PAPER A Growth Model for Root Systems of Virtual Plants with Soil and Moisture Control

Transcription

1 IEICE TRANS. INF. & SYST., VOL.E89 D, NO.5 MAY PAPER A Growth Model for Root Systems of Virtual Plants with Soil and Moisture Control Jijoon KIM a), Member SUMMARY A realistic computer graphics (CG) model of root growth that accounts for the effects of soil obstruction and moisture variations is proposed. While the exposed parts of plants have been modeled extensively in CG, realistic root models have received little attention, and the potential effects of root characteristics on the growth of foliage has yet to be considered in detail. The proposed model represents roots as series of bend points and link points and defines the root systems as a layered structure formed by roots connected via link points. This approach allows for two general types of root systems based on branching probabilities of lateral and adventitious roots: main root systems consisting of a thick main root and thinner lateral roots, and fibrous root systems consisting of adventitious roots of relatively uniform diameter. The model also expresses the behavior of root growth in terms of hydrotropism, gravitropism, flexion and growth inhibition by assigning gravity, moisture and consistency parameters to underground voxels. The model is shown through simulations of various growth conditions to generate individualized root systems that reflect the growth environment and characteristics of the plant. key words: computer graphics, plant, root system, growth model, soil Fig. 1 Example of real plant root structure. 1. Introduction Research on the representation of natural objects and natural phenomena in computer graphics (CG) has become popular in recent years. CG images are used in diverse fields, including industrial design, scientific visualization, movies, advertising, and computer games. In all of these fields, however, improved CG techniques are required to present more realistic images of natural objects and phenomena. The representation of plants by CG is necessary in the visual simulation of buildings, park and garden environments, as well as in wood product design and growth forecasts for garden plants. Research on the representation of plants in CG is therefore an important theme with many applications. The generation of CG images of plants has been studied by many researchers. The most popular area of study is the production of tree structure data, and many tree models have been proposed. Some models generate branching patterns using fractals and/or rules of formula grammar [1] [4], while others construct tree shapes based on stochastic data [5], [6]. Some recent models have achieved more realistic tree shapes by integrating the effects of the environment and plant hormones [7] [14]. Generally, the tree struc- Manuscript received July 27, Manuscript revised December 11, The author is with the Dept. of Information and Telecommunication Engineering, Faculty of Mathematical Sciences and Information Engineering, Nanzan University, Seto-shi, Japan. a) kim@it.nanzan-u.ac.jp DOI: /ietisy/e89 d ture data created by these models are more complex than for other artificial objects, resulting in increased rendering load. Some fast rendering techniques have thus been proposed for generating tree images [15] [18]. A number of models for plant image creation also include swaying due to wind [19] [22], leaf and bark textures, the representation of leaves, flowers, and fruit, and autumnal colors of leaves [23]. Previous research on plant CG has thus mainly been directed toward the above-surface plant structure. In contrast, there is very little research on CG representations of root structures, which are important in the growth and support of plants (Fig. 1) and thus control the overall plant shape. In real plants, water and nutrients absorbed by the roots influence the growth of foliage. By including the root structure in a plant growth model, it is possible to produce more realistic plant representations. Such an approach is expected to be useful in fields such as the simulation of bonsais and garden plants, growth forecasts for roadside trees, and the planning of afforestation. Previous work on root growth includes studies by Mech and Prusinkiewicz [24], who introduced a root model with growth affected by the water environment, and Ohshida et al. [25], who proposed a model to integrate a tree model with a new root model incorporating interaction between the ground and the underground. Neither of these root models takes into account the differences in root pattern between types of plants or the influence of soil properties on the growth of the root system. In this paper, a realistic root system growth model is proposed as an extension of the model introduced previously by the present author [26]. The proposed model considers the differences in root patterns be- Copyright c 2006 The Institute of Electronics, Information and Communication Engineers

