Collision detection of virtual plant based on bounding volume hierarchy: A case study on virtual wheat

TANG Liang, SONG Wei-guo, HOU Tian-cheng, LIU Lei-lei, CAO Wei-xing, ZHU Yan

National Engineering and Technology Center for Information Agriculture/Key Laboratory for Crop System Analysis and Decision Making, Ministry of Agriculture/Jiangsu Key Laboratory for Information Agriculture/Jiangsu Collaborative Innovation Center for Modern Crop Production/Nanjing Agricultural University, Nanjing 210095, P.R.China

1. Introduction

Collision detection in virtual environments is of considerable importance in computer animation, physical simulation, and robotics (Lin and Gottschalk 1998; Jimenezet al. 2001).Several collision detection methods have been proposed,such as space partitioning tree (Nayloret al. 1990; Vaně?ek 1991), biomimetic pattern recognition (Xiaoet al. 2015),distance computations between convex objects (Van Den Bergen 1999), bounding volume hierarchy (BVH) (Van Den Bergen 1997; Klosowskiet al. 1998; Wilsonet al. 1999), and interference detection using graphics hardware and image space (Knot and Pai 2003). BVH is considered an effective collision detection algorithm due to its extensive application and robustness. Different methods have been used for the development of BVHs such as sphere (Wilsonet al. 1999),Axis-Aligned Bounding Box (AABB) (Van Den Bergen 1997),Oriented Bounding Box (OBB) (Gottschalket al. 1996), and K-Discrete Orientation Polytopes (Klosowskiet al. 1998).These BVH methods have strengths and weaknesses, and should be selected according to the intended application(M?lleret al. 2008).

The use of virtual plants was first proposed approximately 20 years ago and can be applied in research, education/extension, and decision support in relation to plant breeding and management, landscape architecture, entertainment,and art (Roomet al. 1996; DeJonget al. 2011). Currently,many studies on plant visualization focus on the development of organ geometric models and 3D visualization (Zhanget al.2013, 2017). The phenomenon of plant organ intersection often occurs during the process of 3D visualization of plant morphology and reduces the realism of plant visualization.Thus, collision detection of plant organs is very important for accurate visualization of the characteristics of plant morphology and canopy structure.

Several previous studies on collision detection of virtual plants that avoid intersection of plant organs have been proposed. Fowleret al. (1992) presented a collision-based model of spiral phyllotaxis for detecting and eliminating collisions between neighboring primordia, however this method cannot be used in a complex plant canopy. Wuet al.(2011) studied collision detection of leaves in gramineous crop using the K-Discrete Orientation Polytopes method,but detection precision still should be improved. Qinet al.(2012) presented a collision detection method for maize using a combined OBB based on triangular data obtained from a digitizer to improve the accuracy and reliability.However, the methods for OBB construction and intersection detection are generally time-consuming. Thus, methods on collision detection among plants canopy need to be improved with regard to accuracy, computing speed, and practical application.

The objectives of this study are: 1) to optimize computing efficiency and detection accuracy by developing a collision detection method based on BVH for plant visualization according to the characteristics of 3D plant morphology and canopy structure; 2) to apply the method of collision detection to wheat visualization at the organ, individual and population scales, and evaluate the effects on the visualization.

2. Materials and methods

2.1. Visualization of wheat crop

The morphological models of wheat used in this study were developed based on simulation dynamics of leaves (Chenet al. 2007), leaf sheaths and internodes (Zhuet al. 2009),panicle morphology (Tanet al. 2006) and stem-sheath angle (Zhanget al. 2011), and leaf color in relation to thermal time (Zhuet al. 2008). Using the OpenGL library,geometric models for visualization of different organs were developed using geometric shapes (e.g., cylinder) and nonuniform rational B-splines (Wuet al. 2009). The dynamic visualization of wheat at the organ, tiller, and population scales was developed (Tanget al. 2008; Wuet al. 2009; Leiet al. 2011) and improved (Tanget al. 2011) by integration of morphological models and geometric models.

2.2. An overview of the procedures of wheat collision detection

A overview of the procedures of wheat collision detection in this study is presented in Fig. 1. First, the triangular data of leaves are generated with the tessellation of non-uniform rational B-splines surfaces (Wuet al. 2009). Second,triangle-triangle intersection detection is performed by the interval overlap method (M?ller 1997), and then BVs and BVHs of leaves were constructed based on the leaf morphological characteristics using the Separating Axis Theorem (Ericson 2004). Third, the collision detection of two leaves was performed by using the depth-first search algorithm (Ericson 2004). Finally, the BVs of other organs(ear, stem, and leaf sheath) in virtual wheat plant were constructed based on their geometric morphology, and the collision detections were conducted at the organ, individual,and population levels.

