ABSTRACT
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 PREVIOUS WORKS
 3 THE MODELING METHOD
 4 IMPLEMENTATION AND RESULTS
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 Biographies
This study proposes a novel procedural modeling method using convolution sums of divisor functions to model a variety of natural trees in a virtual ecosystem efficiently. The basic structure of the modeling method defines the growth grammar, including the branch propagation, a growth pattern of branches and leaves, and a process of growth deformation for various tree generation. Here, the proposed procedural method for trees is to utilize convolution sums of divisor functions as a novel approach. The structure of convolution sums has branch propagation of a uniform pattern, which is controllable, so that it is efficient for realtime virtual ecosystem construction. Furthermore, it can process changes of environment factors or growth deformation for various and unique tree generation simply through the properties of divisor functions. Finally, an experiment is performed in order to evaluate our proposed modeling method whether it can generate natural and various tree models, and a realtime virtual ecosystem of a large area where a variety of trees are presented using the modeling method can be constructed efficiently.Copyright © 2013 John Wiley & Sons, Ltd.
1 INTRODUCTION
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 PREVIOUS WORKS
 3 THE MODELING METHOD
 4 IMPLEMENTATION AND RESULTS
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 Biographies
In computer graphics, realistic and efficient generation, and expression of plants composing a broad terrain or virtual ecosystem is a continuing problem. For that reason, many studies have been conducted to model diverse plants realistically using methods such as defining the growth process of plants with diverse rules, allowing for interactive user control of this process, and analyzing plantrelated input images [14].
Most methods related to tree modeling use rulebased modeling methods for defining tree growth rules, including a method involving branch propagation and the positioning of leaves. These include representative rulebased modeling methods, such as the Lsystem [2], tree modeling from the recursive and repetitive tree structures defined using the geometrical attributes of a tree, such as the growth angle and the branch length ratio [1], and modeling through selforganization based on botanical theory [5]. Other studies have examined diverse, realistic treemodeling methods that control parameters defined based on growth rules or environmental elements, such as light and gravitropism [68].
The propagation rules or grammars of most rulebased modeling were not simple and intuitive, and these method also required the assignment of complex parameters. Furthermore, they did not properly account for the complication of branch distribution in trees. As a result of this, existing rulebased modeling was mostly used for realistic image synthesis than a virtual ecosystem consisting of a large number of trees. Currently, rather than studying new rulebased modeling approaches, the focus of research has been on imagebased tree reconstruction methods using computer vision technique. However, it should be noted that this approach also has its limitations, and it is only applicable for a small number of trees.
Therefore, in this paper, a procedural modeling method is being proposed by which unique and various trees can be grown naturally using small lines of rules. Further, an efficient branch propagation structure based on convolution sums of divisor functions is also being proposed to construct a realtime virtual ecosystem consisting of various trees in a simple and effective way.
First, an existing growth volume algorithm is used with regard to the growth rules that are related to the tree structures and branch propagation.
Next, as for the fundamental formula of the procedural modeling method proposed in this study, the properties of divisor functions and the definition and characteristics of convolution sums based on the combination of those functions are analyzed.
Then, a grammar is defined to understand intuitively the growth pattern of branches and leaves, the growth process, changes in environmental factors such as light, fallen leaves, and the growth deformation for various types of tree growth.
Finally, an experiment is conducted to evaluate the efficiency of constructing an ecosystem in real time where various trees are constructed.
This paper is organized as follows. Section 2 presents a review of the previous works on the expression of the trees, and the proposed procedural modeling method is described in Section 3. In Section 4, the results are analyzed through a growth experiment on modeling and a performance analysis on efficiency. Finally, in Section 5, conclusions are presented.
2 PREVIOUS WORKS
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 PREVIOUS WORKS
 3 THE MODELING METHOD
 4 IMPLEMENTATION AND RESULTS
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 Biographies
It is impossible to express tree modeling using a general model because of the complex structure and diverse forms available. To express such natural objects, Lindernmayer [9] proposed the Lsystem, and diverse methods of tree modeling using this system have been researched. An Lsystem divides a 3D tree growth form into branch length, growth angle, width, and other features. It then assigns string symbols to each of these and determines the state of a tree form from the previous step using a string transposition rule [2]. It creates complex objects by changing simple individuals with such regional rules, in which regional phenomena are analyzed to create a global phenomenon corresponding with the method of modeling plants. This approach has inspired active research using an Lsystem that generates plant forms through the use of internal parameters and rules [7].
