Isoparametric hypersurfaces and hypersurfaces

with constant principal curvatures in Finsler spaces

††${}^{*}\,$ Project supported by AHNSF (No.2108085MA11).
${}^{\dagger}\,$<EMAIL_ADDRESS>
Peilong Dong1, Yali Chen2${}^{\dagger}\,$

1\. School of Mathematics and Statistics, Zhengzhou Normal University,

Zhengzhou Henan 450044, China.

2\. School of Mathematics and Statistics, Anhui Normal University,

Wuhu, Anhui, 241000, China.

Abstract: In this paper, we study the relationship between isoparametric
hypersurfaces and hypersurfaces with constant principal curvatures in Finsler
spaces. We give some examples of isoparametric hypersurfaces with
(non)constant principal curvatures on Randers manifolds with nonconstant flag
curvatures. Furthermore, we construct an example of a conformally flat Randers
manifold which admits a family of nonisoparametric hyperplanes with constant
principal curvatures.

Key words: isoparametric hypersurfaces, constant principal curvatures, mean
curvatures, parallel hypersurfaces, nonconstant flag curvatures

Mathematics Subject Classification (2010): 53C60, 53B40.

##  1 Introduction

In Riemannian geometry, a hypersurface is called isoparametric if itself and
all its sufficiently close parallel hypersurfaces have constant mean
curvature. Since 1938, E. Cartan began to study the isoparametric
hypersurfaces of real space forms systematically and proved that a
hypersurface has constant principal curvatures if and only if it is
isoparametric [1].

In Finsler geometry, the conception of isoparametric hypersurfaces has been
firstly introduced in [2]. Moreover, the analytic definitions of locally
isoparametric hypersurfaces and $d\mu$-isoparametric hypersurfaces are given
in [3, 4]. Let $(N,F,d\mu)$ be an $n$-dimensional Finsler manifold with volume
form $d\mu=\sigma(x)dx$ and $f$ be a non-constant $C^{1}$ function and smooth
on $N_{f}=\\{x\in N~{}|~{}df(x)\neq 0\\}$. Set $J=f(N_{f})$. Then $f$ is said
to be _isoparametric_ (resp. _$d\mu$ -isoparametric_) on $(N,F,d\mu)$, if
there exist a smooth function ${a}(t)$ and a continuous function $b(t)$ on $J$
such that

$\left\\{\begin{aligned} &F(\nabla f)=a(f),\\\ &\Delta
f=b(f),\end{aligned}\right.$ (1.1)

where $\nabla f$ denotes the gradient of $f$ and $\Delta{f}=\Delta^{\nabla
f}f$ (resp. $\Delta{f}=\Delta_{\sigma}f$) is defined by (2.2)(resp. (2.3)).
All the regular level surfaces ${M}_{t}={f}^{-1}(t)$ are named an _(
$d\mu$-)isoparametric family_, each of which is called an _(
$d\mu$-)isoparametric hypersurface_ on $(N,F,d\mu)$. A function $f$ only
satisfying the first equation of (1.1) is said to be _transnormal_.

Similar to Riemannian geometry, [2] proved that a transnormal function $f$ is
isoparametric if and only if each regular level hypersurface of $f$ has
constant ($d\mu$)-mean curvatures. Particularly, if $N$ has constant flag
curvature (and constant $\mathbf{S}$-curvature), then a transnormal function
$f$ is ($d\mu$)-isoparametric if and only if all the principal curvatures of
$M_{t}$ are constant. So the relationship between isoparametric hypersurfaces
and principal curvatures is worth studying.

The previous work mainly focused on isoparametric hypersurfaces with constant
principal curvatures in Finsler space forms (with constant flag curvature).
For some very special Finsler space forms, such as Minkowski spaces (with zero
flag curvature) and Funk spaces (with negative constant flag curvature), the
isoparametric hypersurfaces have been completely classified in [2, 5, 6]. In
[7], the authors studied isoparametric hypersurfaces of Finsler space forms,
obtained the Cartan-type formula and some classifications on the number of
distinct principal curvatures or their multiplicities.

In Riemannian geometry, the equivalence between isoparametric hypersurfaces
and hypersurfaces with constant principal curvatures is no longer true for
more general ambient spaces with nonconstant curvatures. There exist many
examples of isoparametric hypersurfaces with (non)constant principal
curvatures with nonconstant curvatures [8, 9]. In [10], a conformally flat
metric in $\mathbb{R}^{n}$ is constructed, which admits a family of
nonisoparametric hyperplanes. This shows that there is no necessary relation
between the isoparametric hypersurface and the hypersurface with constant
principal curvature.

