on a Shadow Volumetric Binary and Line Space Partitioning. Simone Santini, Stefano Bertini, Agostino Monorchio. Microwave and Radiation Laboratory, ...
An Acceleration Technique for Ray Tracing Simulation Based on a Shadow Volumetric Binary and Line Space Partitioning Simone Santini, Stefano Bertini, Agostino Monorchio Microwave and Radiation Laboratory, Department of Information Engineering, University of Pisa,Via G. Caruso, I-56122 Pisa, Italy, www.iet.unipi.it stefano.bertini (a.monorchio)@iet.unipi.it
Introduction Ray Tracing (RT) techniques are widely employed to estimate the field levels both in urban areas and within building (indoor propagation). The radio propagation is modeled as a set of plane waves, approximated by rays which, undergoing multiple scattering phenomena (reflection, diffraction, transmission), link the transmitting to receiving antennas. Typically, the computational time of RT tools quickly increases as the complexity of the scenario and the maximum order of bouncing allowed grow. Several acceleration techniques have been proposed based on a reduction of the total number of ray-facet intersection tests [1-2-3]. In order to reduce the allowed rays, one can resort to algorithms typically employed in computer graphics: to this end, an evaluation of visibility relations between pair of facets and between facets and transmitting/receiving antennas is needed (two objects - antenna or facets - are visible if and only if a not occluded direct ray can be traced between them). In this paper, we propose a novel and efficient RT acceleration procedure based on a faster ray-facet intersection technique. A combination of Image Theory and Shadow Volumetric Binary Space Partitioning (SVBSP) algorithm [4] is used to determine which facets can support the first and the last propagation phenomena. A simplified version of the Line Space Partitioning (LSP) [5] algorithm is employed to check visibility between pair of facets and thereby reduce the number of the allowed higher order rays. A high computational gain is shown with respect to a pure RT procedure and also with respect to the Space Volumetric Partitioning (SVP) technique without any loss of accuracy in the prediction of the field distribution. Acceleration Technique By performing a ray tracing method based upon the Image Theory (IT), the image point calculation of transmitting and receiving antennas and visibility tests utilize most of the computational time. The amount R of the traced rays for every transmitting-receiving antenna pair, by considering only reflections, results: c
R = ∑ N ( N − 1)
k −1
(1)
k =1
where c is the maximum number of bounces and N is the number of facets in the environment. If any acceleration technique is adopted, every ray has to be checked through N visibility tests, so that the computational time increases at a rate of O(Nc+1), being O(N) the amount due to the visibility tests for each ray and O(Nc) dependent on the number of traced rays. Even if the single visibility test, which determines whether an obstacle intersects the path of a ray or not, is computationally cheap, it is clear that computation time becomes very large slightly increasing N or c. Our approach is based
978-1-4244-2042-1/08/$25.00 ©2008 IEEE
on a-priori determination of not occluded rays which can exist between two facets and between a facet and a transmitting/receiving antenna. These information can be easily derived from visibility relations (for instance, two facets that are not visible one respect the other cannot support a doubly reflected ray). The evaluation of the visibility relation for each pair set up by two facets or by a facet and an observation point (transmitting and receiving antenna) is performed during the preprocessing before the RT computation. The previous data are stored into two integer arrays for the facet-observation point visibility relations and into a (N×N) integer matrix for the facet-facet visibility relations. In such a way, we can lower the per ray visibility tests term, and, more remarkably, the tern O(Nc), reducing the number of traced rays with higher effectiveness than back/face culling and Binary Space Partitioning (BSP). Our approach has been implemented in the RT tool “EMvironment 3.0” that is a fully three dimensional simulator based upon a combination of BSP and IT and developed at the Microwave and Radiation Laboratory of the University of Pisa. In the simulator, a recursive RT core is also implemented to strongly reduce the overflowing due to the calculation of the image points. As in other approaches [6] we employed a 2.5 analysis, that consists in analyzing firstly the environment on the horizontal plane and then on the vertical plane, thus reducing a three dimensional spatial computation into a easier two dimensional one followed by a simple check in the remaining dimension (height). In such an approach we applied the computation of visibility relations only to polygonal facets perpendicular to the ground (in both urban and indoor scenarios a large amount of facets fulfill this characteristics). Facet-Observation Points Visibility The evaluation of the facet-observation points visibility is based on SVBSP algorithm [4] and allow us to determine which facets can support the first or the last contribution of the ray path. Referring to Figure 1a, we generate the two-dimensional shadow region seen from the observation point by the SVSBP trees [4]. Working on the horizontal plane and following a front-to-back ordering of the polygonal facets, we build the elementary SVBSP tree of the first facet called Regional Occlusion Tree (ROT). The space is therefore splitted by the lines passing through the observation point under test and the vertexes of the facet (see Fig. 1a). Once the first ROT is created, we build the SVSBP tree generated by the second facet CD. Filtering down this facet along the SVBSP tree, we are able to verify whether it lays in the shadow region (“in” leaves in Figure 1a) generated by the previous