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Keywords:

  • procedural modeling of trees;
  • convolution sums of divisor functions;
  • real-time virtual ecosystems;
  • growth grammar of trees

ABSTRACT

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. 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 real-time 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 real-time 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

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. 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 plant-related input images [1-4].

Most methods related to tree modeling use rule-based modeling methods for defining tree growth rules, including a method involving branch propagation and the positioning of leaves. These include representative rule-based modeling methods, such as the L-system [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 self-organization based on botanical theory [5]. Other studies have examined diverse, realistic tree-modeling methods that control parameters defined based on growth rules or environmental elements, such as light and gravitropism [6-8].

The propagation rules or grammars of most rule-based 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 rule-based modeling was mostly used for realistic image synthesis than a virtual ecosystem consisting of a large number of trees. Currently, rather than studying new rule-based modeling approaches, the focus of research has been on image-based 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 real-time 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

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. 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 L-system, and diverse methods of tree modeling using this system have been researched. An L-system 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 L-system that generates plant forms through the use of internal parameters and rules [7].

On the basis of such rule-based 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 self-organizing, 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 self-organization.

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 tree-modeling algorithms through the high-level control of a grammar-based procedural model [14], and so on.

Image-based modeling methods for generating on the basis of analyzing the input images of trees in their desired forms, in contrast to rule-based modeling, are also under recent study. This method constructs 3D models from more than one user-provided 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. Lobe-based tree modeling proposed by Livny et al.[16] is a further development in image-based 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 real-time forest.

Therefore, in this study, a procedural tree-modeling method based on the convolution sums of divisor functions is proposed that is appropriate for a real-time virtual ecosystem consisting of various trees.

3 THE MODELING METHOD

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. Biographies

This study uses a growth-volume-based algorithm for effectively managing the growth determination processes such as branch propagation in the tree model with a complex structure [6]. Additionally, using the basic formula in the procedural modeling proposed in this study, the properties of divisor functions and their convolution sums are analyzed. Based on this process, a grammar is defined that describes efficiently the growth process of various trees and their shape determination.

3.1 Tree Structure and Branch Propagation

A basic tree structure is hierarchical, with each branch being affected by the growth elements of its parent branch (Figure 1). Leaves are generated at an arbitrary position on the basis of both the branch in the current generation and its parent branches. Here, branch inline image is a child branch inherited from the parent branch(inline image, inline image), thus inline image are from one branch. Therefore, when a leaf is generated at the inline image step, the position of a leaf can be determined as one of inline image (Figure 1). Here, by using divisor functions for the patterns of branches and leaves newly generated from a parent branch, we observe that we can conduct the growth process efficiently.

image

Figure 1. Hierarchical branch structure by growth step for a tree.

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For more natural growth, a growth process based on self-organization will be applied to branch propagation. Generally, a branch is composed of a bud and leaf, and a bud can be either an apical bud or a lateral bud. Meanwhile, a growth space (pc,r,l) is determined for each bud, and based on that, the growth direction (inline image= λ(pgr − pbud)) for the next branch is determined (Figure 2(a)). Here, external elements such as phototropism(inline image), gravitropism(inline image), and surrounding trees(inline image) are introduced in the candidate bud (pc), a growth space element, to allow for perception of the surrounding environment. The change of the external elements can influence the position of pc and as a result, make it possible for the branches to grow straight and high, or in the specific direction (Figure 2(b)).

  • display math(1)
image

Figure 2. Branch propagation process with respect to the surrounding environment: (a) calculation of the growth direction of a bud through its growth space; and (b) determination of candidate bud (pc) through consideration of its diverse surrounding environment elements.

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Finally, a lateral bud grows with a rotation angle (θi,φi) from the growth direction of a parent branch (inline image) (Figure 2). The detailed growth method and surface construction of a branch are designed on the basis of the growth-volume-based modeling method [6].

3.2 Analysis of Convolution Sums of Divisor Functions

A tree model in this study is a structure for determining the growth pattern of a tree based on convolution sums of divisor functions. First, the mathematical meaning of divisor functions and convolution sums are analyzed, and the advantages when convolution sums of divisor functions are applied to a tree model are discovered.

In mathematics, a divisor function σa(n) is defined as the sum of the ath power of positive divisors (d) of a natural number (n) Equation (2).

  • display math(2)

For example, σ1(3) = 11 + 31 = 4.

