Efficient Computation of Ob ject Boundary Intersection and Error Tolerance in VRCC-3D +

Transcription

1 Proceedings of the 18h International Conference on Distributed Multimedia Systems (DMS 12), Miami, FL, pp , Efficient Computation of Ob ject Boundary Intersection and Error Tolerance in VRCC-3D + Nathan Eloe nwe5g8@mst.edu Jennifer Leopold leopoldj@mst.edu Zhaozheng Yin yinz@mst.edu Missouri University of Science and Technology Chaman Sabharwal chaman@mst.edu Abstract Computational geometry is a field that is relevant to computer graphics rendering, computational physical simulation, and countless other problem domains involving the use of image data. Efficiently determining the intersection of the boundaries, interiors, and exteriors of two 3D objects can mean the difference between a realistic and relevant simulation, and a slow program that produces results to that do not keep pace with user interaction with the object. However, the speed of these calculations is not the only area of concern. Taking into consideration the finite unit of resolution in a computer display (the pixel), the minimum change in distance the human eye can perceive, and error in the floating-point representation of numbers, it may be the case that the perceived correctness of these computations does not necessarily correspond to the accuracy with which the calculations are carried out. In this paper, we examine two of the most well-known methods of determining such intersections, as well as various programming language libraries available to perform these calculations. These existing approaches are considered with respect to limitations in human perception, display resolution, and floating point error. We also propose a new method which lends itself to exploiting the inherently parallel nature of these calculations. 1. Introduction VRCC-3D + [11, 12, 6, 5, 7] is a mathematical model that describes relations between three dimensional objects in space in terms of the connectivity and obscuration between each pair of objects in a scene. The system facilitates knowledge discovery by determining possible intermediate configurations of the object from one state to another. However, in implementing the VRCC-3D mathematical model, computational inefficiencies were exposed, resulting in an unacceptable delay between the initial loading of an image file and the calculation of the spatial relations between all pairs of objects in the scene. Our investigation of a more efficient method to perform these computations focused on determination of the intersections between the interiors, exteriors, and boundaries of the objects. concerned. 2. VRCC-3D + Intersections The VRCC-3D + system distinguishes eight 3D spatial relations computed using eight intersections that 1

2 2

3 compare involve the boundary, interior, and/or exterior of one object and with those of another (see Table 1). The intersection of exteriors is disregarded because we are not bounding the universe of our physical system; the intersection of the exteriors of any two bounded objects always will be non-empty. VRCC-3D + uses a hierarchical structure to determine the three dimensional intersections of the objects [11] (see Figure 1). A pair of objects, A and B, can be either disconnected (DC(A,B)) or connected (C(A,B)). If connected, they may be Externally Connected (EC(A,B)), Partially Overlapping (PO(A,B)), Equivalent (EQ(A,B)), or one of the objects may be a proper part (completely inside) of the other. We define the converse relation Proper Part Converse (PPc(A,B)) such that if B is a proper part of A (PP (B, A)), then P P c(a, B). If P P (A, B) holds, then the objects can either be tangential or non-tangential (TPP(A,B), NTPP(A,B)). If P P c(a, B) holds, then we can have the TPPc(A,B) or NTPPc(A,B) relations. Connectivity in 3D is only one part of each (composite) VRCC-3D + relation. The other part of each 3

4 Table 1. Definition of 3D Spatial Relationships Using Intersection Predicates: Φ denotes empty intersection set, Φ denotes non-empty intersection set. IntInt IntBnd IntExt BndInt BndBnd BndExt ExtInt ExtBnd DC Φ Φ Φ Φ Φ Φ Φ Φ EC Φ Φ Φ Φ Φ Φ Φ Φ EQ Φ Φ Φ Φ Φ Φ Φ Φ NTPP Φ Φ Φ Φ Φ Φ Φ Φ NTPPc Φ Φ Φ Φ Φ Φ Φ Φ PO Φ Φ Φ Φ Φ Φ Φ Φ TPP Φ Φ Φ Φ Φ Φ Φ Φ TPPc Φ Φ Φ Φ Φ Φ Φ Φ Figure 1. Hierarchy of the RCC-8 JEPD Relations [11] 4