On the basis of such rulebased modeling, diverse attempts have been made to apply this method. Honda [10] constructed trees using parameter characteristics such as a specified recursive structure of branch generation and branch rotation angle and ratio in recursive steps of a tree structure. Ulam [11] constructed trees through the process of selforganizing, in which basic elements for generating a branch construct a branch pattern in their respective spaces through competition. Based on this work, Palubicki et al. [5] proposed a modeling method based on the regional control of geometric branches and the definition of rules for competition among buds and branches in space and for internal competition and selforganization.
Research using a statistical approach on the fate of buds composing a branch was first constructed by de Reffye et al. [12]. Takenaka [13] used the effect of lights in the process of growth for controlling the fate of buds and distribution of branches. Furthermore, research related to realistic plant generation, is actively in progress, a treemodeling algorithms through the highlevel control of a grammarbased procedural model [14], and so on.
Imagebased modeling methods for generating on the basis of analyzing the input images of trees in their desired forms, in contrast to rulebased modeling, are also under recent study. This method constructs 3D models from more than one userprovided image. Tan [4] proposed a method of combining input images with user interaction in constructing trees, whereas other studies in progress reconstruct trees from sets of 3D vertices derived from tools such as 3D scanners or computer vision technique [15]. However, these methods require expensive equipment and expert editing in the process of controlling each tree. Lobebased tree modeling proposed by Livny et al. [16] is a further development in imagebased modeling that simplifies the modeling process by calculating sets of 3D vertices in lobe form.
However, most tree modeling methods were focused on improving the deformation or applications of existing rules, rather than finding new rule definitions. Furthermore, when numerous various trees are generated at the same time, the rules of existing methods become complicated further or many more parameters are generated, which limit their application. Moreover, the structure of a tree cannot be optimized to express a realtime forest.
Therefore, in this study, a procedural treemodeling method based on the convolution sums of divisor functions is proposed that is appropriate for a realtime virtual ecosystem consisting of various trees.
4 IMPLEMENTATION AND RESULTS
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 PREVIOUS WORKS
 3 THE MODELING METHOD
 4 IMPLEMENTATION AND RESULTS
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 Biographies
The proposed tree modeling program was developed using VISUAL STUDIO 2008 and DirectX SDK 9.0 (Microsoft, USA, State of Washington, Redmond), and the PC used for performance testing was built with an Intel core i5650 CPU, 4 GB RAM (Intel, USA, State of California, Mountain View), and a GeForce GT 320 GPU (NVIDIA, USA, State of California, Santa Clara). For the rendering of natural terrain during the construction of a virtual ecosystem, a Unity 3D 3.4.0 engine (Unity Technologies, USA, San Francisco) was used.
The experiment was largely divided into two steps. An experiment was performed to analyze whether our proposed method could generate various trees in a virtual ecosystem in an easy and effective way and whether it could create an efficient structure in a realtime system.
Figure 8 shows that various tree models were generated easily using several lines of rules. The figure shows that as a consequence of applying growth volume as well as random factors to the grammar, various kinds of tree models can be generated easily. The types of leaves were specified properly as a billboard form.
Figure 9 shows the result of virtual ecosystems consisting of a number of trees using the succeeding grammar. Here, growth volume and divisor functions were generated randomly. Virtual ecosystems consisting of about 300 and 500 trees are shown in Figure 9, respectively. Here, depending on the size of the system, the branch element (B^{i}(x,y)) of convolution sums can be specified selectively. Through this, the complexity of branches that make up a tree can be controlled and configured to make not damage on realtime processing of a virtual ecosystem.
Next, the efficiency of the growth pattern, which is based on convolution sums of divisor functions, was confirmed. Figure 10(a) shows the result of tree generation using only the growth structure via growth volume and selforganization proposed by Kim et al. [6]. Figures 10(b) and (c) show the result of applying our proposed growth pattern () in addition to the original algorithm. As can be found by comparing the numbers of branch vertices, our method can generate efficient branch patterns while having similar results.
