1 PREPRINT, MARCH 1, Multiimensional Sampling of Isotropically Banlimite Signals Erik Agrell an Balázs Csébfalvi arxiv: v1 [cs.it] 1 Mar 017 Abstract A new lower boun on the average reconstruction error variance of multiimensional sampling an reconstruction is presente. It applies to sampling on arbitrary lattices in arbitrary imensions, assuming a stochastic process with constant, isotropically banlimite spectrum an reconstruction by the best linear interpolator. The lower boun is exact for any lattice at sufficiently high an low sampling rates. The two threshol rates where the error variance eviates from the lower boun gives two optimality criteria for sampling lattices. It is prove that at low rates, near the first threshol, the optimal lattice is the ual of the best sphere-covering lattice, which for the first time establishes a rigorous relation between optimal sampling an optimal sphere covering. A previously known result is confirme at high rates, near the secon threshol, namely, that the optimal lattice is the ual of the best sphere-packing lattice. Numerical results quantify the performance of various lattices for sampling an support the theoretical optimality criteria. I. INTRODUCTION By generalizing the classical sampling theorem to multiple imensions, it has been proven that the Nyquist rate for isotropically banlimite signals, i.e., the lowest sampling rate that allows error-free reconstruction, is etermine by the ensest sphere-packing lattice , . In three imensions, for example, the boy-centere cubic (BCC) lattice is optimal for sampling, because its ual, the face-centere cubic (FCC) lattice, is the optimal sphere packing [3, Ch. 1]. In practice, however, it is not always possible to satisfy the Nyquist criterion. In this case, the spectrum of the original signal is replicate aroun the points of the ual lattice in the frequency omain such that an overlapping between the replicas cannot be avoie. This overlapping causes the typical prealiasing effects, for example, in volume visualization applications . Several papers , [6, Sec. 4.1],  rely on a conjecture that an optimal sphere-covering lattice ensures minimal overlap in the frequency omain. As the BCC lattice is optimal for three-imensional sphere covering [3, Ch. ], its ual, the FCC lattice, is suppose to be optimal for sampling isotropically banlimite signals below the Nyquist rate in the spatial omain. However, Va et al.  emonstrate that the overlap between the replicas of the spectrum epens very much on the sampling rate, which results in ifferent optimality ranges for the BCC an FCC lattices. In this paper, we present an exact expression for the reconstruction error variance for signals with constant, isotropically banlimite spectrum using any sampling lattice, in any imension, an at any rate. A lower boun on the error variance is erive, which is tight at high an low rates. Comparing the error variance with the lower boun allows us to ientify optimal lattices for sampling at any rate. It is shown that the ual of the best sphere-packing lattice is optimal not only just above the Nyquist rate, but also in a range below the Nyquist rate. The ual of the best sphere-covering lattice is optimal at significantly lower sampling rates, where the reconstruction variance is high. To our best knowlege, this is the first time a mathematically precise relationship is establishe between optimal low-rate sampling an the sphere-covering problem. Finally, the reconstruction error variance is numerically compute for lattices in, 3, 4, an 8 imensions, supporting the theoretical results an establishing rate intervals for the optimality of various well-known lattices. II. DEFINITIONS AND PROBLEM STATEMENT Let B R n an A R n be matrices such that n an all elements of AB T /(π) are integers. We enote with Λ(B) the -imensional lattice generate by B, an its ual, scale by π, is Λ(A). The Voronoi cell, packing raius, an covering raius of Λ(B) are enote by Ω(B), ρ(b), an R(B), resp. The lattice Λ(B) is use for sampling a stationary stochastic process. We will refer to the