# On Hadwiger’s covering problem in small dimensions

A. Arman  Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T
2N2, Canada<EMAIL_ADDRESS>,  A. Bondarenko  Department of
Mathematical Sciences, Norwegian University of Science and Technology, NO-7491
Trondheim, Norway<EMAIL_ADDRESS>and  A. Prymak  Department of
Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada
<EMAIL_ADDRESS>
###### Abstract.

Let $H_{n}$ be the minimal number such that any $n$-dimensional convex body
can be covered by $H_{n}$ translates of interior of that body. Similarly
$H_{n}^{s}$ is the corresponding quantity for symmetric bodies. It is possible
to define $H_{n}$ and $H_{n}^{s}$ in terms of illumination of the boundary of
the body using external light sources, and the famous Hadwiger’s covering
conjecture (illumination conjecture) states that $H_{n}=H_{n}^{s}=2^{n}$.

In this note we obtain new upper bounds on $H_{n}$ and $H_{n}^{s}$ for small
dimensions $n$. Our main idea is to cover the body by translates of John’s
ellipsoid (the inscribed ellipsoid of the largest volume). Using specific
lattice coverings, estimates of quermassintegrals for convex bodies in John’s
position, and calculations of mean widths of regular simplexes, we prove the
following new upper bounds on $H_{n}$ and $H_{n}^{s}$: $H_{5}\leq 933$,
$H_{6}\leq 6137$, $H_{7}\leq 41377$, $H_{8}\leq 284096$, $H_{4}^{s}\leq 72$,
$H_{5}^{s}\leq 305$, and $H_{6}^{s}\leq 1292$.

For larger $n$, we describe how the general asymptotic bounds
$H_{n}\leq\binom{2n}{n}n(\ln n+\ln\ln n+5)$ and $H_{n}^{s}\leq 2^{n}n(\ln
n+\ln\ln n+5)$ due to Rogers and Shephard can be improved for specific values
of $n$.

###### Key words and phrases:

Illumination problem, illumination number, covering by smaller homothetic
copies, John’s position, quermassintegrals

###### 2010 Mathematics Subject Classification:

Primary 52A20; Secondary 52A39, 52A40, 52C07, 52C17

The first author was supported by the University of Manitoba Research Grant
Program project “Covering problems with applications in Euclidean Ramsey
theory and convex geometry” and PIMS PDF

The second author was supported in part by Grant 275113 of the Research
Council of Norway.

Corresponding author: Andriy Prymak; email<EMAIL_ADDRESS>
fff The third author was supported by NSERC of Canada Discovery Grant
RGPIN-2020-05357.

##  1\. Introduction

Let ${\mathcal{K}}_{n}$ be the family of all convex bodies in
${\mathbb{E}}^{n}$, i.e., all convex compact sets $K\subset{\mathbb{E}}^{n}$
with a nonempty interior (${\rm int}(K)\neq\emptyset$). For
$A,B\subset{\mathbb{E}}^{n}$, we denote by

$C(A,B):=\min\left\\{N:\exists t_{1},\dots,t_{N}\in{\mathbb{E}}^{n}\text{
satisfying }A\subset\bigcup_{j=1}^{N}(t_{j}+B)\right\\},$

the minimal number of translates of $B$ needed to cover $A$.

Hadwiger [Ha] raised the question of determining the value of

$H_{n}=\min\\{C(K,{\rm int}(K)):K\in{\mathcal{K}}_{n}\\}$

for all $n\geq 3$.

Considering an $n$-cube, one immediately sees that $H_{n}\geq 2^{n}$, and the
well-known Hadwiger’s covering conjecture states that $H_{n}=2^{n}$ for $n\geq
3$ with equality only for parallelepipeds. It was shown that $H_{2}=4$ by Levi
[Le]. Aside from Levi and Hadwiger, the conjecture may be associated with the
names of Boltyanski, who in [Bo] has established an equivalent formulation in
terms of illumination of the boundary of the body by external light sources,
and with Gohberg-Markus [Go-Ma] who asked the question in terms of the minimal
number of smaller homothetic copies of $K$ required to cover $K$. As of today,
the conjecture, which is also known as the illumination conjecture, is wide
open. For details about the history and partial results for special classes of
convex bodies see, e.g., [Be].

In what follows we will outline the current approaches in obtaining the upper
bounds on $H_{n}$ and our modifications allowing to obtain new bounds.

To this end, let ${\mathcal{K}}_{n}^{s}$ is the subfamily of
${\mathcal{K}}_{n}$ consisting of centrally symmetric convex bodies, and
define

$H_{n}^{s}=\min\\{C(K,{\rm int}(K)):K\in{\mathcal{K}}_{n}^{s}\\}.$

The best known _explicit_ upper bound on $H_{n}$ in high dimensions follows
from the results of Rogers and Zong [Ro-Zo]:

(1) $H_{n}\leq\binom{2n}{n}n(\ln n+\ln\ln n+5).$

The asymptotic behavior of $\binom{2n}{n}$ is $\frac{4^{n}}{\sqrt{2\pi n}}$,
so this upper bound is $(4+o(1))^{n}$ while the conjecture states
$H_{n}=2^{n}$. Remarkable sub-exponential improvements of (1) have been
obtained only recently by Huang, Slomka, Tkocz and Vritsiou [HSTV]. Namely,
using “thin-shell” volume estimates they obtained

(2) $H_{n}\leq\exp(-c\sqrt{n})4^{n}.$

This was later improved, using the new breakthrough bound on isotropic
constant, by Campos, van Hintum, Morris and Tiba [CHMT] to

(3) $H_{n}\leq\exp\Bigl{(}\frac{-cn}{\ln^{8}n}\Bigr{)}4^{n}.$

Both constants $c$ in (2) and (3) are independent of $n$ but not given
explicitly. For centrally symmetric bodies, one has

(4) $H_{n}^{s}\leq 2^{n}n(\ln n+\ln\ln n+5),$

which is asymptotically close to the conjectured $2^{n}$ (see [Ro-Zo]).

The first step in obtaining any of the inequalities (1)–(4) is the result of
Rogers [Ro] on covering density of ${\mathbb{E}}^{n}$ by translates of a body
$L$. Namely, let

$\theta(L)=\lim_{R\to\infty}\inf_{\Lambda}\left\\{\frac{|L|\\#\Lambda}{(2R)^{n}}:[-R,R]^{n}\subset\bigcup_{\lambda\in\Lambda}(x_{\lambda}+L)\right\\}$

denote the covering density of space by translates of $L$, where $|L|$ is the
volume of $L$, and $\\#\Lambda$ is the cardinality of $\Lambda$. Rogers [Ro]
showed that for any body $L\in{\mathcal{K}}_{n}$,

(5) $\theta(L)\leq n(\ln n+\ln\ln n+5).$

The second step is the use of the inequality

(6) $C(K,L)\leq\frac{|K-L|}{|L|}\theta(L),$

which is valid for any $K,L\in{\mathcal{K}}^{n}$ and was proved by Rogers and
Zong [Ro-Zo].

For symmetric bodies, one obtains (4) from (6) by choosing
$L=(1-\varepsilon)K$ with $\varepsilon\to 0+$ and using the fact that
$|K-K|=|2K|=2^{n}|K|$ for $K\in{\mathcal{K}}_{n}^{s}$. For general bodies, (1)
follows from (6) by applying the Rogers-Shephard (see [Ro-Sh]) inequality
$|K-K|\leq\binom{2n}{n}|K|$ for $K\in{\mathcal{K}}_{n}$.