A natural question is whether there exist similar results in Finsler spaces.
In this paper, we give some examples of isoparametric hypersurfaces with
(non)constant principal curvatures in Randers manifolds with nonconstant flag
curvatures. Furthermore, we construct an example of a conformally flat Finsler
manifold which admits a family of hypersurface with constant principal
curvatures, but each of them is an nonisoparametric hypersurface, that is, its
sufficiently close parallel hypersurfaces don’t have constant mean curvatures
or $d\mu_{BH}$-mean curvatures. So we have the following theorem.

######  Theorem 1.1.

For Finsler spaces with nonconstant flag curvatures, the relation between
isoparametric hypersurfaces and hypersurfaces with constant principal
curvatures is no longer equivalent. In fact, there exist isoparametric
hypersurfaces without constant principal curvatures and nonisoparametric
hypersurfaces with constant principal curvatures.

##  2 Preliminaries

###  2.1 Finsler manifolds

Let $(N,F)$ be an $n$-dimensional smooth connected Finsler manifold and $TN$
be the tangent bundle over $N$ with local coordinates $(x,y)$, where
$x=(x^{i})$ and $y=y^{i}\frac{\partial}{\partial x^{i}}$. Here and from now
on, we will use the following convention of index ranges unless otherwise
stated:

$1\leq i,j,\cdots\leq n;~{}~{}~{}~{}~{}~{}~{}1\leq a,b,\cdots\leq n-1.$

The fundamental form $g$ of $(N,F)$ is

$g=g_{ij}(x,y)dx^{i}\otimes
dx^{j},~{}~{}~{}~{}~{}~{}~{}g_{ij}(x,y)=\frac{1}{2}[F^{2}]_{y^{i}y^{j}}.$

The projection $\pi:TN\rightarrow N$ gives rise to the pull-back bundle
$\pi^{\ast}TN$. On it there exists a unique _Chern connection_ $\nabla$ with
$\nabla\frac{\partial}{\partial x^{i}}=\omega_{i}^{j}\frac{\partial}{\partial
x^{j}}=\Gamma^{i}_{jk}dx^{k}\otimes\frac{\partial}{\partial x^{j}}$ satisfying
([11])

$dg_{ij}-g_{ik}\omega^{k}_{j}-g_{kj}\omega^{k}_{i}=2C_{ijk}(dy^{k}+N^{k}_{l}dx^{l}),$
$~{}~{}~{}~{}N_{j}^{i}:=\frac{\partial G^{i}}{\partial
y^{j}}=\Gamma^{i}_{jk}y^{k},$

where $C_{ijk}=\frac{1}{2}\frac{\partial g_{ij}}{\partial y^{k}}$ is called
the _Cartan tensor_ and
$G^{i}=\frac{1}{4}g^{il}\left\\{[F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}\right\\}$
are the _geodesic coefficients_ of $(N,F)$. For
$X=X^{i}\frac{\partial}{\partial x^{i}}\in\Gamma(TN)$, the covariant
derivative of $X$ along $v=v^{i}\frac{\partial}{\partial x^{i}}\in T_{x}N$
with respect to a reference vector $w\in T_{x}N\setminus\\{0\\}$ is defined by

$D^{w}_{v}X(x)=\\{v^{j}\frac{\partial X^{i}}{\partial
x^{j}}(x)+\Gamma^{i}_{jk}(w)v^{j}X^{k}(x)\\}\frac{\partial}{\partial x^{i}}.$

Another torsion-free Berwald connection ${}^{b}\nabla$ is defined by

${}^{b}\omega^{i}_{j}=^{b}\Gamma^{i}_{jk}dx^{k}=(\Gamma^{i}_{jk}+L^{i}_{jk})dx^{k}$

where $L^{i}_{jk}$ is the Landsberg curvature of $(N,F)$. For
$X=X^{i}\frac{\partial}{\partial x^{i}}\in\pi^{\ast}TN$, the _horizontal
covariant derivative_ of $X$ with respect to ${}^{b}\nabla$ is defined by

$\displaystyle X^{i}_{~{}|j}=\frac{\delta X^{i}}{\delta
x^{j}}+^{b}\Gamma^{i}_{js}X^{s},$ (2.1)

where $\frac{\delta}{\delta x^{j}}=\frac{\partial}{\partial
x^{j}}-N_{j}^{i}\frac{\partial}{\partial y^{i}}$.