2 1744 IEICE TRANS. INF. & SYST., VOL.E89 D, NO.5 MAY 2006 tween types of plants and the influences of water and soil. Specifically, the model can represent the following behaviors: hydrotropism gravitropism and plagiogravitropism flexion and growth inhibition by soil branching of lateral and adventitious roots This paper is organized into 4 sections, as follows. Section 2 discusses the general features of roots and introduces the root model. Section 3 presents the results of simulations using the proposed model, and the paper is concluded in Sect Root Model The roots are a plant organ that supports the plant body and absorbs water and nutrients for plant growth [27]. The model considers a root system consisting of individual roots and in some cases the root environment. The structure of the root system is classified into two broad types: main root systems (Fig. 2 (a)) consisting of a thick main root and thinner lateral roots, and fibrous root systems (Fig. 2 (b)) consisting of adventitious roots of relatively uniform diameter. The present root model is defined such that both types of root system structures can be represented. 2.1 Root Structure Figure 3 shows the general structure of a root [28]. Roots grow by cell division on the meristem, which is covered by the root cap, and by cell elongation in the elongation zone. After elongation of the root, root hairs germinate from cells (a) Main root system (b) Fibrous root system Fig. 2 Two types of root systems. on the root surface. Root hairs expand the surface area of roots significantly and thus fulfill an important role in the absorption of water and nutrients. After the root is fully grown, the root hairs fall out and the root undergoes a transition from absorption to the transportation of water and nutrients, as well as anchoring the plant to the ground. Following the model proposed previously [26], a part of root is modeled as a series of bend points terminated between two link points, and the root system is defined as a layered structure formed by parts of roots connected via link points (Fig. 4). Each part of the root system performs growth and branching in every generation, and root hairs exist only in the newest generation. In this paper, the word root is defined as a series of bend points terminated between two link points hereafter. Bend points are the shift positions of the growth directions of roots, and link points are the bend points becoming the boundary position of growing generation of roots. Roots of the model may shift the growth direction at bend points and/or link points only. The intervals between two points are connected by straight line. Link points on the tips of the roots of the newest generation are also the root caps that will grow new roots of the next generation. Generally, the initial root germinating from a seed is referred to as the main root, roots branching from the main root are labeled lateral roots, and roots germinating from the nodes of a stem or a branch are referred to as adventitious roots. As there are few organic or functional distinctions between these roots types in the proposed model, not every root is classified, and the same parameters are used for all types. The main parameters and symbols of this model are listed in Table Root Growth Plant growth above ground undergoes obvious seasonal change, including periods of dormancy, germination of buds, spreading of leaves, flowering, and fruiting. In contrast, seasonal changes in root growth are unobservable. The model proposed in the present study does not consider seasonal change. Root system growth is modeled by the following steps: Step 1 Germinate an initial main root from a seed for a few length by predefined parameters. Fig. 3 General structure of a root. Fig. 4 Root system model.

3 KIM: A GROWTH MODEL FOR ROOT SYSTEMS OF VIRTUAL PLANTS WITH SOIL AND MOISTURE CONTROL 1745 Table 1 Parameters and symbols. Parameter Soil consistency Moisture level Max. number of lateral roots for each bend point Max. number of adventitious roots Branching angle of lateral roots Variation range of branching angle of lateral roots Number of bend points Spacing between bend points Branching probability of lateral roots on new roots Branching probability of lateral root on old roots Initial germination probability of adventitious roots Later germination probability of adventitious roots Max. flexion angle Angle of plagiogravitropism Coefficient of gravitropism Coefficient of hydrotropism Symbol C Q N lmax N amax θ dθ n l P dn P do P ai P al φ ψ α β Fig. 5 Moisture level at root apex. Step 2 Calculate moisture level Q and hydrotropism target vector v w for all new root apices individually (see Sects. 2.3, 2.4.2). Step 3 Calculate the number of bend points n and the spacing between bend points l for all new roots (see Sect. 2.3). Step 4 Calculate the growth direction target vector v s for all new roots (see Sect )). Step 5 Elongate all new roots until the length nl with shifting the growth direction on the bend points (see Sects , 2.4.4). Step 6 Calculate the moisture level Q for every root, then determine the number, the positions, and the directions of new lateral and adventitious roots (see Sect. 2.5). Step 7 Cease the apex growth of main root (fibrous root system only). Step 8 Germinate new lateral and adventitious roots. Step 9 Return to Step Root Elongation Below ground, root growth is affected not only by the presence of water and nutrients but also by the soil conditions. Growth mechanism of a real root is divided into two steps: cell division on a meristem, and cell elongation in a elongation zone, i.e. a length of a root is determined by these two steps. In real root, it is assumed that cell division rate is controlled by internal factors such as water, nutrients, and plant hormones and after elongated cell size may become genetically predetermined value if the root elongation isn t inhibited by the external factors such as a soil consistency. Based on these assumptions, the elongation length of a root is controlled in the proposed model by both of these factors: moisture controls the number of bend points of a root, and the soil consistency controls the spacing between bend points. The underground region is divided into voxels. Each voxel has a soil consistency C, and each vertex has a moisture level. These parameters are normalized to 0 1. The voxel containing the root apex is identified, and the mois- Fig. 6 Root elongation. ture level of the apex (Q) is calculated by linear interpolation of the moisture levels of the voxel vertices (Fig. 5). The number of bend pointsn is then determined by the following equation (Fig. 6 (a)): n 0 (0 Q < Q 0 ) n = n0 Q 1 n 1 Q 0 + (n 1 n 0 )Q Q 1 Q (Q 0 Q < Q 1 ) 0 n 1 (Q 1 Q 1) (1) The spacing between bend points l is then calculated by (Fig. 6 (b)) l = l 1 (1 C) + l 0 C (2) 2.4 Root Flexion Root growth flexes in response to the surrounding environment. Root flexions are categorized into two groups; those due to external factors (e.g., obstacle in the soil or collision between roots), and those due to internal factors (e.g., gravity and moisture). The calculation of root flexion in the present model accounts for the consistency of the soil as an external factor, and gravity and moisture as internal factors Gravitropism Roots exhibit gravitropic behavior, that is, roots grow downward. The main root usually displays positive gravitropism