process as being a function of (multiimensional) time, although the theory applies equally well to spatial or other processes. Thus, the omain where Λ(B) resies is time an the omain of Λ(A) is (angular) frequency. The lattice ensity 1/ vol Ω(B), where vol enotes volume, gives the number of samples per unit volume an can be regare as a multiimensional analogy of the sampling rate. We therefore efine the sampling rate in ra/s as r π(vol Ω(B)) 1/ (1) = (vol Ω(A)) 1/. () An isotropically banlimite stochastic process is characterize by a power spectral ensity being uniform over a - imensional sphere S 0, ω, S(ω) = (3) 0, ω /, where ω R : ω ω 0 } (4) an ω 0 is the (angular) banwith of the process. The process variance is σ = 1 (π) S(ω)ω (5) R = S 0, (6) (π)
2 PREPRINT, MARCH 1, 017 where we recall that the volume of a -imensional sphere is = π/ ω0 Γ ( (7) + 1). This paper eals with the problem of minimizing the average reconstruction error variance σ e with the best linear interpolator [1, Sec. VI], , for given ω 0 an B. III. THE RECONSTRUCTION ERROR The main results in this paper are an exact expression for the average reconstruction error variance when the signal has a constant, isotropically banlimite spectrum (3) an a lower boun thereon, given by Theorems 1 an, resp. Theorem 1: When an isotropically banlimite process is sample on the lattice Λ(B) an reconstructe using the best linear interpolator, the error variance is σ vol ( \ Ω(A)) e = σ. (8) Proof: The theorem can be erive as a special case of the last equation in [10, Sec. III-B]. More precisely, it follows from [10, Eq. (13)] that σ e = σ 1 λ Λ(A) S (ω + λ) (π) ω (9) Ω(A) where a value of 0/0 shoul be interprete as 0. Consier first any pair of points ω Ω(A) \ an λ Λ(A). Their sum satisfies ω + λ ω > ω 0, where the first inequality follows from the efinition of Ω(A) an the secon from the efinition (4) of. Thus, by (3) (4), = 0 an the numerator an enominator of (9) are both 0. It follows that points ω Ω(A) \ o not contribute to the integral, an (9) can therefore be rewritten as σ e = σ 1 (π) Ω(A) λ Λ(A) S (ω + λ) ω. (10) Since S(ω) is uniform, S (ω) = S 0 S(ω) for all ω an σ e = σ 1 λ Λ(A) S 0 (π) ω (11) Ω(A) = σ S 0 vol(ω(a) ) (1) (π) = σ S 0 [ vol( \ Ω(A))] (13) (π) vol( \ Ω(A)) = σ, (14) where the last step follows from (6). In the next section, we will evaluate (8) as a function of the sampling rate for ifferent lattices, which in general involves -imensional numerical integration. A close-form lower boun, which epens only on the normalize sampling rate r/ω 0, can be erive as follows. Theorem : For any lattice, the reconstruction error variance satisfies σ e σ lb, where σ lb σ max 0, 1 Γ( + 1) π / ( ) } r. (15) ω 0 TABLE I THE RECONSTRUCTION ERROR VARIANCE DEVIATES FROM ITS LOWER BOUND WHEN r/ω 0 IS BETWEEN THESE TWO THRESHOLDS, WHERE THE FIRST DEPENDS ON THE DUAL LATTICE S COVERING RADIUS AND THE SECOND ON ITS PACKING RADIUS. Lattice Λ(B) (vol Ω(A)) 1/ /R(A) (vol Ω(A)) 1/ /ρ(a) Integers Z 1 Square Z = 1.41 Hexagonal A 3 3/4 / = /4 = 1.86 Cubic Z 3 3 / 3 = 1.15 BCC A 3 3 1/3 = 1.6 5/6 = 1.78 FCC A 3 3 5/3 / 5 = 1.4 5/3 / 3 = 1.83 Z D 4 4 1/4 = /4 = 1.68 A /8 / = /8 = 1.83 Z 8 8 1/ = 0.71 E = 1.41 A /8 / 0 = /8 / = 1.85 Further, σ e = σ lb if an only if ω 0 ρ(a) or ω 0 R(A). Proof: Bouning the numerator of (8) using vol( \ Ω(A)) 0 an vol( \ Ω(A)) vol Ω(A) yiels σ e σ max 0, 1 vol Ω(A) }. (16) The right-han sie of (16) can be evaluate using (7) an (), which yiels (15). To establish the if an only if conitions for σ e = σ lb, we observe that ω 0 ρ(a) Ω(A) \ Ω(A) = an ω 0 R(A) Ω(A) vol( \ Ω(A)) = vol Ω(A). Example 1: As a sanity check of Theorem 1, we consier the one-imensional case. The spectrum is flat in = [ ω 0, ω 0 ] an the sampling instants are Λ(B) =..., 0, π/r, 4π/r,...}. The ual lattice is Λ(A) =..., 0, r, r,...