# Monte Carlo computer investigations of higher generation ideal dendrimers

M. Jura  $|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, M. Bishop  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, B. Thrope  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, R. de Regt  $|$
${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?}}}}$ Corresponding
author<EMAIL_ADDRESS>
(Received August 16, 2020, in final form February 5, 2021)

###### Abstract

The properties of ideal tri-functional dendrimers with forty-five, ninety-
three and one hundred and eighty-nine branches are investigated. Three methods
are employed to calculate the mean-square radius of gyration, $g$-ratios,
asphericity, shape parameters and form factor. These methods include a
Kirchhoff matrix eigenvalue technique, the graph theory approach of Benhamou
et al. (2004), and Monte Carlo simulations using a growth algorithm. A novel
technique for counting paths in the graph representation of the dendrimers is
presented. All the methods are in excellent agreement with each other and with
available theoretical predictions. Dendrimers become more symmetrical as the
generation and the number of branches increase.

Key words: soft matter, dendrimer, analytical approach, MC simulation

###### Abstract

Äîñëäæåíî âëàñòèâîñò äåàëüíèõ òðè-ôóíêöîíàëüíèõ äåíäðèìåðâ ç ñîðîêà ï’ÿòüìà,
äåâ’ÿíîñòî òðüîìà òà ñòî âñìäåñÿò äåâ’ÿòüìà ãëêàìè. Äëÿ ðîçðàõóíêó
ñåðåäíüîêâàäðàòèíîãî ðàäóñà ãðàö¿, $g$-ñïââäíîøåííÿ, àñôåðèíîñò, ïàðàìåòðâ
ôîðìè òà ôîðì-ôàêòîðà âèêîðèñòîâóþòüñÿ òðè ðçí ìåòîäè. Äî íèõ íàëåæàòü: òåõíêà
âëàñíèõ çíàåíü ìàòðèö Êðõãîôà, ôîðìàëçì â ðàìêàõ òåîð¿ ãðàôâ Áåíàìó òà í.
(2004), à òàêîæ ìîäåëþâàííÿ Ìîíòå-Êàðëî ç âèêîðèñòàííÿì àëãîðèòìó çðîñòàííÿ.
Çàïðîïîíîâàíî íîâó ìåòîäèêó ïäðàõóíêó òðàåêòîðé ïðè ïðåäñòàâëåíí äåíäðèìåðâ ó
âèãëÿä ãðàôâ. Óñ ìåòîäè âäìííî óçãîäæóþòüñÿ ìæ ñîáîþ òà ç íàÿâíèìè òåîðåòèíèìè
ïåðåäáàåííÿìè. Äåíäðèìåðè ñòàþòü áëüø ñèìåòðèíèìè ïðè çáëüøåíí ïîðÿäêó ãëîê òà
¿õ êëüêîñò.

Ключов слова: ì’ÿêà ðåîâèíà, äåíäðèìåð, àíàëòèíèé ïäõä, ìîäåëþâàííÿ Ìîíòå-
Êàðëî

##  1 Introduction

The architecture of a polymer is determined by how its units are linked
together. The simplest polymer is a linear chain. Three or more linear chains
can be grown from a single junction to form a star polymer. If the ends of
each chain are then used in turn repeatedly as junctions for a further growth
one obtains a dendrimer. These polymers have the shape of a branching tree and
can be classified by their functionality (number of linear branches growing
from a junction), the spacer size (the number of units in a linear branch),
and the generation. A generation is defined by the number of rings of branches
starting from a single central junction.

Figure 1: 93-branch dendrimer (fourth generation).

Figure 1 illustrates a typical dendrimer. The diagram displays a 93-branch
dendrimer. This is the fourth generation in its family. The circles represent
the junctions. The first three branches attached to the central junction
represent a star polymer. The next ring, containing six more branches,
represents the first dendrimer generation, a molecule with nine branches. The
second generation has twenty-one branches, the third, forty-five and the
fourth, ninety-three. The fifth generation will have one hundred and eighty-
nine branches so that its structure would include another ring of branches.
Two more branches would extend from each circle, adding 96 branches to the
structure for a total of 189. The reason for the 9-21-45-93-189 branch numbers
is that the number being added doubles; 12 branches are added to 9 to get 21,
24 to 21 to get 45, etc. The number of units in each branch is uniform
throughout the entire structure and can be any value. The total number of
units, $N$, in each dendrimer is $N=fm+1$ where $f$ is the number of branches
defined by the dendrimer generation and $m$ is the number of units in each
branch. In our study, $N$ ranged from 901 to 4915.

The mean-square radius of gyration [1], $\langle S^{2}\rangle$, is used to
measure the total size of a dendrimer. The mean-square radius of gyration of
an object composed of $N$ identical units, around its center of mass, is

$\langle
S^{2}\rangle=\frac{1}{N}\sum_{i}^{N}\left\langle\left(X_{i}-X_{CM}\right)^{2}+\left(Y_{i}-Y_{CM}\right)^{2}+\left(Z_{i}-Z_{CM}\right)^{2}\right\rangle,$
(1.1)

where $X_{i},Y_{i}$ and $Z_{i}$ are the components of the position of the
$i$-th unit and the center of mass is defined by

$X_{CM}=\frac{1}{N}\sum_{i}^{N}X_{i};\ \ Y_{CM}=\frac{1}{N}\sum_{i}^{N}Y_{i};\
\ Z_{CM}=\frac{1}{N}\sum_{i}^{N}Z_{i}.$ (1.2)

The braces $\langle\quad\rangle$ indicate an average over multiple dendrimer
constructions.

Sheng et al. [2] determined an exact equation for the radius of gyration of
ideal tri-functional dendrimers:

$\frac{\langle S^{2}\rangle N^{2}}{\langle l^{2}\rangle}=3m^{3}\left[-1+6\cdot
2^{G1}+(3G1-5)2^{2G1}\right]-0.5m(m-1)(2^{G1}-1)(9\cdot 2^{G1}m-7m+2)\,,$
(1.3)

where $N=3m(2^{G1}-1)+1$, $\langle l^{2}\rangle$ is the mean-square bond
length between units [1 for the cubic lattice used in the Monte Carlo (MC)
simulations], and $G1$ is one more than the generation number.

For linear chains, $\langle S^{2}\rangle$, there follows the well-known [3]
scaling law,

$\langle S^{2}\rangle=CN^{2\nu}.$ (1.4)

In this equation, the value of $C$ depends on the polymer model used, but the
exponent, $2\nu$, is universal and is equal to 1 for ideal linear chains.
Sheng et al. [2] found that for a given generation, as the spacer length is
increased, the size of an ideal dendrimer has the same exponent value. Earlier
investigations by Burchard et al. [4] (see their figure 4) showed that for a
fixed generation the radius of gyration scales linearly with the spacer
length, but with a fixed spacer length it displays a logarithmic scaling.
Equation (1.4) was originally obtained in terms of the number of bonds, $N-1$,
and for the finite systems studied in the Monte Carlo (MC) calculations this
factor is used instead of $N$.

In order to measure the compactness of the dendrimer’s structure, the
$g$-ratio was calculated. This calculation involves the ratio of the radius of
gyration of a dendrimer and the corresponding linear polymer chain. Both
polymers must contain the same number of units:

$g=\frac{\langle S^{2}\rangle}{\langle S^{2}\rangle_{l}}.$ (1.5)

There is an exact solution for the $g$-ratio determined by Wawrzyriska et al.
[5] :

$g=\frac{6(W_{1}-W_{2})}{(n/m)^{3}}.$ (1.6)

Here, $n$ is the number of bonds and

$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle\frac{F_{1}}{(F_{2}-1)^{3}}\left[F_{2}^{2G}(F_{1}F_{2}G-F_{1}G-F_{1}-F_{2})\right.$
(1.7) $\displaystyle+$
$\displaystyle\left.F_{2}^{G}(2F_{1}+F_{1}G-F_{1}F_{2}G-G+GF_{2}^{2})-F_{1}+F_{2}\right],$

whereas

$W_{2}=\frac{F_{1}^{2}(F_{2}^{G}-1)^{2}}{2(F_{2}-1)^{2}}+\frac{F_{1}(F_{2}^{G}-1)}{3(F_{2}-1)}.$
(1.8)

The tri-functional dendrimers have $F_{1}=3$ and $F_{2}=2$.

The overall shape of the dendrimer can be found using the radius of gyration
tensor. This tensor is composed of different eigenvalues. In three dimensions,
they are $\lambda_{3}\leqslant\lambda_{2}\leqslant\lambda_{1}$. These
eigenvalues are the principal moments of gyration, and there is one for each
direction along the principal orthogonal axes [6]. $\langle S^{2}\rangle$ is
equal to the average trace of the radius of gyration tensor,
$\lambda_{1}+\lambda_{2}+\lambda_{3}$. The average asphericity $\langle
A\rangle$ of a dendrimer in three dimensions is defined by Rudnick and Gaspari
[7, 8] as

$\langle
A\rangle=\left\langle\frac{\sum_{i>j}^{3}(\lambda_{i}-\lambda_{j})^{2}}{2(\sum_{i=1}^{3}\lambda_{i})^{2}}\right\rangle,$
(1.9)

$\langle A\rangle$ is a number between 0 and 1, with 0 representing a
perfectly spherical object, while an asphericity of 1 indicates that the units
form a rod shape. The prolateness, $\langle P\rangle$, is also used to
identify the overall shape. It is defined as

$\langle
P\rangle=\left\langle\frac{27(\lambda_{1}-\overline{\lambda})(\lambda_{2}-\overline{\lambda})(\lambda_{3}-\overline{\lambda})}{(\sum_{i=1}^{3}\lambda_{i})^{3}}\right\rangle,$
(1.10)

where $\overline{\lambda}$ is

$\overline{\lambda}=\frac{\lambda_{1}+\lambda_{2}+\lambda_{3}}{3}.$ (1.11)

It should be noted that $\langle S^{2}\rangle$, the $g$-ratio$,\langle
A\rangle$, and $\langle P\rangle$ are all rotationally invariant universal
quantities which can be expressed [9] in terms of the trace, Tr, and
determinant, Det, of the gyration tensor, Q.