Let ${\mathcal{L}}:TN\rightarrow T^{\ast}N$ denotes the _Legendre
transformation_ , satisfying ${\mathcal{L}}(\lambda
y)=\lambda{\mathcal{L}}(y)$ for all $\lambda>0,~{}y\in TN$. For a smooth
function $f:N\rightarrow\mathbb{R}$, the _gradient vector_ of $f$ at $x$ is
defined as $\nabla f(x)={\mathcal{L}}^{-1}(df(x))\in T_{x}N$. Set
$N_{f}=\\{x\in N|df(x)\neq 0\\}$ and $\nabla^{2}f(x)=\nabla^{\nabla f}(\nabla
f)(x)$ for $x\in N_{f}$. We define the _Laplacian_ of $f$ by

$\Delta^{\nabla f}f=\textmd{tr}_{g_{{}_{\nabla f}}}(\nabla^{2}f).$ (2.2)

And the _Laplacian_ of $f$ with respect to the volume form
$d\mu=\sigma(x)dx=\sigma(x)dx^{1}\wedge dx^{2}\wedge\cdots\wedge dx^{n}$ can
be represented as

$\displaystyle\Delta_{\sigma}f=\textmd{div}_{\sigma}(\nabla
f)=\frac{1}{\sigma}\frac{\partial}{\partial x^{i}}(\sigma g^{ij}(\nabla
f)f_{j})=\Delta^{\nabla f}f-S(\nabla f),$ (2.3)

where

$\displaystyle S(x,y)=\frac{\partial G^{i}}{\partial
y^{i}}-y^{i}\frac{\partial}{\partial x^{i}}(\ln\sigma(x))$ (2.4)

is the _$\mathbf{S}$ -curvature_.

Let $\phi:M\to N$ be an embedded hypersurface of $(N,F)$. For any $x\in M$,
there exist exactly two unit _normal vectors_ $\mathbf{n}_{\pm}$. Let
$\mathbf{n}$ be a given normal vector of $N$. Set
$\hat{g}=\phi^{*}g_{\mathbf{n}}$. From [2], we have the following _Gauss-
Weingarten formulas_

$\displaystyle{D}^{\textbf{n}}_{X}Y$
$\displaystyle={{\hat{\nabla}}}_{X}Y+\hat{h}(X,Y)\textbf{n},$ (2.5)
$\displaystyle D^{\textbf{n}}_{X}\textbf{n}$
$\displaystyle=-{A}_{\textbf{n}}X,~{}~{}~{}~{}~{}\quad\forall
X,~{}Y\in\Gamma(TM),$ (2.6)

where $\hat{\nabla}$ is the induced connection on $(M,\hat{g})$ and
$\hat{h}(X,Y):=g_{\textbf{n}}(\textbf{n},D^{\textbf{n}}_{X}Y)$. The
eigenvalues of the shape operator ${A}_{\mathbf{n}}$,
$\mu_{1},\mu_{2},\cdots,\mu_{n-1}$, and
$\hat{H}_{\mathbf{n}}=\sum\limits_{a=1}^{n-1}\mu_{a}$ are called the
_principal curvatures_ and the _anisotropic mean curvature_ with respect to
$\mathbf{n}$, respectively.

######  Lemma 2.1.

[2] Let $M$ be an embedded hypersurface of $(N,F,d\mu)$, then

$H_{\textbf{$\mathbf{n}$}}=\hat{H}_{\textbf{$\mathbf{n}$}}+S(\textmd{{$\mathbf{n}$}}),$
(2.7)

where $H_{\textbf{$\mathbf{n}$}}$ is the $d\mu$-mean curvature induced by
variation of the volume in $(N,F,d\mu)$ (see (3.3) in [2] for detail).

Similar to Riemannian geometry, we can also give the geometric definitions of
isoparametric hypersurface on Finsler manifolds as following.

######  Definition 2.2.

A hypersurface $M$ on Finsler manifold $(N,F)$ is called isoparametric if it
and its sufficiently close parallel hypersurfaces have constant anisotropic
mean curvatures. Moreover, a hypersurface $M$ in Finsler manifold $(N,F,d\mu)$
is named $d\mu$-isoparametric if it and all its sufficiently close parallel
hypersurfaces have constant $d\mu$-mean curvatures.

######  Remark 2.3.

Obviously, when $S(\textmd{{$\mathbf{n}$}})=\textmd{constant}$, a hypersurface
$M$ is $d\mu$-isoparametric if and only if it is isoparametric.

##  3 The examples of isoparametric hypersurfaces on Randers manifolds with
nonconstant flag curvatures

In Riemannian geometry, there are many examples of isoparametric hypersurfaces
with (non)constant principal curvatures on some Riemannian manifolds with
nonconstant curvature [8, 9, 12, 13]. The first examples were found by Wang
[12], who constructed some inhomogeneous isoparametric hypersurfaces with
nonconstant principal curvatures in the complex projective space, by
projecting some of the inhomogeneous hypersurfaces in spheres via the Hopf
map. In [13], the examples of isoparametric real hypersurfaces with
nonconstant principal curvatures in complex hyperbolic spaces
$\mathbb{C}H^{n}$ are constructed by Lie groups and Lie algebras.

In this section, we plan to give some examples of isoparametric hypersurfaces
with (non)constant principal curvatures in Randers manifolds with nonconstant
flag curvatures.

