(t, m, s)-nets and Maximized Minimum Distance, Part II

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1 (t,, s)-nets and Maxiized Miniu Distance, Part II Leonhard Grünschloß and Alexander Keller Abstract The quality paraeter t of (t,, s)-nets controls extensive stratification properties of the generated saple points. However, the definition allows for points that are arbitrarily close across strata boundaries. We continue the investigation of (t,, s)-nets under the constraint of axiizing the utual distance of the points on the unit torus and present two new constructions along with algoriths. The first approach is based on the fact that reordering (t, s)-sequences can result in (t,, s + 1)-nets with varying toroidal distance, while the second algorith generates points by perutations instead of atrices. 1 Introduction Iportant probles in iage synthesis like e.g. anti-aliasing, heispherical integration, or the illuination by area light sources can be considered as low-diensional nuerical integration probles. Aong the ost successful approaches to coputing this kind of integrals are quasi-monte Carlo and randoized quasi-monte Carlo ethods [9, 13, ] based on the two-diensional Larcher-Pillichshaer points [10], which expose a coparatively large iniu toroidal distance. These points belong to the class of (t,, s)-nets in base q (for an extensive overview of the topic we refer to [1, Ch. 4, especially p. 48]), which is given by Leonhard Grünschloß ental iages GbH, Fasanenstraße 81, 1063 Berlin, Gerany. eail: leonhard@gruenschloss.org Alexander Keller ental iages GbH, Fasanenstraße 81, 1063 Berlin, Gerany. eail: alex@ental.co 1

2 L. Grünschloß, A. Keller Definition 1. For integers 0 t, a (t,, s)-net in base q is a point set of q points in [0, 1) s such that there are exactly q t points in each eleentary interval E with volue q t. Figure 1 shows an instance of a (0, 4, )-net in base and illustrates the concept of eleentary intervals E = s [ ai q bi, (a i + 1)q bi) [0, 1) s i=1 as used in the definition, where a i, b i Z, b i 0 and 0 a i < q bi. The concept of (t,, s)-nets can be generalized for sequences of points as given by Definition. For an integer t 0, a sequence x 0, x 1,... of points in [0, 1) s is a (t, s)-sequence in base q if, for all integers k 0 and > t, the point set x kq,..., x (k+1)q 1 is a (t,, s)-net in base q. Obviously, the stratification properties of (t,, s)-nets and (t, s)-sequences are best for the quality paraeter t = 0, because then every eleentary interval contains exactly one point, as illustrated in Figure 1. The conception does not consider the utual distance of the points, which allows points to lie arbitrarily close together across shared interval boundaries. In this paper we continue previous work fro [6] that used the iniu toroidal distance d in ({x 0,..., x N 1 ) := in 0 u<v<n x u x v T to classify point sets {x 0,..., x N 1, where the toroidal distance of two points x = (x 1,..., x s ) [0, 1) s and y = (y 1,..., y s ) [0, 1) s is defined as x y T := s (in{ x i y i, 1 x i y i ). i=1 Maxiizing the shift-invariant easure of toroidal distance further increases unifority, allows one to tile the resulting point sets, and to consider periodic integrands, which is especially useful in the aforeentioned graphics applications. In Section we therefore extend the construction of Larcher and Pillichshaer to s = 3 diensions resulting in a (0,, 3)-net in base, which exhibits a large iniu toroidal distance. In Section 3 we then present a new perutation-based construction for (0,, )-nets in base, which often have a larger iniu toroidal distance than can be obtained by any digital net in base.

3 (t,, s)-nets and Maxiized Miniu Distance, Part II 3 Fig. 1 The Larcher-Pillichshaer points as an exaple of a (0, 4, )-net in base superiposed on all possible eleentary intervals. There is exactly one point in each eleentary interval. A new (0,, 3)-Net in Base with Large Miniu Toroidal Distance Although both the Haersley and Larcher-Pillichshaer points are constructions of (0,, )-nets in base, the latter have a uch larger iniu toroidal distance [8]. In fact already in [10, 11] the Haersley points have been identified to be the worst construction with respect to certain other easures, too. The extension of the Larcher-Pillichshaer points to s = 3 diensions has been an open proble for long. In the following, we present a construction of a (0,, 3)-net in base, where one two-diensional projection equals the Larcher-Pillichshaer points [9] and therefore benefits fro their large iniu toroidal distance..1 Digital Nets and Sequences We briefly suarize necessary notation and algorithic facts on digital nets and sequences fro [1] in a siplified anner. While in general digital nets and sequences are defined using a finite field F q, with q being a prie power, we only consider the case q = in the following. Given suitable -atrices C 1,..., C s over F, the i-th coponent of the n-th point for 1 i s and 0 n < can be generated by x (i) n = 1. d 0 (n) C i. [0, 1), d 1 (n) T where the atrix-vector ultiplication has to be perfored in F and the digits d k (n) are defined by the binary expansion of n = 1 k=0 d k (n) k.

