants and trees. All above mentioned methods [1], Smith [39], Bloomenthal [4], Niklas [30], Demko, focus mostly on the parameters involved in the development 1. LABRI, D6partement d'Informatique, Universit6 de Bordeaux I, process, rather than on the direct control of the final shape. 33405 Talence, FRANCE - T61.: (33) 56 84 60 85 The fact that the shape is implicitly contained in its history, is 2. LIB, d6partement d'Informatique, Universit6 de Franche-comt6, a well known concept : D'Arcy Thompson [6], Hall~, Oldeman 25030 Besan~on, FRANCE - T61.: (33) 81 66 64 63 and Tomlinson [13]. In this paper a new approach, different from previous ones, is proposed, separating shape from deve- lopment. In order to obtain better control of the final shape, Permission to copy without fee all or part of this material is granted we first stress on the necessity of defining numerical para- provided that the copies are not made or distributed for direct meters which allow a certain measure of shape of the tree, i.e. commercial advantage, the ACM copyright notice and the title of the allowing the numerical evaluation of features as thorny, dense, publication and its date appear, and notice is given that copying is by slender, well built or bushy. permission of the Association for Computing Machinery. To copy Herein, the shape of tree is measured by introducing the new otherwise, or to republish, requires a fee and/or specific permission. notion of ramification matrix. This matrix is a triangular ©1989 ACM-0-89791-312-4/89/007/0031 $00.75 31 ,'¢.,,~~SIGGRAPH'89, Boston, 31 July-4 August, 1989 stochastic one, associated with each tree or ramified structure. node the distinction between the so-called "straight edge" and Depending only on the underlying topological tree, this "lateral edge", as in [36], is arbitrary. Herein neither deve- matrix is defined from the combinatorial notions of branch lopment history nor arbitrary convention are required. order and branching node biorder. These notions constitute a 2,3. Horton-Strahler analysis in hydrogeology refinement of some concepts introduced by the hydrogeolo- and other sciences. gists Horton [15] and Strahler [41] in the morphological study A river network is supposed to be without islands and triple of river networks. Our study shows that many visual characte- (or multiple) junction points. Thus, the underlying topologi- ristics connected with the shape of the tree, are reflected in the cal tree is a binary tree. Many studies have been carried out in associated matrix. Very recently, this notion has been used in hydrogeology in order to examine river network morphology. physical study of' fraetal ramified structures, Vannimenus, In particular, the concept of bifurcation ratio 1~, of order k has Viennot [43], [44]. been introduced. If b k is the number of order k segments, then Our method consists in the choice of a ramification matrix 13k is the ratio bk.1/bk. For example, the binary tree in Figure 1 and then random generation of a combinatorial tree whose has bifurcation ratios 1~z=8/2=4, B3=2/1=2. These ratios give ramification matrix is the one chosen (or very similar). At last certain information about the "shape" of the binary tree. For an elementary 2D geometry is defined. The main idea is to con- example the trees in Figures 2, 3 are two extreme cases. The trol the width and length of branches, the angles of branching perfect tree of height 3 (each terminal node has the same nodes by linear, polynomial or exponential laws in terms of height, Fig. 2) has all segments reduced to the edges: each the order mad biorder parameters. The advantage is in a better bifurcation ratio is equal to 2 (this is the minimum possible control of the shape. The method is simple and fast while value). For the very thin tree (Fig. 3) there is one segment of keeping a great variety of possible forms. Although this theo- order 2, each terminal edge is a segment of order 1, the retical model moves away from the botanical ones, a realistic bifurcation ratio of order 2 is equal to the number of terminal rendering can be obtained. For this purpose, we introduce a fast nodes and can be arbitrarily large. leaf drawing algorithm. Our method also seems well adapted for obtaining figurative trees as could be painted by artists. 2. Horton-Strahler analysis for binary trees 2.1. Binary trees and botanical trees In order to avoid confusion between botanical tree, Figure 2. Perfect tree Figure 3. "Very thin" tree. topological tree or geometrical tree, we recall a few concepts Geologists have observed that bifurcation ratios are usually from theoretical computer science, see for example Knuth [18]. between 3 and 5. We define a random binary tree as a binary A binary tree is defined by a set of vertices or nodes, joined by tree chosen at random with uniform probability among the C n edges. There are two kinds of vertices : the internal nodes (or = (2n!)/n!/(n+l)! binary trees with n branching nodes. In branching nodes) and the external nodes (or terminal nodes). particular, it is known that all the bifurcation ratios of a Each internal node has two sons : a left and a right son, the random binary tree approach 4 as the number of nodes becomes external nodes having no sons. An internal node is called the larger, Meier & Moon [25].