$Q_{ij}=\frac{1}{N}\sum_{n}^{N}(X_{n}^{i}-X_{CM}^{i})(X_{n}^{j}-X_{CM}^{j})\ \
\ i,j=1...3.$ (1.12)

Then,

$\langle S^{2}\rangle=\langle\text{Tr}\ \textbf{Q}\rangle,$ (1.13)

and

$\langle
A\rangle=\left\langle\dfrac{3\text{Tr}\left(\textbf{Q}-\overline{\lambda}\textbf{I}\right)^{2}}{2(\text{Tr}\
\textbf{Q})^{2}}\right\rangle,$ (1.14)

where I is the identity matrix. Also,

$\langle
P\rangle=\left\langle\dfrac{27\text{Det}\left(\textbf{Q}-\overline{\lambda}\textbf{I}\right)}{(\text{Tr}\
\textbf{Q})^{3}}\right\rangle.$ (1.15)

Another important polymer property is the scattering function [10], $S(k)$.
Formally, this is defined as

$S(k)=\frac{1}{N^{2}}\sum_{l}^{N}\sum_{m}^{N}\mathrm{e}^{\mathrm{i}\mathbf{k}\
\bullet\ (\mathbf{R}_{m}-\mathbf{R}_{l})}\,.$ (1.16)

Here, $N$ is the number of units in the polymer, $\mathbf{k}$ is the
scattering vector and $\mathbf{R_{l}}$ and $\mathbf{R_{m}}$ are the respective
positions of the $l$-th and $m$-th units. After averaging over the angles in
three dimensions, the scattering function becomes

$S(k)=\frac{1}{N^{2}}\left\langle\sum_{l}^{N}\sum_{m}^{N}\dfrac{\sin
x}{x}\right\rangle,$ (1.17)

where $x=k^{2}\langle S^{2}\rangle$. In the case of linear polymers, the
theoretical values for the scattering function follow the Debye equation [3],

$S(k)=\frac{2(x-1+\mathrm{e}^{-x})}{x^{2}}.$ (1.18)

Dendrimers are not linear and each structure will have its own equation for
$S(k)$. The linear equation is included in one of the figures for comparison
with the behavior of the dendrimers.

Dendrimers have received a lot of attention [11] because they have
applications in medical research, such as in pharmaceutical drug development.
Their branched structure permits these molecules to be folded up into cages
which can carry drugs into the body. In earlier work de Regt et al. [12]
studied the properties of low generation tri-functional ideal structures
containing nine or twenty-one branches. In this paper we extend their
investigations to three higher generations, containing forty-five, ninety-
three, and one hundred and eighty-nine branches, respectively.

##  2 Methods

Universal ratios for Gaussian (ideal) tree-branched polymers such as the
$g$-ratio, $\langle A\rangle$, and $\langle P\rangle$ were computed with
methods developed by Wei [13, 14] using the Kirchhoff matrix and its
corresponding eigenvalues. These ratios are determined by extrapolation to
infinite $N$.

To calculate these shape parameters, the $N-1$ non-zero eigenvalues
$\lambda_{1},\ldots,\lambda_{N-1}$, of the $N\times N$ Kirchhoff matrix
$\hat{K}$ for a random walk polymer with $N$ monomers is used whereby the
characteristic polynomial of $\hat{K}$ can be expressed as

$P_{N}(x)=\text{Det}[\Lambda_{N}(-x)]=\prod_{j=1}^{N-1}(\lambda_{j}-x).$ (2.1)

From here the following functions; $D_{N}(x)$, $S_{1,N}(x)$ and $S_{2,N}(x)$
can be defined which are essential for the calculation of the shape parameters
$\langle A\rangle$, $\langle P\rangle$ and the $g$-ratio. Using the notation
$y=x/N$,

$D_{N}(x)=\text{Det}[\Lambda_{N}^{-1}(0)]\text{Det}[\Lambda_{N}(y^{2})]=P_{N}^{-1}(0)P_{N}\left(-\frac{x^{2}}{N^{2}}\right)=\prod_{j=1}^{N-1}\frac{\lambda_{j}+y^{2}}{\lambda_{j}}\
,$ (2.2)

and

$S_{1,N}(x)=\frac{1}{N^{2}}\text{\text{Tr}}\left[\Lambda_{N}^{-1}(y^{2})\right]=\frac{1}{N^{2}}\sum_{j=1}^{N-1}(\lambda_{j}+y^{2})^{-1}\
.$ (2.3)

Following the work of Wei, the functions for $S_{k,N}$ for $k=2,3,..$
correspond to traces of higher powers of $\Lambda_{N}^{-1}$,

$S_{2,N}(x)=\frac{1}{N^{4}}\sum_{j=1}^{N-1}\left(\lambda_{j}+y^{2}\right)^{-2}\
,$ (2.4)

and

$S_{3,N}(x)=\frac{1}{N^{6}}\sum_{j=1}^{N-1}\left(\lambda_{j}+y^{2}\right)^{-3}\
.$ (2.5)

The asphericity can then also be defined as

$\langle
A\rangle=\frac{15}{2}\int_{0}^{\infty}x^{3}D_{N}^{-3/2}(x)S_{2,N}(x){\mathrm{d}}x=\frac{15}{2}\int_{0}^{\infty}\sum_{j=1}^{N-1}\frac{y^{3}}{(\lambda_{j}+y^{2})^{2}}\left[\prod_{k=1}^{N-1}\frac{\lambda_{k}}{\lambda_{k}+y^{2}}\right]^{3/2}{\mathrm{d}}y$
(2.6)

and the prolateness as

$\langle
P\rangle=\frac{105}{8}\int_{0}^{\infty}x^{5}D_{N}^{-3/2}(x)S_{3,N}(x){\mathrm{d}}x=\frac{105}{8}\int_{0}^{\infty}\sum_{j=1}^{N-1}\frac{y^{5}}{(\lambda_{j}+y^{2})^{3}}\left[\prod_{k=1}^{N-1}\frac{\lambda_{k}}{\lambda_{k}+y^{2}}\right]^{3/2}{\mathrm{d}}y.$
(2.7)

The $g$-ratio is given as

$g=S_{1,N}^{\text{dendrimer}}(0)/S_{1,N}^{\text{linear}}(0)=\sum_{j=1}^{N-1}\lambda_{j}^{-1}/\sum_{k=1}^{N-1}\tilde{\lambda}_{k}^{-1},$
(2.8)

where $\lambda_{j}$ and $\tilde{\lambda}_{k}$ are the non-zero eigenvalues of
the Kirchhoff matrix for a dendrimer and linear polymer with $N$ monomers.

The exact 9-branch and 21-branch dendrimer $S(k)$’s have previously been
determined [12]:

$S(k)_{9}=\frac{2}{x^{2}}\left(3+x-3\mathrm{e}^{-x/9}-12\mathrm{e}^{-x/3}+12\mathrm{e}^{-4x/9}\right)$
(2.9)

and

$S(k)_{21}=\frac{2}{x^{2}}\left(9+x-3\mathrm{e}^{-x/21}+6\mathrm{e}^{-2x/21}-12\mathrm{e}^{-x/7}-48\mathrm{e}^{-5x/21}+48\mathrm{e}^{-2x/7}\right).$
(2.10)

We have applied the general algorithm detailed in the Appendix in combination
with the Benhamou et al. [15] formalism to obtain results for 45, 93 and
189-branched systems. Their method expresses $S(k)$ as

$S(k)=\dfrac{1}{f}\left(x-1+\mathrm{e}^{-x}+\dfrac{2(\mathrm{e}^{2x}-2\mathrm{e}^{x}+1)}{fx^{2}}\sum_{k\subset
G,|k|\geqslant 2}\mathrm{e}^{-|k|x}\right).$ (2.11)

Here, $k$ is the length of a connected path between any two junctions in the
graph $G$ and $x=k^{2}\langle S^{2}\rangle$ calculated for the linear chain.
The algorithm developed in the Appendix first determines the number of walks
of length $k$ connecting two junctions and then prunes this set to obtain the
number of paths. A walk may visit a junction more than once whereas a path
cannot.