4 4 L. Grünschloß, A. Keller The eleents of the i-th generator atrix are denoted by c (i) j,r. The theoretical construction of (t, s)-sequences requires generator atrices of infinite size. However, in practice this does not pose a proble when enuerating the points, since d k (n) = 0 for all sufficiently large k and, in addition, we require the atrix entries c (i) j,r = 0 for all sufficiently large j [1, p. 7, (S6)]. Therefore, only finite upper left subatrices have to be considered when generating points. Finally, due to the finite precision of integer coputation, is finite in any case. Based on the results in [10, 11, 4], we will consider the (0, 1)-sequence in base generated by the non-singular upper triangular atrix ({ ) C 1 1 if j r, := (1) 0 otherwise j,r=0 in order to generate the Larcher-Pillichshaer points [9].. Reordering the Sobol -Sequence We now take a look at the first two coponents of the Sobol -sequence [15], which are a (0, )-sequence in base, and are defined by the infinite upper triangular atrices (( ) ) C 1 := (δ j,r ) r j,r=0 and C := od, j j,r=0 where δ j,r = 1 for j = r and zero otherwise. In the above definition and the reainder of this work, by a od we ean the coon residue. That is the nonnegative value b <, such that a b od. Suppose that the infinite atrices C 1,..., C s over F generate a (0, s)- sequence in base. Furtherore suppose that the infinite atrix C 1 over F generates a (0, 1)-sequence in base. This iplies that C 1 is a nonsingular upper triangular atrix. Then a (0, s)-sequence in base is generated by C 1, C D,..., C s D, where D := C1 1 C 1. This new sequence consists of the sae points as before, however, in a different order. This result is proven in ore general for in [5, Prop. 1]. Note that the first generator atrix C 1 is the one of Larcher-Pillichshaer. Since C 1 is the identity, we have D = C1 1 C 1 = C 1 and obtain

5 (t,, s)-nets and Maxiized Miniu Distance, Part II 5 (( ) ) r C D = od C 1 j j,r=0 ( r ( ) ) k = od j k=0 j,r=0 (( ) ) r + 1 = od =: C j + 1, () j,r=0 as the second generator atrix, where the last equality follows fro the Christas Stocking Theore. Note that the sae reordering can be applied to the Faure-sequence [3], which can be regarded as a generalization of the Sobol -sequence for any prie base q. An even ore general construction for any prie power base can be found in [1]..3 Construction n Cobining the coponent with the first points of a (0, )-sequence in base yields a (0,, 3)-net in base (see [1, Lea 4., p. 6]). The relationship of C 1 and C as expressed in Equation () in fact is a property of all (0, )-sequences in base generated by non-singular upper triangular atrices [7, Prop. 4]. Consequently fixing the first generator atrix uniquely deterines the second generator atrix and thus the (0,, 3)-net. In the previous section we reordered the Sobol -sequence in base such that one coponent atches the (0, 1)-sequence as defined in [10]. For the construction of the new (0,, 3)-net in base, it is sufficient to consider the atrices C 1, C fro Section., because for s = b = Sobol s construction is identical to Faure s construction. By construction two diensions of the resulting three-diensional net are the Larcher-Pillichshaer points [9]. In Figure we copared the iniu toroidal distance of the new construction to the one that uses the original Sobol generator atrices C 1 and C. Except for = 5 and = 6, the new construction is by far superior for all. We did not copute the iniu toroidal distance for >, though..3.1 Ipleentation The following code in C99 (the current ANSI standard of the C language) returns the n-th point of the (0,, 3)-net generated using C 1 and C (see Equations (1) and ()) for < 3 in O(). The vectorized ipleenta-

