In this Section, we describe the voxel patterns themselves and calculate their spectra. The basic element of VPs is the symmetric rectangular unit pulse of the unit width commonly defined as,

$$\text{rect}\left(x\right)=\left\{\begin{array}{c}0, \left|x\right|>\frac{1}{2}\\ \frac{1}{2}, \left|x\right|=\frac{1}{2}\\ 1, \left|x\right|<\frac{1}{2}\end{array}\right.$$

1

The single pulse of the width *w*, phase *θ*, and the amplitude *a* is expressed as

$${u}_{0}=a\text{rect}\left(\frac{\text{x}}{w}\right)$$

2

Let the integer *d* ≠ 0 be a distance from the screen, as in [14]; the distance in front of the screen of ASD is positive, behind it – negative. In space, the distance corresponds to a plane. The zeroth plane is intentionally omitted (zero partition means an absence of the pattern). The first depth plane (the sign does not matter) is the screen itself. Each VP corresponds to certain discrete depth plane, and the *d*-th pattern consists of *n* = |*d*| non-negative pulses of the identical amplitude and width; *n* = 1, 2, … In other words, according to [13], VP is a finite series of rectangular pulses with the unit height; the width of each pulse is *c*/*n*, where *c* is the size of the logical cell. Similar to [13], [14], VP can be written as a linear combination of the identical rectangular pulses as follows,

$${P}_{n}\left(x\right)=\sum _{i=1}^{n}\text{rect}\left(\frac{nx-{ip}_{n}-{\text{s}}_{n}}{c}\right), n>1$$

3

where *n* is the index of the discrete depth plane, *c* is the size of the cell, *x* is the coordinate, and (*ip**n* + *s**n*)/*c* is the phase shift of the *i*-th pulse; in the latter expression, the initial phase of the first pulse (i.e., the position of its center) is

$${s}_{n}=\left(1\pm \frac{1}{2n}\right)c=c\bullet \left\{\begin{array}{c}1+\frac{1}{2n}\equiv \frac{1}{2n}, d>0\\ 1-\frac{1}{2n}, d<0\end{array}\right.$$

4

and the period of pulses (the distance between the centers of repeated pulses) is

$${p}_{n}=\left(1\pm \frac{1}{n}\right)c$$

5

The graph of the period as a function of *n* is shown in Fig. 1.

The interval (gap) between the pulses of VP is

$${\delta }_{n}=\left(1-\frac{1}{n}\pm \frac{1}{n}\right)c=c\bullet \left\{\begin{array}{c}1, d>0\\ 1-\frac{2}{n}, d<0\end{array}\right.$$

6

Note that in the positive depth planes, the gap does not depend on the distance.

Substituting the period and phase into Eq. (3), we get the formula for the voxel patterns,

$${P}_{n}\left(x\right)=\sum _{i=0}^{n-1}\text{rect}\left(n\frac{x}{c}-i\left(1\pm \frac{1}{n}\right)-\left(1\pm \frac{1}{2n}\right)\right), n>1$$

7

.

The examples of series of the rectangular pulses comprising VPs are shown in Fig. 2. The implied intensity range of MV images is [0, 1], which can be easily transformed into the typical range of digital images [0, 255].

The main properties of VPs [4], [5] in MV images are as follows. In the 1D case, the summed length of all pulses is equal to the size of the cell. In the 2D case, the voxel patterns are the products of two 1D functions in the orthogonal directions, and the summed area of all pulses is equal to the area of the cell. With the same *n* (i.e., across the same number of the cells) the pulses of VPs for positive and negative planes are distributed differently, see Fig. 1. Depending on the sign of *d*, the left/right edge of the first/last pulse coincides with the left/right edge of the cell. Specifically, the first and last partitions are aligned either to the outer or to the inner boundaries of the outmost (leftmost and rightmost) cells of the pattern. By Eq. (6), there are two cases: a wide gap *c* and the maximum separated pulses at the positive distances in front of the screen, and a narrow gap and the maximum close pulses at the negative distances behind the screen. We may call these cases as separated and close pulses, resp.

All patterns including the patterns with *n* = 1, 2 obey the general rules; although, for *n* = 1, neither the period nor the gap has a physical meaning, and effectively, the negative sign is only used in Eqs. (5), (6) in this case. Besides, for *n* = 2, the gap between the pulses may be zero. These features might look uncertain, and to avoid confusion, we explain these cases separately.