} with Ω(A) = [ r/, r/]. It is easily verifie that vol( \ Ω(A)) = ω 0 r if r < ω 0 an 0 otherwise. Hence, Theorem 1 yiels σ σ rσ e = ω 0, if r < ω 0, (17) 0, if r ω 0, which is as expecte from the stanar (one-imensional) sampling theorem. The lower boun in Theorem is tight everywhere in the one-imensional case, since ρ(a) = R(A) = r/. Theorem explains why multiimensional sampling of isotropically banlimite processes enables perfect reconstruction also for some rates below the stanar (one-imensional) Nyquist rate. The reconstruction error is zero whenever ω 0 ρ(a), in other wors when r/ω 0 (vol Ω(A)) 1/ /ρ(a). This threshol rate is liste the right column of Table I for various lattices. It equals for the cubic lattice in any imension, which agrees with the stanar (one-imensional) sampling theorem. It is lower for several other lattices in imensions, reflecting the fact that their Voronoi cells are more spherical than the -imensional cube. The minimum threshol rate over all possible lattices in a given imension can be regare as the Nyquist rate in that imension. For example, whereas r ω 0 is require if = 1, r 1.86ω 0 is sufficient if = an only r 1.41ω 0 if = 8.
3 PREPRINT, MARCH 1, Σ Square Hexagonal Boun Σ 4 D 4 A 4 Boun Σ Cubic BCC FCC Boun Σ 8 E 8 A 8 Boun Fig. 1. The normalize reconstruction error variance accoring to Theorem 1 for -imensional sampling lattices Λ(B). Fining the maximum ρ(a) for a given imension an a given volume vol(a) is known as the sphere-packing problem in the lattice literature, an the optimal lattices in imensions, 3, 4, an 8 are A, A 3, D 4, an E 8 [3, Ch. 1]. Note, however, that Table I lists the lattices Λ(B) use for sampling in the time omain, whereas the sphere-packing problem applies to their uals Λ(A) in the frequency omain. This analogy between sampling isotropically banlimite processes in imensions at relatively high rates (efine as rates near the Nyquist rate) was observe alreay in ,  1. Theorem furthermore inicates the existence of a lower threshol rate, which is also inclue in Table I. Whenever ω 0 R(A) or, equivalently, when r/ω 0 (vol Ω(A)) 1/ /R(A), the reconstruction error variance is the minimal possible for any lattice. At these lower rates, the optimal sampling lattice is therefore the ual of the lattice with the minimum R(A). This connection between multiimensional low-rate sampling an the sphere-covering problem has not, to our best knowlege, been reporte previously. In summary, the optimal sampling lattices for signals with isotropically banlimite spectra at high an low rates are the uals of the best sphere-packing lattice an sphere-covering lattice, resp. These results are somewhat unexpecte in view of , where asymptotically optimal lattices for signals with isotropical, exponentially ecaying spectra were erive. The 1 Some of the results in  were creite to Miyakawa, whose paper  is unfortunately unaccessible to us. optimal high-rate lattice is in both cases the ual of the best sphere-packing lattice, but at low rates, the optimal lattice for the scenario of  was foun to be the best sphere-packing lattice. This lattice is in general ifferent from the ual of the best sphere-covering lattice [3, Ch. 1, Tab. 1.1], which as shown above is optimal in the banlimite case. IV. NUMERICAL RESULTS The reconstruction error variance accoring to Theorem 1 was numerically calculate for selecte low-imensional lattices. To this en, vol( \ Ω(A)) neee to be estimate for banwiths ρ(a) ω 0 R(A). We are particularly intereste in banwiths near these two threshols, where the gap to the lower boun is expecte to be small. Regular Monte-Carlo integration i not give sufficient accuracy, ue to the intricate geometry of multiimensional polytopes. In particular, the vertices of the Voronoi cells, which etermine the covering raius R(A), resemble narrow spikes an account for a negligible fraction of the total volume in high imensions [9, Sec ]. To aress this problem, we generate ranom vectors uniformly on a -imensional sphere of a given raius ω an ecoe these vectors using well-known lattice ecoing algorithms  to etermine the fraction of the vectors that belonge to Ω(A). Repeating this process for an appropriately chosen sequence of ω values enable accurate estimation of vol( \ Ω(A)) for a given Λ(A), also for banwiths ω 0 near ρ(a) an R(A).