The improvements (2) and (3) are both based on choosing $L$ to be the largest
(by volume) centrally symmetric subset of $K$. With such choice, we have
$|K-L|/|L|\leq|2K|/|L|\leq 2^{n}\Delta_{KB}(K)^{-1}$, where

$\Delta_{KB}(K):=\max_{x\in{\mathbb{E}}^{n}}\frac{|K\cap(x-K)|}{|K|}$

is the Kövner-Besicovitch measure of symmetry of $K$. Simple averaging gives
$\Delta_{KB}(K)\geq 2^{-n}$ for any $K\in{\mathcal{K}}_{n}$, while [HSTV] and
[CHMT] obtain the corresponding sub-exponential improvements to the lower
bound on $\Delta_{KB}(K)$, yielding (2) and (3).

For low dimensions, Lassak [La] showed that

(7) $H_{n}\leq(n+1)n^{n-1}-(n-1)(n-2)^{n-1},$

which outperforms (1) for $n\leq 5$. For $n=3$ this gives $H_{3}\leq 34$,
which was improved to $H_{3}\leq 20$ by Lassak [La2], then to $H_{3}\leq 16$
by Papadoperakis [Pa], and then to $H_{3}\leq 14$ by Prymak [Pr]. For slightly
larger dimensions, it was shown in [Pr-Sh] that $H_{4}\leq 96$, $H_{5}\leq
1091$ and $H_{6}\leq 15373$ improving both (1) and (7) for $n=4,5,6$. Then
Diao [D] obtained $H_{5}\leq 1002$ and $H_{6}\leq 14140$. All of these results
in low dimensions were based on comparing the body with a suitable
parallelepiped. For the symmetric case, Lassak [La3ds] obtained the sharp
result $H_{3}^{s}=8$, but for $n\geq 4$ no estimate better than (4) or than
the corresponding bound on $H_{n}$ (obviously, $H_{n}^{s}\leq H_{n}$) was
known.

Our main result is the following new bounds.

######  Theorem 1.

$H_{5}\leq 933$, $H_{6}\leq 6137$, $H_{7}\leq 41377$, $H_{8}\leq 284096$,
$H_{4}^{s}\leq 72$, $H_{5}^{s}\leq 305$, and $H_{6}^{s}\leq 1292$.

The main idea of the proof is to utilize (6) with $L$ being the maximal (by
volume) inscribed ellipsoid into $K$ (such ellipsoids were characterized by
John [J]). After an appropriate affine transform, we can assume that $L$ is
the unit ball $B_{2}^{n}$ in ${\mathbb{E}}^{n}$, while $K$ is in the so-called
John’s position, so applying (6) we get

(8) $C(K,{\rm int}{K})\leq\frac{|K+B_{2}^{n}|}{|B_{2}^{n}|}\theta(B_{2}^{n}).$

Next we make use of several geometric results which allow us to obtain an
upper bound on $|K+B_{2}^{n}|$. One ingredient in such estimates is the fact
that the mean width and the volume of a body in John’s position are largest
for the regular simplex (general case) or for the cube (symmetric case), the
results due to Ball [Ba], Barthe [Bar], Schechtman and Schmuckenschläger
[Sc-S]. Another ingredient in our estimates is a Bonnnesen-type inequality by
Bokowski and Heil [Bo-He] on quermassintegrals of $K$. Finally, we use upper
bounds on $\theta(B_{2}^{n})$ for specific small $n$ which arise from known
lattice coverings.

Theorem 1 is proved in Sections 2–4. Of possibly independent interest are
estimates of the mean width of regular simplex in dimensions $5\leq n\leq 8$
given in Section 3.

In Section 5 we show how one can improve (5) for each fixed $n$ by optimizing
choices of certain parameters in the original proof of Rogers [Ro] (the
original proof provides a succinct bound valid for all $n$). Consequently, (1)
and (4) can be somewhat improved for $n$ larger than those covered by Theorem
1.

To finalize, in Tables 1 and 2 we provide the best known upper bounds on
$H_{n}$ and $H_{n}^{s}$ for $3\leq n\leq 14$.

$n$ | $H_{n}\leq$ | reference | $n$ | $H_{n}\leq$ | reference
---|---|---|---|---|---
$3$ | $14$ | [Pr] | $9$ | $2\,064\,332$ | Prop. 7
$4$ | $96$ | [Pr-Sh] | $10$ | $8\,950\,599$ | Prop. 7
$5$ | $933$ | Th. 1 | $11$ | $38\,482\,394$ | Prop. 7
$6$ | $6\,137$ | Th. 1 | $12$ | $164\,319\,569$ | Prop. 7
$7$ | $41\,377$ | Th. 1 | $13$ | $697\,656\,132$ | Prop. 7
$8$ | $284\,096$ | Th. 1 | $14$ | $2\,947\,865\,482$ | Prop. 7
Table 1. Best known upper bounds on $H_{n}$ for $3\leq n\leq 14$. $n$ | $H_{n}^{s}\leq$ | reference | $n$ | $H_{n}^{s}\leq$ | reference
---|---|---|---|---|---
$3$ | $8$ | [La3ds] | $9$ | $21\,738$ | Prop. 7
$4$ | $72$ | Th. 1 | $10$ | $49\,608$ | Prop. 7
$5$ | $305$ | Th. 1 | $11$ | $111\,721$ | Prop. 7
$6$ | $1\,292$ | Th. 1 | $12$ | $248\,895$ | Prop. 7
$7$ | $3\,954$ | Prop. 7 | $13$ | $549\,506$ | Prop. 7
$8$ | $9\,370$ | Prop. 7 | $14$ | $1\,203\,936$ | Prop. 7
Table 2. Best known upper bounds on $H_{n}^{s}$ for $3\leq n\leq 14$.

##  2\. Preliminaries

In what follows, $B_{2}^{n}=\\{x\in{\mathbb{E}}^{n}\;:\|x\|\leq 1\\}$, and
${\mathbb{S}}^{n-1}=\\{x\in{\mathbb{E}}^{n}\;:\|x\|=1\\}.$ The $n$-dimensional
volume is denoted by $|\cdot|_{n}$, and subscript is usually dropped if the
value of $n$ is clear from the context.

###  2.1. John’s position

Let ${\mathcal{J}}_{n}$ be the family of convex bodies $K\in{\mathcal{K}}_{n}$
which are in John’s position, i.e. $B_{2}^{n}$ is the maximal volume ellipsoid
of $K$ (see, e.g., [isotropic-book]*Sect. 1.5.1). Similarly, define
${\mathcal{J}}_{n}^{s}$ to be the family of centrally symmetric convex bodies
$K\in{\mathcal{K}}_{n}^{s}$ which are in John’s position. For any
$K\in{\mathcal{K}}_{n}$ (or $K\in{\mathcal{K}}_{n}^{s}$) there exists an
affine image of $K$ in ${\mathcal{J}}_{n}$ (or in ${\mathcal{J}}_{n}^{s}$).