To confirm the growth pattern more specifically, the number of branches generated due to the increase in the growth step was analyzed. As shown in Table 1, we checked the average number of branches randomly generated in each step using the convolution sum of divisor functions used in this study. In rules such as those used in rulebased modeling [6], over two to four branches are generated repeatedly from the previous branch, increasing the number of branches in powers (n^{k}). In contrast, a growth rule that uses a divisor function is an efficient structure with continuous growth, and because it further considers the growth of leaves, we see that this rule is capable of a more efficient expression as the virtual environment increases in size.
Table 1. Comparison between the average numbers of branches increasing by growth step (tree DivFunc(D,1){branches}).  Growth step(g_{s})  5  8  11  Kim 

Average number of branches  D(k) :  σ_{1}(k)  38  190  1,276  6,124 g_{s}(11) 
σ_{1,b}(k; 2)  23  104  695  
 31  151  1,013  
 27  149  1,085  
 53  305  1,920  
In particular, the convolution sum of can control the number of branches depending on the value of y of the branch increase/decrease ratio B^{i}(2,y) (approximately multiples of 2). The reason for this control is due to the characteristics of the divisor functions (Table 1, Figure 10).
Table 2 shows the results of the performance tests in complicated virtual ecosystems consisting of 500, 1000, 1500, and 2000 trees, based on the rendering method [6] in the growthvolumebased algorithm. Kim's method, in which a difference between the numbers of increasing branches becomes larger as a growth step progresses, could affect the frames per second (FPS) when the number of growth steps decreases. Therefore, even when the number of trees increases, growth steps and the number of branches that one single tree has decreases thereby increasing performance. However, this method only allows up to 2000 trees due to the limitation of the minimum number of growth steps (g_{s} : 5). On the other hand, the number of branches in a single tree in our proposed method is not influenced by the number of growth steps, so that the FPS decreases when the number of trees increases. Even so, the growth pattern (B(x,0) ≡ I) of the proposed method has a more efficient structure than that of the existing selforganizationbased growth volume method, thereby showing better performance relatively.
Table 2. Rendering speed for a large number of trees using the proposed method.Number  Growth  Average number of  FPS 

of trees  step (g_{s})  nodes  Our  Kim 


500  7 ∼ 9  194  49.83  20.07 
1000  7  155  34.73  24.71 
1500  6  99  34.20  31.03 
2000  5  59  24.17  — 
Finally, the growth step in Table 2 shows the boundary value, for example, a tree in a group of around 1000 to 1500 trees will have six to seven growth steps overall. As seen in the experiment results, the FPS was continuously kept over 24 fps, and our proposed method was more suitable for a realtime system.
5 CONCLUSION
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 PREVIOUS WORKS
 3 THE MODELING METHOD
 4 IMPLEMENTATION AND RESULTS
 5 CONCLUSION
 ACKNOWLEDGEMENTS
 REFERENCES
 Biographies
In this study, a procedural modeling method based on convolution sums of divisor functions generated various trees easily and efficiently among plants constructed in a virtual ecosystem. We used a growth volume in which a growth result can be estimated intuitively, and many parameters that determine tree growth were calculated automatically. In addition, properties of convolution sums of divisor functions were analyzed, and a method to apply the findings was studied to perform an efficient management of complicated tree structures. Additionally, a growth grammar of tree models, using which the generation of various trees can be understood intuitively, was used, that made use of factors such as a growth pattern of branches and leaves, growth process, the application of environmental factors, and the definition of leaves. Based on the procedural method, the possibility of the efficient generation of natural and varied trees in a virtual ecosystem was evaluated through an experiment.
The growth rule proposed herein is presented in the form of a convolution sum with a generalized rule on two divisor functions. In the future, other ways to model diverse and realistic trees may become possible through research on methods using the convolution sum of divisor functions, including the scalar product, or convolution sums composed of a combination of a number of divisor functions. This study deals with a modeling method related to tree growth rules without considering realistic rendering via lighting technology suited for tree structure or technology for natural animation of the surrounding environment. Research on technologies for expressing realistic rendering and animation technologies in combination with a principle of divisor functions will be able to generate more natural tree models.