6 6 L. Grünschloß, A. Keller 1 (0,,3)-net fro Sobol -sequence (0,,3)-net fro LP-atched sequence 0.1 iniu toroidal distance e-05 1e Fig. A plot of the toroidal distance in [0, 1) 3 for both the Sobol (0,, 3)-net as well as the (0,, 3)-net constructed using C 1, C, where one of the two-diensional projections equals the Larcher-Pillichshaer point set. We would like to stress that the iniu distance is scaled logarithically, so the absolute difference is very significant for increasing. In contrast to the Sobol -net, the iniu distance decreases soothly for the new construction. tion is based on the fact that addition in F corresponds to the exclusive or operation. void x_n(unsigned int n, const unsigned int, float x[3]) { // first coponent: n / ^ x[0] = (float) n / (1U << ); // reaining coponents by atrix ultiplication unsigned int r1 = 0, r = 0; for (unsigned int v1 = 1U << 31, v = 3U << 30; n; n >>= 1) { if (n & 1) { // vector addition of atrix colun by XOR r1 ^= v1; r ^= v << 1; // update atrix coluns v1 = v1 >> 1; v ^= v >> 1; // ap to unit cube [0,1)^3 x[1] = r1 * (1.f / (1ULL << 3)); x[] = r * (1.f / (1ULL << 3));

7 (t,, s)-nets and Maxiized Miniu Distance, Part II 7 3 Perutation-Generated (0,, )-Nets in Base with Larger Miniu Toroidal Distance Taking the integer part of the coordinates of a (0,, s)-net in base q ultiplied by the nuber of points q results in each coponent being a perutation fro the syetric group S q. For s = diensions and base q = this is visualized in Figure 1, where each colun and each row (the leftost and rightost set of eleentary intervals in the figure) contain exactly one point. Using Heap s efficient perutation generation algorith [14] allows one to enuerate all perutations π : {0,..., 1 {0,..., 1 that 1 represent (t,, )-nets in base with the points (n, π(n)) [0, 1). We extended this basic backtracking algorith by pruning the search tree whenever the eleentary-interval property t = 0 was violated or already a net with larger iniu toroidal distance than the current one had been found. Verifying the t = 0 property is siple using the code published in [6, Section.]. Perforing such an exhaustive cobinatorial coputer search sees hopeless given the nuber of possible perutations S = ( )! and in fact we did not succeed in running an exhaustive search for > 5. In these cases we siply generated rando perutations, but kept the backtracking approach entioned above. However in [6], we were able to find very regular looking (0, 5, )-nets with a iniu toroidal distance of 3/ , which is larger than the iniu toroidal distance of all possible digital (0, 5, )-nets, where the axiu is 9/ In continuation of these findings we now present a first construction (see Figure 3) of such perutation-based nets. In Table 1 we copare the iniu toroidal distance of different (0,, )-nets in base. The new perutation construction clearly features the largest iniu toroidal distance. Due to the novelty of the approach, we present two different derivations and provide ore interpretations than usually necessary. 3.1 Iterative Construction by Quadrupling Point Sets Given the search results as displayed in Figure 3, an iterative construction procedure can be inferred. This procedure repeatedly quadruples an initial point set until the desired nuber of points is reached. We therefore need to distinguish the cases of even and odd as depicted by the exaples in Figure 4. We represent the (0, k, )-nets in base by

8 8 L. Grünschloß, A. Keller = 3 = 5 = 7 = 4 = 6 = 8 Fig. 3 Exaples of the perutation-generated (0,, )-nets in base for = 3,..., 8. Haersley Larcher-Pillichsh. Opt. atrices Perutation Table 1 Coparison of the Haersley points, the Larcher-Pillichshaer points, the points resulting fro the optiized atrices given in [6], and the new perutation construction with respect to iniu toroidal distance. Note that for easier coparison all distances have been ultiplied by the nuber of points of a (0,, )-net in base and squared afterwards. { 1 k (u od k, v) : (u, v) P k, (3) where the set P k contains k integer coordinates (u, v).