In the case of *n* = 1, the same pattern matches both positive and negative distances ± 1; neither period nor gap have a meaning. The initial phase is,

Therefore,

$${P}_{1}\left(x\right)=\text{rect}\left(\frac{\text{x}}{c}-\frac{1}{2}\right)$$

9

The initial phase and the period of the pulses by Eqs. (4), (5) are,

$${s}_{2}=\text{c}\bullet \left\{\begin{array}{c}\frac{1}{4}, d>0\\ \frac{3}{4}, d<0\end{array}\right.$$

10

$${p}_{2}=\text{c}\bullet \left\{\begin{array}{c}\frac{3}{2}, d>0\\ \frac{1}{2}, d<0\end{array}\right.$$

11

Besides, the gap *δ*2- at the second negative distance is 0,

$${\delta }_{2}=\left\{\begin{array}{c}c, d>0\\ 0, d<0\end{array}\right.$$

12

i.e., two adjacent pulses of the width *c*/2 compose a single pulse of the total width *c*, as shown in Fig. 3(b),

Therefore,

$${P}_{2}\left(x\right)=\left\{\begin{array}{c}rect\left(2\frac{x}{c}-\frac{1}{4}\right)+rect\left(2\frac{x}{c}-\frac{7}{4}\right), d>0\\ rect\left(2\frac{x}{c}-\frac{3}{4}\right)+rect\left(2\frac{x}{c}-\frac{5}{4}\right)\equiv rect\left(\frac{x}{c}-1\right), d<0\end{array}\right.$$

13

.

Two particular cases of VPs with *n* = 1 and *n* = 2 are shown in Fig. 3.

Let’s calculate the Fourier transform (FT) of 1D VPs. Note that we use the definition of the Fourier transform without 2π,

$$F\left(k\right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(x\right){e}^{-ikx}dx$$

14

.

Also, recall that FT of a shifted function *f*(*x*-*s*) is the FT of the original function multiplied by the exponential,

$${F}_{s}\left(k\right)={e}^{-iks}F\left(k\right)$$

15

Then, FT of the single pulse Eq. (2) is,

$$F\left(\omega \right)=aw\bullet \text{sinc}\frac{kw}{2}$$

16

where *k* is the wavenumber, and the unnormalized sinc function is

$$\text{sinc}z=\frac{\text{sin}z}{z}$$

17

In three specific cases, FT is as follows.

(a) an elementary pulse with *w* = *c*/*n*, *a* = *A*,

$${F}_{1}=2\frac{A}{k}\text{sin}\frac{kc}{2n}=\frac{Ac}{n}\text{sinc}\frac{kc}{2n}$$

18

(b) a typical combination of the elementary pulses (a pair of narrow pulses) with *w* = *c*/*n*, *a* = *A*, symmetrically displaced around the origin by ± *s*. Thus,

$${F}_{2}=4\frac{A}{k}\text{sin}\frac{kc}{2n}\text{cos}\left(ks\right)=2\frac{Ac}{n}\text{sinc}\frac{kc}{2n}\text{cos}\left(ks\right)$$

19

(c) a bias of a wavelet (a single “shallow” pulse) with *w* = *qc*, *a* = *A*/*q*,

$${F}_{-}=2\frac{A}{qk}\bullet \text{sinc}\frac{qkc}{2}=Ac\bullet \text{sinc}\frac{qkc}{2}$$

20

Consequently, the FT of VP at the first plane is,

$${F}_{p1}=Ac\bullet \text{sinc}\frac{kc}{2}$$

21

Then, there are two pulses with *w* = *c*/2 in VP of the second order, which are displaced either by *c*/4 or 3*c*/4, as shown in Figs. 3(a), (c). In the case of the adjacent pulses (*s* = *c*/4), by Eq. (19), FT is,

$${F}_{p2-}=2\frac{A}{k}\text{sin}\frac{kc}{2}=Ac\bullet \text{sinc}\frac{kc}{2}$$

22

Note that Eq. (22) is identical to Eq. (21), i.e., *F**p*2- = *F**p*1. In the case of separated pulses (*s* = 3*c*/4), FT of the pattern is,

$${F}_{p2+}=2\frac{A}{k}\text{sin}\frac{kc}{2}\left(4{\text{cos}}^{2}\frac{kc}{4}-3\right)$$

23

The graphical illustrations of these three particular cases of FT are shown in Fig. 4.

FP of VPs of higher orders can be calculated similarly, as the sum of pairs Eq. (116), conditionally plus the central pulse Eq. (18) for odd *n*,

$${F}_{pn}=\left\{\begin{array}{c}\sum _{i=0}^{\frac{n}{2}}{F}_{2}-{F}_{-}, n even\\ \sum _{i=0}^{\frac{n}{2}}{F}_{2}-{F}_{-}+{F}_{1}, n odd\end{array}\right.$$

24

To illustrate Eq. (24), FT of VPs up to 16th order is shown in Fig. 5.

Note that the minimum (the second extremum after DC term) of the FT of the close pulses is on the right side from the dotted vertical line *k* = 2π in Fig. 5, while the second extremum of the separated pulses on the left side. This corresponds to the periods of VPs by Eq. (5) shown in Fig. 1.