Sampling and Aliasing Properties of Three-Dimensional Point-Lattices

Robin Strand Centre for Image Analysis, Uppsala University, Sweden Email: [email protected]

Abstract—Sampling properties of some three-dimensional lattices are examined. The Cartesian cubic lattice is compared with the non-Cartesian face- and body-centered cubic lattices. Numerical computations using analytical functions are used to compare rotational dependency and aliasing effects.

NTRODUCTION I.I Fig. 1. Voronoi regions (voxels) in (from left to right) the cubic, fcc, There are many imaging techniques that generate three- and bcc lattices. dimensional volume images today. With higher precision in the image acquisition equipment, storing and process- by sampling a function is analyzed in frequency domain. ing these images require an increasing amount of data By using Plancherel theorem, the energy can be calculated processing capacity. Traditionally, three-dimensional im- by the energy in one period in frequency domain. Rota- ages are represented by cubic (or cuboid) picture elements tional dependency will be compared by using function on a cubic lattice1. values on a one-dimensional linear subsets in spatial In this report, sampling and aliasing properties of some domain. Also, results on sampling a three-dimensional three-dimensional lattices are presented. The use of non- version of the Shepp-Logan phantom are presented. standard lattices, such as the 2D hexagonal lattice, in image analysis, image processing, and computer graphics II. (DISCRETE SPACE) FOURIER TRANSFORM is often motivated by the high sampling efﬁciency of these lattices [1]–[3]. A lattice is generated by a matrix V is The following deﬁnition of the Fourier transform (f(x)) of a function f(x) will be used obtained by linear integer combinations of the colums of F V. uT x F (u) = f (x) e−2πi dx for any u Rn. (1) In the one-dimensional case, given a function f ban- Rn ∈ dlimited by B (the Fourier transform of f, F is such Z The inverse Fourier transform −1 (F (u)) is that F (u) = 0 if u > B), the optimal sampling is to F | | | | 1 1 sample the function slightly below , i.e., ∆x = ǫ. xT u 2B 2B − f(x) = F (u) e2πi du for any x Rn. This is the well known Shannon/Nyquist sampling theo- Rn ∈ rem. In higher dimensions, given an isotropic function Z f bandlimited by B, F is also isotropic which implies Change of variables in spatial domain gives that it is non-zero only in the interior of a ball of radius uT t (f(Vt) = f (Vt) e−2πi dt B. Sampling f on points according to a sampling lattice F Rn Z generated by V gives spectra replicated according to the T −1 1 −2πiu V x V V−T V−1 T = f (x) e dx inverse of the transpose of , i.e., = . det V Rn Z The lattice is optimal in frequency domain if the balls of | | T −1 T T 1 −2πiu ((V ) ) x radius B that contain the spectrum are as densely packed = f (x) e dx det V Rn as possible. Thus, a dense lattice in frequency domain is | | Z 1 V−1 Tu Tx needed. The densest point-lattice in three dimensions is = f (x) e−2πi(( ) ) dx det V Rn the face-centered cubic (fcc) lattice [4]. If a point-lattice | | Z − 1 generated by V is such that V T is an fcc lattice, then = F V−T u . V det V the lattice generated by is a body-centered cubic (bcc) | | lattice [2], [3], [5]. In other words, we have the following Fourier pair: The Voronoi regions of the cubic, fcc, and bcc lattices 1 are shown in Figure 1. f(Vx) F V−T u . ←→ det V In this report, sampling properties of the cubic, fcc, and | | bcc lattices are compared. The energy that is represented The Dirac comb, or Shah function, is

1A (point) lattice is a discrete subgroup of Euclidean space, which XV (x) = δ (x Vn) , contains the origin. n∈Zn − X where δ(x) is the n-dimensional Dirac delta function. The The effect of sampling is simulated by using (4). Cubic, Fourier transform of the Dirac comb is fcc, and bcc sampling lattices in with equal sampling den- 1 sity will be compared by considering reciprocal lattices δ (x Vn) δ u V−T n . − ←→ det (V) − with equal volume of the Voronoi regions. The plot in n∈Zn | | n∈Zn X X Figure 2 shows the aliasing error (y-axis) that is covered Convolution is denoted by , i.e., by sampling using sampling lattices of increasing density ∗ (x-axis). (f g)(x) = f(t)g(x t)dt ∗ Rn − Z 0 The convolution theorem yields −1