Table 1: Number of paths $p^{f}_{k}$ of length $k$ in an $f$-branch dendrimer. $k$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---
$p^{45}_{k}$ | 66 | 84 | 120 | 144 | 192 | 192 | 192 | 0 | 0 | 0 | 0
$p^{93}_{k}$ | 138 | 180 | 264 | 336 | 480 | 576 | 768 | 768 | 768 | 0 | 0
$p^{189}_{k}$ | 282 | 372 | 552 | 720 | 1056 | 1344 | 1920 | 2304 | 3072 | 3072 | 3072

Using the data in table 1 (see the Appendix for how these numbers were
obtained) we find that

$\displaystyle S(k)_{45}$ $\displaystyle=$
$\displaystyle\dfrac{2}{x^{2}}\left(21+x-3\mathrm{e}^{-x/45}+18\mathrm{e}^{-2x/45}-12\mathrm{e}^{-x/15}+24\mathrm{e}^{-4x/45}\right.$
(2.12) $\displaystyle-$
$\displaystyle\left.48\mathrm{e}^{-x/9}-192\mathrm{e}^{-7x/45}+192\mathrm{e}^{-8x/45}\right).$

This has a first-order Taylor expansion

$S(k)_{45}=1-\frac{3089x}{30375}+...\,.$ (2.13)

The $g$-ratio can then be determined by dividing the linear term in $x$ from
$S(k)_{45}$ by the corresponding term in the Taylor expansion for a linear
chain:

$S(k)_{\ell}=1-\frac{x}{3}+...\,.$ (2.14)

This gives a $g$-ratio value of $3089/10125\approx 0.3051$.

The $S(k)$ expressions for 93 and 189 branch dendrimers are

$\displaystyle S(k)_{93}$ $\displaystyle=$
$\displaystyle\dfrac{2}{x^{2}}\left(45+x-3\mathrm{e}^{-x/93}+42\mathrm{e}^{-2x/93}-12\mathrm{e}^{-x/31}+72\mathrm{e}^{-4x/93}\right.$
(2.15) $\displaystyle-$
$\displaystyle\left.48\mathrm{e}^{-5x/93}+96\mathrm{e}^{-2x/31}-192\mathrm{e}^{-7x/93}-768\mathrm{e}^{-3x/31}+768\mathrm{e}^{-10x/93}\right)$

and

$\displaystyle S(k)_{189}$ $\displaystyle=$
$\displaystyle\dfrac{2}{x^{2}}\left(93+x-3\mathrm{e}^{-x/189}+90\mathrm{e}^{-2x/189}-12\mathrm{e}^{-x/63}+168\mathrm{e}^{-4x/189}\right.$
(2.16) $\displaystyle-$ $\displaystyle
48\mathrm{e}^{-5x/189}+288\mathrm{e}^{-2x/63}-192\mathrm{e}^{-x/27}+384\mathrm{e}^{-8x/189}-768\mathrm{e}^{-x/21}$
$\displaystyle-$
$\displaystyle\left.3072\mathrm{e}^{-11x/189}+3072\mathrm{e}^{-4x/63}\right).$

After Taylor expanding and dividing by the linear chain result, we find that
the $g$-ratios for 93 and 189 branched structures are approximately 0.2009 and
0.1271, respectively.

The MC simulations employed the following algorithm to generate random
configurations. The dendrimers are grown on a three-dimensional simple cubic
lattice. The central unit is placed at the origin of the coordinate system.
Then, a random number is generated to select one of the six possible
directions for placing the second unit a distance one apart from the central
unit. Each successive placement of units is performed by this procedure. Since
ideal dendrimers are the focus of this research, the units are allowed to
overlap. Each completed dendrimer is an independent configuration and
properties were computed by averaging over each of these random samples. In
all cases, 100 000 random samples were employed.

##  3 Results

Tables 2, 3, and 4 show the property values found for the different branch
dendrimers as a function of the number of units $N$; $\langle\lambda\rangle$
values for each of the eigenvalues are included, as well as, $\langle
A\rangle$ and $\langle P\rangle$. The radius of gyration is presented for both
the dendrimer and the corresponding linear polymer chain. The error deviation
is shown as a number in parenthesis (one standard deviation from the mean).

The tables contain some interesting results. For any dendrimer generation,
$\langle A\rangle$ and $\langle P\rangle$ level off as the number of total
units in the dendrimer increases. Another observation is that $\langle
S^{2}\rangle$ is always higher for the linear chains than it is for the
dendrimers. This is to be expected because the dendrimer is less spread out
than a linear chain and would therefore have an average shorter distance from
its center of mass to an outer unit.

Table 2: Effect of the number of units, $N$, for 45-branch dendrimers. Property/$N$ | 901 | 1531 | 1756 | 2206
---|---|---|---|---
$\langle\lambda_{1}\rangle$ | 26.79(3) | 45.45(5) | 52.11(6) | 65.54(8)
$\langle\lambda_{2}\rangle$ | 12.78(1) | 21.60(2) | 24.81(3) | 31.09(3)
$\langle\lambda_{3}\rangle$ | 6.57(1) | 11.09(1) | 12.70(1) | 15.93(2)
$\langle A\rangle$ | 0.160(1) | 0.161(1) | 0.161(1) | 0.162(1)
$\langle S^{2}\rangle$ | 46.14(4) | 78.13(7) | 89.62(8) | 112.56(10)
$\langle P\rangle$ | 0.095(1) | 0.096(1) | 0.096(1) | 0.097(1)
$\langle S^{2}\rangle_{\ell}$ | 149.51(24) | 254.05(42) | 291.51(47) | 366.89(60)
Table 3: Effect of the number of units, $N$, for 93-branch dendrimers. Property/$N$ | 931 | 1861 | 2791 | 3721
---|---|---|---|---
$\langle\lambda_{1}\rangle$ | 17.23(2) | 34.28(4) | 51.36(5) | 68.39(7)
$\langle\lambda_{2}\rangle$ | 9.15(1) | 18.12(2) | 27.11(2) | 36.12(3)
$\langle\lambda_{3}\rangle$ | 5.23(1) | 10.32(1) | 15.44(1) | 20.54(2)
$\langle A\rangle$ | 0.121(1) | 0.123(1) | 0.123(1) | 0.123(1)
$\langle S^{2}\rangle$ | 31.61(2) | 62.73(5) | 93.91(7) | 125.05(10)
$\langle P\rangle$ | 0.059(1) | 0.060 (1) | 0.060(1) | 0.060(1)
$\langle S^{2}\rangle_{\ell}$ | 154.54(25) | 309.18(50) | 465.57(76) | 619.62(101)
Table 4: Effect of the number of units, $N$, for 189-branch dendrimers. Property$/N$ | 946 | 2080 | 3592 | 4915
---|---|---|---|---
$\langle\lambda_{1}\rangle$ | 10.53(1) | 22.94(2) | 39.55(4) | 54.10(5)
$\langle\lambda_{2}\rangle$ | 6.12(5) | 13.26(1) | 22.76(2) | 31.12(2)
$\langle\lambda_{3}\rangle$ | 3.83(1) | 8.26(1) | 14.14(1) | 19.32(1)
$\langle A\rangle$ | 0.091(1) | 0.092(1) | 0.093(1) | 0.094(1)
$\langle S^{2}\rangle$ | 20.48(1) | 44.47(3) | 76.45(5) | 104.53(7)
$\langle P\rangle$ | 0.036(1) | 0.037(1) | 0.038(1) | 0.038(1)
$\langle S^{2}\rangle_{\ell}$ | 157.12(26) | 345.21(56) | 599.47(98) | 819.05(134)

Table 5 demonstrates that the radius of gyration data reported in tables 2, 3,
and 4 is in excellent agreement with the Sheng et al. [2] exact equation. The
scaling exponent for the radii of gyration, $2\nu$, was obtained by fitting
the $\langle S^{2}\rangle$ data to a power function. Figure 2 presents a log-
log plot of the data for each distinct generation. The exponents found are:
0.995(1), 0.992(1), and 0.988(1) for 45, 93 and 189 branches, respectively.
These are close to the expected theoretical exponent of 1.0.

Figure 2: Log-log plot of the radius of gyration vs. $N$ for each dendrimer generation: circles 45-branch, squares 93-branch, triangles 189-branch, lines best linear fit. Table 5: Comparison of $\langle S^{2}\rangle$ MC to Sheng et al.[2] exact predictions. 45 Branches | 93 Branches | 189 Branches
---|---|---
$N$ | MC | Exact | $N$ | MC | Exact | $N$ | MC | Exact
901 | 46.14(4) | 46.161 | 931 | 31.61(2) | 31.578 | 946 | 20.48(1) | 20.469
1531 | 78.13(7) | 78.195 | 1861 | 62.73(5) | 62.723 | 2080 | 44.47(3) | 44.484
1756 | 89.62(8) | 89.636 | 2791 | 93.91(7) | 93.868 | 3592 | 76.45(5) | 76.503
2206 | 112.56(10) | 112.517 | 3721 | 125.05(10) | 125.014 | 4915 | 104.53(7) | 104.520

The data in the tables are for finite $N$ whereas most of the theories are for
infinite $N$. Thus, the data for $\langle A\rangle$ and $\langle P\rangle$
have been extrapolated using a linear function in $1/N$. The $g$-ratios were
extrapolated after first determining their error by relating the error in a
ratio to the error in the numerator and the error in the denominator. As the
number of units increases, $1/N$ gets smaller, so that when $1/N\rightarrow
0$, $N$ becomes infinite. Hence, extrapolation should remove all finite $N$
effects.

Table 6 compares the extrapolation MC $g$-ratios to Eq. (1.6), the Wei and
BGGB methods. All the results are in excellent agreement. The MC
extrapolations for $\langle A\rangle$ and $\langle P\rangle$ are compared to
the Wei method in table 7. Again excellent agreement is found.