2.3. Leaf tessellation

Tessellation, also called surface subdivision, is the subdivision of concave polygons or polygons with intersecting edges into convex polygons (M?lleretet al.2008; Shreiner 2009). Using the OpenGL utility library, nonuniform rational B-splines objects are compartmentalized into lines and triangles (or polygons). Vertices data can be obtained through non-uniform rational B-splines surface mesh division with a callback function in OpenGL. Fig. 2-A and B show a normal wheat leaf and a meshed leaf based on existing leaf morphological modeling (Wuet al. 2009).

2.4. Efficiency of triangle-triangle intersection detection

Collisions among leaves detected by meshed triangles on the leaf surface are highly accurate at the final stage of collision detection based on BVHs. The interval overlap method presented by M?ller (1997) is used to detect triangletriangle collision. However, the detection efficiency between plant organs still needs to be improved. For two leaves,supposing that the numbers of triangles are N1and N2after tessellation and the detection time for a pair of triangles is Costi, the total detection time (TDT) is calculated by the following equation:

Fig. 1 An overview of the procedures of collision detection in this study. BVs, bounding volumes; BVHs, bounding volume hierarchies.

Fig. 2 Tessellation of leaf. A, a normal wheat leaf. B, a meshed wheat leaf.

If the value of Costi is fixed, the speed of collision detection is limited by N1and N2. To enhance the efficiency of collision detection, the number of triangles is minimized after tessellation. The reduced number of triangles is checked and non-intersecting triangles are removed.

2.5. Bounding volume hierarchy of leaf

The leaf-nodes in BVH wrap all geometric models in the space and each parent node contains the geometric spacing of successor nodes. In the process, BV detection is performed before conducting triangle detection, which can improve detection efficiency by significantly reducing the time complexity by incorporating BVs into the tree structure(Ericson 2004).

2.6. Selection of bounding volume hierarchy method

The type of BVH to select depends upon the characteristics of the object being visualized. The basic cost function wasfirst formulated by Weghorstet al. (1984), first applied by Gottschalket al. (1996) and later enhanced by Klosowskiet al. (1998):

Where, T is the total execution time for detecting interference,NVis the number of BV pairs for intersection tests,CVis the time required for testing a pair of bounding volume intersections,NPis number of primitive pairs overlapping in the tests,CPis the time required for testing a pair of primitives for interference;NUis the number of nodes that need to be updated, andCUis the time cost of updating each node.

In the process of plant organ assembly, different organs(leaf, stem, and ear) and their BVs rotate and move one or more times. Hence, the preferred BV should have: (1) a relatively low calculation of creation; (2) low calculation of updating after rotation; (3) better tightness; and (4) faster overlap testing.

2.7. Development of BVs and BVHs for leaves

BVH development requires two steps: 1) building a BV for each triangle; and 2) building a tree structure and incorporating BVs into the tree structure. AABBs are created for triangles or organs by taking the maximum and minimum extents of these objects along each axis. A rotation matrix is used for the BVs rotation in order to reduce the computation time of the AABB.

The algorithm for tree structure development used in this study is the top-down algorithm (M?lleret al. 2008). After leaf tessellation, the BVs of triangles (S) are calculated to determine whether the BV is a leaf-node (minimum BV). If it is, we set the node type and save the triangle data of that node. Otherwise, the cutting point is computed to divide the set S into two subsets, and then a cutting point is computed for each subset (Fig. 3).

We assumed that a set (Si) with a number (j) of triangles is obtained from leaf tessellation. Among thejtriangles there are a certain number (k) of the mass points which have differentx-coordinates. The two segmented triangle setsSi′ andSi′′ derived fromSiare as follows:

Where,p1,p2, andp3are thex-coordinates of the vertices of the triangle set and t is the average value of thosex-coordinates of thektriangles.