1 −2 f(x)XV (x) (F (u) XV−T (u)) . ←→ det (V) ∗ | | −3 Let V be the Voronoi region of the lattice point 0 in −4 the lattice deﬁned by V. The Voronoi region on the cubic, −5 Q fcc, and bcc lattices are cubes, rhombic dodecahedra, and −6 truncated octahedra, respectively, [1]–[3], [5]. Let also −7

x Aliasing error, logarithmic scale 1 if V −8 V(x) = ∈ ⊓ 0 otherwise. Q −9 −10 Let φV be the ideal lowpass function deﬁned as: 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Sampling density 1 x −T u (φV( )) = V V ( ). (2) F det ( ) ⊓ Fig. 2. The aliasing error as a function of sampling density when | | 2 2 2 sampling f(x,y,z) = e−(x +y +z ) on a cubic (3), fcc (2), and If f is bandlimited (i.e., F is zero outside −T ), then V bcc (◦) lattice. f is completely represented by its samples since Q uT x For the test function, numerical experiments show the f(x) = F (u)e2πi du = Rn sampling that covers 98% of the relative energy by ad- Z T justing the density of the sampling lattice in (4). Compare 2πiu x F (u)e du = (f(x)XV (x)) φV(x). (3) with the plot in Figure 2. The volume of V−T , i.e., Q −T ∗ Z V the period in frequency domain, needed to cover 98% III. SAMPLING AND ALIASING of the relative energy for the cubic, fcc, andQ bcc lattices are 0.64, 0.56, and 0.55, respectively. This corresponds Let g(x) = f(x) φV(x), i.e., the reconstructed function from the samples∗ of f using (3). As performance to sampling densities 1.55, 1.80, and 1.81, respectively, measure, the aliasing error, , (the relative error of the in spatial domain. energy that is represented byE the function reconstructed Now the sampling density in spatial domain is set to from the samples, see [6]), is used. The Plancherel theo- 1.55, the relative energy covered by the sampling in the rem (the L2 norm is preserved by the Fourier transform) cubic lattice is 98%. The corresponding values for the fcc is used to get and bcc lattices are 98.7% and 98.8%, respectively. 2 2 A. Rotational dependency n g(x) dx n (g(x)) du = 1 R | | = 1 R |F | 2 2 For a number of rotations, a 1D subset in spatial domain E − Rn f(x) dx − Rn (f(x)) du R | | R |F | is analyzed. We describe the formula when the 1D subset F (u) 2du R RQV−T = 1 | | . (4) is aligned with the x-azis. The rotation is obtained by u 2 u − Rn F ( ) d rotating the domain of integration. R | | Analogous to the Fourier slice theorem, a slice in the The following Gauss functionR will be used as test function spatial domain corresponds to a projection in frequency 2 2 2 domain: f(x,y,z) = e−(x +y +z ) with 2 2 2 2 (f(x,y,z)) = F (u,v,w) = π(3/2) e−π (u +v +w ). f(x, 0, 0) F (u,v,w) dv dw. (5) ←→ R2 F Z The total energy is The total energy in the 1D subset of a separable 2 function f(x,y,z) = f(x)f(y)f(z) with real Fourier − 2 2 2 e (x +y +z ) dx dy dz = transform F is Rn Z 2 2 2 2 2 2 3 −π (u +v +w ) (f(x)f(0)f(0)) dx dy = π e du dv dw = R Rn Z Z 2 √2 2 π(3/2) 1.97. F (u) F (v)F (w)dv dw du. R R2 4 ≈ Z Z The reconstructed function is now analyzed, i.e., the formula in (3). For the analysis in frequency domain, (5) will be used. 3 Given C, a compact set in R , let W1(u,v), W2(u,v), V1(u), V2(u), U1, U2 bound the set, i.e.,

C = (u,v,w) : W (u,v) w W (u,v), 1 ≤ ≤ 2 V (u) u V (u),U u U . . 1 ≤ ≤ 2 1 ≤ ≤ 2 Rotated versions of V−T are used as domain C. Re- member that the cubic, fcc, and bcc lattices correspond to the Voronoi regionQ in the reciprocal lattice (Brillouin zones) in frequency domain, i.e., cubic, truncated octahe- dral, and rhombic dodecahedral domain, respectively. The effect of sampling a one-dimensional linear subset on the three lattices is now considered. The test function will be used. In general, the aliasing error along one- (a) dimensional structures in a 3D volume is simulated by this procedure. Consider 2 2 2 f(x,y,z) = e−(x +y +z ) with z = 0,y = 0. 2 2 2 The Fourier transform of f(x) = e−x is √πe−π u and the total energy is √2π/2. By the Fourier slice theorem, the energy in the 1D proﬁle is 2 U2 V2(u) W2(u,v) F (u)2 F (v)F (w) dw dv du. ZU1 ZV1(u) ZW1(u,v) ! (6) For the numerical computations, the integral is rewritten using Fubini’s theorem and get

U2 V2(u) W2(u,v) F (u)2 F (v)F (w) dw dv · (b) ZU1 ZV1(u) ZW1(u,v) !