Table 6: Comparsion of extrapolated MC $g$-ratios to theoretical predictions. Branches | MC | Eq. 1.6 | Wei | BGGB
---|---|---|---|---
45 | 0.306(2) | 0.305086 | 0.305088 | 0.3051
93 | 0.201(1) | 0.200937 | 0.200938 | 0.2009
189 | 0.128(1) | 0.127060 | 0.127062 | 0.1271
Table 7: Comparison of extrapolated MC $\langle A\rangle$ and $\langle
P\rangle$ to Wei method results.

Branches |  $\langle A\rangle$ MC | Wei |  $\langle P\rangle$ MC | Wei
---|---|---|---|---
45 | 0.163(2) | 0.1623100 | 0.098(1) | $0.0967390^{a}$
93 | 0.124(1) | 0.1237990 | 0.061(1) | $0.0609874^{a}$
189 | 0.094(1) | 0.0944898 | 0.039(1) | $0.0389064^{a}$

$a$ Note that the Wei method defines $\langle P\rangle$ with an additional
factor of $1/2$.

The values of the asphericity reflect how spherical and symmetric a dendrimer
is; an $\langle A\rangle$ value of 0.0 represents a perfect sphere. The data
show that, as the number of branches increases, the values for $\langle
A\rangle$ and $\langle P\rangle$ decrease. De Regt et al. [12] found that
$\langle A\rangle$ changed from 0.266(1) to 0.211(1) when going from the first
generation 9-branch dendrimer to the second generation 21-branch dendrimer;
similarly $\langle P\rangle$ changed from 0.233(3) to 0.153(2). These values
indicate that higher generation dendrimers have a more symmetric and spherical
shape.

In table 8 the $\langle A\rangle$ and $g$-ratio results for ideal dendrimers
are compared to Wawrzyriska et al. [16] values for a MC self-avoiding walk
(SAW) model. One notes that the excluded volume effects in the SAW become more
and more important as the generation number increases. Indeed, the excluded
volume model is forced to be more symmetrical because the units cannot
overlap.

It is interesting to compare the shape properties of dendrimers to multi-
branched comb polymers. Casassa and Berry [17] have derived a general equation
for the $g$-ratio of uniform three-functional ideal comb polymers:

$g_{\text{comb}}=r-\dfrac{r^{2}(1-r)}{f+1}+\dfrac{2r(1-r)^{2}}{f}+\dfrac{(3f-2)(1-r)^{3}}{f^{2}},$
(3.1)

where $r$ is the ratio of the number of branches in the backbone to the total
number of branches. This equation yields 0.638, 0.566, 0.532, 0.516 and 0.508
when $f$ is 9, 21, 45, 93 and 189, respectively. Comparing these values to the
dendrimer results in table 8 one sees a very different behavior. The
$g$-ratios of the comb polymers appear to level off to 0.5 as the number of
branches increase whereas the dendrimer $g$-ratios get smaller and smaller.
Van Ferber et al [18] have computed $\langle A\rangle$ for an ideal 9 branch
comb polymer. They found that $\langle A\rangle$ = 0.295(2) which is larger
than the 9 branch dendrimer value of 0.266(1). The $g$-ratio and $\langle
A\rangle$ value indicate that dendrimers are more symmetrical than the
corresponding comb polymers.

Table 8: Comparison of ideal and SAW dendrimers.

Branches | Ideal $\langle A\rangle$ |  $g$-ratio | SAW $\langle A\rangle$ |  $g$-ratio
---|---|---|---|---
9 | 0.266(1)a | 0.606(1)a | 0.2682 | 0.6068
21 | 0.211(1)a | 0.443(1)a | 0.1968 | 0.4485
45 | 0.163(2) | 0.306(2) | 0.1390 | 0.3180
93 | 0.124(1) | 0.201(1) | 0.0972 | 0.2199
189 | 0.094(1) | 0.128(1) | 0.0685 | 0.1509

$a$ see reference [12].

Figure 3: Number dependence for the 93-branch dendrimer scattering function.

A Kratky plot [19] of $S(k)$ is often used to investigate branched polymers.
Figure 3 shows the variation of a Kratky plot for the MC 93-branch dendrimer
$S(k)$ calculations as the total number of units in all the branches is
increased from 931 to 3727. Note that we have used $x=k^{2}\langle
S^{2}\rangle$ throughout this paper instead of the usual $x^{2}=k^{2}\langle
S^{2}\rangle$ to conform to the notation employed by Benhamou et al. [15]. It
is clear that number dependent effects are quite small if one uses a
sufficiently large number of units in the simulations.

Figure 4 presents the results for the largest $N$, $S(k)$ MC calculations for
linear chains and 9, 21, 45, 93 and 189-branch dendrimers compared to the
exact equations. In order to compare the structures with different branching
arrangements, the data have been scaled with the appropriate $g$-ratio values
as described in Toporowski and Roovers [20]. As reported by Burchard [19],
“Linear randomly coiled chains result in an angular asymptote which has a
value of 2 for monodisperse coils… The striking feature with branched chains
is the appearance of a maximum for stars and other regularly branched chains.
This maximum becomes more and more pronounced with an increasing branching
density.” This plot confirms the observation that higher generation dendrimers
are more symmetrical. Even higher dendrimer generations are needed to see
sphere-like behavior [21].

Figure 4: Variation of the scattering function with dendrimer generation and
number of units: diamond linear 2080, X 9-branch 1342, up triangle 21-branch
3130, square 45-branch 2206, down triangle 93-branch 3727, circle 189-branch
4915, lines exact equations.

##  4 Conclusions

Three different ideal dendrimer generations have been examined using a variety
of methods. Multiple properties were computed, and excellent agreement was
found between the Monte Carlo simulations and the theoretical predictions. All
the properties studied demonstrate that ideal dendrimers become more
symmetrical as the generation number and the number of branches increase.

As we have shown, excluded volume effects become more and more important at
higher generations. Coarse grained simulation models reviewed by Klos and
Sommer [22] have established that the radius of gyration of isolated excluded
volume dendrimers has different scaling behaviors in good, theta and poor
solvents. Even more realistic models of isolated charged dendrimers have
demonstrated the important influence of parameters such as pH on the
structural properties. The ideal systems examined here form a first
approximation to these more complicated models and permit exact calculations
for many dendrimer properties.

## Acknowledgements

We wish to thank the Manhattan College Computer Center and the Kakos Center
for Scientific Computing for large grants of computer time. We would also like
to thank Professors Mark DeBonis and Paula Whitlock for helpful discussions.
Brandon Thrope was supported by a Manhattan College Summer Research grant.

## Appendix

The structure of a dendrimer can be considered as an abstract graph, $G$, in
which the polymer junctions are represented by nodes (vertices, $V$) and the
branches as edges, $E$. A walk in such a graph $G=(V,E)$ is then a sequence of
nodes $v_{1},v_{2},\ldots v_{n}$ so that $E(v_{i},v_{i+1})$ for each $i$ with
$1\leqslant i\leqslant n-1$. A path in $G$ is a walk in which the nodes are
pairwise distinct (so, a walk may visit a node more than once, while a path
may not).

In order to be able to compute the theoretical $S(k)$ and thus the $g$-ratio
for a given polymer $T$, we need to be able to count the number of paths of
each length in $T$. Since each of the polymers considered here are dendrimer
polymers (having a tree structure), we know that between any two nodes, there
is a unique path (trees have no cycles, i.e., they have no paths which begin
and end at the same node). Therefore, if $T$ is a dendrimer polymer with $N$
nodes, then $T$ has exactly $N\choose 2$ total paths. However, we are
interested in finding the number of paths of any given length.

We define the adjacency matrix $A$ of the graph $G$ with node set
$V=\\{1,\ldots,N\\}$ to be the $N\times N$ matrix $(a_{ij})$ so that
$a_{ij}=1$ if $\\{i,j\\}\in E$, and $a_{ij}=0$ if $\\{i,j\\}\not\in E$, where
$a_{ij}$ denotes the entry in the $i$-th row and $j$-th column of the matrix.
Let $M=(m_{ij})$ be a matrix. We denote $M(i,j)=m_{ij}$ for each $i,j$, and
$M^{n}$ to be the $n$-th power of the matrix $M$ (i.e., $M$ multiplied by
itself $n$-many times). From [23], we see that the number of walks in $G$ from
node $i$ to node $j$ of length $k$ is given by $A^{k}(i,j)$ (i.e., the entry
in the matrix $A^{k}$ contained in the $i$-th row and $j$-th column).

Therefore, one way to count the number of paths of length $k\in\mathbb{N}$ in
a given dendrimer polymer $T$ (of the type under consideration) with adjacency
matrix $A$, would be to do the following:

  1. 1.

Case $k=2$:

    1. (a)

Compute $A^{2}$ and initialize $A_{2}=A^{2}$.

    2. (b)

Change all main diagonal entries in $A_{2}$ to 0.

    3. (c)

Sum all entries in $A_{2}$.

    4. (d)

Divide the result by 2.

  2. 2.

Case $k>2$ with $k\in\mathbb{N}$:

    1. (a)

Compute $A^{k}$ and initialize $A_{k}=A^{k}$.

    2. (b)

Change all entries of $A_{k}$ that are greater than $1$ to $0$.

    3. (c)

Sum all entries in $A_{k}$.