Based on the basic divisor functions of Equation (2), various divisor functions can be defined, as shown in Equation (3). Case 1 is a divisor function that adds only those divisors with a remainder b when divided by m, from among the divisors of n. In general, studies have been conducted on identity rules with regard to general numbers with divisors as even or odd numbers. Case 2 is a function in which the signs of the even divisors from the divisors are changed into negative signs prior to then summing them, which results in a negative value. Case 3 is a function that represents the sum over the divisor numbers (d) where the complement numbers (n ∕ d) among the divisors are odd, which also results in creating a unique pattern.

  • display math(3)

The convolution sums of a divisor function was first mentioned in a letter sent by Besge to Liouville in 1862, and research for finding generation rules for divisor functions has been in progress ever since. Research on generalizing the convolution sum of divisor functions (generating functions such as the discrete sum for an arbitrary N) is currently being actively pursued [17, 18].

The expression later is a simple example of a convolution sum of divisor functions that express the basic formula for tree growth proposed in this study. Here, N is assumed to be a natural number greater than 1.

  • display math(4)

Because a tree model consists of branches and leaves, the study of convolution sums of two divisor functions to a growth model to take into consideration these two elements is applicable.

Recently, studies on the convolution sums of divisor functions have been carried out in mathematical and computational areas with a view to utilizing them. In particular, many studies have taken advantage of the characteristics of the convolution sums of divisor functions that can provide consistent and generalized patterns, and induce accurate values to improve the performance in a system where complicated structures and processing are involved. For example, in Internet technology, divisor functions have been used in sliding window sizing, which is used for the control of (TCP) transmission control protocol congestion.

In the case of a tree model, both the structure as well as the number of increasing branches in the growth steps have irregular and complicated patterns. Due to this, if existing rules are applied to a number of trees, it becomes impossible to construct a complicated virtual ecosystem efficiently. However, if we assume that the convolution sums of the divisor functions, in which arbitrary N is a growth step, this could become a growth pattern of the tree, then a complicated structure of branch growth can be expressed effectively (Figure 3). This can be carried out once the number of trees in a virtual ecosystem or the size of a system is determined approximately.

image

Figure 3. Comparison of convolution sums of divisor functions inline image by changes in m and branch growth pattern of growth volume (Kim et al.[6]).

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To apply the convolution sums of divisor functions to the growth structure of a tree model, Equation (4) is modified and expressed as

  • display math(5)

Where D represents various divisor functions (e.g., σ1), i is the current growth step (gs), and N − 1 is the final iteration number of the ith growth step. Here, Di(Bi(x,y)k) is a divisor function that determines the pattern of the number of branches, and Di(Li(x,y)(N − k) is a divisor function that determines the number of leaves. Further, Bi(x,y) and Li(x,y) are the ith exponential function of branches and leaves (e.g., B(3,2) = 32), respectively.

3.3 Growth Grammar of Tree Model

This study aimed to generate a large number of various trees in an easy and intuitive way based on the efficient structure of trees and propagation patterns. The growth rule controls the growth of a tree, intuitively defines the trees shape based on the growth volume, and efficiently processes growth pattern in a branch propagation through convolution sums of divisor functions. It also proposes a new growth grammar to enable growth control for multiple trees. Further, to understand the procedural method easily, the computer graphics architecture shape grammar [19] structure was used. The growth grammar of the proposed tree model is a sequential grammar, which includes definitions from the notation of tree growth to the growth volume, a determination of the numbers of branches and leaves via operations of convolution sums of divisor functions, and so on.

Tree structure

A basic tree structure has a hierarchical branch structure as described in Section 3.1. A new branch is generated from a bud followed by the generation of leaves.

Growth process

Tree growth starts from the trunk. When the number of buds and its locations are determined, branches and leaves are propagated step by step. To this end, growth factors such as the length or width of branches, the number of leaves generated from the branch, and the number of buds to be grown into branches have to be determined. Therefore, the tree growth in this study is processed as follows. (1) Growth volume is set up to calculate the growth factors automatically using a simple intuitive method; (2) then, while the number of branches and leaves is determined, convolution sums of divisor functions are chosen to have efficient and controllable distributions; (3) once the number of current buds is determined via the process of (2), branches are propagated based on (1). (4) Finally, the generation and fall of leaves are defined by (2). Unlike existing methods, it took into account the fate of branches and leaves at the same time.

Notation

In this study, a growth structure can be explained from the fact that a predecessor is replaced by a successor, which has symbols for factors such as branches, leaves, and buds that determine a tree model [19].