′ V2(u) W2(u,v ) F (v′)F (w′) dw′ dv′ du = ′ ZV1(u) ZW1(u,v ) !

′ U2 V2(u) W2(u,v) V2(u) W2(u,v ) F (u)2 ′ · ZU1 ZV1(u) ZW1(u,v) ZV1(u) ZW1(u,v ) F (v)F (w)F (v′)F (w′) dw′ dv′ dw dv du =

F (u)2 ′ ′ · Z(u,v,w)∈C and (u,v ,w )∈C F (v)F (w)F (v′)F (w′) dw′ dv′ dw dv du This integral can be computed numerically by standard methods. A Monte-Carlo method with bounds on the (c) estimated error is used. In Figure 3, the aliasing error 1 I/(√2π/2), where Fig. 3. Aliasing error of a 1D proﬁle as function of direction. The √ − sampling density 1.55 is used for all lattices. The spheres are viewed 2π/2 is the total energy and I is the integral (6), for from the y-direction and the z-direction is up. A linear scale between some different rotations are shown. 16.2% (black) and 21.8% (white) is used. Minimum/maximum error is 18.6%/21.1% (cubic lattice, a), 16.2%/17.0% (bcc lattice, b), and 16.2%/17.0% (fcc lattice, c), 0.35

0.3

0.25

0.2

0.15

0.1 relative difference in represented energy

0.05

0 −4 −3 −2 −1 0 1 sampling density, logarithmic scale

Fig. 6. The relative difference between the fcc (2)/bcc(◦) lattice and the cubic lattice in represented energy as a function of sampling density Fig. 4. A three-dimensional version of the Shepp-Logan phantom. when sampling the 3D Shepp Logan phantom in Figure 4.

IV. CONCLUSIONS B. Sampling an analytical 3D Phantom As expected, the fcc and bcc lattices are more efﬁcient than the cubic lattice when it comes to sampling functions. In this section, some experiments on a three- It is known that if all frequencies should be represented dimensional version of the Shepp-Logan phantom, see when sampling a band-limited function, almost 30% Figure 4, are performed. The 3D Shepp-Logan phantom fewer samples can be used when a bcc lattice is used is composed of the union of a number of ellipsoids. Since compared to when a cubic lattice is used [1]–[3]. When the Fourier transform of a general ellipsoid is known, sampling the non-bandlimited test function used here the Fourier transform of the 3D Shepp Logan can be such that 2% of the relative energy is represented, then calculated, [7]. 1 0.55/0.64 14% fewer samples can be used when − ≈ The energy represented by sampling at a given density a bcc lattice is used instead of a cubic lattice. For the is again simulated by integration in frequency domain. In Shepp-Logan phantom, the gain is almost 30% when Figure 5, the energy in that is represented by sampling the sampling density is low, see Figure 6. When the the 3D phantom independent of rotation of the sampling sampling density is high enough, then all frequencies are grid is given as a function of sampling density. represented independent of the choice of sampling lattice. This can also be seen in Figure 6. Figure 3 illustrates that both the effect of aliasing and the rotational dependency is lower on the fcc and bcc 4.5 lattices compared to the cubic lattice on one-dimensional 4 linear subsets of the three-dimensional Gauss-function

3.5 used here.

3 REFERENCES

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0.5 lattices: with focus on the 3D face- and body-centered cubic grids,” Ph.D. dissertation, Uppsala University, Sweden, 2008. 0 [4] T. Hales, “Cannonballs and honeycombs,” Notices of the American −4 −3 −2 −1 0 1 sampling density, logarithmic scale Mathematical Society, vol. 47, pp. 440–449, 2000. [5] J. H. Conway, N. J. A. Sloane, and E. Bannai, Sphere-packings, lattices, and groups. New York, NY, USA: Springer-Verlag New Fig. 5. The aliasing error as a function of sampling density when York, Inc., 1988. sampling the 3D Shepp Logan phantom in Figure 4 on a cubic (3), fcc [6] J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, (2), and bcc (◦) lattice. “Selection of a convolution function for fourier inversion using gridding,” IEEE Transactions on Medical Imaging, vol. 10, no. 3, pp. 473–478, 1991. [7] C. G. Koay, J. E. Sarlls, and E. Özarslan, “Three-dimensional In Figure 6, the relative difference between the fcc/bcc analytical magnetic resonance imaging phantom in the fourier lattice and the cubic lattice in represented energy as a domain,” Magnetic Resonance in Medicine, vol. 58, no. 2, pp. 430– function of sampling density is plotted. 436, 2007.