A Model for the Expected Running Time of Collision Detection using AABB Trees

Transcription

1 Eurographics Symposium on Virtual Environments (26) Roger Hubbol an Ming Lin (Eitors) A Moel for the Expecte Running Time of Collision Detection using AABB Trees Rene Weller2 an Jan Klein an Gabriel Zachmann2 MeVis, Center for Meical Diagnostic Systems an Visualization, Bremen, Germany 2 Department of Computer Science, Clausthal University, Germany Abstract In this paper, we propose a moel to estimate the expecte running time of hierarchical collision etection that utilizes AABB trees, which are a frequently use type of bouning volume (BV). We show that the average running time for the simultaneous traversal of two binary AABB trees epens on two characteristic parameters: the overlap of the root BVs an the BV iminishing factor within the hierarchies. With this moel, we show that the average running time is in O(n) or even in O(log n) for realistic cases. Finally, we present some experiments that confirm our theoretical consierations. We believe that our results are interesting not only from a theoretical point of view, but also for practical applications, e. g., in time-critical collision etection scenarios where our running time preiction coul help to make the best use of CPU time available.. Introuction Bouning volume hierarchies (BVHs) have proven to be a very efficient ata structure for collision etection (CD), even for (reuce) eformable moels [JP4]. The iea of BVHs is to partition the set of object primitives (e. g. polygons or points) recursively until some leaf criterion is met. In most cases, each leaf contains a single primitive, but the partitioning can also be stoppe when a noe contains less than a fixe number of primitives. Each noe in the hierarchy is associate with a subset of the primitives an a BV that encloses this subset. Given two BVHs, one for each object, virtually all CD approaches traverse the hierarchies simultaneously by an algorithm similar to Algorithm. It conceptually traverses a bouning volume test tree (BVTT; see Figure 2) until all overlapping pairs of BVs have been visite. It allows to quickly zoom in to areas of close Figure : Some moels of our test suite: Infinity Triant ( lock (courtesy by BMW) an pipes. proximity an stops if an intersection is foun or if the traversal has visite all relevant sub-trees. Most ifferences between hierarchical CD algorithms lie in the type of BV, the overlap test, an the algorithm for constructing the BVH. There are two conflicting constraints for choosing an appropriate BV. On the one han, a BV-BV overlap test uring the traversal shoul be one as fast as possible. On the other han, BVs shoul enclose their subset of primitives as tight as possible so as to minimize the number of false positives with the BV-BV overlap tests. As a consequence, a wealth of BV types has been explore in the past, such as spheres [Hub96, PG95], OBBs [GLM96], DOPs [KHM 98, Zac98], Boxtrees [Zac2,ABG ], AABBs [vb97,lam], spherical shells [KGL 98] an convex hulls [EL]. In orer to capture the characteristics of ifferent approaches an to estimate the time require for a collision query, the cost function T = NvCv + N pc p + NuCu + Ci was propose [GLM96, KHM 98, He99], where Nv,Cv = N p,c p = Nu,Cu = Ci = num. an avg. costs of BV overlap tests, resp. num./avg. costs of primitive intersection tests num. an avg. costs of BV upates, resp. initialization costs An example of a BV upate is the transformation of the BV into a ifferent coorinate system. During a simultaneous traversal of two BVHs, the same