Growth volume

For natural growth of a tree, many parameters such as the length and width of a branch exist. In this paper, a growth volume algorithm is used to calculate these parameters automatically using an intuitive structure [6]. Before processing the procedural modeling for tree growth, a growth volume (cnt,id,wb,h,wt,grad) for each tree is assigned, where grad represents the degree of inclination of the growth volume.

  • display math

Prior to the start of growth of a tree model, the growth volume is defined (Figure 4).

image

Figure 4. Grammar-based growth volume generation.

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Growth pattern rule

Along with a determination of growth information, one of the important elements to be defined is the number of buds that should be assigned from the current branches to multiple branches. This study uses the convolution sums of divisor functions to define a growth pattern that is suitable for a real-time virtual ecosystem (Figure 3).

On the basis of Equations (2), (3), and  (5), the convolution sums of divisor functions for the current tree model are specified.

  • display math

The grammar earlier is one example of defining a divisor function as inline image. Additionally, using DivFunc(type,param,...), the required divisor functions can be specified. As another example, DivFunc( “ exp”,3,1){branch} defined a current branch element of divisor function (B(3,1)) as D(31k).

Based on this, in general, a growth pattern for branches and leaves can be approached using two steps. The convolution sum of divisor functions is inline image; thus, if a current growth step is gs, then the first step is the generation of new branches from the trunk as well as the leaves from the branches. Here, if k ≡ N − 1, inline image is used, which is calculated in the last iteration step of the convolution sums, where inline image is a new branch element generated from a trunk, and inline image is a new leaf element generated from the new branch.

The second step is the propagation of child branches from the newly generated parent branches in the previous step. This is a case where k < N − 1. Here, on the basis of the branch element of the convolution sum Di(k), the difference between the divisor functions calculated in the previous step, inline image, is calculated. It is then defined as a child branch element. Similarly, inline image, which is the difference of leaf elements inline image, is defined as the leaf determination element of a child branch.

The processes earlier can be summarized as follows. First, for branches,

image

where ndv is a value calculated from the convolution sums, and nbr is the number of generated branches induced from ndv. Next, a rule for creating new lateral buds from current branches up to nbr was formulated.

  • display math
Leaves

Leaves are defined in the same way as branches. That is, depending on k and gs, new leaves are generated according to inline image or inline image from the current branches, where nlf is the number of leaves generated.

  • display math

In particular, if the value (nlf) for a leaf is negative, this leaf is assigned as a falling leaf.

If nlf value is − 3, three leaves that are generated in the previous step are assigned as falling leaves. Additionally, when physical force is applied to a tree, some of the leaves from a selection of candidate leaves fall. The falling leaves can be defined with a single-line grammar and the rule of growth pattern, without a separate calculation.

Branch propagation

For branch propagation, branches are propagated from the generated buds up to the calculated value nbr.

Environment

Finally, environmental elements such as phototropism can be specified or modified to influence tree growth. Next, attribute values are defined as environmental factors; for example, if the height of growth volume is less than 5, gravitropism is set as 0.2.

  • display math

Figure 5 shows the result of the growth of a tree by steps based on the convolution sums of the divisor function(D = σ1) via the defined growth grammar. In the figure, the red leaves represent leaves to be fallen, and assuming a physical force is applied to the tree, some of the candidate leaves are shown to have fallen. As shown in the figure, using a simple and small number of rules, various trees can be generated intuitively and easily, and the growth pattern of branches can be managed efficiently.

image
image

Figure 5. Result of step-by-step growth with the proposed modeling method using a small number of rules.

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3.4 Application of Growth Grammar

Based on the growth grammar of a tree model, two methods are proposed for unique and varied tree growth. First, a new tree form can be generated by creating a child trunk, not a child branch, based on the current tree. Second, a unique tree model can be created by independently deforming the growth of some branches.

Child trunk generation

Based on the trunk in the current tree, a new child trunk can be created. It is intended to create a new trunk following the growth information of the current tree.

  • display math

For example, if a branch(ndv) newly generated from the current trunk is 0, a child trunk is generated with a probability 0.1. The size of the growth volume of the generating trunk is then set on the basis of the growth volume size of the parent trunk, where a random size is determined between [1,GrowV ol.size ∕ 2]. The growth volume information of a bud at the case of ndv ≡ 0 is referred to when a number of growth volumes are applied to the current tree.

In general, this method can be used in a divisor function σ1,x(k; 2) consisting of the sums of even numbers where the number of branches in a trunk may be 0.