2.8. Collision detection and response

After leaf BVH development, BVs of root-node (the largest BV) need to be traversed to detect collisions. The traversal method used in this study is the depth-first search algorithm,which traverses or searches a tree or graph. The algorithm begins at the root (selecting some node as the root in the graph case) and explores as far as possible along each branch before backtracking (Knuth 2005). The 2 BVs maintain the same depth during the traverse, which is also called synchronous depth-first traversal (Ericson 2004).Separating Axis Theorem is used to detect the intersection of leaf BVs (Gottschalket al. 1996; Ericson 2004). The response method after collision is to redraw the model by rotating the leaf by a certain angle around the stem according to the positions of leaves, which would not change the morphology of organs (Wuet al. 2011).

Fig. 3 Building bounding volume hierarchy of a leaf.

3. Results

3.1. Collision detection between leaves

Based on an existing wheat morphological model and visualization technique (Wuet al. 2009; Tanget al. 2011),a virtual experiment is conducted for evaluation of the new collision detection method using the cultivar Yangmai 9 and weather data from 1997 to 1998 in Nanjing, China under the condition of potential growth (assuming no water or nutrient limitation). The row spacing was 18 cm×18 cm. A pair of identical triangular data obtained from the fourth leaf on the main stem is to apply the collision detection algorithm. Two normal leaves (light and deep black for clear distinction)are used for the collision detection process (Fig. 4-A).First, the BVHs are built for the two leaves (Fig. 4-B), then intersecting BVs are detected (Fig. 4-C). All triangles in leaf-nodes of the intersecting BVs are obtained (Fig. 4-D)providing a pair of intersecting triangles (Fig. 4-E). When intersecting triangles are detected (Fig. 3-F), we conclude that the leaves intersect.

3.2. Performance test for collision detection between leaves

Intersection detection speedThe performance of the proposed Leaf Collision Detection (LCD) method was tested on the graphic workstation HPZ600 (HP, USA) and compared with the traditional AABB tree algorithm (M?lleret al. 2008), and with the package of Robust and Accurate Polygon Interference Detection based on the OBB tree algorithm (Gottschalket al. 1996). The test results are listed in Table 1.

AABB tree is the traditional Axis-Aligned Bounding Box hierarchy algorithm; OBB is Oriented Bounding Boxes tree algorithm; LCD is the algorithm of leaf collision detection developed in this study.

Initialization time is the time for building BVs and BVHs.Intersection detection time is for traversing two BVHs and intersection detection of the BVs. Update re-computation after collision and response is the time for one more collision test after building the BVH by rotating the leaf once. Nonintersection detection time is the time for one intersection detection computation of the root nodes of two BVHs.

Overall, LCD detection speed is faster than the original AABB tree and OBB tree methods (Table 1). During the initialization phase, the original AABB tree and LCD required more time to calculate the longest axis of the triangle set and dividing the set into two subsets. OBB took more time to calculate the minimum BV of the triangle set.

The LCD and AABB tree take the longest axis of a BV as thex-axis when creating the BV. Although LCD and AABB tree lose some tightness compared with OBB, their initialization times are reduced by about two thirds of OBB.In the intersection test stage, LCD and AABB have the same number of leaf-node triangles which are greater than OBB.Therefore, the speed of intersection detection of AABB and LCD is slightly slower than that of OBB, respectively. In the process of updating and detection after collision, the LCD and OBB are faster than AABB because neither is required to build hierarchies again.

With regard to the non-intersection detection, AABB tree requires detection of two intersecting BVs at XYZ axes. The BVs in LCD perpendicular to the plane XOY requires five potential separating axes, which is more complex than that in AABB tree and less than that in OBB tree which requires 15 potential separating axes (Gottschalket al. 1996).

Overall, LCD has better initialization time and recomputation time after collision as well as similar performance on intersection detection time and noninterception detection time. These factors make LCD faster than the AABB and OBB methods.

Tightness of bounding volume The volume of the leaf in question reflects the tightness of the BV of that leaf (M?lleret al. 2008). Unlike OBB and LCD, the volume of AABB varies with the leaf’s rotation angle around thez-axis. For example, a leaf has a length of 19.67 cm, a maximum breadth of 1.60 cm, and an angle between the stem and leaf of 30°. When the leaf rotates to 45°, the volume is 14.6 cm3as calculated by the minimum OBB computation method proposed by Barequet and Har-Peled (2001) compared with 104.1 and 25.5 cm3for AABB and LCD, respectively(Fig. 5). Thus, from the characteristics of the three detection methods, the order of the BV tightness of LCD is significantly higher than that of AABB and slightly lower than that of OBB.However, LCD does not compute the minimum BV which is commonly used for OBB. The speed of building a BV and BVH in LCD was faster than that in OBB tree, respectively.Overall, the LCD method had a better balance between tightness and efficiency.