    4. (d)

Divide the result by 2.

Verification:

Case $k=2$. Every main diagonal entry in $A^{2}$ will be greater than $0$, but
these entries do not correspond to paths. They correspond to walks from a node
back to the same node, and therefore they are not valid paths; each of these
entries therefore counts the degree of the corresponding node. The degrees of
each leaf node is $1$ (the degree of a node is the number of nodes adjacent to
it, and a leaf node in a tree is a node that is only adjacent to one other
node), and, therefore, there will be some main diagonal entries that are
exactly $1$. Since, in this case, we are counting length-two paths, no other
entries in $A^{2}$ will be greater than 1. It is easy to see that the non-
main-diagonal entries of 1 in $A^{2}$ correspond to valid paths.

After changing the main diagonal entries to $0$, and summing the remaining
entries, we will have double the number of valid paths, because if there is a
path from $i$ to $j$, then there is also a path from $j$ to $i$ (they are the
same path), so that we will have $A_{2}(i,j)=1=A_{2}(j,i)$.

Case $k>2$, $k\in\mathbb{N}$. We do not need to treat main diagonal entries
separately in this case. If $k$ is odd, then the main diagonal entries will
all be $0$ (because there are no odd-length walks from a node back to itself),
and if $k$ is even with $k>2$, then the main diagonal entries corresponding to
leaf nodes will be greater than $1$, since the nodes adjacent to the leaf
nodes have degree larger than $1$ (so, there will always be more than $1$ walk
from every node back to itself).

First, we claim that if $A^{k}(i,j)=0$, then there is no path between $i$ and
$j$. Suppose $A^{k}(i,j)=0$. Then, there is no walk between $i$ and $j$, and
therefore there can be no path between $i$ and $j$, proving the claim.

Next, we claim that if $A^{k}(i,j)>1$, then there is no path of length $k$
between $i$ and $j$. Dendrimer polymers have a tree structure, in which there
is not more than one distinct path between two given nodes. So, suppose that
$A^{k}(i,j)>1$ and there is exactly one path between $i$ and $j$. Let $p$
denote the path and let $w$ denote a walk between $i$ and $j$ that is
different from $p$. It cannot be that both $p$ and $w$ are paths, because then
we could form a cycle, which is impossible because of the tree structure of
the polymer. Thus, $w$ must repeat some node. Then, from this fact, we can
argue that we could form some path $p^{\prime}$ from $i$ to $j$ which has the
length less than $k$. Clearly, $p$ cannot be the same as $p^{\prime}$, and
therefore we could again form a cycle, which is impossible. This proves the
claim.

Now, we claim that if $A^{k}(i,j)=1$, then there is exactly one valid path
between $i$ and $j$ of length $k$. Suppose that $A^{k}(i,j)=1$, but there is
not exactly one path between $i$ and $j$ of length $k$. (There cannot be more
than one such path because there is only one walk.) Because $A^{k}(i,j)=1$,
there must be a unique walk $w$ (of length $k$), consisting, say, of nodes
$\\{v_{1},\ldots,v_{k}\\}$ with $v_{1}=i$ and $v_{k}=j$, which passes through
some $v_{q}$, with $q\in\\{1,\ldots,k\\}$, more than once. But then we can
find a walk $w^{\prime}\neq w$ from $i$ to $j$. It cannot be that there is a
cycle starting at $v_{q}$ because the polymer has a tree structure. Therefore,
it must be the case that $w$ passes from $v_{q}$, then through a line of
nodes, then traverses back through the same line of nodes to reach $v_{q}$
once again. Without loss, assume that $w$ has the form
$v_{1},\ldots,v_{q},v_{q^{\prime}},v_{q},\ldots,v_{k}$. Because of the
structure of the dendrimer polymers under consideration (and the fact that
each non-leaf node has degree $3$), there must be some $v_{\ell}$ in $w$ which
is adjacent to some node $v^{\prime}$ which is not in $w$. Define $w^{\prime}$
to consist of nodes $v_{1},\ldots,v_{\ell},v^{\prime},v_{\ell},\ldots v_{k}$
(where $v_{q^{\prime}}$ is excluded, and $v_{\ell}$ could be either $v_{1}$ or
$v_{k}$ as well). Thus, there is more than one distinct walk between $i$ and
$j$, contradicting the assumption that $A^{k}(i,j)=1$. This proves the claim.

Finally, after changing the entries in $A^{k}$ which are larger than $1$ to
$0$, and summing the remaining entries, we will have double the number of
valid paths.

This ends the verification.

The above algorithm was implemented in MATLAB for the dendrimer polymers under
consideration. If $p^{f}_{k}$ denotes the number of paths of length $k$ in the
$f$-branch dendrimer polymer, we obtain the data presented in table 1 to use
in equation (2.11).

## References

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doi:10.1002/pol.1982.180200201.

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2014, 23, 288, doi:10.1002/mats.201300159.

  * [6] Solc K., Stockmeyer W. H., J. Chem. Phys., 1971, 54, 2756, doi:10.1063/1.1675241.
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  * [9] Blavatska V., Janke W., J. Chem. Phys., 2010, 133, 184903, doi:10.1063/1.3501368.
  * [10] Flory P. J., Statistical Mechanics of Chain Molecules, Hanser, Munich, 1989.
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  * [12] De Regt R., von Ferber C., Bishop M., Hamling T., Physica A, 2019, 516, 50, doi:10.1016/j.physa.2018.09.196.
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doi:10.5488/CMP.7.1.179.

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doi:10.1002/mats.201500036.

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doi:10.1063/1.4905101.

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# ATPPNet: Attention based Temporal Point cloud Prediction Network

Kaustab Pal∗1, Aditya Sharma∗1, Avinash Sharma2, K. Madhava Krishna1 ∗ denotes
equal contribution.1 are with RRC, IIIT Hyderabad, India.{kaustab21,
<EMAIL_ADDRESS><EMAIL_ADDRESS>is with IIT Jodhpur, India.
<EMAIL_ADDRESS>is available at https://tinyurl.com/atppnet

###### Abstract

Point cloud prediction is an important yet challenging task in the field of
autonomous driving. The goal is to predict future point cloud sequences that
maintain object structures while accurately representing their temporal
motion. These predicted point clouds help in other subsequent tasks like
object trajectory estimation for collision avoidance or estimating locations
with the least odometry drift. In this work, we present ATPPNet, a novel
architecture that predicts future point cloud sequences given a sequence of
previous time step point clouds obtained with LiDAR sensor. ATPPNet leverages
Conv-LSTM along with channel-wise and spatial attention dually complemented by
a $3D$-CNN branch for extracting an enhanced spatio-temporal context to
recover high quality fidel predictions of future point clouds. We conduct
extensive experiments on publicly available datasets and report impressive
performance outperforming the existing methods. We also conduct a thorough
ablative study of the proposed architecture and provide an application study
that highlights the potential of our model for tasks like odometry estimation.

##  I Introduction

Autonomous navigation is a widely explored research direction in the robotics
domain with applications in autonomous aerial/aquatic drones, vehicles, mobile
robots, etc. Recent advancements in $3D$ sensing led by commercial LiDAR
(Light Detection and Ranging) technology have reinvigorated interest in this
field as LiDAR sensors yield large-scale real-time sequential point clouds (
also represented as range images), providing high-fidelity perception of the
$3D$ world in comparison to traditional monocular/stereo based vision
solutions. The availability of such large-scale data [1, 2] has enabled
researchers to explore relevant complex tasks such as Localization [3, 4],
Place Recognition [5], Segmentation [6, 7] and Obstacle Trajectory Prediction
[8]. The majority of existing methods attempting to solve these tasks rely on
captured sequential point clouds available in a given temporal window of the
recent past. Interestingly, predicting the future $3D$ point cloud that the
sensor is likely to see, can immensely enhance the performance of autonomous
navigation tasks like active localization [9].

Nevertheless, the task of predicting future point clouds comes with its own
set of challenges. One key challenge is that the point clouds are unordered in
the space dimension (albeit ordered temporally) and vary in sampling size
hence it is difficult to model spatio-temporal coherence among them. As a
result, conventional architectures for feature encoding (e.g., CNNs) and
sequence prediction (e.g., LSTMs) cannot be directly employed as they cannot
process spatially unordered data. Another key challenge is that the LiDAR
point clouds are extremely sparse making it difficult to capture the
geometrical structures of the objects in the scene and hence predicting them
in the future timesteps is extremely difficult. The noise in sensing puts
additional challenges in the perception of real-world scenes where objects are
largely cluttered. Moreover, each full-scale point cloud contains more than
$100,000$ points. Extracting features from these sequences of full-scale point
clouds becomes a memory-intensive task.

Figure 1: (a) Predicted range images by our ATPPNet and existing methods in
comparison to ground truth and, (b) the $3D$ rendering of the predicted point
cloud by ATPPNet (blue) and ground-truth (red). Green circle/rectangle
highlights regions where ATPPNet’s predictions are superior.

Traditionally, $3D$ data is processed with deep learning encoders using
volumetric [10, 11, 12], point-cloud [13, 14] and multi-view projection [15,
16, 17] methods. In regard to future point cloud prediction, primarily two
lines of work exist, focusing on point cloud and range image representation.
The existing point cloud prediction methods either reformulate the task as
scene flow estimation [18] or employ RNN kind of temporal prediction [19, 20].
The former predicts just a translation of the $3D$ points and hence does not
represent the future point cloud accurately. At the same time, the latter
works on down-sampled point clouds (for memory efficiency reasons) thereby
limiting the resolution of $3D$ data.