John’s theorem (see e.g. [aga-book]*Thm. 2.1.3, Remark 2.1.17) implies that

(9) $K\subset nB_{2}^{n}{\quad\text{for
any}\quad}K\in{\mathcal{J}}_{n},{\quad\text{and}\quad}K\subset\sqrt{n}B_{2}^{n}{\quad\text{for
any}\quad}K\in{\mathcal{J}}_{n}^{s}.$

###  2.2. Quermassintegrals

In order to bound $H_{n}$ and $H_{n}^{s}$ we will use the inequality (8), and
so we start by discussing upper bounds on $|K+B_{2}^{n}|$, where $K$ is in
John’s position.

By the Steiner’s formula, for any $K\in{\mathcal{K}}_{n}$

(10) $|K+tB_{2}^{n}|=\sum_{j=0}^{n}\binom{n}{j}W_{j}(K)t^{j},$

where

$W_{j}(K)=V(\underset{n-j\text{
times}}{\underbrace{K,\dots,K}},\underset{j\text{
times}}{\underbrace{B_{2}^{n},\dots,B_{2}^{n}}})$

is the $j$-th quermassintegral of $K$ and $V(\cdot)$ is the mixed volume, see,
e.g., [aga-book]*Sect. 1.1.5 or [isotropic-book]*Sect. 1.4.2. Additionally,
$W_{0}(K)=|K|$, $W_{1}(K)=\partial(K)/n$, where $\partial(K)$ is the surface
area of $K$, and $W_{n}(K)=|B_{2}^{n}|$. Also note that if $B_{2}^{n}\subset
K$, then due to the monotonicity of mixed volumes we have $W_{i}(K)\geq
W_{j}(K)$ for $i\leq j$.

Next, we will discuss upper bounds on $W_{i}(K)$ for $K\in{\mathcal{J}}_{n}$
and $K\in{\mathcal{J}}_{n}^{s}$ respectively.

Let $T^{n}$ be a regular simplex in ${\mathbb{E}}^{n}$ of unit edge length,
and let $\Delta^{n}$ be a dilation of $T^{n}$ for which $B_{2}^{n}$ is the
inscribed ball, then $\Delta^{n}=\sqrt{2(n+1)n}\;T^{n}$. Also, let
$C^{n}:=[-1,1]^{n}$ be a cube circumscribed about $B^{2}_{n}$. Ball [Ba]
proved that among all convex bodies in ${\mathbb{E}}^{n}$ simplexes have
maximal volume ratio (volume ratio measures how much of the volume of the
whole body can be contained in the largest inscribed ellipsoid) while for the
symmetric ones such a maximizer is the cube [Ba-sym]. These results can be
stated in our terms as follows:

(11) $W_{0}(K)\leq\begin{cases}W_{0}(\Delta^{n}),&\text{if
}K\in{\mathcal{J}}_{n},\\\ W_{0}(C^{n}),&\text{if
}K\in{\mathcal{J}}_{n}^{s}.\end{cases}$

For $K\in{\mathcal{K}}_{n}$ and a direction $u\in{\mathbb{S}}^{n-1}$, the
support function is defined as $h_{K}(u)=\sup\\{\langle x,y\rangle:y\in K\\}$.
Let $\sigma$ be the rotationally invariant probability measure on
${\mathbb{S}}^{n-1}$, then the mean width of $K$ is defined by

$w(K)=\int_{{\mathbb{S}}^{n-1}}h_{K}(u)\,d\sigma(u).$

We have $W_{n-1}(K)=|B_{2}^{n}|w(K)$, which is a partial case of the Kubota’s
formula [aga-book]*eq (1.1.1). Barthe [Bar]*Thm. 3 proved 222The result of
Barthe is stated in terms of $\ell$-norm, and can be translated in the
language of mean width using the formula [Bar]*p. 685 for the $\ell$-norm of
the dual body. that among the bodies from ${\mathcal{J}}_{n}$, the mean width
is maximized for $\Delta^{n}$, while Schechtman and Schmuckenschläger [Sc-S]
(the proof is also included in [Bar]*Thm. 2) remarked that $C^{n}$ maximizes
the mean width among the bodies from ${\mathcal{J}}_{n}^{s}$, i.e.,

(12) $W_{n-1}(K)\leq\begin{cases}W_{n-1}(\Delta^{n}),&\text{if
}K\in{\mathcal{J}}_{n},\\\ W_{n-1}(C^{n}),&\text{if
}K\in{\mathcal{J}}_{n}^{s}.\end{cases}$

We estimate $w(\Delta^{n})$ for the required values of $n$ in the next
section. It is known [Fi], [DCG-book]*Sec. 13.2.3 (or can be obtained by a
straightforward computation) that
$w(C^{n})=2\frac{|B_{2}^{n-1}|_{n-1}}{|B_{2}^{n}|_{n}}$ for $n\geq 2$, and so
$W_{n-1}(C^{n})=2|B_{2}^{n-1}|$.

######  Remark 2.

The results [Ba]*Thm. 1, Thm. $1^{\prime}$ imply that $W_{1}(K)\leq
W_{1}(\Delta^{n})$ for any $K\in{\mathcal{J}}_{n}$ (recall that
$W_{1}(K)=\partial(K)/n$, where $\partial(K)$ is the surface area of $K$).
This would not lead to any improvements in our context as
$W_{1}(\Delta^{n})=W_{0}(\Delta^{n})$ and we get the same upper bound on
$W_{1}(K)$ as from $W_{1}(K)\leq W_{0}(K)$. A similar remark regarding the
inequality $W_{1}(K)\leq W_{1}(C^{n})$ also holds for
$K\in{\mathcal{J}}^{s}_{n}$, and follows from [Ba]*Thm. 2, [Ba-sym]*Thm. 3.

######  Remark 3.

It is natural to conjecture that $W_{j}(K)\leq W_{j}(\Delta^{n})$ (and that
$W_{j}(K)\leq W_{j}(C^{n})$) for any $K\in{\mathcal{J}}_{n}$ (respectively,
$K\in{\mathcal{J}}_{n}^{s}$) and all $0\leq j\leq n$. This is an obvious
equality for $j=n$ and is valid for $j\in\\{0,1,n-1\\}$ as described above
((11), Remark 2, (12)).

Finally, we need a Bonnesen-style inequality by Bokowski and Heil [Bo-He]. If
$K\in{\mathcal{K}}_{n}$ satisfies $K\subset RB_{2}^{n}$, then for all $0\leq
i<j<k\leq n$

(13)
$W_{j}(K)\leq\frac{(k-j)(i+1)R^{i}W_{i}(K)+(j-i)(k+1)R^{k}W_{k}(K)}{(k-i)(j+1)R^{j}}=:B_{R,i,j,k}(W_{i}(K),W_{k}(K)).$

Recall that in our settings we can choose $R$ according to (9).

###  2.3. Density of coverings by balls

One of the approaches to construct specific efficient coverings of the space
by balls is to use lattices, see [Co-Sl]*Ch. 2. Considering the $A_{n}^{*}$
lattice yields the following estimate (valid for all $n$):

(14)
$\theta(B_{2}^{n})\leq|B_{2}^{n}|\sqrt{n+1}\left(\frac{n(n+2)}{12(n+1)}\right)^{n/2}.$

It turns out that the above is optimal (smallest possible _lattice_ covering
density) for $2\leq n\leq 5$ and provides the best known upper bound on
$\theta(B_{2}^{n})$ in most dimensions $10\leq n\leq 21$. Better lattices were
found by Schürmann and Vallentin [SchVall] for $6\leq n\leq 8$, which is
important for our applications. To the best of our knowledge, there have been
no improvements after the work [SchVall], where an interested reader can find
a brief survey of the topic. We list the corresponding lattice covering
densities of ${\mathbb{E}}^{n}$ by balls in Table 3.