2 traverse(a, B) if A an B o not overlap then return if A an B are leaves then return intersection of primitives enclose by A an B else for all chilren A i an B j o traverse(a i,b j ) Algorithm : Most hierarchical collision etection methos implement this algorithm to traverse two given BVHs. BVs might be visite multiple times. However, if the BV upates are not save, then N v = N u. This cost function was introuce by [WHG84] to analyze hierarchical methos for ray tracing an later aapte to hierarchical collision etection methos by [GLM96, KHM 98, He99]. In practice, N v, the number of overlap tests, usually ominates the running time, i. e., T (n) N v(n), because N p = 2 Nv in a binary tree an Nu Nv. While it is obvious that N v = n 2 in the worst case, it has long been notice that, in practice, this number seems to be linear or even sub-linear. However, until now there is no rigorous average-case analysis for the running time of simultaneous BVH traversals. Therefore, the goal of this paper is to present a moel, with which one can estimate the average number N v, the number of overlap tests in the average case. In this paper, we restrict ourselves to AABB trees (axis-aligne bouning box trees), which allows us to estimate the probability of an overlap of a pair of bouning boxes by simple geometric reasoning. 2. Relate Work In the last few years, some very interesting theoretical results on the collision etection problem have been shown. One of the first results was presente by Dobkin an Kirkpatrick [DK85]. They have shown that the istance of two convex polytopes can be etermine in time O ( log 2 n ), where n = max{ A, B }, an A an B are the number of faces of object A an B, respectively. For two general polytopes whose motion is restricte to fixe algebraic trajectories, [ST95] have shown that there is an O ( n 5 3 +ε) algorithm for rotational movements, an an o(n 2 ) algorithm for a more flexible motion that still has to be along fixe, known trajectories [ST96]. [SHH98] prove that for n convex, well-shape polytopes (with respect to aspect ratio an scale factor), all A B A,B A A 2 B B 2 A,B A,B 2 A 2,B A 2,B 2 Figure 2: The BV test tree (BVTT) shows all possible pairs of BVs that might nee to be teste for overlap. All hierarchical CD algorithms, such as the one in Algorithm, basically perform a traversal through this (conceptual) tree. intersections can be compute in time O ( (n+k)log 2 n ), where k is the number of intersecting object pairs. They have generalize their approach to first averageshape results in computational geometry [ZS99]. Uner mil coherence assumptions, [VCC98] showe linear expecte time complexity for the CD between n convex objects. They use well-known ata structures, namely octrees an heaps, along with the concept of spatial coherence. The Lin-Canny algorithm [LC9] is base on a closest-feature criterion an makes use of Voronoi regions. Let n be the total number of features, the expecte run time is between O ( n ) an O(n) epening on the shape, if no special initialization is one. In [KZ3], an average-case approach for CD was propose. However, no analysis of the running time was given. 3. Analyzing Simultaneous Hierarchy Traversals In this section, we will erive a moel that allows to estimate the number N v, the number of BV overlap tests. This is equivalent to the number of noes in the BVTT (see Fig. 2) that are visite uring the traversal. The orer an, thus, the exact traversal algorithm are irrelevant. For the most part of this section, we will eal with 2-imensional BVHs, for sake of illustration. At the en, we exten these consierations to 3D, which is fairly trivial. The general approach of our analysis is as follows. For a given level l of the BVTT, we estimate the probability of an overlap by recursively resolving it to similar probabilities on higher levels. This yiels a proucct of conitional probabilities. Then, we estimate the conitional probabilities by geometric reasoning. Let Ñ v (l) be the expecte number of noes in the BVTT that are visite on level l. Clearly, Ñ (l) v = 4 l P[A (l) B (l) ] () where P[A (l) B (l) ] enotes the probability that any pair of boxes on level l overlaps. In orer to rener the text more reaable, we will omit the part an just write P[A (l) B (l) ] henceforth.

3 a y a' y a x a' x A 2 A A L a y + b y P L ay + by a y L a y + b y L ay + by P B B B 2 o b' x b' y b y ay a y b y by b y b x Figure 3: Left: general configuration of the boxes, assume throughout our probability erivations. For sake of clarity, boxes are not place flush with each other. Mile: The ratio of the length of segments L an L equals the probability of A overlapping B. Right: itto for p 2. Overall, the expecte total number of noes we visit in the BVTT is Ñ v(n) = Ñ (l) v = 4 l P[A (l) B (l) ] (2) l= l= where = log 4 (n 2 ) = lg(n) is the epth of the BVTT (equaling the epth of the BVHs). In orer to erive a close-form solution for P[A (l) B (l) ], we recall the general equations for conitional probabilities: an, in particular, if X Y P[X Y ] = P[Y ] P[X Y ] (3) P[X] = P[Y ] P[X Y ] (4) where X an Y are arbitrary events (i. e., subsets) in the probability space. Let o (l) x enote the overlap of a given pair of bouning boxes when projecte on the x-axis, which we call the x-overlap. Then, P[A (l) B (l) ] = P[A (l) B (l) A (l ) B (l ) o (l) x > ] by Eq. 4, an then, by Eq. 3, P[A (l ) B (l ) o (l) x > ] P[A (l) B (l) ] = P[A (l) B (l) A (l ) B (l ) o (l) x > ] P[A (l ) B (l ) ] P[o (l) x > A (l ) B (l ) ] Now we can recursively resolve P[A (l ) B (l ) ], which yiels P[A (l) B (l) ] = l P[A (i) B (i) A (i ) B (i ) o (i) x > ] i= 3.. Preliminaries l P[o (i) x > A (i ) B (i ) ] (5) i= Before proceeing with the erivation of our estimation, we will set forth some enotations an assumptions. Let A := A (l) an B := B (l). In the following, we will, at least temporarily, nee to istinguish several cases when computing the probabilities from Eq. 5, so we will enote the two chil boxes of A an B by A,A 2 an B,B 2, resp. For sake of simplification, we assume that the chil boxes of each BV sit in opposite corners within their respective parent boxes. Furthermore, without loss of generality, we assume an arrangment of A, B, an their chilren accoring to Figure 3, so that A an B overlap before A 2 an B o (if at all). Finally, we assume that there is a constant BV iminishing factor throughout the hierarchy, i.e., a x = α xa x, a y = α ya y, etc. Only for sake of clarity, we assume that the scale of the boxes is about the same, i. e., b x = a x, b x = a x, etc. Accoring to our experience, this is a very mil assumption [Zac2].