On the other hand, range image based representations project the point cloud
data to a $2D$ virtual image plane of the LiDAR sensor, thereby retaining only
the single (closest, farthest, or average) depth of the scene for every pixel.
Early work with this representation [21] used LSTMs to process the temporal
sequences and predict a sequence of future range images. [22] used $3D$-CNNs
with circular padding and skip-connections to predict a sequence of future
range images while [23] used Conv-LSTMs on each of the features from the
convolution encoder for the prediction task. However, their network is
cumbersome and they use the auto-regressive approach for prediction of range
images. Recent work in [24] uses the self-attention mechanism of Transformers
along with a semantic-based loss function. This method compresses the $3D$
tensors into height and width dimensions and processes each of them separately
using two separate transformer blocks. As a result, they are using self-
attention only on the channels and since they are compressing the feature
tensor into height and width they are also losing the spatial context.
Additionally, their model size in terms of the number of parameters is large.

In this paper, we propose a novel architecture for predicting future point
clouds from a given sequence of past point clouds represented as LiDAR range
images. More specifically, we propose ATPPNet: Attention based Temporal Point
cloud Prediction Network that leverages Conv-LSTM [25] blocks along with
channel-wise and spatial attention modules for extracting an enhanced spatio-
temporal context for the task of future point cloud prediction. Further, we
also leverage a complimentary $3D$-CNN branch to spatio-temporally encode the
global feature embeddings of the range images. Additionally, we also predict
the re-projection mask associated with the predicted range images to retain
only the valid range values when re-projecting to the point cloud. Compared to
[24], we show that processing the range image sequences using Conv-LSTM and
using spatial and channel-wise attention directly on learned spatio-temporal
$3D$ features works better without the need for a separate semantic-based loss
function. Our proposed architecture achieves state-of-the-art performance on
two publicly available datasets. Our method yields real-time future point
cloud prediction (faster than a typical rotating $3D$ LiDAR sensor point cloud
rate i.e., 10Hz). We conduct thorough qualitative and quantitative evaluations
as well as provide a detailed ablation study to validate the effectiveness of
our proposed architecture. To summarize, our main contributions are as
follows:

  * •

We proposed a novel architecture (ATPPNet) that leverages Conv-LSTM and
spatial and channel-wise attention for predicting future point clouds from a
sequence of past point clouds.

  * •

ATPPNet achieves SOTA performance on various publicly available datasets while
beating the existing methods by $8-10\%$ margin.

  * •

We empirically show that ATPPNet improves the performance of downstream tasks,
like odometry estimation.

##  II Our Approach

We provide details of our novel ATPPNet (Attention based Temporal Point cloud
Prediction Network) that leverages Conv-LSTM blocks along with channel-wise
and spatial attention modules complemented by a 3D-CNN branch to process a
past sequence of $3D$ point clouds to predict future point clouds.

Figure 2: ATPPNet Architecture. ATPPNet leverages Conv-LSTM along with
channel-wise and spatial attention dually complemented by a 3D-CNN branch for
extracting an enhanced spatio-temporal context to recover high quality fidel
predictions of future point clouds.

Let $S_{P}=\\{S_{t-M+1},S_{t-M+2},\dots S_{t}\\}$ be the set of input $3D$
point cloud sequence of $M$ time steps in a temporal window where
$S_{\tau}\in\mathbb{R}^{3}$ is set of $3D$ points captured at specific time
step $\tau$. The goal of our ATPPNet is to predict the future sequence of $3D$
point clouds for a temporal window of $N$ time steps represented as
$S_{F}=\\{S_{t+1},S_{t+2},\dots,S_{t+N}\\}$. Similar to [22], we adopt the
range image representation by first converting the point clouds into the
spherical coordinate system and then projecting the corresponding point cloud
($S_{\tau}\in\mathbb{R}^{3}$) to the virtual image plane of the LiDAR sensor,
represented as $R_{\tau}\in\mathbb{R}^{2}$. Let
$R_{P}=\\{R_{t-M+1},R_{t-M+2},\dots R_{t}\\}$ be the sequence of past range
images in a fixed temporal window (obtained from $S_{P}$ ) and similarly
$R_{F}=\\{R_{t+1},R_{t+2},\dots,R_{t+N}\\}$ be the sequence of predicted range
images associated with $S_{F}$.

###  II-A Overall Architecture

Figure 2 provides the overview of the proposed ATPPNet architecture. A shared
convolution encoder processes the input range images $R_{P}$ and generates $L$
number of multi-scale feature tensors for each of the range images.
Subsequently, the first $L-1$ feature tensors are fed to Conv-LSTMs to model
the spatio-temporal relationships across $R_{P}$. Further, we exploit spatial
as well as channel-wise attention on the outputs of Conv-LSTMs to obtain the
$L-1$ context tensors (i.e., the consolidated spatio-temporal encoding of
$R_{P}$). Additionally, for the final $L$-th feature tensor, we use a $3D$-CNN
layer to process the spatio-temporal relationship and generate the $L$-th
feature tensor for $N$ future time steps. On the decoder side, we feed the
context tensor along with the hidden state of the last time step to $L-1$
Conv-LSTMs and generate the feature tensors for each of the $L-1$ layers for
$N$ time steps into the future. All these $L$ feature tensors on the decoder
side are subsequently processed to generate the range image sequence $R_{F}$
along with their corresponding re-projection masks $M_{F}$ for all the $N$
future time steps where each pixel of $M_{\tau}\in M_{F}$ can be interpreted
as the probability for each of the range image pixels to be valid or invalid.
This re-projection mask is used while back-projecting a range image to a point
cloud where we only retain the range values corresponding to probabilities
greater than $0.5$. The construction of specific architectural blocks is given
below.

###  II-B Convolution Encoder & Decoder block

The convolution encoder block takes the range image and first performs a
$2D$-convolution operation, resulting in a tensor with an increased number of
channels but the same spatial resolution. This tensor is further processed
using $L$ convolutional sub-blocks. Each sub-block takes as input a feature
tensor and applies a combination of $2D$-convolution, $2D$-batch
normalization, and leaky-ReLU operation while keeping the tensor dimensions
the same. A strided-convolution operation is subsequently performed, resulting
in a down-sampled tensor.

The convolution decoder block follows the reversed structure of the
convolution encoder block. There are $L$ sub-blocks, each of which takes an
input tensor and passes it through a $2D$-transposed convolution, $2D$-batch
normalization, and leaky-ReLU operation resulting in a spatially scaled-up
tensor while keeping the number of channels the same. Another $2D$-convolution
operation is then performed, to decrease the channel size of the tensor. The
output of the $L$-th layer is finally passed through another
$2D$-convolutional layer resulting in the predicted range images and the
associated re-projection mask.

###  II-C Conv-LSTM encoder

We propose to use $L-1$ Conv-LSTMs to exploit the spatio-temporal context of
input sequences. Similar to the S2Net [23] architecture, we used multiple
Conv-LSTM layers for each of the feature tensors. This helps us in preserving
the high-frequency details across the range image sequences. The Conv-LSTMs
for each of the $L-1$ features generate a hidden state for each of the $M$
time steps.

###  II-D Attention Module

Let $\tau\in[t-M+1,\dots,t-1,t]$ be a specific time step in the given input
temporal window, and the feature tensors for the layer $l$ at every time step
be represented as $f_{\tau}^{l}$. Similarly, let the output of $l$-th Conv-
LSTM at time step $t$ denoted as $g_{t}^{l}$ where $l\in[1,\dots L-1]$. As
part of the attention module, we first compute the joint embedding
$\mathcal{J}_{\tau}^{l}$ of $f_{\tau}^{l}$ and $g_{t}^{l}$ as (using the
formulation [26]):

$\mathcal{J}_{\tau}^{l}=\sigma_{1}(W_{f}f_{\tau}^{l}\oplus W_{g}g_{t}^{l}),$

where $\sigma_{1}$ is a non-linear activation function chosen as ReLU and
$\oplus$ is the concatenation operation. $W_{f}$ and $W_{g}$ are implemented
as $2D$-convolution operations with $1\times 1$ kernel. The use of
$\sigma_{1}$, $W_{f}$, and $W_{g}$ allows the network to learn non-linear
relationships between the features, which is especially important when the
image is noisy like our range images. The resulting tensors
$\mathcal{J}_{\tau}^{l}$ are passed through the spatial and channel-wise
attention module [27] to find the $3D$ attention map
$\mathcal{M}(\mathcal{J}_{\tau}^{l})\in R^{C_{l}\times H_{l}\times W_{l}}$.
The refined feature tensor for layer $l$ at time step $\tau$ is computed as:

$\hat{f}_{\tau}^{l}=f_{\tau}^{l}\otimes\mathcal{M}(\mathcal{J}_{\tau}^{l}).$

Here $\otimes$ is the element-wise multiplication. To compute the $3D$
attention map $\mathcal{M}(J_{\tau}^{l})$, we compute the channel-wise
attention and spatial attention separately and then combine them as

$\mathcal{M}(\mathcal{J}_{\tau}^{l})=\sigma(M_{c}(\mathcal{J}_{\tau}^{l})\otimes
M_{s}(\mathcal{J}_{\tau}^{l}))a,$

where $\sigma$ is the Sigmoid function and $\otimes$ is the element-wise
multiplication operation. The $M_{c}$ function first applies the global
average pooling operation on the $3D$ tensor $\mathcal{J}_{\tau}^{l}$ to get
the channel tensor which is then passed through an MLP layer to get the
channel-wise attention values. The $M_{s}$ function applies $2D$-convolution
operation on the $3D$ tensor $\mathcal{J}_{\tau}^{l}$ and returns a single
channel tensor which represents the spatial attention values.