$n$ | least known lattice covering density | $n$ | least known lattice covering density
---|---|---|---
$2$ | $1.209199$ | $8$ | $3.142202$
$3$ | $1.463505$ | $9$ | $4.340185$
$4$ | $1.765529$ | $10$ | $5.251713$
$5$ | $2.124286$ | $11$ | $5.598338$
$6$ | $2.464801$ | $12$ | $7.510113$
$7$ | $2.900024$ | $13$ | $7.864060$
Table 3. Least known lattice covering densities, as in [SchVall].

##  3\. Estimates on mean width of regular simplex

The values of $2w(T^{n})$ for $2\leq n\leq 6$ expressed as certain integrals
and numerically evaluated with high precision can be found in [Fi]. We remark
that in [Fi] expected values of widths of simplexes are considered, while the
mean width as we defined here (and as commonly defined in the literature on
geometry, see [aga-book, isotropic-book]) is the expected value of the support
function, this discrepancy results in the mean width of $T^{n}$ from [Fi] (and
[Su]) being equal to $2w(T^{n})$. We need upper estimates of $w(T^{n})$ for
$n=7,8$, which we were unable to find in the literature. We present a simple
computational technique to obtain upper and lower estimates on $w(T^{n})$
which will suffice for our purposes.

The starting point is the following representation that follows the work [Fi]
by Finch and references therein, in particular, Sun [Su].

Let $F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\;dt$ be the
comulative distribution function of the standard normal distribution, then
$F(x)=\frac{1}{2}+\frac{1}{2}{\rm erf}(\frac{x}{\sqrt{2}})$, where ${\rm
erf}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\,dt$ is the error function.

For $n\geq 0$ let

$\displaystyle g_{n+1}(x)$
$\displaystyle:=1-F(x)^{n+1}-(1-F(x))^{n+1}=1-\left(\frac{1+{\rm
erf}(\tfrac{x}{\sqrt{2}})}{2}\right)^{n+1}-\left(\frac{1-{\rm
erf}(\tfrac{x}{\sqrt{2}})}{2}\right)^{n+1}.$

The following formula for mean width follows from [Su] and was derived in
[Fi]:

(15) $\displaystyle w(T^{n})=$
$\displaystyle\frac{\Gamma(\tfrac{n}{2})}{2\Gamma(\tfrac{n+1}{2})}\int_{0}^{\infty}g_{n+1}(x)\,dx.$

Next, we estimate the integral in the formula.

######  Proposition 4.

For any $a>2$ and any positive integers $n$, $N$, the following estimates
hold:

(16)
$\frac{a}{N}\sum_{k=1}^{N}g_{n+1}\left(\tfrac{ak}{N}\right)\leq\int_{0}^{\infty}g_{n+1}(x)\,dx\leq\frac{a}{N}\sum_{k=0}^{N-1}g_{n+1}\left(\tfrac{ak}{N}\right)+\frac{n+1}{\sqrt{2\pi}}\exp\left(-a\right).$

###### Proof.

We directly estimate the “tail” of the integral and use simple endpoint
Riemann sums for the “main” part of the integral in (15).

Recalling that $\frac{d}{dx}F(x)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ it is
straightforward to verify that $g_{n+1}$ is a positive strictly decreasing
function on $[0,\infty)$. Hence, considering the upper and the lower Riemann
sums for $\int_{0}^{a}g_{n+1}(x)\,dx$ and the uniform partition of $[0,a]$
into $N$ subintervals, we obtain

(17) $\frac{a}{N}\sum_{k=1}^{N}g_{n+1}\left(\tfrac{a\cdot
k}{N}\right)\leq\int_{0}^{a}g_{n+1}(x)\,dx\leq\frac{a}{N}\sum_{k=0}^{N-1}g_{n+1}\left(\tfrac{a\cdot
k}{N}\right).$

For any $x>a\geq 2$

$\displaystyle g_{n+1}(x)$ $\displaystyle\leq
1-F(x)^{n+1}=1-\left(1-F(-x)\right)^{n+1}=1-\left(1-\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}\,dt\right)^{n+1}$
$\displaystyle\leq\frac{n+1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}\,dt\leq\frac{n+1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t}\,dt=\frac{n+1}{\sqrt{2\pi}}\exp\left(-x\right),$

so

$\int_{a}^{\infty}g_{n+1}(x)\,dx\leq\frac{n+1}{\sqrt{2\pi}}\exp\left(-a\right).$

Taking this inequality and the evident $\int_{a}^{\infty}g_{n+1}(x)\,dx\geq 0$
into account, we deduce (16) from (15) and (17). ∎

Since the error function can be computed numerically with any given precision,
employing a simple SageMath ([sagemath]) computation [github], we obtain the
following corollary.

######  Corollary 5.

The following inequalities hold:

$\displaystyle 0.4208\leq$ $\displaystyle w(T^{5})\leq 0.4215,$ $\displaystyle
0.4067\leq$ $\displaystyle w(T^{6})\leq 0.407,$ $\displaystyle 0.39425\leq$
$\displaystyle w(T^{7})\leq 0.39427,$ $\displaystyle 0.383\leq$ $\displaystyle
w(T^{8})\leq 0.38301.$

These estimates for $n=5,6$ are consistent with the values obtained in [Fi];
we include them here for completeness. While our computational method allows
to obtain a tighter gap in the estimates, we only derived what was necessary
for our application of estimating $H_{n}$ from above ensuring that no further
improvement is possible (even if the value of the lower bound on $w(T^{n})$ is
used, the upper bound on the integer value $H_{n}$ does not improve). The
computations take less than two hours on a modern personal computer.

##  4\. Proof of Theorem 1

Let us begin with the general (not necessarily symmetric) case. Since
$C(K,{\rm int}(K))$ is invariant under affine transforms, we can assume that
$K\in{\mathcal{J}}^{n}$. Then using $B_{2}^{n}\subset K$, compactness and (6),
we have

$\displaystyle C(K,{\rm int}(K))$ $\displaystyle\leq C(K,{\rm
int}(B_{2}^{n}))\leq\lim_{r\to 1^{-}}C(K,rB_{2}^{n})$
$\displaystyle\leq\lim_{r\to
1^{-}}\frac{|K-rB_{2}^{n}|}{|rB_{2}^{n}|}\theta(B_{2}^{n})=\frac{|K+B_{2}^{n}|}{|B_{2}^{n}|}\theta(B_{2}^{n}).$

Hence, applying (10),

(18) $C(K,{\rm
int}(K))\leq\frac{\theta(B_{2}^{n})}{|B_{2}^{n}|}\sum_{j=0}^{n}\binom{n}{j}W_{j}(K).$

The upper bounds for $\theta(B_{2}^{n})$ come from Table 3, so it remains to
estimate $W_{i}(K)$. As $\Delta^{n}=\sqrt{2n(n+1)}T^{n}$, by (11),

(19) $W_{i}(K)\leq W_{0}(K)\leq
W_{0}(\Delta^{n})=(2n(n+1))^{n/2}|T^{n}|=\frac{n^{n/2}(n+1)^{(n+1)/2}}{n!}$

for any $0\leq i\leq n$. Using (12),

(20) $W_{n-1}(K)\leq
W_{n-1}(\Delta^{n})=|B_{2}^{n}|w(\Delta^{n})=|B_{2}^{n}|\sqrt{2n(n+1)}w(T^{n}).$

Trivially, $W_{n}(K)=|B_{2}^{n}|$.