4 1746 IEICE TRANS. INF. & SYST., VOL.E89 D, NO.5 MAY 2006 Fig. 9 Direction of root Growth. Fig. 7 Gravitropism target vector. Fig. 10 Flexion due to soil consistency. Fig. 8 Hydrotropism target vector. and grows down parallel with gravity, whereas lateral roots may grow at a fixed angle with respect to the gravity direction, which is called plagiogravitropism. In the present model, the direction of a fixed angle from the gravity direction is defined as a gravitropism target vector v g (Fig. 7) Hydrotropism Roots also exhibit hydrotropic behavior, that is, roots bend toward moisture. In the present model, the direction of increasing moisture is determined on each vertex of every cell from the gradient of moisture between adjoining vertices, and the hydrotropism target vector v w of a root is calculated by linear interpolation around the humid directions of the vertices of the voxel including the tip of the root (Fig. 8) Direction of Root Growth The growth direction target vector v s for a root is determined by the following equation. v s = αv g + βv w (3) where α : gravitropism coefficient, β : hydrotropism coefficient. As gravitropism appears at every root, α always has a given fixed value. However, as hydrotropism is weak in regions of sufficient moisture and strong when moisture is scarce, β is given by β = β 1 (1 Q) + β 0 Q (β 0 <β 1 ) (4) The growth direction v of the root is then determined by the following equation (Fig. 9). v = rot(v 0, v s, min( v s, 1)), (5) where rot(a, b, c) : rotate a to b by the ratio of c. v 0 : initial direction of root growth Flexion due to Soil Consistency In contrast to plant growth above ground, which is rarely obstructed, roots must avoid stones and other obstacles, resulting in complex flexion during growth. In the present model, flexion due to the soil properties is represented by the following steps: 1. Calculate the maximum flexion angle φ by the soil consistency C (Fig. 10 (a)). 2. Calculate the shift length and the shift direction by uniform random numbers within the shift area defined by φ. 3. Shift the growth direction v (Fig. 10 (b)). 2.5 Root Branching Root branching is categorized into two groups: the branching of new lateral roots from existing roots, and the germination of new adventitious roots from stem nodes above and below ground. The main root system is primarily organized by the branching of lateral roots, while the fibrous root system is organized by the germination of adventitious roots Branching of Lateral Roots Plant branches have nodes, from which new branches and leaves germinate. In contrast, roots do not have nodes