In section 2, we study the principal curvatures of anisotropic submanifolds
and hypersurfaces in a Randers space with the navigation datum. Here we will
use this relationship to construct examples. Let $(N,h)$ be an $n$-dimensional
Riemannian manifold and $v$ be a vector field. By navigation problem, $(h,v)$
can define a Randers metric

$F=\frac{\sqrt{\lambda h^{2}+v_{0}^{2}}-v_{0}}{\lambda}=\frac{\sqrt{\lambda
h_{ij}y^{i}y^{j}+(v_{i}y^{i})^{2}}-v_{i}y^{i}}{\lambda},$ (3.1)

where $\lambda=1-b^{2},~{}b=\|v\|_{h},~{}v=v^{i}\frac{\partial}{\partial
x^{i}},~{}v_{i}=h_{ij}v^{j}$. Let $\phi:M\to(N,F)$ be an embedded hypersurface
and the unit normal vector field of $M$ with respect to $F$ and $h$ be
$\mathbf{n}$ and $\bar{\mathbf{n}}$, respectively. From [3], we know

$\mathbf{n}=\bar{\mathbf{n}}+v$ (3.2)

and the relationship between the principal curvatures of $M$ in $(N,F)$ and
$(N,h)$ is as follows.

######  Lemma 3.1.

[3] Let $M$ be a hypersurface in a Randers space $(N,F,d\mu_{BH})$ with the
navigation datum $(h,v)$. If $F$ has isotropic $\mathbf{S}$-curvature
$S=(n+1)k(x)F$, then for any unit normal vector field $\mathbf{n}$, the shape
operators of $M$ in Randers space $(N,F)$ and Riemannian space $(N,h)$,
${A}_{\mathbf{n}}$ and $\bar{A}_{\bar{\mathbf{n}}}$, have the same principal
vectors and their principal curvatures satisfy

$\displaystyle\mu=\bar{\mu}+k(x),$ (3.3)

where $\mu$ and $\bar{\mu}$ are the principal curvatures of $M$ in Randers
space $(N,F)$ and Riemannian space $(N,h)$, respectively.

Later, Xu and his coworkers got the local correspondence between isoparametric
functions or isoparametric hypersurfaces for homothetic navigation in a
Finsler manifold.

######  Lemma 3.2.

[14] Let $v$ be a homothetic vector field on Finsler manifold $(N,\bar{F})$,
and $F$ the metric defined by navigation from the datum $(\bar{F},v)$. Then
locally around any point $x_{0}$ with $\bar{F}(x_{0},-v(x_{0}))<1$, a
hypersurface is isoparametric for $(\bar{F},d\mu^{\bar{F}}_{BH})$ if and only
if it is isoparametric for $(F,d\mu^{F}_{BH})$.

According to [15], Randers metric $F$ with respect to the Busemann-Hausdorff
(BH) measure has constant $\mathbf{S}$-curvature if and only if $v$ is a
homothetic vector field. From Lemma 3.1 and Lemma 3.2, we know that if $v$ is
a homothetic vector field, the ($d\mu^{F}_{BH}$-)isoparametric hypersurface of
$(N,h)$ is also ($d\mu^{F}_{BH}$-)isoparametric for the Randers manifold
$(N,F)$, and the principal curvatures of $M$ in $(N,F)$ are all (non)constant
if and only if its principal curvatures in $(N,h)$ are all (non)constant.

######  Remark 3.3.

By using homothetic navigation, we can construct
($d\mu^{F}_{BH}$-)isoparametric hypersurfaces with (non)constant principal
curvatures on some Randers manifolds with nonconstant flag curvatures.

From Lemma 3.1, Lemma 3.2 and [13], we have the following example.

######  Example 3.4.

Let $\mathbb{C}H^{n}$ be a complex hyperbolic space, $AN$ be the connected and
simply connected subgroups of $G=SU(1,n)$, $S_{\varpi}$ be a connected
subgroup of $AN$, $M^{r}$ be the tube of radius $r$ around the submanifold
$W_{\varpi}$, where $W_{\varpi}$ is the orbit $S_{\varpi}\cdot o$ of the group
$S_{\varpi}$ through the point $o\in\mathbb{C}H^{n}$. Then for every $r>0$,
$M^{r}$ is an isoparametric real hypersurface which has, in general,
nonconstant principal curvatures. If given a (locally) homothetic vector field
$v$ in $\mathbb{C}H^{n}$, then by (3.1), we can get a Randers metric $F$ with
nonconstant flag curvatures. So $M^{r}$ is also the isoparametric hypersurface
of $(\mathbb{C}H^{n},F)$, and the principal curvatures of $M^{r}$ in
$(\mathbb{C}H^{n},F)$ are all nonconstant.

In Riemannian spaces of constant curvatures, a hypersurface is isoparametric
if and only if it has constant principal curvatures. But in Finsler manifolds
with constant flag curvatures, for any given volume form $d\mu$ on $(N,F)$, if
$F$ doesn’t have constant $\mathbf{S}$-curvature, then the existence of
isoparametric hypersurfaces with nonconstant principal curvatures is still an
open problem.