5 VRCC-3D+ relation characterizes the obscuration between the two objects of interest, which can be none, partial, or complete (as well as converse relations for each). Obscuration in VRCC-3D+ is determined by the overlap of the projections into the image plane as well as the object depth. While beyond the scope of this paper due to space limitations, the techniques (Triangle Triangle intersection, AABB Bounding Boxes) and considerations (error tolerance and speed) discussed in this paper will also apply to the calculations of the obscuration term. Definition of the spatial relations using Table 1 results in an over-determined system. At most three intersection algorithms must be implemented to uniquely determine the 3D spatial relationship between two objects [11]. For example, the intersection of the interior of object A and the boundary of object B is the same as the intersection of the boundary of object B and the interior of object A; that is, IntBnd(A,B)=BndInt(B,A). It is not necessary to determine the interior and boundary of each of the two objects. Thus, an efficient algorithm for IntBnd will provide an efficient algorithm for BndInt, and vice versa. The same holds for the intersections of boundaries and exteriors. However, those four intersections are not sufficient to determine the VRCC-3D+ relationship between two objects. For example, DC and EC are still degenerate; the only difference between the two relations is the intersection of the boundaries. Bearing in mind that the only geometric information available (in our implementation) is the triangular faces that comprise the surface of an object, it makes sense to calculate the intersection of the boundaries rather than the enclosed volumes or the rest of the universe. Adding the computation of the boundaries allows us to uniquely determine the 3D relationships of two objects while not performing additional unnecessary computation. of the edges intersect or if one triangle is completely inside the other. More sophisticated methods such as those utilizing AABB trees take a different approach by using nested axis-aligned bounding boxes to decrease the 3. Computation of Boundary Intersection A naive method of computing the intersection of the boundaries is to use an algorithm such as that presented in [9] to determine whether every pair of faces between the two objects intersects. The algorithm determines segments of the line of intersection between a triangle and the plane supporting the other triangle. If the union intersection of the two segments is non-empty, the triangles are considered to intersect. If the triangles are coplanar, the algorithm determines whether any 5

6 according to the computational intersection of boundaries, the wings and the fuselage do not intersect, and as such they cannot be considered externally connected. While the calculations are being carried out accurately, they do not reflect the cognitive perception of the image. This could be attributed to several factors Figure 2. Two objects that are disconnected (DC). The purple sphere does not obscure the red sphere and is further away number of faces that require the intersection calculation, as well as other calculation optimizations. CGAL [2] is a computational geometry toolkit that is based on the use of AABB trees, together with intersection and distance algorithms. Table 2 shows the runtimes of the triangle intersection and the CGAL methods on two spheres (see Figure 2), each of which has approximately 2000 faces. All timing was done on an AMD Bulldozer processor running at 3.1Ghz. The triangle intersection algorithm was implemented as a C module for Python, and the Python bindings for CGAL were used to obtain the runtimes. All runtimes were averaged over 100 runs. Table 2. Intersection Test Runtimes Implementation Average Runtime (s) C Triangle Intersection CGAL AABB Human Perception and Floating Point Error In a situation such as that shown in Figure 2 there is no ambiguity in the intersection of the boundaries. However, in a more complicated case such as that shown in Figure 3, more rigorous calculations are required. insufficiencies in these calculations arise. Upon cursory examination of the rendering of the airplane, it appears that all of the wings are indeed attached to the fuselage. However, 6

7 Figure 3. Model of an airplane. Note that the wings appear to be externally connected (EC) to the fuselage. such as errors in the representation of floating point numbers in both the viewer display and model generation software, or the minimum distance that can be represented in rendering to a monitor with finite resolution. For example, CGAL follows an exact computing paradigm, a computing model in which calculations can be carried out with arbitrary precision. Even the smallest difference in parallel planes is significant and they will not be considered to be intersecting. To remedy this, we must allow some tolerance: the shortest distance between the two planes must be less than some small value, E. Implementing this in the direct triangle intersection test is trivial. However, using the CGAL AABB implementation requires a significant change in the algorithm itself. Instead of determining whether the triangles intersect, we need to find the minimum distance between the two closest faces. One way to do this is to determine all of the segments that make up the faces of the object and find the closest distance to a face in the other object. The CGAL bindings do not have the closest distance between triangles, but a shortest squared distance function is implemented for triangles and segments. Table 3 shows the timings for these modified epsilon tolerance tests. Of note is the fact that while the airplane image is a more complex model in terms of having Figure 4. Nut and Bolt model. The surfaces are smooth and sharply defined with a large number of faces. man eye to perceive changes in position, and the finite nature of the pixels that comprise the image display. Of the three of these factors, we can both calculate and control the effect of the finite nature of the pixel at the viewer level. Knowing the distance in world units from the virtual camera to the viewing plane (v in world units), the horizontal width in pixels of the scene (h in world units, w in pixel count), and the field of view angle (θ) allows the calculation of the physical size of the pixel at the image plane (see Figure 5); we can determine this value as follows: Let px be pixel uinits and units be world units h = v tan(θ/2) E v = h units / (w/2) px E v = 2h/w units/px E v = 2h/w units If px is unity, E v = 2v/w tan(θ/2) Thus, the physical size of the pixel in the scene at the image plane is The current defaults for the visualizer are as follows: θ = 40 degrees, v = 1, w = 511. This yields a value / of E more objects, it has significantly fewer faces than the v , which is a greater tolerance than the 4 that was found through manual experimentapair of spheres shown in Figure 2. Both algorithms also were used to determine the intersection of a high Once a tolerance test was developed for the resolution depiction of a nut and bolt (Figure 4). intersection, it was necessary to determine an appropriate value of epsilon. Using trial and error, the smallest value epsilon that produced a display of 5. E Determination the airplane similar to that shown in Figure 3 was