5 KIM: A GROWTH MODEL FOR ROOT SYSTEMS OF VIRTUAL PLANTS WITH SOIL AND MOISTURE CONTROL 1747 that germinate lateral roots. Root systems tend to reduce the frequency of root branching under conditions of sufficient moisture, and increase the frequency when moisture is scarce. Although real root branching is also affected by nutrient availability and plant hormones, the present root model determines the lateral root branching probability P d for each bend point solely in terms of moisture, as follows (Fig. 11). P d = P d1 (0 Q < Q 0 ) P d1 Q 1 P d0 Q 0 (P d1 P d0 )Q Q 1 Q 0 (Q 0 Q < Q 1 ) P d0 (Q 1 Q 1) (6) The difference between the main root system and the fibrous root system is represented by a difference in the branching probability (P d0 and P d1 ). The main root system has a high branching probability, while the fibrous root system has a low probability. If a new root germinate as a result of trial with probability P d at the bend point B i, the position of germination is determined by uniform random number within a segment between B i and an above bend point B i 1. The direction of new root growth is shifted by uniform random number from the predetermined branching angle θ within a fixed shift area (Fig. 12). However, in this case, the main root system has a low probability of branching while the fibrous root system has a high probability. 3. Growth Simulation The proposed model was used to simulate the growth of root systems using a trial implementation. Images of the generated root systems are shown in Figs. 13 and 14. Each root system was grown for 10 generations, and the root hairs are not displayed. In the main root system (Fig. 13), many lateral roots branch from the thick main root. In the fibrous root system (Fig. 14), almost all of the roots sprout as adventitious roots from the base of the stem, and all have a similar thickness. Very few lateral root branches can be seen in this example. The parameters employed for these root systems Germination of Adventitious Roots The germination probability of adventitious roots P a is also controlled by moisture, as given by P a1 (0 Q < Q 0 ) P a1 Q 1 P a0 Q 0 (P a1 P a0 )Q Q 1 Q (Q 0 Q < Q 1 ) 0 P a = P a0 (Q 1 Q 1) (7) Fig. 13 Main root system. Fig. 11 Lateral root branching probability. Fig. 12 Direction of new root growth. Fig. 14 Fibrous root system.

6 1748 IEICE TRANS. INF. & SYST., VOL.E89 D, NO.5 MAY 2006 are listed in Table 2. Similar parameters were used in both models except for the maximum number of branches and the probabilities of lateral and adventitious root branching. 3.1 Variation in Root Growth due to Changes in Moisture and Soil Consistency The variation of root growth due to changes in moisture and soil consistency is shown below. All examples in this section were generated using the same basic set of parameters as used for the models in Figs. 13 and 14. In each example, the set of parameters related to either moisture or soil changes was varied, as listed in Table Variation in Root Length with Moisture and Soil Consistency The variation of root length is shown in Figs. 15 and 16. In the variation of soil moisture (Fig. 15), the number of bend points n increases with Q because branching is calculated for all bend points. In the case of variations in soil consistency (Fig. 16, the spacing between bend points l decreases with increasing C. The total number of roots remains fixed and the shape of the root system is unchanged because the number of nodes does not change Variation in Root Flexion with Changes in Gravitropism The variation of root flexion due to gravitropism is shown in Figs. 17 and 18. In Fig. 17, the roots follow the direction of gravitropism more closely as the coefficient of gravitropism α is increased. In the case of plagiogravitropism ψ (Fig. 18), an increase in ψ with α fixed at 0.2 results in a more horizontal direction of root growth Variation in Root Flexion with Changes in Hydrotropism The effect of hydrotropism on root flexion is illustrated in Fig. 19. In Fig. 19, the soil moisture varies linearly from high on the left to zero on the right. The direction of root growth v varies spatially with Q; spreading widely from left to right as Q decreases Variation in Root Flexion with Changes in Soil Consistency The variation in root flexion as a result of changes in the consistency of the soilc is illustrated in Fig. 20. The results show that growth becomes more randomly as C increases. Fig. 16 Variation in root length with C. Table 2 Parameters of root system models. Parameter Main root Fibrous root Coefficient C Q N lmax 3 2 N amax θ dθ n Q l C P dn Q P do Q P ai Q P al Q φ C ψ 0 0 α β Q (a) 0.0 (b) 0.1 (c) 0.2 (d) 0.3 Fig. 17 Variation in root flexion with α. (a) 0 (b) 30 (c) 60 (d) 90 Fig. 18 Variation in root flexion with ψ. Fig. 15 Variation in root length with Q. (a) (b) (c) Fig. 19 Variation in root flexion with spatial gradient of Q (left to right).