##  4 The example of nonisoparametric hypersurfaces with constant principal
curvatures

Let $N=\mathbb{R}^{n}$, $h=\sqrt{\delta_{ij}y^{i}y^{j}}$ and
$v=b\frac{\partial}{\partial x^{n}}$ be a vector field, where $b$ is a
constant and satisfies $b<1$. Consider a Randers-Minkowski metric

$F(y)=\alpha+\beta=\frac{\sqrt{\lambda
h^{2}+v^{2}_{0}}}{\lambda}-\frac{v_{0}}{\lambda},$ (4.1)

where $v_{0}=by^{n},\lambda=1-\delta_{ij}v^{i}v^{j}=1-b^{2}$. Set
$\alpha=\sqrt{\alpha_{ij}y^{i}y^{j}},~{}\beta=b_{i}y^{i}$, then

$\alpha_{ab}=\frac{\delta_{ab}}{\lambda},~{}\alpha_{an}=0,~{}\alpha_{nn}=\frac{\lambda+b^{2}}{\lambda^{2}}=\frac{1}{\lambda^{2}},$
$\alpha^{ab}=\lambda\delta_{ab},~{}\alpha^{an}=0,~{}\alpha^{nn}=\lambda^{2},$
$b^{a}=\alpha^{aj}b_{j}=0,~{}b^{n}=-\lambda
b,~{}\parallel\beta\parallel^{2}_{\alpha}=b^{2}.$

Hence

$\alpha_{y^{a}}=\frac{y^{a}}{\lambda\alpha},~{}\alpha_{y^{n}}=\frac{y^{n}}{\lambda^{2}\alpha},$
$g_{ab}=\frac{F}{\lambda\alpha}\delta_{ab}-\frac{\beta
y^{a}y^{b}}{\alpha^{3}\lambda^{2}},~{}g_{an}=-\frac{by^{a}}{\lambda^{2}\alpha}-\frac{\beta
y^{a}y^{n}}{\lambda^{3}\alpha^{3}},$ (4.2)
$g_{nn}=\frac{F}{\lambda^{2}\alpha}\left(1-\frac{(y^{n})^{2}}{\lambda^{2}\alpha^{2}}\right)+(\frac{y^{n}}{\lambda^{2}\alpha}-\frac{b}{\lambda})^{2},$

and

$g^{ab}=\frac{\alpha}{F}\lambda\delta^{ab}+\frac{b^{2}\alpha+\beta}{F^{3}}y^{a}y^{b},~{}g^{an}=\frac{b\lambda\alpha}{F^{2}}y^{a}+\frac{b^{2}\alpha+\beta}{F^{3}}y^{a}y^{n},$
$g^{nn}=\frac{\lambda^{2}\alpha}{F}+2\frac{b\lambda\alpha}{F^{2}}y^{n}+\frac{b^{2}\alpha+\beta}{F^{3}}(y^{n})^{2}.$

###  4.1 A conformally flat Randers manifold

Define a new metric

$\widetilde{F}(x,y)=e^{\rho(x)}F(y)$

and the corresponding fundamental tensor is

$\tilde{g}_{ij}(x,y)=e^{2\rho(x)}g_{ij}(y),$ (4.3)

where $\rho(x)=\sum\limits_{a=1}^{n-1}\ln(2+cos(\pi x^{a}))$,
$x=(x^{i})\in\mathbb{R}^{n}$. It is easy to know that $\tilde{g}$ is (locally)
conformally flat. From [16], we get the relation of spray coefficient
$\tilde{G}^{i}$ and $G^{i}$ with respect to $\widetilde{F}$ and $F$,
respectively,

$\tilde{G}^{i}=G^{i}+\rho_{k}y^{k}y^{i}-\frac{1}{2}F^{2}\rho^{i},$ (4.4)

where $\rho_{k}=\frac{\partial\rho}{\partial x^{k}},\rho^{i}=g^{ij}\rho_{j}$.
By $\rho_{n}=0$, $G^{i}=0$,

$\rho^{a}=g^{ai}\rho_{i}=g^{ab}\rho_{b}=\frac{\alpha}{F}\lambda\rho_{a}+\frac{b^{2}\alpha+\beta}{F^{3}}y^{a}y^{b}\rho_{b}$

and

$\rho^{n}=g^{nb}\rho_{b}=(\frac{b\lambda\alpha}{F^{2}}y^{b}+\frac{b^{2}\alpha+\beta}{F^{3}}y^{b}y^{n})\rho_{b}.$