9 (t,, s)-nets and Maxiized Miniu Distance, Part II 9 For odd the initial point set P 1 consists of the points (0, 0) and (1, 1). The quadrupling rule to construct a point set P k+ := P k,0 P k,1 P k, P k,3 with k+ points fro a point set P k with P k = k points is as follows: Lower left: P k,0 := {(u, v) : (u, v) P k Lower right: P k,1 := { ( k+1 + u + 1, v + 1) : (u, v) P k Upper left: P k, := { ( k+1 + u, k+1 + v) : (u, v) P k Upper right: P k,3 := { ( k+ + u + 1, k+1 + v + 1) : (u, v) P k For even, the iterative construction is ore involved. We start out with four diagonals, each consisting of four points: P,0 := {(w + 1, 4w) : w = 0,..., 3, P,1 := {(w + 5, 4w + ) : w = 0,..., 3, P, := {(w + 9, 4w + 1) : w = 0,..., 3, P,3 := {(w + 13, 4w + 3) : w = 0,..., 3. The quadrupling rule for even k constructs P k+ := P k,0 P k,1 P k, P fro sets P k,l for l = 0,..., 3 as follows: 1. Multiply all points by two and add an offset for the points in P 1 and P 3 : P k,l := {(u + (l od ), v) : (u, v) P k,l for l = 0,..., 3.. Extend the diagonals: P k,l := P k,l { ( k 1 + u, k+1 + v) : (u, v) P k,l for l = 0,..., Cobine diagonals: 4. Append shifted copies: P k,0 := P k,0 P k,1, P k,1 := P k, P k,3. P k, := { ( k+1 + u, u + 1) : (u, v) P k,0, P k,3 := { ( k+1 + v, v + 1) : (u, v) P k,1. k,3 3. Direct Construction As these nets consist of points lying on parallel diagonals siilar to rank-1 lattices [], a direct construction is possible by coputing a single point on

10 10 L. Grünschloß, A. Keller a) Exaple for odd : Construction of a (0, 9, )-net in base. b) Exaple for even : Construction of a (0, 8, )-net in base. Fig. 4 Iterative construction of (0,, )-nets in base by quadrupling point sets, where the final net results fro wrapping the points such that they atch the unit square. This wrapping corresponds to the odulo operation in Equation (3). each diagonal. Since the points are evenly spaced along the diagonals, the reaining points follow iediately. These diagonals are to be understood with a odulo wrap-around for the first coordinate Construction for Odd The position of the point with the sallest second coordinate on the d-th diagonal is given by ( φ (d) + d, d), where 0 d < and φ (d) [0, 1) denotes the van der Corput radical inverse in base. Theore 1. For odd, the points { (u d,w, v d,w ) : 0 d <, 0 w <, where u d,w = ( φ (d) + d + w ) od, (4) v d,w = d + w, (5) constitute a (0,, )-net in base with a iniu toroidal distance of. This distance is easured using coponents ultiplied by, i.e. on integer scale.

11 (t,, s)-nets and Maxiized Miniu Distance, Part II 11 Case 1: width > height. Case : width < height. Fig. 5 The two cases in the proof corresponding to certain kinds of eleentary intervals, illustrated here for = 5. Proof. First, we will show that the eleentary interval property holds. For each kind 0 h of eleentary intervals, there ust be exactly one point in each integer-scaled eleentary interval [ x h, (x + 1) h) [ y h, (y + 1) h), where 0 x < h and 0 y < h. Partitioning the set of eleentary intervals as depicted in Figure 5, we need to consider two cases: 1. We first consider the case h. The width of these kinds of eleentary intervals is larger than their height. Looking at Equation (5), we can see that the least significant bits of vd,w are equal to the bits of d, while the reaining ost significant bits of vd,w are equal to the bits of w. Considering one horizontal strip of eleentary intervals, y deterines the h ost significant bits of v d,w. Since h, w is copletely deterined by y. What reains to show is that for this fixed w, the h ost significant bits of u d,w differ for suitable values for d. Analyzing Equation (4), we see that the ter w is constant, while the addition of d only odifies the least significant bits. However { φ (d) : 0 d < = { d : 0 d <, (6) thus due to the addition of φ (d) it is possible to guarantee that the h ost significant bits of ud,w are different by selecting suitable values for d. Thus the corresponding points fall into different eleentary intervals.. Now we consider the case 0 h. The width of these kinds of eleentary intervals is saller than their height. We consider one vertical strip of eleentary intervals of width h. Looking at Equation (4), we can see that the the only way to achieve consecutive values u d,w equal to x, x + 1,..., x + h 1 is by using consecutive values for d. That is because the ters φ (d) and w do not odify the h least significant bits of u d,w. However h consecutive values for d ean that the h ost significant bits of the values φ (d) are different for each d (cf. to Equation (6)). In order to stay in the vertical strip of eleentary intervals deterined by x, the h ost significant bits of w thus ust be