The refined feature tensors for all the $M$ time steps for each of the $L-1$
layers are then used to compute the context tensors, that are subsequently
served as input to the decoder Conv-LSTMs.

$context_{l}^{t}=\sum_{\tau=t}^{t-M+1}\hat{f}_{\tau}^{l}$

###  II-E Conv-LSTM decoder

The Conv-LSTM decoder follows a similar structure as the Conv-LSTM encoder.
$L-1$ Conv-LSTM decoders are used to predict $L-1$ feature tensors for each of
the $N$ future time steps. Let $\tau\in[t+1,\dots,N]$ be a specific time step
in the predicted future temporal window. For the $\tau$-th time step, the
$l$-th Conv-LSTM decoder takes as input the context tensor
$context_{l}^{\tau-1}$ where $l\in[1,\dots,L-1]$ along with the hidden state
of the Conv-LSTM for time step $\tau-1$ to compute the output feature tensor.
This output feature tensor along with the hidden states of the previous time
steps are used to re-compute the context tensor $context_{l}^{\tau}$ to be
used for the next time step. These output feature tensors on the Conv-LSTM
decoder side are used by the convolutional decoder to generate the predicted
range images $R_{F}$

###  II-F $3D$-CNN block

The feature tensors for the $L^{th}$ layer from the convolutional encoder
block for all the past $M$ time steps are concatenated to create a $4D$
tensor. A $3D$-CNN layer is used to process this feature tensor and generate
$N$ feature maps for the $L$-th convolutional decoder block.

It is important to note that since $3D$-CNNs place their spatial focus on
fewer, contiguous areas in the feature tensors [28], we employ $3D$-CNN on the
last $L-th$ layer of the feature tensor (obtained with $2D$-CNN) as it tends
to capture global structures in the range image [29], [30]. Thus, the $3D$-CNN
block extracts only the complementary spatio-temporal context as the primary
spatio-temporal context is already obtained by applying Conv-LSTM’s on the
initial layers of the convolutional encodings as they tend to capture the high
frequency details in the range images. This also gives us the additional
advantage of speeding up our inference time.

###  II-G Loss Function

We use a combination of losses when training the network. Since our ground
truth point clouds are projected onto $2D$ range images of dimension $H\times
W$, we can use $2D$ image-based losses.

Firstly, we use the average range loss $\mathcal{L}_{R}$ to compute the error
between the predicted range values $\hat{r}_{\tau,i,j}\in\mathbb{R}^{N\times
H\times W}$ and the ground-truth range values
$r_{\tau,i,j}\in\mathbb{R}^{N\times H\times W}$. The average range loss can be
formulated as:

$\mathcal{L}_{R}=\frac{1}{N\times H\times
W}\sum_{\tau=t+1}^{t+N}\sum_{i=1}^{H}\sum_{j=1}^{W}\mid\mid\hat{r}_{\tau,i,j}-r_{\tau,i,j}\mid\mid_{1},$

where $\mid\mid\bullet\mid\mid_{1}$ represents the $L_{1}$ norm. The range
loss $\mathcal{L}_{R}$ is computed only for the valid ground truth points. To
train the re-projection mask output, we use the Binary Cross-Entropy loss
between the predicted mask values $\hat{m}_{\tau,i,j}\in\mathbb{R}^{N\times
H\times W}$ and the ground truth mask values
$m_{\tau,i,j}\in\mathbb{R}^{N\times H\times W}$. The average mask loss
$\mathcal{L}_{M}$ is computed as:

$\displaystyle\mathcal{L}_{M}$ $\displaystyle=\frac{1}{N\times H\times
W}\sum_{\tau=t+1}^{t+N}\sum_{i=1}^{H}\sum_{j=1}^{W}-m_{\tau,i,j}\log\hat{m}_{\tau,i,j}$
(1) $\displaystyle-(1-m_{\tau,i,j})\log(1-\hat{m}_{\tau,i,j}),$ (2)

where $\hat{m}_{\tau,i,j}$ is the predicted probability of whether the range
value is valid. $m_{\tau,i,j}$ is $1$ if the ground-truth range value is valid
and $0$ otherwise. A masked range image is generated by taking only the range
values from the range image whose corresponding mask values are greater than
$0.5$. Since we are re-projecting the predicted masked range images into point
clouds, we use Chamfer distance [31] represented as $\mathcal{L}_{C}$ for
evaluating fidelity of the predicted point clouds.

The combined loss function is given as

$\mathcal{L}=\mathcal{L}_{R}+\mathcal{L}_{M}+\alpha_{C}\mathcal{L}_{C}.$

$\alpha_{C}$ is the weight associated with the Chamfer distance.

Figure 3: Qualitative comparison conducted on sequence 10 of the KITTI odometry dataset. The predicted points (blue) and the ground truth points (red) are combined for a better visual comparison. The top row shows the point clouds at prediction step $t+1$ and the bottom row shows the point clouds at prediction step $t+5$. The areas of interest are circled in green. Prediction Step |  TCNet [22] |  PCPNet [24] |  PCPNet-Semantic [24] |  ATPPNet (Ours)
---|---|---|---|---
1 |  0.554 |  0.543 |  0.503 |  0.468
2 |  0.671 |  0.662 |  0.620 |  0.570
3 |  0.779 |  0.773 |  0.727 |  0.667
4 |  0.878 |  0.872 |  0.825 |  0.760
5 |  0.974 |  0.973 |  0.920 |  0.851
Mean |  0.771 |  0.765 |  0.719 |  0.663
TABLE I: Range Loss results on the KITTI odometry test set verifies that ATPPNet has a performance improvement of $4.026\%$ over SOTA on the mean range loss. Bold values correspond to the best performing model in that corresponding time step. | Sampled Point Cloud | Full-Scale Point Clouds
---|---|---
Prediction Step |  PointLSTM [19] |  MoNet [20] |  TCNet [22] |  PCPNet [24] |  PCPNet-Semantic [24] |  ATPPNet (Ours) |  TCNet [22] |  PCPNet [24] |  PCPNet-Semantic [24] |  ATPPNet (Ours)
1 |  0.332 |  0.278 |  0.290 |  0.285 |  0.280 |  0.258 |  0.253 |  0.252 |  0.242 |  0.225
2 |  0.561 |  0.409 |  0.357 |  0.341 |  0.340 |  0.311 |  0.309 |  0.301 |  0.298 |  0.270
3 |  0.810 |  0.549 |  0.441 |  0.411 |  0.412 |  0.375 |  0.377 |  0.362 |  0.354 |  0.326
4 |  1.054 |  0.692 |  0.522 |  0.492 |  0.495 |  0.445 |  0.448 |  0.435 |  0.427 |  0.391
5 |  1.299 |  0.842 |  0.629 |  0.580 |  0.601 |  0.523 |  0.547 |  0.514 |  0.503 |  0.461
Mean |  0.811 |  0.554 |  0.448 |  0.422 |  0.426 |  0.382 |  0.387 |  0.373 |  0.365 |  0.335
TABLE II: Chamfer distance results on KITTI Odometry test sequence with the sampled point clouds on the left and full-scale point clouds on the right. ATPPNet has a performance improvement of $10.328\%$ over SOTA for the sampled point clouds and $8.219\%$ over SOTA for the full-scale point clouds. Bold values correspond to the best performing model in that corresponding time step. Evaluation metric |  TCNet [22] |  PCPNet-Semantic [24] |  ATPPNet (Ours)
---|---|---|---
Mean Chamfer Distance |  1.389 |  1.360 |  0.932
Mean Range Loss |  0.719 |  0.704 |  0.598
TABLE III: Mean Range Loss and Chamfer distance on the nuScenes test set. ATPPNet is making an improvement of $15.056\%$ over SOTA on the mean Range loss and $31.470\%$ over SOTA on the mean Chamfer distance. | A) Attention Module | B) Conv-LSTM (CLSTM) layers |
---|---|---|---
Evaluation metric |  No Attention |  S-Attention |  C-Attention |  $L-1$ CLSTM |  $L-1$ & $L-2$ CLSTM |  all $L$ CLSTM |  ATPPNet (Ours)
Chamfer distance |  0.365 |  0.359 |  0.356 |  0.405 |  0.378 |  0.366 |  0.335
Range loss |  0.719 |  0.687 |  0.690 |  0.770 |  0.698 |  0.717 |  0.663
TABLE IV: Results of Ablation study on Attention Module and Conv-LSTM layers. Bold values correspond to the best performing model. Window size |  3 |  5 |  7
---|---|---|---
T+1 |  0.221 |  0.255 |  0.258
T+2 |  0.274 |  0.270 |  0.306
T+3 |  0.344 |  0.326 |  0.363
T+4 |  NA |  0.391 |  0.426
T+5 |  NA |  0.461 |  0.498
T+6 |  NA |  NA |  0.580
T+7 |  NA |  NA |  0.657
TABLE V: An ablation study on how the performance changes as we vary the sequence length. Pose error |  TCNet [22] |  PCPNet [24] |  PCPNet-Semantic [24] |  ATPPNet (Ours)
---|---|---|---|---
$\mathcal{L}_{P}^{t+1}$ |  0.1342 |  0.1363 |  0.1280 |  0.1209
$\mathcal{L}_{P}^{t+2}$ |  0.2412 |  0.2282 |  0.2235 |  0.2065
$\mathcal{L}_{P}^{t+3}$ |  0.3670 |  0.3388 |  0.3343 |  0.3038
$\mathcal{L}_{P}^{t+4}$ |  0.5084 |  0.4630 |  0.4558 |  0.4128
$\mathcal{L}_{P}^{t+5}$ |  0.6736 |  0.6022 |  0.5878 |  0.5328
Mean |  0.3849 |  0.35374 |  0.3459 |  0.3154
TABLE VI: LOAM pose error. We adopt LOAM [36] and evaluate the disparity
between the motion estimates on ground truth and predictions.