8 ( to be consistent later 1e-4). It should be noted that acceptable values of this epsilon tolerance will change based on the precision with which the values are stored in the model, the ability of the hu tion, and is a sufficient tolerance to handle the precision in the test object files (6 decimal places). This value of E also determines that the wings of the airplane are externally connected to the fuselage of the plane in Figure 3, the issue that made using epsilon testing necessary. This value will change if the viewer allows resizing of the objects; as long as the camera parameters are static, the only thing that will change will be the value of the pixel width of the scene. Objects farther away from the image plane will have a larger minimum distance than will be visible due to the pixel width. This scenario can be broken into two cases: the two objects have equal depth from the image plane, and the two objects have differing depths. The 8

9 Table 3. Intersection Runtimes using E Tolerance Airplane Spheres Nut and Bolt Obj1 Name / Face Count Obj2 Name / Face Count C Triangle Intersection Runtime(s) Wings 4 / 18 Prop 4 / A / 1984 B / Helix01 / Helix02 / CGAL AABB Runtime (s) Figure 5. Camera and Image Plane first case is trivial: given two objects of depth d from the camera, we use similar triangles to obtain: Note that E v /v is independent of v, so is E d /d E d /d = E v /v E d = d/v E v E d = 2d/w tan(θ/2) 6. Future work The inherently parallel nature of this solution clearly stands out. The intersection of any pair of faces is independent of the intersection of any other pair of faces. The fact that there is a large number of similar calculations on a large, but static, set of data should be enough E c = E w f d E c 2f θ = w tan( 2 ) f f to overcome the primary downfall of using general pur- 2 θ = tan( ) pose GPU computing technologies (e.g., OpenCL [3], w 2 NVIDIAS CUDA [4], or AMDs Stream [1]); namely, Thus the time it takes to transfer the data to the graph- E c = 2 tan( θ ) = E w ics card. Of note is the fact that CUDA actually f w 2 d can be slower than a sequential algorithm for small 2d θ enough problem sets [8]. It already has been shown E w = tan( ) w 2 that CUDA is proficient at some kinds of geometrical The case involving objects of differing depth is more between the objects as if they were of equal depth. To complex: we need to determine the effective gap do this we need to consider the lines of sight to the 9

10 boundaries of the projections of the objects, and determine the distance between the lines of sight as if they were on the same plane. The specifics and implementation of this are outside the space scope of this paper. calculations [10]. In order for the intersection of triangles method to be as fast as CGALs AABB implementation, we require a speed up of approximately 50X ( for the spheres, the example that benefitted most from AABB Trees), which is well within the plausible speedups reported in [10]. A heavily parallelized AABB algorithm may also work, but would not necessarily be as cost efficient 1 0

11 due to the recursive nature of trees. Exploration of an OpenCL implementation of these algorithms may even allow these computations to be done efficiently on mobile devices with sufficiently new integrated graphics. journal of graphics, gpu, and game tools, 2(2):25 30, Summary Being able to efficiently calculate the intersections between the boundaries, interiors, and exteriors of 3D objects introduces new ways to program simulations, collisions, and model transformations over time. Exploiting the ability of VRCC-3D+ to identify impossible states and eliminating the need to do some of the calculations would lead to more efficient collision detection in game engines, simulation engines, and modeling software. In this paper, we have explored two methods of calculating these intersections efficiently and discussed methods of dynamically determining the error tolerance. These strategies will help to make spatial reasoning applications such as VRCC-3D+ more practical for knowledge validation and discovery in real-time. References [1] Ati stream technology. [2] Cgal, Computational Geometry Algorithms Library. [3] Chronos Group: OpenCL. [4] Nvidia technologies: Cuda. one. [5] J. Albath, J. Leopold, and C. Sabharwal. Visualization of Spatio-Temporal Reasoning Over 3D Images. In The 2010 International Workshop on Visual Languages and Computing (in conjunction with the 16th International Conference on Distributed Multimedia Systems (DMS 10)), pages KSI, Oct [6] J. Albath, J. Leopold, C. Sabharwal, and A. Maglia. RCC-3D: Qualitative Spatial Reasoning in 3D. In The 23rd International Conference on Computer Applications in Industry and Engineering (CAINE 2010), pages CAINE, Nov [7] J. Albath, J. Leopold, C. Sabharwal, and K. Perry. Efficient Reasoning with RCC-3D. In The 4th International Conference on Knowledge Science, Engineering, and Management (KSEM 2010), pages KSEM, Sept [8] D. B. Kirk and W.-m. W. Hwu. Programming Massively Parallel Processors: A Hands-on Approach. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1st edition, [9] T. Möller. A Fast Triangle-Triangle Intersection Test. 1 1

12 [10] A. J. R. Ruiz and L. M. Ortega. Geometric algorithms on cuda. In GRAPP, pages , [11] C. Sabharwal and J. Leopold. Reducing 9-Intersection to 4-Intersection for Identifying Relations in Region Connection Calculus. In The 24th International Conference on Computer Applications in Industry and Engineering (CAINE 2011), pages CAINE, Nov [12] C. Sabharwal, J. Leopold, and N. Eloe. A More Expressive 3D Region Connection Calculus. In The 2011 International Workshop on Visual Languages and Computing (in conjunction with the 17th International Conference on Distributed Multimedia Systems (DMS 11)), pages KSI, Aug