We have

$\tilde{G}^{a}=\rho_{b}y^{b}y^{a}-\frac{1}{2}F^{2}\rho^{a}=(1-\frac{b^{2}\alpha+\beta}{2F})\rho_{b}y^{b}y^{a}-\frac{1}{2}\lambda\alpha
F\rho_{a},$ (4.5)
$\tilde{G}^{n}=\rho_{b}y^{b}y^{n}-\frac{1}{2}F^{2}\rho^{n}=(1-\frac{b^{2}\alpha+\beta}{2F})\rho_{b}y^{b}y^{n}-\frac{b\lambda\alpha}{2}\rho_{b}y^{b}.$
(4.6)

Then by (4.4), it followed that

$\tilde{N}^{i}_{j}=\frac{\partial\tilde{G}^{i}}{\partial
y^{j}}=\rho_{k}y^{k}\delta^{i}_{j}+\rho_{j}y^{i}-\frac{1}{2}(F^{2})_{y^{j}}\rho^{i}+F^{2}C^{ik}_{j}\rho_{k},$
(4.7)

where

$\displaystyle-2C^{ik}_{j}=\frac{\partial g^{ik}}{\partial
y^{j}}=(\frac{\alpha}{F})_{y^{j}}\alpha^{ik}-(\frac{\alpha}{F^{2}})_{y^{j}}(b^{i}y^{k}+b^{k}y^{i}),$
$\displaystyle-\frac{\alpha}{F^{2}}(b^{i}\delta^{k}_{j}+b^{k}\delta^{i}_{j})+(\frac{b^{2}\alpha+\beta}{F^{3}})_{y^{j}}y^{i}y^{k}+\frac{b^{2}\alpha+\beta}{F^{3}}(y^{i}\delta^{k}_{j}+y^{k}\delta^{i}_{j}).$
(4.8)

From [16], we have

$\tilde{S}(y)=S(y)+F^{2}\rho^{k}I_{k},$

where $I_{k}=g^{ij}C_{ijk}$ is the Cartan form, $\tilde{S}$ and $S$
respectively are the $\mathbf{S}$-curvature of $\tilde{F}$ and $F$. It is well
known that $S=0$ with respect to the Busemann-Hausdorff volume form. Then by
(4.1), $\rho_{n}=0$ and $\rho_{k}b^{k}=0$, we have

$\displaystyle\tilde{S}=$ $\displaystyle F^{2}\rho_{k}C^{ik}_{i}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}F^{2}\rho_{k}\left[(\frac{\alpha}{F})_{y^{i}}\alpha^{ik}-(\frac{\alpha}{F^{2}})_{y^{i}}b^{i}y^{k}+(\frac{b^{2}\alpha+\beta}{F^{3}})_{y^{i}}y^{i}y^{k}+(n+1)y^{k}\frac{b^{2}\alpha+\beta}{F^{3}}\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\rho_{k}\left[(\alpha_{y^{i}}\beta-\alpha\beta_{y^{i}})\alpha^{ik}-b^{i}y^{k}\frac{\alpha_{y^{i}}(\beta-\alpha)-2\alpha\beta_{y^{i}}}{F}+(n-1)y^{k}\frac{b^{2}\alpha+\beta}{F}\right].$
(4.9)

###  4.2 The geodesics with respect to $\widetilde{F}$

By (4.1), $\tilde{F}$ is produced by navigation datum $(\tilde{h},\tilde{v})$,
where $\tilde{h}=e^{\rho}h,\tilde{v}=e^{\rho}v$. Then
$\tilde{v}_{a}=0,\tilde{v}_{n}=be^{\rho}$,
$\tilde{v}^{a}=0,\tilde{v}^{n}=be^{-\rho}$. Let
$M=\\{x\in\mathbb{R}^{n}|x^{n}=x_{0}^{n}\in\mathbb{R}\\}$, $\bar{\mathbf{n}}$
and $\tilde{\mathbf{n}}$ be the unit normal vector field of $M$ with respect
to $\tilde{h}$ and $\tilde{F}$, respectively. According to [3], we have

$\tilde{\mathbf{n}}=\bar{\mathbf{n}}+\tilde{v}.$

From [10], we know that $\bar{\mathbf{n}}=(0,0,\cdots,e^{-\rho})$, then
$\tilde{\mathbf{n}}=(1+b)\bar{\mathbf{n}}$.

Define

$\Omega:=\\{(m_{1},m_{2},\cdots,m_{n-1},x^{n})\in N|~{}m_{a}\in\mathbb{Z}\\}.$

Let $p=(m_{1},m_{2},\cdots,m_{n-1},x_{0}^{n})\in\Omega$,
$\\{e_{1},e_{2},\cdots,e_{n-1},\tilde{\textbf{n}}\\}$ be a local orthogonal
frame in the neighbourhood of $p$ with respect to $\tilde{F}$. Let $\gamma$ be
an unit-speed geodesic with respect to $\tilde{F}$ and start at $p$ with
initial direction $\tilde{\textbf{n}}$. By (4.5) and (4.6), we know that
$\tilde{F}$ is not projectively flat. So the geodesics with respect to
$\tilde{F}$ are not necessarily straight lines. Set

$(1)~{}\Lambda_{a}:x=(x^{1},\cdots,x^{a},\cdots,x^{n})\rightarrow\hat{x}=(x^{1},\cdots,-x^{a},\cdots,x^{n})$,

$(2)~{}\Psi_{m_{a}}:x=(x^{1},\cdots,x^{a},\cdots,x^{n})\rightarrow\check{x}=(x^{1},\cdots,x^{a}+2m_{a},\cdots,x^{n})$.