4 4 PREPRINT, MARCH 1, Square Hexagonal 4 D 4 Σlb Σ 01 Σlb Σ 01 A Σlb Σ 01 Cubic BCC FCC Σlb Σ 01 8 E 8 A r Ω r Ω 0 Fig.. The gap to the lower boun in Fig. 1 for various sampling lattices Λ(B). The threshol rates in Table I are marke with ashe vertical lines, illustrating the fact that the reconstruction variance of a lattice iffers from the lower boun only when ρ(a) < ω 0 < R(A). The results of Theorem 1 are shown in Fig. 1 as functions of the normalize sampling rate r/ω 0. This normalization is chosen because Ω(A) in (8) scales proportionally to r an scales proportionally to ω 0. The lower boun in Theorem is also shown. As expecte from theory, all lattices yiel zero error for high enough r/ω 0. When r/ω 0 is ecrease below the secon threshol in Table I, aliasing occurs an the error variance begins to increase. For very low r/ω 0, below the first threshol in Table I, the lattices will eventually have the same performance again, following the lower boun exactly. To better illustrate the range of sampling rates for which the error variance iffers from the lower boun, Fig. shows the normalize ifference σ e σ lb in logarithmic scale. The theoretical threshols are shown as ashe vertical lines an it is clearly seen that σ e σ lb only when ρ(a) < ω 0 < R(A). In imension, sampling on the hexagonal lattice is clearly superior to the square lattice for any rate. In imension 3, the BCC lattice A 3 is better at high rates an the FCC lattice A 3 at low rates. This reflects the well-known fact that the ual of the BCC lattice (which is FCC) is the optimal 3- imensional packing lattice an the ual of the FCC lattice is the optimal covering lattice. The crossover point lies at r/ω 0 = The situation is similar in imension 4, where D 4 is optimal at high rates an A 4 at low rates, again ue to the optimality of their uals for sphere packing an covering, resp. In imension 8, the optimal sphere covering is not known, but it is conjecture to be A 8 [3, Chs. 1, 4]. Its covering raius R(A) is however only about 1% smaller than that of E 8 (see Table I), which leaves some oubts about the best lattice for low-rate sampling. The numerical results in Figs. 1 show that for practical purposes, E 8 shoul be the preferre sampling lattice at any rate, espite its marginally weaker covering raius. Finally, for any imension, cubic sampling is significantly weaker than sampling on the optimal lattice, being more than 3 times farther away from the lower boun for all rates. V. CONCLUSIONS The new lower boun on the reconstruction error variance for sampling multiimensional, isotropically banlimite signals is exact for sufficiently low an high sampling rates. From this attractive property, two optimality criteria are erive for the choice of sampling lattices. The theoretical an numerical results confirm that the optimal lattice for multiimensional sampling at high rates (near the multiimensional Nyquist rate, as efine in Sec. III) is the ual of the best sphere-packing lattice . The optimal lattice at low rates turns out to be the ual of the best sphere-covering lattice, which was conjecture by Entezari et al. , [6, Sec. 4.1],  but, to our best knowlege, has never before been establishe mathematically. REFERENCES  D. P. Petersen an D. Mileton, Sampling an reconstruction of wave-number-limite functions in n-imensional Eucliean spaces, Inf. Control, vol. 5, no. 4, pp , Dec. 196.
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