12 1 L. Grünschloß, A. Keller chosen accordingly for each point inside this strip in order to copensate using the ter w. As a consequence of the different values for w it follows fro Equation (5) that the h ost significant bits of vd,w are different for each point inside this strip, thus they fall into different eleentary intervals. We now consider the achieved iniu toroidal distance of( the point set. Points on the diagonals are placed with a ultiple of the offset, ), so their squared iniu toroidal distance to each other is ( ) =. The squared distance between the diagonals with slope 1 is ( ) =. In conclusion, the iniu toroidal distance is. As the diagonals can be tiled sealessly, the result is identical for the toroidal distance easure. On the unit square [0, 1), we need to divide the point coordinates by, so the iniu toroidal distance on the unit square is =, which concludes the proof. The theore allows for an additional interpretation of the structure: Using Equation (4), we can solve for d and w given u d,w : d = u d,w od, w = (u d,w φ (d) d) od. Inserting these equations into Equation (5) yields ( v d,w = d + (u d,w φ u d,w od ) ) d od ( = (u d,w φ u d,w od )) od, (7) which can be regarded as replicating a (0,, ) integer Haersley-net ties horizontally, scaling each one vertically by and finally adding a linear coponent to the cobination of nets. The resulting points are wrapped around the unit square.

13 (t,, s)-nets and Maxiized Miniu Distance, Part II 13 Fig. 6 On the left, the odified tiling for the perutation construction is shown: is even and each row of the tiling is shifted by (, 0) relative to its adjacent row below. Using an axis aligned tiling on the right reveals that all tiles in a row share the sae saple pattern, while the pattern differs by row by a odulo wrap-around in direction of the x-axis. 3.. Construction for Even For even 6 the perutation search did not reveal an iproveent of the iniu toroidal distance in coparison to the full atrix search. We would like to note that the perutation search for = 6 did not finish, though. However, when allowing a distance easure that accounts for a slightly irregular tiling, an approach very uch resebling the previous construction can be taken, yielding the nets shown in Figure 3. Using a tiling where each row is shifted by (, 0) relative to its adjacent row below (see Figure 6), we can get a sealess tiling using diagonals, each with points. Again, the points are spaced evenly on each diagonal, this tie by the offset vector (, ). The position of the point with the sallest second coordinate on the d-th diagonal, where d { 0,..., 1, is given by ( φ (d) + d 4 +, d). With the odified tiling described above, these diagonals are continued sealessly. These nets cannot be generated via the classical way of using generator atrices as the point (0, 0) is not included. While we believe that a siilar proof that t = 0 is possible as for the odd case, we only verified the t = 0 property using a coputer progra for all even. The iniu toroidal distance equals 41 for = 8 and 41 8 / for all even where 10. See Table 1 for iniu toroidal distance values for < 8. We would like to stress that the odified iniu toroidal distance easure that respects the shifts of the rows in the tiling has been used for these easureents. Considering a superiposed axis-aligned tiling as illustrated in Figure 6 reveals that in a row each tile contains the sae set of saples, while the set of saples varies by row. This in turn allows for constructing a larger pattern that consists of shifted copies of the original pattern as shown in Figure 7. Four rows and four coluns of such odulo-wrapped patterns were cobined