Next we will prove $\hat{\gamma}(t)=\Lambda_{a}(\gamma(t))$ and
$\check{\gamma}(t)=\Psi_{m_{a}}(\gamma(t))$ are all geodesics. So
$\hat{\gamma}(t)$ and $\check{\gamma}(t)$ should satisfy

$\ddot{\gamma}^{i}+\widetilde{\Gamma}^{i}_{jk}\dot{\gamma}^{k}\dot{\gamma}^{j}=0.$
(4.10)

For $\Lambda_{a}$, we get

$\ddot{\hat{\gamma}}^{a}=-\ddot{\gamma}^{a},~{}\alpha(\dot{\hat{\gamma}})=\alpha(\dot{\gamma}),~{}F(\dot{\hat{\gamma}})=F(\dot{\gamma}),$
$\rho_{a}(\hat{\gamma})=-\rho_{a}(\gamma),~{}\rho_{b}(\hat{\gamma})\dot{\hat{\gamma}}^{b}=\rho_{b}(\gamma)\dot{\gamma}^{b}.$

So by (4.5), we have

$\displaystyle\ddot{\hat{\gamma}}^{b}+2\widetilde{G}^{b}(\hat{\gamma},\dot{\hat{\gamma}})$
$\displaystyle=\ddot{\hat{\gamma}}^{b}+2\left[\left(1-\frac{b^{2}\alpha(\dot{\hat{\gamma}})+\beta(\dot{\hat{\gamma}})}{2F(\dot{\hat{\gamma}})}\right)\rho_{c}(\hat{\gamma})\dot{\hat{\gamma}}^{c}\dot{\hat{\gamma}}^{b}-\frac{1}{2}\lambda
F(\dot{\hat{\gamma}})\alpha(\dot{\hat{\gamma}})\rho_{b}(\hat{\gamma})\right]$
$\displaystyle=\left\\{\begin{aligned}
&-(\ddot{\gamma}^{b}+2\widetilde{G}^{b}(\gamma,\dot{\gamma})),~{}a=b,\\\
&\ddot{\gamma}^{b}+2\widetilde{G}^{b}(\gamma,\dot{\gamma}),~{}~{}~{}~{}~{}~{}~{}a\neq
b.\end{aligned}\right.$

Then $\hat{\gamma}(t)=\Lambda_{a}(\gamma(t))$ is a geodesic.

For $\Psi_{m_{a}}$, we also get

$\ddot{\check{\gamma}}^{a}=\ddot{\gamma}^{a},~{}F(\dot{\check{\gamma}})=F(\dot{\gamma}),~{}\rho_{a}(\check{\gamma})=\rho_{a}(\gamma).$

Then

$\displaystyle\ddot{\check{\gamma}}^{a}+2\widetilde{G}^{a}(\check{\gamma},\dot{\check{\gamma}})$
$\displaystyle=\ddot{\check{\gamma}}^{a}+2\left[\left(1-\frac{b^{2}\alpha(\dot{\check{\gamma}})+\beta(\dot{\check{\gamma}})}{2F(\dot{\check{\gamma}})}\right)\rho_{c}(\check{\gamma})\dot{\check{\gamma}}^{c}\dot{\check{\gamma}}^{a}-\frac{1}{2}\lambda\alpha(\dot{\check{\gamma}})F(\dot{\check{\gamma}})\rho_{a}(\check{\gamma})\right]$
$\displaystyle=\ddot{\gamma}^{a}+2\widetilde{G}^{a}(\gamma,\dot{\gamma}).$

So $\hat{\gamma}(t)=\Psi_{a}(\gamma(t))$ is also a geodesic.