Fig. 4 Collision detection of leaf-leaf. A, two intersecting leaves. B, the bounding volumes and hierarchies of two leaves. C, the intersecting bounding volume of two leaves. D, the intersecting bounding volumes in leaf-node. E, a pair of intersecting triangles.F, all the intersecting triangles.

Table 1 Performance comparisons of different algorithms for leaf collision detection

3.3. Building hierarchies for all organs on a single tiller of wheat

According to the wheat canopy structure, most of collisions happen between leaves and other organs (leaf sheath,stem, and ear). Collisions between leaf sheaths, ears,and stems are infrequent. Based on the output parameters from the morphological model (Tanet al. 2006; Zhuet al.2009), the BVs are simply built for wheat ears (Fig. 6-A),internodes (Fig. 6-B), and leaf sheaths (Fig. 6-C). No BVHs are built for them. Therefore, the intersection detection between leaf and other organs is the intersection detection of BVHs and BVs.

Bounding volumes of ears, stems, and leaf sheaths and the hierarchies of leaves in tillers are created to detect collision between tillers (Fig. 7-A and B). In order to rapidly determine a potential collision between organs, BVHs of tiller is created which regards different organs as basic graphic elements, and the longest axis hierarchy classification method is used to create the BVs of different organs on a tiller (Fig. 7-C and D) (M?lleret al. 2008).

Fig. 5 Bounding volumes with the methods of Axis-Aligned Bounding Box (AABB, A), Oriented Bounding Box (OBB, B)and Leaf Collision Detection (LCD, C) when leaf rotates to 45°C from z-axis.

Fig. 6 Bounding volumes of different organs (A, ear; B,internode; C, leaf sheath).

3.4. Collision detection and response between tillers

The process of collision detection and response of two tillers are presented in Fig 7. Fig. 8-A assumes there are two colliding tillers. The leaves on a tiller collide with the organs (ear, leaf, and stem) on another tiller (Fig. 8-B-1, 2,and 3, respectively). BVs and hierarchies of two tillers are created (Fig. 8-C). BVs of organs on the single tillers are traversed and detected (Fig. 8-D). Then collision detection between organs is performed (Fig. 8-E) and the colliding organs are identified (Fig. 8-F). The collision response method performs and renews leaf matrices (Fig. 8-G) (Wuet al. 2011). Fig. 8-H shows the effects of the visualization of two single tillers after collision.

3.5. Collision detection of wheat population

Collisions among wheat populations are closely correlated with plant number and planting space. The leaf-sheath angle and the leaf-stem angle are set to random values to check the universality of the collision algorithm in a wheat population that has six rows, six columns, and three tillers on each plant (Fig. 9-A). The entire hierarchies of tillers,hierarchies of leaves, and BVs of ears and stems are presented in Fig. 9-B–E. Collision detection is performed and intersecting BVs and triangles are shown in Fig. 9-F and G. The response method is to rotate the colliding leaf to a certain angle around the stem according to the positions of leaves. Due to complexity of the topological structure of wheat, the process of leaf rotation might lead to other collisions. Therefore, we continued renewing leaf matrices and performing collision detection until there are no or few collisions. The possibility of collision is reduced significantly after collision detection and redrawing compared to the original collisions (Fig. 9-H).

Fig. 7 Bounding volume hierarchy of a wheat tiller. A, a wheat tiller. B, bounding volumes of ears, stems, and leaf sheaths and the hierarchies of leaves in the tiller. C, tiller hierarchies regarding different organs as basic graphic elements. D, all bounding volumes and hierarchies in the tiller.

Fig. 8 Collision detection between two wheat tillers. A, two tillers. B, collision positions of two tillers. C, bounding volumes and hierarchies of two tillers. D, collisions of the bounding volumes and the hierarchies of two tillers. E, performing collision detection. F, identifying the colliding organs. G,response of collision. H, effects of collision response.

Fig. 9 Collision detection among wheat population. A, a wheat population. B, bounding volume hierarchies (BVHs) and bounding volumes (BVs) of single tillers. C, hierarchies of the leaves. D, bounding volumes of the ears. E, bounding volumes of the stems.F, identifying the colliding organs. G, the intersecting bounding volumes. H, effects after collision response.