processed facet, but to mark it as occluded a further check on the vertical plane is needed. Processing in front-to-back order all the facets and refreshing the ROT, we can efficiently mark all the facets supporting an interaction with the observation point under test. The SVBSP algorithm has to be executed anew whenever the positions of the observation points (transmitting and receiving antenna) have been moved; however, it is important to highlight that this procedure is extremely efficient from a computational point of view. Facet-Facet Visibility The visibility relations between pairs of facets is evaluated by a simplified version of the Line Space Partitioning (LSP) algorithm [5]. In this case front-to-back order of the polygonal facets is generated in the same manner as for SVBSP, but now it is built with respect to the center of the facet under test. On the other hand, the structure of the ROT is
very different because it depicts a dual unlimited space called Line Space subdivided into polygonal cells. This kind of dual projective oriented space allows to represent complex shadow regions generated by facets with respect to a reference one by simple polygons. In this space, points of the two-dimensional real space are projected onto lines and vice versa, by using the so called Plüker coordinates of a line [7]. Referring to Figure 1b, in Line Space planes that support the faces of the ROT, polygonal cell are represented by half-lines crossing the origin of the dual space [5]; these lines correspond to two vertexes of one line out of the four describing the shadow region in the real space. Fixing a reference facet in the real space, the other facets generate new polygonal cells in the Line Space. Therefore by applying Plüker coordinates and describing the polygonal subdivided Line Space by a BSP tree, we are able to easily represent the real twodimensional shadow region generated by the reference facet. The steps of the algorithm are very similar to the ones described in the previous section: build the front-to-back order with respect the reference facet, generate for each facet the line space polygon and describe it by a BSP tree, filter the tree down the ROT to test the visibility, refresh the ROT. It is important to highlight that LSP algorithm works on the static part of the scenario, i.e. if the only changes are in the positions of transmitting or receiving antennas we can use for the new simulation the previous saved visibility matrix without perform a new LSP process. Results To verify the effectiveness of our RT acceleration technique we defined the percentage gain as G = (1 − TA T ) × 100 where TA is the RT time obtained by employing the SVBSP-LSP algorithm, and T is the time obtained through a “brute force” ray tracing. Additionally, we compared our method against the Space Volumetric Partitioning (SVP) also implemented in the simulator. We analyzed several urban scenarios from district of Lucca (Figure 2a) with only one transmitting antenna and one receiver, changing the number of facets N and the order of the contribution c. Referring to Figure 2b and Figure 3a, we can observe that, in opposite with respect to the SVP technique, the percentage gain performed by our algorithm increases with an asymptotic rate reaching the best at 70-80%. In Figure 3b, we show the processing times experienced by the SVBSP and LSP algorithm. Although LSP time becomes very long as the parameter N is increased, it is worth highlighting that LSP can be performed once and its results saved for further simulations. Moreover the memory usage is very small because all the visibility relations can be stored in integer arrays for every facet-observation points pair and in a single integer matrix for the facet-facet pairs. References [1] M. F. Catedra, J. Perez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: applications to picocell and microcell scenarios”, IEEE AP Mag., vol.2, pp.15-28, Apr. 1998. [2] R. P. Torres, L. Valle, M. Domingo, S. Loredo, “An efficient ray-tracing method for radio propagation based on the modified BSP algorithm”, IEEE Vehicular Technology Conference, vol.4, pp. 1967-1971, Sept. 1999. [3] W. C. Thibault, B. F. Naylor, “Set operations on polyhedra using binary space partitioning trees”, SIGGRAPHS ’87 Proc., vol. 21, pp. 153-162, July 1987. [4]. N. Chin, S. Feiner, “Near Real-Time Shadow Generation Using BSP Trees”, Computer Graphics, 23:3, pp. 99-106, August 1989.
[5] J. Bittner, P. Wonka, M. Wimmer, “Visibility Preprocessing for Urban Scenes using Line Space Partitioning”, IEEE 9th Pacific Conference on Computer Graphics and Applications, Oct. 2001. [6] G. E. Athanasiadou, R. Nix, “A Novel 3-D Indoor Ray-Tracing Propagation Model: The Path Generator and Evaluation of Narrow-Band and Wide-Band Predictions”, IEEE Transaction on Vehicular Technology, vol. 49, no. 4 , pp. 1152-1167, July 2000. [7] J. Stolfi, Oriented Projective Geometry: a framework for geometric computations, Academic Press, San Diego, 1991.
(a)
(b)
Fig. 1 a) ROT construction: “in” and “out” leaves describe internal and external areas of shadow region respectively; b) half-lines and planes in two-dimensional real space and dual polygon mapped in Line Space. 100
SVP S V B S P -L S P
Percentage Gain
80
60
40
20
0
360
720
1080
N °, n u m b e r o f fa c e ts
(a)
1440
1800
(b)
Fig. 2 a) A district of Lucca simulated with EMvironment 3.0; b) percentage gain for increasing number of facets (contribution order c=3): SVP (continuous line), SVBSP-LSP (dotted line). 100
SV P
5
10
4
SVBSP
SV BS P-LSP
Preprocessing Time (sec)
80
Percentage Gain
10
60
40
20
LSP
1 00 0 10 0 10 1 0 .1 0 .0 1
0 2
3
4
c, contribution order
(a)
5
0.00 1 0
5 00
10 0 0
1500
2 00 0
25 0 0
3000
P , n u m b e r o f B S P tre e 's n o d e s
(b)
Fig. 3 a) Percentage gain for increasing contribution order c (number of facets N=206): SVP (continuous line), SVBSP-LSP (dotted line); b) Preprocessing time for increasing number of nodes of BSP tree which describes the urban scenarios