Next we combine the above, and use (13) (valid with $R=n$ due to (9)) for
appropriate parameters. The calculations were performed in the script
[github].

Bound on $H_{5}$. We apply (18) and estimate the quermassintegrals as follows.
For $0\leq i\leq 2$, use (19); (20) and (13) give $W_{4}(K)\leq
W_{4}(\Delta^{5})$ and $W_{3}(K)\leq
B_{5,2,3,4}(W_{0}(\Delta^{5}),W_{4}(\Delta^{5}))$; recall that
$W_{5}(K)=|B_{2}^{5}|$. Combining the above, using Corollary 5 and calculating
the actual value of the bound yields $H_{5}\leq 933$.

Bound on $H_{6}$. The arguments are similar to the previous case. The required
application of (13) is $W_{4}(K)\leq
B_{6,3,4,5}(W_{0}(\Delta^{6}),W_{5}(\Delta^{6}))$. The result is $H_{6}\leq
6137$.

Bounds on $H_{n}$ for $n=7,8$. We proceed similarly with the only difference
that $W_{j}(K)\leq B_{n,n-4,j,n-1}(W_{0}(\Delta^{n}),W_{n-1}(\Delta^{n}))$ is
used for $j\in\\{n-3,n-2\\}$ yielding $H_{7}\leq 41377$ and $H_{8}\leq
284096$.

When the body is centrally symmetric, we follow a similar route, with the
following differences. In place of (19) we have

(21) $W_{i}(K)\leq W_{0}(K)\leq W_{0}(C^{n})=2^{n}$

for any $0\leq i\leq n$. Using (12) we get

(22) $W_{n-1}(K)\leq W_{n-1}(C^{n})=2|B_{2}^{n-1}|.$

Finally, by (9), the inequality (13) can be used with $R=\sqrt{n}$.

Bound on $H_{4}^{s}$. Apply (18) and bound the quermassintegrals as follows.
For $i=0,1$ use (21); use (22) to get $W_{3}(K)\leq 2|B_{2}^{3}|$; we have
$W_{4}(K)=W_{4}(C^{4})=|B_{2}^{4}|$; by (13) $W_{2}(K)\leq
B_{2,1,2,3}(W_{1}(C^{4}),W_{3}(C^{4}))$. Combining the above and calculating
the value of the bound yields $H_{4}^{s}\leq 72$.

Bounds on $H_{n}^{s}$ for $n=5,6$. We use similar arguments: (21) for
$i=0,1,2$; $W_{n}=W_{n}(C^{n})=|B_{2}^{n}|$; (22) for $i=n-1$; and
$W_{j}(K)\leq B_{\sqrt{n},2,j,n-1}(W_{0}(C^{n}),W_{n-1}(C^{n}))$ for $3\leq
j\leq n-2$, which imply $H_{5}^{s}\leq 305$ and $H_{6}^{s}\leq 1292$.

######  Remark 6.

In the above computations, we used the exact value for the density arising
from $A_{n}^{*}$ lattice from the estimate (14) for $n=4,5$. For $n=6,7,8$ we
used the values of covering densities from [SchVall] presented in Table 3
increased by $5\cdot 10^{-6}$ as they were given to 6 decimal places, while we
require an upper bound. Even if such an increase is not performed, the
resulting _integer_ valued upper bounds in Theorem 1 would not change, so the
accuracy given in [SchVall] is more than sufficient for our needs.

##  5\. Estimates via optimized Rogers bound

######  Proposition 7.

Suppose $n\geq 3$. If

(23)
$r_{n}=\min_{x\in(0,1/n)}f_{n}(x),{\quad\text{where}\quad}f_{n}(x)=(1+x)^{n}(1-n\ln(x)),$

then

(24) $\theta(K)\leq r_{n}{\quad\text{for any}\quad}K\in{\mathcal{K}}_{n},\quad
H_{n}\leq\binom{2n}{n}r_{n},{\quad\text{and}\quad}H_{n}^{s}\leq 2^{n}r_{n}.$

###### Proof.

The first inequality in (24) is established in [Ro]*p. 5. (The bound (5) was
obtained in [Ro] by taking $x=\frac{1}{n\ln n}$ in (23).) The other two
inequalities in (24) follow from (6) in the same way as (1) and (4) do. ∎

For our purposes, it suffices to use the straightforward
$r_{n}\leq\min\\{f_{n}(\frac{j}{Nn}):1\leq j\leq N-1\\}$ with $N=1000$. The
resulting bounds on $r_{n}$ for $3\leq n\leq 14$ are given in Table 4 (the
computed values of $\min\\{f_{n}(\frac{j}{Nn}):1\leq j\leq N-1\\}$ are rounded
up in the sixth digit, so they represent actual upper bounds), and the
estimates on $H_{n}$ and $H_{n}^{s}$ can be found in Tables 1 and 2. For the
computations, see [github]. We remark that $\max\\{\theta(K),\
K\in{\mathcal{K}}_{2}\\}=\frac{3}{2}$ was established by Fáry [Fary].

$n$ | $\max\\{\theta(K),\ K\in{\mathcal{K}}_{n}\\}\leq$ | $n$ | $\max\\{\theta(K),\ K\in{\mathcal{K}}_{n}\\}\leq$
---|---|---|---
$3$ | 10.064123 | $9$ | 42.458503
$4$ | 14.916986 | $10$ | 48.445515
$5$ | 20.024359 | $11$ | 54.551530
$6$ | 25.362768 | $12$ | 60.765566
$7$ | 30.898293 | $13$ | 67.078451
$8$ | 36.603890 | $14$ | 73.482436
Table 4. Upper bounds on $\max\\{\theta(K),\ K\in{\mathcal{K}}_{n}\\}$, for
$3\leq n\leq 14$.

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11institutetext: United Imaging Intelligence, Cambridge MA, USA
22institutetext: University at Buffalo, Buffalo NY, USA
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Self-supervised 3D Patient Modeling with Multi-modal Attentive
Fusion††thanks: Corresponding author: Meng Zheng, email<EMAIL_ADDRESS>
Meng Zheng 11 0000-0002-6677-2017 Benjamin Planche 11 0000-0002-6110-6437 Xuan
Gong 22 0000-0001-8303-633X Fan Yang 11 0000-0003-1535-447X Terrence Chen 11
Ziyan Wu 11 0000-0002-9774-7770

###### Abstract

3D patient body modeling is critical to the success of automated patient
positioning for smart medical scanning and operating rooms. Existing CNN-based
end-to-end patient modeling solutions typically require a) customized network
designs demanding large amount of relevant training data, covering extensive
realistic clinical scenarios (_e.g_., patient covered by sheets), which leads
to suboptimal generalizability in practical deployment, b) expensive 3D human
model annotations, _i.e_., requiring huge amount of manual effort, resulting
in systems that scale poorly. To address these issues, we propose a generic
modularized 3D patient modeling method consists of (a) a multi-modal keypoint
detection module with attentive fusion for 2D patient joint localization, to
learn complementary cross-modality patient body information, leading to
improved keypoint localization robustness and generalizability in a wide
variety of imaging (_e.g_., CT, MRI etc.) and clinical scenarios (_e.g_.,
heavy occlusions); and (b) a self-supervised 3D mesh regression module which
does not require expensive 3D mesh parameter annotations to train, bringing
immediate cost benefits for clinical deployment. We demonstrate the efficacy
of the proposed method by extensive patient positioning experiments on both
public and clinical data. Our evaluation results achieve superior patient
positioning performance across various imaging modalities in real clinical
scenarios.