4. Discussion

The phenomenon of intersection among different organs in the process of plant visualization decreases the realism.Proper simulation of collisions is crucial for realistic animation of plant development (Fowleret al. 1992).Computational efficiency and detection accuracy are also important factors in visualization of large plant populations,due to the large number of collisions between different organs in crop canopies. Previous studies and methods on collision detection for crop visualization have efficiency and accuracy disadvantages and limitations (Fowleret al.1992; Wuet al. 2011; Qinet al. 2012).

With the aim of increasing computational efficiency and detection accuracy, this study addresses the selection and creation of BVs and BVHs for collision detection. Few of the commonly-used BV can meet the requirements of the basic cost function (eq. (2)). For example, spheres have a poor compactness for wheat organs; AABBs require updating after rotation, and the worst compactness is observed when a leaf rotates 45° along they-axis. Additionally, the computation for the initialization and detection of the OBBs are relatively slow. The K-Discrete Orientation Polytopes method requires conversion of all triangle vertices to new spaces (Klosowskiet al. 1998; Ericson 2004), i.e., all triangle vertices will have a matrix multiplication. Here, we take advantage of different types of BVs and BVHs according to the characteristics of wheat morphology and canopy structure.

At the organ level, AABB is used for creating BVs, which is more efficient than OBB in the initialization phase. When tillers are assembled, a rotation matrix is computed and saved to avoid repeating computation of AABB and also to increase the tightness of BV (Fig. 5). There are three main algorithms for building a tree structure: A bottom-up method,an incremental tree insertion, or a top-down approach(M?lleret al. 2008). Among them, the top-down algorithm,which is used by the majority of hierarchy construction algorithms, is the fastest (Knuth 2005). After creation of leaf BVHs in which BVs at the root-node are intersected,they need to be traversed. Commonly traversal methods include the breadth-first search algorithm which start at the tree root and explores the neighbor nodes, before moving to the next level neighbors, and the depth-first search algorithm (Knuth 2005). Compared with the breadth-first search algorithm, the depth-first search algorithm used in this study is the most common option in the system of collision detection, due to its significantly lower demand on memory space during the traverse (M?lleret al. 2008).At the intersection detection phase, utilizing Separating Axis Theorem proposed by Gottschalket al. (1996) and the traversal hierarchy principle, the intersection detection was implemented at leaf, tiller, and population levels. The Separating Axis Theorem to detect collisions between OBBs requires computing 2N-1 direction matrices and N times of matrix inversion computation (N is the height of OBB tree).LCD, using the same direction BV (Fig. 3), only needs to save one rotation matrix and compute the matrix inversion once. In particular, Gottschalket al. (1996) tested 15 potential separation axes to determine whether two OBBs intersect by using Separating Axis Theorem. With the LCD method, only five potential separation axes are needed due to most of leaves and sheaths rotating and moving along thez-axis (Fig. 3).

At the tiller and population levels, regarding organs as basic graphic elements, the longest axis hierarchy classification method is used to develop BVHs of tillers to enhance detection efficiency, which efficiently excludes non-intersecting organs (M?lleret al. 2008). A simple method for the response to collision detection is to rotate a leaf to a certain angle (Wuet al. 2011). After this rotation,the number of clashes decreases significantly but did not totally disappear due to high density of plant organs in a plant population (Fig. 9-H). The response method should be further improved to enhance the efficiency and accuracy.

Several methods for collision detection have been developed for different purposes, but only two classic methods (AABB and OBB) are compared in this study.More algorithms and technologies should be compared and developed for plant collision in future studies. Although this method for collision detection has improved efficiency,the speed of leaf triangular data acquisition also should be further improved when there are a large number of leaves in a large population to enhance the efficiency of visualization(Duet al. 2017; Yonget al. 2017).

5. Conclusion

With the aim of accurate visualization of crops, the phenomenon of organ cross-drawing was addressed with a method for collision detection of plant organs based on BVHs. The collision detection method is applied on wheat and presented at the leaf, tiller, and population levels which showed a good balance between computing efficiency and detection accuracy for improving the realism of 3D visualization of wheat crops.

Acknowledgements

This work was supported by the National High-Tech Research and Development Program of China (2013AA102404), the National Science Fund for Distinguished Young Scholars,China (31725020), the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD), China and the 111 Project, China (B16026). We would also like to thank Dr. Alison Beamish at the University of British Columbia for her assistance with English language and grammatical editing of the manuscript.

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