##  III Experiments and Results

###  III-A Experimental Settings

We train ATPPNet in a self-supervised manner in the sense that we use only
sequential point cloud data sans no manually annotated labels. For our
experiments, we keep the temporal window size $M=N=5$. In the convolutional
encoder block, the initial convolutional operation outputs $16$ channels while
retaining the spatial dimension. We use $L=4$ sub-blocks where the channel
size increases by a factor of $2$ for every successive sub-block obtained by
the convolutional encoder. In each of the sub-blocks, the first convolutional
operation uses a kernel size of $3\times 3$ with stride $(1,1)$, and the
second convolution operation uses a kernel size of $2\times 4$ with stride
$(2,4)$. All the convolutional operations use circular padding [22]. Each
Conv-LSTM block uses $3$ layers.

Similar to the trend in the literature [22, 24], we train our architecture for
$50$ epochs with $\alpha_{C}=0$ and then fine-tuned with the Chamfer distance
loss for the next $10$ epochs by setting $\alpha_{C}=1$. We train our model on
a system with an Intel Xeon E5-2640 CPU and $3$ Nvidia RTX 2080 GPUs using the
Distributed Data Parallel strategy. While training, we have used the ADAM
optimizer [32] with default parameters and an initial learning rate of
$0.0003$ and the StepLR learning rate scheduler with gamma as $0.99$.

####  III-A1 KITTI Odometry dataset [2]

We use sequences $00-05$ for training, $06-07$ for validation, and $08-10$ for
testing. The LiDAR used in the KITTI dataset [2] has $64$ channels, so we have
used range images of size $64\times 2048$.

####  III-A2 nuScenes dataset [1]

We trained our network on this dataset with the same training strategy we used
on the KITTI dataset. Following PCPNet [24], we used sequence $00-69$ for
training, scenes $70-84$ for validation, and $85-99$ for testing. We trained
our architecture on range images of size $32\times 1024$ since the LiDAR used
here has $32$ channels.

###  III-B Qualitative Analysis

Figure 3 (and Figure 1) shows a qualitative comparison of the predicted point
clouds generated using our proposed ATPPNet and the other methods: TCNet [22],
PCPNet and PCPNet-semantic [24]. The areas of interest are highlighted with
numbered green circles.

In the predicted sequence $t+1$ shown in Figure 3, we can observe over the
circles $a,b,c$ & $d$ that our ATPPNet outperforms the other methods by
generating point clouds that are less noisy and structurally more similar to
the ground truth. We can observe in time step $t+5$ (bottom row) that the
circles numbered $a,b,c,d,e$ & $f$ in the point cloud generated by ATPPNet is
more fidel to the ground truth compared to the predicted point clouds from the
other methods that have large visible deviations from the ground truth and are
more noisy.

###  III-C Quantitative Analysis

In this section, we perform a quantitative analysis of our proposed ATPPNet
with two point based methods (PointLSTM [19], MoNet [20]) on the KITTI [2]
test set, and three range image based methods (TCNet [22], PCPNet and PCPNet-
semantic [24]) on the KITTI [2] and the nuScenes [1] test set. The point based
methods [19, 20] use down-sampled point clouds to $65536$ points and this is
also adopted by us and other methods i.e., [22, 24]. We use the range loss and
Chamfer distance to evaluate the predicted range images and the point clouds,
respectively.

Table I shows the quantitative results of the range loss for all the methods
on the KITTI test set. Compared to the other methods, ATPPNet generates better
range images as the prediction time step increases, which can also be verified
by the improvement of $4.026\%$ over SOTA (PCPNet-semantic [24]) in the mean
range loss.

In Table II, we evaluate the Chamfer distance on the sampled point clouds
(left column) and full-scale point clouds (right column) on the KITTI test
set. As we can observe, our method is having an improvement of $10.328\%$ over
SOTA on sampled point clouds and an improvement of $8.219\%$ over SOTA on
full-scale point clouds. It can also be observed that the margin of Chamfer
distance between ATPPNet and other methods increases as the prediction time
step increases (i.e., farther in future). This indicates a more stable
prediction of the point clouds across all the time steps as depicted in Figure
3.

In Table III, we report the quantitative analysis of our model trained on the
nuScenes dataset. ATPPNet is improving $15.05\%$ on the mean range loss and
$31.47\%$ on the mean Chamfer distance over SOTA. Our inference time on the
KITTI and nuScenes dataset is $89.5$ and $70.7$ ms respectively.

###  III-D Ablation Study

In this section, we conduct a thorough investigation of the relevance of
different blocks of our architecture on the KITTI test set and demonstrate the
effectiveness of our method.

Impact of Attention Module: In Table IV column $A$, we show the results of our
ablation study on the attention module. To conduct this study, we set up $3$
experiments: (1) removing the attention module (column “No Attention”), (2)
using just spatial attention (column “S-Attention”) and (3) using just
channel-wise attention (column “C-Attention”). We observe in all the 3
experiments that the range loss and Chamfer distance deteriorates as compared
to our original method.

Effects of Spatio-Temporal Modelling: In Table IV column $B$, we demonstrate
the impact of varying the number of feature tensors from the convolutional
encoder to be modelled spatio-temporally. For this, we adopt three
experimental setups: (1) modelling only the $L-1$-th feature tensor with Conv-
LSTM and $L$-th tensor with $3D$ CNN. (column “$L-1$ CLSTM”), (2) modelling
only the $L-1$-th and $L-2$-th feature tensor with Conv-LSTM and $L$-th tensor
with $3D$ CNN. (“$L-1$ & $L-2$ CLSTM”), and (3) modelling all the $L$ layers
from the convolutional encoder with Conv-LSTM (column “all $L$ CLSTM”). For
the aforementioned experiments we observe a deterioration in the performance
compared to our original method. We can also conclude the importance of the
$3D$-CNN layer for the $L$-th layer from experiment (3). This verifies that
$3D$ CNNs are better at modelling contiguous areas in the feature tensors [28]
which tend to appear at the lower level features from the convolutional
encoder [29], [30].

Impact of Sequence Length: In Table V, we report the results by varying the
temporal window size. It is important to note that the temporal window size is
kept the same for both input and output. We can observe a decrease in the
Chamfer distance as we increase the window size from $3$ to $5$. However,
further increasing the window size from $5$ to $7$ leads to an increase in the
Chamfer distance. A possible explanation for this is that the window size of
$3$ is too short to model the spatio-temporal context of the scene while, for
the window size of $7$, the context length and the prediction horizon is too
long.

###  III-E Results on Downstream Task

In this section, we analyze the impact of our model on a downstream task of
generating motion estimates (i.e., odometry) for the ego vehicle. We adopt
LOAM[33], and evaluate the disparity between the motion estimates on ground
truth and predictions. Let $\hat{p}^{\tau}\in\mathbb{R}^{2}$ denote the
trajectory pose obtained using predicted point clouds and
$p^{\tau}\in\mathbb{R}^{2}$ denote the trajectory pose obtained using ground
truth point clouds at time step $\tau$, where $\tau\in[t+1,\dots,t+N]$. The
pose error $\mathcal{L}_{P}^{\tau}$ is given as:

$\mathcal{L}_{P}^{\tau}=\mid\mid\hat{p}^{\tau}-p^{\tau}\mid\mid_{2}^{2}.$

As reported in Table VI, the pose error ($\mathcal{L}_{P}^{\tau}$) for ATPPNet
is the least as compared to the other methods. This verifies that the improved
point cloud prediction from our proposed method translates to a tangible
outcome in the form of improved localization vis-a-vis other methods.
Additionally, such prediction of localization error can be effectively
leveraged by active localization strategies [9] that steer the vehicle to
regions where the localization is expected to be better.

##  IV Conclusion

In this paper, we present a novel self-supervised approach for predicting
future point cloud sequences based on given past point cloud sequences. We
leverage the spatial and channel-wise attention with Conv-LSTMs dually
complemented by a 3D-CNN branch to model spatio-temporal information to
successfully predict future point cloud sequences while operating at a
frequency less than that of a LiDAR. We evaluate our method on different real-
world datasets and conduct thorough ablations on our approach. Our experiments
suggest that ATPPNet outperforms all other methods. We also present an
application study of the predicted point clouds which highlights the potential
of our approach. In future work, point clouds could be predicted by querying a
location at a particular time, to determine areas with minimal drift in
motion.

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