14 14 L. Grünschloß, A. Keller Fig. 7 By taking 4 4 odulo-wrapped copies of the sae pattern (see Figure 6), we obtain a larger pattern that is periodic and atches the pixel arrangeent. It can be tiled regularly without a decrease in iniu toroidal distance. to generate a larger pattern. This pattern can be tiled regularly without a decrease in iniu toroidal distance. 3.3 Ipleentation for General Exploiting the fact that we are generating nets in base, all operations needed to ipleent the perutation net constructions described above can be ipleented very efficiently without any ultiplications. A sall C++ class that generates the point sets directly for even and odd 4 is available at In ters of perforance, it is coparable to the generation of rank-1 lattice points and also includes code to generate points with the odified tiling explained above. As an exaple, we would like to show the ipleentation of Equation (7). Given the function phi to copute the van der Corput radical inverse in base [9], the ipleentation y has a one line body. inline unsigned int phi(unsigned int bits) { // for 3 bits bits = (bits << 16) (bits >> 16); bits = ((bits & 0x00ff00ff) << 8) ((bits & 0xff00ff00) >> 8); bits = ((bits & 0x0f0f0f0f) << 4) ((bits & 0xf0f0f0f0) >> 4); bits = ((bits & 0x ) << ) ((bits & 0xcccccccc) >> ); bits = ((bits & 0x ) << 1) ((bits & 0xaaaaaaaa) >> 1); return bits; inline unsigned int y(const unsigned int x, const unsigned int ) { return (x - (phi(x & ~-(1 << ( >> 1))) >> (3 - ))) & ~-(1U << ); For successive point requests this can be optiized by precoputing the bitasks and the different radical inverse values.

15 (t,, s)-nets and Maxiized Miniu Distance, Part II 15 4 Conclusion We constructed new (0,, s)-nets in base for s =, 3, which are constrained by axiizing the iniu toroidal distance. Especially the (0,, 3)-net has any applications in coputer graphics such as coputing anti-aliasing and otion blur in the Reyes architecture [1]. Acknowledgeents We would like to thank the reviewers and editors for their very helpful coents. We would also like to thank Sabrina Daertz, Johannes Hanika, Sehera Nawaz, Matthias Raab, and Daniel Seibert for helpful discussions. References 1. Cook, R., Carpenter, L., Catull, E.: The Reyes iage rendering architecture. SIG- GRAPH Coputer Graphics 1(4), (1987). Daertz, S., Keller, A.: Iage synthesis by rank-1 lattices. In: A. Keller, S. Heinrich, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 006, pp Springer (008) 3. Faure, H.: Discrépance de suites associées à un systèe de nuération (en diension s). Acta Arithetica 41(4), (198) 4. Faure, H.: Discrepancy and diaphony of digital (0, 1)-sequences in prie base. Acta Arithetica 117, (005) 5. Faure, H., Tezuka, S.: Another rando scrabling of digital (t, s)-sequences. In: K. Fang, F. Hickernell, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 000, pp Springer-Verlag, Berlin (00) 6. Grünschloß, L., Hanika, J., Schwede, R., Keller, A.: (t,, s)-nets and axiized iniu distance. In: A. Keller, S. Heinrich, H. Niederreiter (eds.) Monte Carlo and Quasi-Monte Carlo Methods 006, pp Springer (008) 7. Hofer, R., Larcher, G.: On existence and discrepancy of certain digital Niederreiter- Halton sequences (009). To appear in Acta Arithetica 8. Keller, A.: Myths of coputer graphics. In: H. Niederreiter (ed.) Monte Carlo and Quasi-Monte Carlo Methods 004, pp Springer (006) 9. Kollig, T., Keller, A.: Efficient ultidiensional sapling. Coputer Graphics Foru 1(3), (00) 10. Larcher, G., Pillichshaer, F.: Walsh series analysis of the L -discrepancy of syetrisized point sets. Monatsheft Matheatik 13, 1 18 (001) 11. Larcher, G., Pillichshaer, F.: Sus of distances to the nearest integer and the discrepancy of digital nets. Acta Arithetica 106, (003) 1. Niederreiter, H.: Rando Nuber Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (199) 13. Pharr, M., Huphreys, G.: Physically Based Rendering: Fro Theory to Ipleentation. Morgan Kaufann Publishers Inc., San Francisco, CA, USA (004) 14. Sedgewick, R.: Perutation generation ethods. ACM Coputing Surveys 9(), (1977) 15. Sobol, I.: On the distribution of points in a cube and the approxiate evaluation of integrals. Zhurnal Vychislitelnoi Mateatiki i Mateaticheskoi Fiziki 7(4), (1967)