Figure 6 shows the tree model result generated by considering the previous grammar.

  • display math
image

Figure 6. Tree model of generation of a child trunk structure using stochastic grammar inline image.

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Independent branch control

Next, as an independent control of branches, the growth environment applied to all the branches is changed independently depending upon the conditions specified for that growth environment.

  • display math

If the value of the divisor function calculated from the current branch is negative, for example, only the corresponding branch decreased its gravitropism by 0.2 and its length 2 × to have independent deformation. Also, it could use the property of negative value generation in the divisor function (inline image). Figure 7 shows the tree model generated by applying the previous grammar.

image

Figure 7. Tree model using independent branch control (inline image).

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By applying the mathematical properties of divisor functions, the modification of complicated rule or addition of parameters for various and unique trees generation is simplified.

4 IMPLEMENTATION AND RESULTS

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. 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 i5-650 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 real-time 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.

image

Figure 8. Result of growth of various trees based on procedural grammar using random values (gs: 7 ∼ 8, footprint[RIGHTWARDS ARROW]GrowV ol(cnt.rand(3),size.rand(10,8,12,60°))...):(D(k), number of vertices): (a) (σ1,1(k; 2), 6352) , (b) (σ1(k), 8423), (c) (inline image, 9437), and (d) (inline image, 7212).

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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 (Bi(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 real-time processing of a virtual ecosystem.

  • display math
image

Figure 9. Result of virtual ecosystems constructed with tree models generated using a divisor function-based growth rule: (number of trees, frames per second (FPS), Bi(x,y)) (a) (265, 32.15 fps, inline image) and (b) (478, 41.18 fps, inline image).

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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 self-organization proposed by Kim et al.[6]. Figures 10(b) and (c) show the result of applying our proposed growth pattern (inline image) 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.

image

Figure 10. Verification of efficiency by comparing our method with existing methods. Number of vertices: (a) 29 200, (b) inline image: 10 346, and (c) inline image: 5147.

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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 rule-based modeling [6], over two to four branches are generated repeatedly from the previous branch, increasing the number of branches in powers (nk). 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[RIGHTWARDS ARROW]DivFunc(D,1){branches}).
 Growth step(gs)5811Kim
Average number of branchesD(k) :σ1(k)381901,2766,124 gs(11)
σ1,b(k; 2)23104695 
inline image311511,013 
inline image271491,085 
inline image533051,920 

In particular, the convolution sum of inline image can control the number of branches depending on the value of y of the branch increase/decrease ratio Bi(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 growth-volume-based 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 (gs : 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 self-organization-based growth volume method, thereby showing better performance relatively.

Table 2. Rendering speed for a large number of trees using the proposed method.
NumberGrowthAverage number ofFPS
of treesstep (gs)nodesOurKim
  1. a

    FPS, frames per second.

5007 ∼ 919449.8320.07
1000715534.7324.71
150069934.2031.03
200055924.17

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 real-time system.

5 CONCLUSION

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. 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.

ACKNOWLEDGEMENTS

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. Biographies

The second author was supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korea Government (B21303).

REFERENCES

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. Biographies
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Biographies

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 PREVIOUS WORKS
  5. 3 THE MODELING METHOD
  6. 4 IMPLEMENTATION AND RESULTS
  7. 5 CONCLUSION
  8. ACKNOWLEDGEMENTS
  9. REFERENCES
  10. Biographies
  • Image of creator

    Jinmo Kim is a full-time researcher at research institute for image & cultural content, Dongguk university. His research interests include computer graphics and game engineering. He received B.E. degree in Multimedia Engineering from Dongguk University in 2006 and his M.E. & Ph.D. degrees in Multimedia from Dongguk university in 2008, 2012 respectively.

  • Image of creator

    Daeyeoul Kim is a senior researcher of department of Mathematical Modelling, National Institute of Mathematical Sciences in Korea. He received M.E. & Ph.D degrees from Chonbuk National University in 1994, 1998 respectively.His research interests include the graphic representation of ecosystem as a scenery of game, number theory.

  • Image of creator

    Hyungje Cho is a professor and head of department of multimedia, Dongguk university in Korea. He received B.E. degree from Busan national university and M.E. & Ph.D degrees from KAIST(Korea Advanced Institute of Science and Technology) in 1978, 1986 respectively. His research interests include the graphic representation of ecosystem as a scenery of game, computer vision and sound processing.