###### Keywords:

3D mesh patient positioning patient modeling.

##  1 Introduction

The automatic patient positioning system and algorithm design for intelligent
medical scanning/operating rooms has attracted increasing attention in recent
years [17, 34, 14, 15, 32], with the goals of minimizing technician effort,
providing superior performance in patient positioning accuracy and enabling
contactless operation to reduce physical interactions and disease contagion
between healthcare workers and patients. Critical to the design of such a
patient positioning system, 3D patient body modeling in medical environments
based on observations from one or a group of optical sensors (_e.g_.,
RGB/depth/IR) is typically formulated as a 3D patient body modeling or pose
estimation problem [2, 16, 18] in the computer vision field, defined as
follows. Given an image captured from an optical sensor installed in the
medical environment, we aim to automatically estimate the pose and shape—and
generate a digital representation—of the patient of interest. Here we consider
3D mesh representations among several commonly used human representations
(_e.g_., skeleton, contour etc. [8]), which consist of a collection of
vertices, edges and faces and contain rich pose and shape information of the
real human body, as demonstrated in Figure 1(a). The 3D mesh estimation of a
patient can be found suitable for a wide variety of clinical applications. For
instance, in CT scanning procedure, automated isocentering can be achieved by
using the patient thickness computed from the estimated 3D mesh [21, 17].
Consequently, there has been much recent work from both algorithm [6, 18, 9]
as well as system perspectives [17, 7].

State-of-the-art patient mesh estimation algorithms [38] typically rely on
end-to-end customized deep networks, requiring extensive relevant training
data for real clinical deployment. For example, training the RDF model
proposed in [38] requires pairs of multi-modal sensor images and 3D mesh
parameters (which are particularly expensive to create [26, 25, 28]).
Moreover, conventional end-to-end 3D mesh estimation methods [19, 16, 38, 39]
assume a perfect person detection as preprocessing step for stable inference,
_i.e_., relying on an efficient person detection algorithm to crop a person
rectangle covering the person’s full body out of the original image. Hence,
any error during this first person detection step propagates and further
impacts the mesh estimation process itself (see Fig. 1(b)), and such detection
errors are especially likely to occur when the target patient is under-the-
cover (_i.e_., occluded by sheets) or occluded by medical devices.

Figure 1: (a) Mesh representation of a patient in MRI scanning room. (b)
Failure cases of state-of-the-art mesh regressors (SPIN [19]) in challenging
clinical scenarios, _e.g_., simulated hospital environment [24], MRI and CT
scanning rooms.

We thus propose a multi-modal data processing system that can (a) perform both
person detection and mesh estimation, and (b) be trained over inexpensive data
annotations. This system comprises several modules (_c.f_. Figure 2). First,
we train a multi-modal fused 2D keypoint predictor to learn complementary
patient information that may not be available from mono-modal sensors. We then
process these 2D keypoints with a novel 3D mesh regressor designed to
efficiently learn from inexpensively-produced synthetic data pairs in a self-
supervised way. Besides technical contributions within each module, _e.g_.,
cross-modal attention fusion for improved joint localization and self-
supervised mesh regression (_c.f_. Section 2), we demonstrate the robustness
and generalizability of our overall system over numerous imaging and clinical
experiments (_c.f_. Section 3).

##  2 Methodology

Figure 2: Proposed framework to localize 2D keypoints and infer the 3D mesh.

(1) Multi-modal 2D Keypoint Detector with Attention Fusion. Most recent works
in 2D pose estimation [4, 10, 35] are essentially single-source (_i.e_., RGB
only) architectures. Consequently, while they work reasonably in generic
uncovered patient cases, they fail in more specific ones, _e.g_., when the
patient is covered by a cloth – a prevalent situation in numerous medical
scanning procedures and interventions. As existing methods fail to
ubiquitously work across imaging modalities and applications, we propose a
multi-sensory data processing architecture that leverages information from
multiple data sources to account for both generic as well as specialized
scenarios (_e.g_., cloth-covered patients). We first introduce how to
individually train 2D keypoint detectors on single modalities (_e.g_., RGB or
depth), then how to learn complementary information from multiple modalities
to improve detection performance and generalizability.

Given an RGB or depth person image, the 2D keypoint detection task aims to
predict a set of $N_{J}$ joint (usually predefined) locations of the person in
the image space, which is typically achieved by learning a deep CNN network in
most recent works. Here we adopt HRnet [35] as the backbone architecture which
takes the RGB or depth image as input and outputs $N_{J}$ 2D joint heatmaps,
with the peak location of each heatmap $i=1,...,N_{J}$ indicating the
corresponding joint’s pixel coordinates, as illustrated in Figure 3
(orange/blue block for RGB/depth).

Figure 3: Proposed RGBD keypoint detection framework with attention fusion.

While the training of RGB-based keypoint detector can leverage many publicly
available datasets [22, 29, 12], the number of depth/RGBD image datasets
curated for keypoint detection is much more limited, mainly due to lower
sensor accessibility and image quality. Thus directly training depth-based
keypoint detectors over such limited data can easily result in overfitting and
poor detection performance during testing. To alleviate this, we propose to
first utilize an unsupervised pretraining technique [5] to learn a generalized
representation from unlabeled depth images,which is proved to have better
generalizability for downstream tasks like keypoint detection; and then to
finetune with labeled training data for improved keypoint detection accuracy.
This way, we can leverage a larger number of public depth or RGBD image
datasets collected for other tasks for a more generic model learning. For
further details w.r.t. the unsupervised pretraining for representation
learning, we refer the readers to [5].

Color and depth images contain complementary information of the patient, as
well as complementary benefits over different scenarios. _E.g_., when the
patient is covered by a surgical sheet (Figure 1(b)), RGB features will be
heavily affected due to the cover occlusion, whereas depth data still contain
rich shape and contour information useful for patient body modeling. We seek
to design an attentive multi-modal fusion network, to effectively aggregate
complementary information across RGB and depth images by enforcing intra- and
inter-modal attentive feature aggregation for improved keypoint detection
performance. Specifically, we propose a two-branch score-based RGBD fusion
network as shown in Figure 3. In the proposed fusion network, we take the last
stage features of the HRnet backbone from RGB and depth branches respectively,
and forward them into a fusion module with intra-modal and inter-modal
attentive feature aggregation, for a binary classification score prediction.
This classifier aims to determine (_c.f_. output score) which modality (RGB or
depth) results in the most reliable prediction, based on the prediction error
from RGB and depth branches during training. For example, if the RGB
prediction error is larger than depth prediction error, we set the classifier
label to 0, and vice-versa (setting to 1 if RGB-based error is lower). After
the binary classifier is learned, it will produce a probability score (range
from 0 to 1) indicating the reliability of RGB and depth branch predictions,
which is then utilized to weight keypoint heatmap predictions from each branch
before their fusion. In this way, the proposed module is able to fuse
complementary information from single modalities and learn enhanced feature
representations for more accurate and robust keypoint detection.