7 KIM: A GROWTH MODEL FOR ROOT SYSTEMS OF VIRTUAL PLANTS WITH SOIL AND MOISTURE CONTROL 1749 will be influenced by the root structure, the author also intends to combine the root growth model with a previously proposed tree growth model [14]. Fig. 20 Variation in root flexion with C. Acknowledgments The author would like to thank Prof. Shigeru Masuyama of the Toyohashi University of Technology and Prof. Norishige Chiba of Iwate University for many valuable comments and suggestions. This research is supported in part by a Pache Research Subsidy (No.I-A-2) from Nanzan University. References Fig. 21 Variation in probability of lateral roots branching with Q. Fig. 22 Variation in probability of adventitious root germination with Q Variation in Branching of Lateral Roots with Changes in Moisture As shown in Fig. 21, the probability of lateral root branching increases as moisture Q decreases. The probability of adventitious root germination varies with the moisture level Q as shown in Fig. 22. The number of germinations varies in a similar manner the change in branching of lateral roots. The difference between the two root systems due simply to differences in the positions of new roots can be readily seen in these two figures. 4. Conclusion A root growth model that accounts for many of the factors affecting real root systems was proposed, and the effectiveness of the model was demonstrated through root growth simulations. This model generates main root and fibrous root patterns, reflecting differences seen in real root systems, by defining two root branching modes: lateral root branching and adventitious root germination. This model also has the ability to create root shapes that reflect growth inhibition and flexion by considering the consistency of the soil, which has seldom been included in previous root models. Furthermore, the proposed model accounts for variations in soil moisture. The implementation of nutrient and plant hormones as growth factors and collision avoidance for avoiding other roots and obstacles will be the focus of future research. As the individuality of a tree, particularly long-lived species, [1] P.E. Oppenhemer, Real time design and animation of fractal plants and trees, Comput. Graph., vol.20, no.4, pp.55 64, [2] M. Aono and T.L. Kunii, Botanical tree image generation, IEEE Comput. Graph. Appl., vol.4, no.5, pp.10 34, May [3] P. Prusinkiewicz, A. Lindenmayer, and J. Haman, Developmental models of herbaceous plants for computer imagery purposes, Comput. Graph., vol.22, no.4, pp , Aug [4] P. Prusinkiewicz, M. James, and R. Mech, Synthetic topiary, SIG- GRAPH94, pp , [5] P. Reffye, C. Edelin, J. Francon, M. Jaeger, and C. Puech, Plant models faithful to botanical structure and development, Comput. Graph., vol.22, no.4, pp , Aug [6] G.V. Xavier, E. Georges, J. Nicolas, and A. Didier, Combinatorial analysis of ramified patterns and computer imagery of trees, Comput. Graph., vol.23, no.3, pp.31 40, July [7] M. Holton, Strands and gravity and botanical tree imagery, Computer Graphics Forum, vol.13, no.1, pp.57 67, [8] T. Agui, A. Fukuda, and M. Nakajima, Botanical tree generation method for scenery rendering, IPSJ Trans., vol.32, no.5, pp , May [9] N. Kanamaru, K. Takahashi, N. Chiba, and N. Saito, Cg simulation of natural shapes of botanical trees based on heliotropism, IEICE Trans. Inf. & Syst. (Japanese Edition), vol.j75-d-ii, no.1, pp.76 85, Jan [10] K. Ohsaki and T. Suzuki, A growth model of botanical trees having abilities to interact with the light environment, IPSJ Graphics & CAD Technical Report, 93-CG-65, pp.37 44, Oct [11] N. Chiba, K. Ohshida, K. Muraoka, M. Miura, and N. Saito, A growth model having abilities of growth-regulations for simulating visual nature of botanical trees, Comput. Graph., vol.18, no.4, pp , [12] N. Chiba, S. Ohkawa, K. Muraoka, and M. Miura, A growth model of botanical trees Generation of natural shapes of trees based on an imaginary plant hormone, IEICE Trans. Inf. & Syst. (Japanese Edition), vol.j76-d-ii, no.8, pp , Aug [13] N. Chiba and K. Ohshida, A growth model of botanical trees toward visual simulation of bonsai, J. 10th NICGORAPH Paper Contest, pp.1 10, [14] C. Kanayama, S. Sakata, and S. Masuyama, A growth model of botanical trees having abilities to simulate the branching rule, light-environment, and plaht hormone, IEICE Trans. Inf. & Syst. (Japanese Edition), vol.j79-d-ii, no.8, pp , Aug [15] K. Tadamura, K. Kaneda, E. Nakamae, F. Katoh, and T. Noguchi, A display method of trees by using photo images, J. Inf. Process., vol.15, no.4, pp , [16] N. Kuwahara, S. Shiwa, N. Tetsutani, and F. Kishino, High speed 3-d tree image generation by using a hierachical tree shape representation, IEICE Trans. Inf. & Syst. (Japanese Edition), vol.j78-d-ii, no.7, pp , July [17] J. Weber and J. Penn, Creation and rendering of realistic trees, SIGGRAPH95, pp , 1995.

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