4 This assumption allows us some nice simplifications in Equations 6 an, but it is not necessary at all Probability of Box Overlap In this section, we will erive the probability that a given pair of chil boxes overlaps uner the conition that their parent boxes overlap. Since we nee to istinguish, for the moment, between 4 ifferent cases, we efine a shorthan for the four associate probabilities: p i j := P[A i B j A B o x > ] One of the parameters of our probability function is the istance o x () := δ, by which the root box B () penetrates A () along the x axis from the right. Our analysis consiers all arrangments as epicte in Figure 3, where δ is fixe but B is free to move vertically, uner the conition that A an B overlap. First, let us consier p (see Figure 3). By preconition, A overlaps B, so the point P (efine as the upper left (common) corner of B an B ) must be on a certain vertical segment L that has the same x coorinate as the point P. Its length is a y +b y. Note that for sake of illustration, segment L has been shifte slightly to the right from its true position in Figure 3 (center). If, in aition, A an B overlap, then P must be on segment L. Thus, p = Length(L ) Length(L) = a y + b y a y + b y = α y. (6) Next, let us consier p 2 (see Figure 3; for sake of clarity, we re-use some symbols, such as a x). For the moment, let us assume o 2,x > ; in Section 3.3 we estimate the likelihoo of that conition. Analogously as above, P must be anywhere on segment L, so p 2 = α y = p an, by symmetry, p 2 = p 2. Very similarly, we get p 22 = α y. At this point, we have shown that p i j α y in our moel Probability of X-Overlap We can trivially boun P[o (i) x > A (i ) B (i ) ] Actually, P can be chosen arbitrarily, uner the conition that stays fixe on B as B assumes all possible positions. L woul be shifte accoringly, but its length woul be the same. α x α y T (n) < /4 O ( ) /4 O ( lgn ) ( ) /8.35 O n 3/4 O ( n.58) O ( n 2) Table : Effect of the BV iminishing factor α y on the running time of a simultaneous hierarchy traversal. Plugging this into Equation 2, an substituting that in Equation 5 yiels Ñ v(n) 4 l α l y = (4αy)+ 4α l= y (4α y ) O ( (4α y) ) = O ( n lg(4αy)). (7) The corresponing running time for ifferent α y can be foun in Table. For α y > /4, the running time is in O ( n c), < c 2. In orer to erive a better estimate for P[o (l) x > A (l ) B (l ) ], we observe that the geometric reasoning is exactly the same as in the previous section, except that we now consier all possible loci of point P when A an B are move only along the x-axis. Therefore, we estimate P[o (l) x > A (l ) B (l ) ] α x. (8) Plugging this into Equations 2 an 5 yiels an overall estimate Ñ v(n) 4 l α l x α l y O ( n lg(4αxαy)). (9) l= This results in a table very similar to Table The 3D Case As mentione above, our consierations can be extene to 3D straight-forwarly. In 3D, L an L in Equation 6 are not line segments any longer, but 2D rectangles in 3D lying in the y/z plane. The area of L can be etermine by (a y +b y)(a z +b z) an the area of L by (a y + b y)(a z + b z). Thus, p = area(l ) area(l) = (a y + b y)(a z + b z) (a y + b y)(a z + b = 4a ya z = α yα z. z) 4a ya z () The other probabilities p i j can be etermine analogously as above, so that p = p 2 = p 2 = p 22 = α yα z. Overall, we can estimate the number of BV overlap tests by Ñ v(n) 4 l α l x α l y α l z O ( n lg(4αxαyαz)). () l= where = log 4 (n 2 ) = lg(n). Note that Table is still vali in the 3D case.