(2) Self-supervised 3D Mesh Regressor. After producing the 2D keypoints, we
aim to recover the 3D mesh representation of the patient for complete and
dense patient modeling. Note that we use the Skinned Multi-Person Linear
(SMPL) model [26], which is a statistical parametric mesh model of the human
body, represented by pose $\bm{\theta}\in\mathbb{R}^{72}$ and shape
$\bm{\beta}\in\mathbb{R}^{10}$ parameters. Unlike prior works [38] that
require both images and the corresponding ground-truth 3D mesh parameters for
training, our proposed method does not need such expensive annotations and
relies only on synthetically generated pairs of 2D keypoint predictions and
mesh parameters. Our method thus does not suffer from the biased distribution
and limited scale of existing 3D datasets.

Specifically, to generate the synthetic training data, we sample SMPL pose
parameters from training sets of public datasets (_i.e_., AMASS [27], UP-3D
[20] and 3DPW [37]), and shape parameters from a Gaussian distribution
following [30]. We then render the 3D mesh given the sampled $\bm{\theta}$ and
$\bm{\beta}$ and project the 3D joints determined by the rendered mesh to 2D
keypoint locations given randomly sampled camera translation parameters. The
$N_{J}$ 2D keypoint locations then can be formed into $N_{J}$ heatmaps (as
described in Section 2(1)) and passed to a CNN for $\bm{\theta}$ and
$\bm{\beta}$ regression. Here we use a Resnet-18 [11] as the baseline
architecture for mesh regression. In our experiments, we extensively sampled
the data points and generate 330k synthetic data pairs to train the mesh
regressor. During testing, the 2D keypoint heatmaps inferred from the RGBD
keypoint detection model (_c.f_. Section 2(1)) are directly utilized for 3D
mesh estimation.

##  3 Experiments

Datasets, implementation, and evaluation. To demonstrate the efficacy of our
proposed method, we evaluate on the public SLP dataset [24] (same train/test
splits from the authors) and proprietary RGBD data collected (with approval
from ethical review board) from various scenarios: computed tomography (CT),
molecular imaging (MI), and magnetic resonance imaging (MRI).

To collect our proprietary MI dataset, 13 volunteers were asked to take
different poses while being covered by a surgical sheet with varying covering
areas (half body, 3/4 body, full body) and facial occlusion scenarios
(with/without facial mask). We use 106 images from 3 subjects to construct the
training set, and 960 images from the remaining 10 subjects as test set. For
the dataset collected in MRI room, we captured 1,670 images with varying
scanning protocols (_e.g_., wrist, ankle, hip, _etc_.) and patient bed
positions, with the volunteers being asked to show a variety of poses while
being covered by a cloth with different level of occlusions (similar to MI
dataset). This resulted in 1,410 training images and 260 testing ones. For our
proprietary CT dataset, we asked 13 volunteers to lie on a CT scanner bed and
exhibit various poses with and without cover. We collected 974 images in
total. To test the generalizability of the proposed mesh estimator across
imaging modalities, we use this dataset for testing only.

During training stage, we use all data from the SLP, MI and MRI training
splits, along with public datasets COCO [22] and MPII [29] to learn our single
RGB keypoint detector, with the ground-truth keypoint annotations generated
manually. For our depth detector, we pretrain its backbone over public RGBD
datasets, _i.e_., 7scene [31], PKU [23], CAD [36] and SUN-RGBD [33]. We then
finetune the model over SLP, MI and MRI training data with keypoint
supervision. We apply the commonly-used 2D mean per joint position error
(MPJPE) and percentage of correct keypoints (PCK) [1] for quantifying the
accuracy of 2D keypoints, and the 3D MPJPE, Procrustes analysis (PA) MPJPE
[16] and scale-corrected per-vertex Euclidean error in a neutral pose
(T-pose), _i.e_., PVE-T-SC [30] (all in mm) for 3D mesh pose and shape
evaluation.

###  3.1 2D Keypoint Prediction

(1) Comparison to State-of-the-art. In Table 1 (first row), we compare the 2D
MPJPE of our keypoint prediction module with competing 2D detectors on the SLP
dataset. Here, “Ours (RGB)" and “Ours (Depth)" refer to the proposed single-
modality RGB and depth keypoint detectors, which achieve substantial
performance improvement, including compared to the recent RDF algorithm of
Yang _et al_. [38]. In Table 4, we compare the<EMAIL_ADDRESS>of proposed RGBD
keypoint detector with off-the-shelf state-of-the-art 2D keypoint detector
OpenPose [4] on SLP and MI datasets. We notice that our solution performs
significantly better than OpenPose across different data domains, which
demonstrates the superiority of the proposed method. We present more<EMAIL_ADDRESS>(torso) evaluations of the proposed multi-modal keypoint detector with
attentive fusion in Table 5 on MRI and CT (cross-domain) dataset, proving the
efficacy of the proposed multi-modal fusion strategy.

(2) Ablation Study on Multi-Modal Fusion. Table 2 (A) contains results of an
ablation study to evaluate the impact of utilizing single (RGB/depth) and
multi-modal fused (RGBD) data for keypoint detection on SLP, MI, MRI and CT
(cross-domain) data. We evaluated on different CNN backbones, _i.e_. HRNet
[35] and ResNet-50 [11] (see supplementary material), and we observe
consistent performance improvement across all datasets with multi-modal
fusion, demonstrating the efficacy of our fusion architecture. See Figure 4
for a qualitative illustration of this aspect.

(3) Ablation Study on Unsupervised Pretraining of Depth Keypoint Detector. To
demonstrate the advantage of utilizing unsupervised pretraining strategy for
generalized keypoint detection, another ablation study is performed w.r.t. our
single depth-based detector on SLP and MI data, pretrained with varying number
of unannotated data, then finetuned with a fix amount of labeled samples (SLP,
MI and MRI data). We can see from Table 3 that the keypoint detection
performance generally increases along with the quantity of pretraining data,
proving the efficacy of the proposed unsupervised pretraining strategy.