5 num. BV overlap tests lock car pipes n Figure 4: The number of visite BVTT noes for moels shown in Fig. at istance δ = Experimental Support Intuitively, not only α shoul be a parameter of the moel of the probabilities (Eqs. 6 an 8), but also the amount of penetration of the root boxes. This is not capture by our moel, so in this section we present some experiments that provie a better feeling of how these two parameters affect the expecte number of BV overlap tests. We have implemente a version of Algorithm using AABBs as BVs (in 3D, of course). As we are only intereste in the number of visite noes in the BVTT, we switche off the intersection tests at the leaf noes. For the first experiment, we use a set of CAD objects, each of them with varying numbers of polygons (Fig. ). Fig. 4 shows the number of BV overlap tests for our moels epening on their complexities for a fixe istance δ =.4. Clearly, the average number of BV overlap tests behaves logarithmically for all our moels. For our secon experiment, we use artificial BVHs where we can ajust the BV iminishing factors α x,y,z. As above, the chil BVs of each BV are place in opposite corners. In aition, we varyie the root BV penetration epth δ. We plotte the results for ifferent choices of α an n, average over the range..9 for δ (see Fig. 5). For larger α s, this seems to match our theoretical results. For smaller α, our moel seems to unerestimate the number of overlapping BVs. However, it seems, that the asymptotical running-time oes not epen very much on the amount of overlap of the root BVs, δ (see Fig. 7). 5. Application to Time-Critical Collision Detection As observe in [KZ3], almost all CD approaches use some variant of Algorithm, but often, there is no special orer efine for the traversal of the hierarchy, which can be exploite to implement time-critical computing. Our probability moel suggests one way how to prioritize the traversal. for a given BVH, we can measure the average BV iminishing factor for each subtree an store this with the noes. Then, uring run-time, a goo heuristic coul be to traverse the subtrees with lower α-values first, because in these subtrees the expecte number of BV pairs we have to check is asymptotically smaller than in the other subtrees. In aition, we coul tabulate the plots in Figure 7 (or fit a function), an thus compute a better expecte number of BV overlaps uring run-time of time-critical collision etection. 6. Conclusion an Future Work We have presente an average-case analysis for simultaneous AABB tree traversals, uner some assumptions about the AABB tree, that provies a better unerstaning of the performance of hierarchical collision etections, which has been observe in the past. Our analysis is inepenent of the orer of the traversal. In aition, we have performe several experiments to support the correctness of our moel. Moreover, we have shown that the running time behaves logarithmically for real worl moels, even for a large overlap between the root BVs. Several existing methos for hierarchical collision etection may benefit from our analysis an our moel. Especially in time-critical environments or real-time applications it coul be very helpful to preict the running-time of the collision etection process only with the help of two parameters that can be etermine on-the-fly. We will try to spee up probabilistic collision etection by the heuristics mentione in this paper. We have alreay trie to erive a theoretical moel of the probabilities that epens on the BV iminishing factor as well as the penetration istance of the two root BVs. This woul, hopefully, lea to a probability ensity function escribing the x-overlaps, thus yieling a better estimat of Ñ v (l). However, this challenge seems to be ifficult. Furthermore, a particular challenge will be a similar average-case analysis for BVHs utilizing other types of BVs, such as DOPs or OBBs. The geometric reasoning woul probably have to be quite ifferent from the one presente in this paper. Finally, it woul be very interesting to apply our technique to other areas, such as ray tracing. An, finally, we believe one coul exploit these ieas to obtain better bouning volume hierarchies.

6 # BV overlap tests / 6 exp. 5 theor exp. 2 theor. # BV overlap tests / Figure 5: For larger values of α, our theoretical moel seems to match the experimental finings fairly well (left: α =.7, right: α =.9. # BV overlap tests / δ=.2 δ=.4 δ=.6 δ=.8 # BV overlap tests / δ=.2 δ=.4 δ=.6 δ=.8 # BV overlap tests / δ=.2 δ=.4 δ=.6 δ= Figure 6: The asymptotic number of overlapping BVs epens mainly on α, the BV iminishing factor, an only to a minor extent on δ, the penetration epth of the root BVs. The plots from left to right show α =.6,.7,.8. # BV overlaps / δ α.6.5 # BV overlaps / δ α.6.5 Figure 7: For each number of leaves in the BVH, the istribution of overlapping BVs seems to be nearly the same. Left: Each BVH has n = 52 leaves, right: each has n = 892 leaves. Acknowlegement We woul like to exten our thanks to one anonymous reviewer (you know who you are), who has aske us a frienly an seemingly artless question an thus, we believe, spurre a significant improvement of this work. This work was partially supporte by DFG grant ZA292/-. References [ABG ] Agarwal P. K., e Berg M., Gumunsson J., Hammar M., Haverkort H. J.: Box-trees an r-trees with near-optimal query time. In Proc. Seventeenth Annual Symposium on Computational Geometry (SCG 2) (2), pp [DK85] Dobkin D. P., Kirkpatrick D. G.: A linear algorithm for etermining the separation of convex polyhera. J. Algorithms 6, 3 (985), [EL] Ehmann S. A., Lin M. C.: Accurate an fast proximity queries between polyhera using convex surface ecomposition. Computer Graphics Forum (Proc. of EUROGRAPHICS 2) 2, 3 (2), 5 5. [GLM96] Gottschalk S., Lin M., Manocha D.: OBB- Tree: A hierarchical structure for rapi interference etection. ACM Transactions on Graphics (SIGGRAPH 996) 5, 3 (996), 7 8. [He99] He T.: Fast collision etection using quospo trees. In SI3D 99: Proceeings of the 999 symposium on Interactive 3D graphics (999), pp [Hub96] Hubbar P. M.: Approximating polyhera with

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