Table 1: Comparison on SLP [24] to existing methods, w.r.t. 2D keypoint
detection and 3D mesh regression (modalities: “RGB" color, “T" thermal, “D"
depth). Grey cells indicate numbers not reported in the references.

| Methods: |  SPIN [19] |  OP [4] | HMR [16] | RDF [38] | Ours
---|---|---|---|---|---|---
|  |  |  RGB |  T |  RGBT |  RGB |  T |  RGBT |  RGB |  D |  RGBD
2D | MPJPE (px)$\downarrow$ |  |  293.8 |  37.2 |  |  |  36.6 |  |  |  17.1 |  14.2 |  13.2
MPJPE (cm)$\downarrow$ |  |  163.9 |  20.8 |  |  |  20.4 |  |  |  9.5 |  7.9 |  7.4
3D | MPJPE (mm)$\downarrow$ |  236 |  |  155 |  149 |  143 |  144 |  138 |  137 |  123 |  118 |  115

Table 2: Ablation study and evaluation on different imaging modalities w.r.t.
2D keypoint detector (A) and 3D mesh regressor (B). (${\dagger}$ $=$ cross-
domain evaluation)

| (A) 2D detector ablation study. | (B) 3D mesh regressor evaluation.
---|---|---
Data | 2D MPJPE (px)$\downarrow$ | 3D PA MPJPE (mm)$\downarrow$ | 3D PVE-T-SC (mm)$\downarrow$
RGB |  D |  RGBD |  RGB |  D |  RGBD |  RGB |  D |  RGBD
SLP |  17.1 |  14.2 |  13.2 |  83.4 |  78.3 |  77.3 |  17.3 |  14.5 |  13.3
MI |  13.0 |  13.6 |  12.6 |  97.0 |  101.5 |  93.1 |  22.9 |  26.6 |  17.7
MRI |  7.7 |  15.6 |  7.2 |  103.1 |  99.3 |  94.3 |  19.8 |  17.8 |  15.1
CT† |  23.3 |  22.5 |  21.2 |  110.9 |  107.5 |  104.3 |  17.3 |  20.2 |  17.3

Table 3:  Impact of pretraining data (7scene [31], PKU [23], CAD [36], SUNRGBD [33]) on MPJPE accuracy (px) of proposed depth-based keypoint detector.  Pretrain Datasets: | [31] |  [31]+[23] |  [31]+[23]+[36] |  [31]+[23]+[36]+[33]
---|---|---|---|---
SLP | 17.8 | 14.5 | 15.1 | 13.5
MI | 14.5 | 13.6 | 13.8 | 13.3
Table 4<EMAIL_ADDRESS>evaluation of our proposed 2D keypoint detector with
competing methods on SLP (top) and MI (bottom).

Methods |  R.Ak. |  R.Kn. |  R.H. |  L.H. |  L.Kn. |  L.Ak. |  R.Wr. |  R.Eb. |  R.Sh. |  L.Sh. |  L.Eb. |  L.Wr. |  Avg
---|---|---|---|---|---|---|---|---|---|---|---|---|---
OpenPose[4] |  13.0 |  38.2 |  74.6 |  73.9 |  34.6 |  11.1 |  54.9 |  74.6 |  95.7 |  95.7 |  73.3 |  52.6 |  57.7
Proposed |  98.4 |  98.4 |  100.0 |  100.0 |  99.6 |  98.2 |  92.5 |  97.2 |  99.9 |  99.3 |  96.1 |  94.7 |  97.9

Methods |  R.Ak. |  R.Kn. |  R.H. |  L.H. |  L.Kn. |  L.Ak. |  R.Wr. |  R.Eb. |  R.Sh. |  L.Sh. |  L.Eb. |  L.Wr. |  Avg
---|---|---|---|---|---|---|---|---|---|---|---|---|---
OpenPose[4] |  0.0 |  0.0 |  2.7 |  3.3 |  0.0 |  0.0 |  20.0 |  34.4 |  85.3 |  86.7 |  34.5 |  18.1 |  23.7
Proposed |  97.6 |  99.3 |  99.9 |  99.7 |  97.2 |  95.4 |  91.6 |  97.8 |  100.0 |  99.8 |  98.7 |  92.5 |  97.5

Table 5<EMAIL_ADDRESS>(torso) evaluation of our proposed 2D keypoint detector on
MRI and CT† (cross-validation: no CT training data used in model learning)
testing data.

|  R.Ak. |  R.Kn. |  R.H. |  L.H. |  L.Kn. |  L.Ak. |  R.Wr. |  R.Eb. |  R.Sh. |  L.Sh. |  L.Eb. |  L.Wr. |  Avg
---|---|---|---|---|---|---|---|---|---|---|---|---|---
MRI |  97.0 |  98.7 |  99.4 |  99.4 |  98.7 |  98.5 |  96.8 |  98.1 |  99.4 |  99.4 |  98.7 |  96.8 |  98.4
CT† |  91.3 |  93.2 |  93.9 |  94.0 |  92.5 |  91.3 |  84.9 |  88.6 |  93.6 |  93.9 |  88.9 |  86.2 |  91.0

###  3.2 3D Mesh Estimation

We next discuss the performance of our 3D mesh estimation module. To generate
ground-truth 3D SMPL pose and shape annotations for all testing data, we apply
an off-the-shelf 3D mesh predictor [13, 28] followed by manual refinement.
Given this testing ground truth, we use the 3D MPJPE and PVE-T-SC metrics to
quantify performance. Comparison to other 3D mesh estimation technique is
shown in Table 1 (bottom row). Again, we observe substantial performance
improvement in terms of 3D joint localization (_c.f_. 3D MPJPE) and per-vertex
mesh accuracy (_c.f_. 3D PVE-T-SC) across a wide variety of cover conditions,
despite purely relying on synthetic training data (whereas competitive methods
require expensive 3D annotations). The proposed solution shines (on the SLP
data) over the state-of-the-art, _e.g_., recent method by Yang _et al_. [38]
and one of the most commonly used 3D mesh estimator SPIN [19]. Table 2 (B)
further shows that these improvements are generally consistent across all
imaging modalities. Qualitative mesh estimation results are shared in Figure
5.

###  3.3 Automated Isocentering with Clinical CT Scans

To demonstrate the clinical value of the proposed method, we evaluate the
isocentering accuracy in a clinical CT scanning setting. To do so, we mount an
RGBD camera above the CT patient support and calibrate it spatially to the CT
reference system. With the RGBD images captured by the camera, our proposed
method can estimate the 3D patient mesh and compute the thickness of the
target body part, which can then be used to adjust the height of the patient
support so that the center of target body part aligns with the CT isocenter.
We conducted this evaluation with 40 patients and 3 different protocols, and
calculated the error based on the resulting CT scout scan as shown in Table 6.
Compared to the currently deployed automated CT patient positioning system
[3], our pipeline automatically aligns the center of target body part and
scanner isocenter with mean errors of 5.3/7.5/8.1mm for abdomen/thorax/head
respectively vs. 13.2mm median error of radiographers in [3], which clearly
demonstrates the advantage of our proposed positioning workflow.

Figure 4: Performance comparison between proposed RGB and RGBD model. Figure
5: Visualization of reconstructed mesh results on CT, MI, MRI and SLP. Table
6: Evaluation on ISO-center estimation with clinical CT scans.

Protocol | Abdomen | Thorax | Head
---|---|---|---
Error |  Mean |  STD |  Mean |  STD |  Mean |  STD
(mm) |  5.3 |  2.1 |  7.5 |  2.9 |  8.1 |  2.2

##  4 Conclusion

In this work, we considered the problem of 3D patient body modeling and
proposed a novel method, consisting of a multi-modal 2D keypoint detection
module with attentive fusion and a self-supervised 3D mesh regression module,
being applicable to a wide variety of imaging and clinical scenarios. We
demonstrated these aspects with extensive experiments on proprietary data
collected from multiple scanning modalities as well as public datasets,
showing improved performance when compared to existing state-of-the-art
algorithms as well as published clinical systems. Our results demonstrated the
general-purpose nature of our proposed method, helping take a step towards
algorithms that can lead to scalable automated patient modeling and
positioning systems.

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