From the above,
$\tilde{\gamma}(t)=\Psi_{m_{a}}\Lambda_{a}(\gamma(t))=(\gamma^{1}(t),\cdots,-\gamma^{a}(t)+2m_{a},\cdots,\gamma^{n}(t))$
is also a geodesic with respect to $\tilde{F}$. But $\tilde{\gamma}(t)$ and
$\gamma(t)$ have the same initial condition. Hence, by the uniqueness of
geodesic, we get $\gamma^{a}(t)=m_{a}\in\mathbb{Z}$. Note that
$\rho(\gamma(t))=c\ln 3$, where $c$ is the number of even entries of
$(m_{1},m_{2},\cdots,m_{n-1})$. Since

$\rho_{a}(\gamma)=\frac{-\pi sin(\pi x^{a})}{2+cos(\pi x^{a})}(\gamma)=0.$
(4.11)

Then by (4.5) and (4.6), we get

$\widetilde{G}^{a}({\gamma},\dot{\gamma})=\widetilde{G}^{n}({\gamma},\dot{\gamma})=0.$

From (4.10), we have

$\ddot{\gamma}^{n}=0.$ (4.12)

According to initial condition, then

$\gamma(t)=(m_{1},m_{2},\cdots,m_{n-1},x_{0}^{n}+(1+b)e^{-\rho}t),$ (4.13)

where $t$ is arc length.

###  4.3 Parallel hypersurfaces

For $x_{0}\in M$, there exist a neighbourhood $U(x_{0})\subset M$ and a
distance function $r(x)$ to $M$, such that $r(x)=t$. Then
$\tilde{\mathbf{n}}=\nabla r$ and all the integral curves of $\nabla r$ are
all normal geodesics (see [2] for detail). So we can define the parallel
hypersurfaces of $M$ by

$M_{t}=\\{x\in M~{}|~{}r(x)=t,t>0\\}.$

For $X=X^{i}\frac{\partial}{\partial x^{i}}\in TM$, note that $X^{n}=0$ and
$\tilde{F}_{y^{j}}X^{j}=F_{y^{j}}X^{j}=\alpha_{y^{j}}X^{j}=0$. Then by
$\alpha(\tilde{\mathbf{n}})=\frac{1}{1-b}e^{-\rho},\beta(\tilde{\mathbf{n}})=\frac{-b}{1-b}e^{-\rho}$,
(4.7) and (4.1), we get

$\displaystyle D^{\tilde{\mathbf{n}}}_{X}\tilde{\mathbf{n}}=$
$\displaystyle(\tilde{\mathbf{n}}^{i}_{x^{j}}+\tilde{N}^{i}_{j}(\tilde{\mathbf{n}}))X^{j}\frac{\partial}{\partial
x^{i}}$ $\displaystyle=$
$\displaystyle\left[\tilde{\mathbf{n}}^{i}_{x^{a}}+\rho_{a}\tilde{\mathbf{n}}^{i}+\frac{1}{2}F^{2}(\tilde{\mathbf{n}})\rho_{a}\left(\frac{\alpha(\tilde{\mathbf{n}})}{F(\tilde{\mathbf{n}})^{2}}b^{i}-\frac{b^{2}\alpha+\beta}{F(\tilde{\mathbf{n}})^{3}}\tilde{\mathbf{n}}^{i}\right)\right]X^{a}\frac{\partial}{\partial
x^{i}}$ $\displaystyle=$
$\displaystyle(\tilde{\mathbf{n}}^{i}_{x^{a}}+\rho_{a}\tilde{\mathbf{n}}^{i})X^{a}\frac{\partial}{\partial
x^{i}}$ $\displaystyle=$ $\displaystyle 0.$

By Weingarten formula (2.6), $M$ has constant principal curvatures
$\tilde{\theta}=0$ with respect to $\tilde{F}$. From [2] (Lemma 4.4), we have

$\frac{dk_{a}(t)}{dt}=\tilde{K}(\tilde{\mathbf{n}},e_{a})+k^{2}_{a}(t),$

where $k_{a}(t)$ are the principal curvature of $M_{t}$, $k_{a}(0)=\mu_{a}$
and $\tilde{K}$ is the flag curvature of $(N,\tilde{F})$. Take the trace

$\frac{d}{dt}\sum\limits_{a=1}^{n-1}k_{a}(t)=\widetilde{Ric}(\tilde{\mathbf{n}})+\sum\limits_{a=1}^{n-1}k_{a}^{2}(t),$
(4.14)

where $\widetilde{Ric}$ is the Ricci curvature of $(N,\tilde{F})$.

Now we prove that $M_{t}$ is neither isoparametric nor $d\mu$-isoparametric.
By [17], we have

$\displaystyle\widetilde{Ric}(y)=$ $\displaystyle
Ric(y)-F^{2}\\{2\rho_{k}J^{k}+g^{ij}\rho_{i|j}+[\rho^{j}I_{j}]_{|k}y^{k}+2\theta\rho^{j}I_{j}\\}$
(4.15) $\displaystyle+(n-2)(\Xi-F^{2}\parallel
d\rho\parallel^{2}_{F})+F^{4}\rho^{j}\rho^{k}\\{2I_{s}C^{s}_{ij}-[I_{j}]_{y^{k}}-C^{s}_{ij}C^{i}_{ks}\\},$

where $Ric(y)$ is the Ricci curvature of $(N,F)$,
$J^{k}=g^{ij}L^{k}_{ij},~{}\theta(y)=\rho_{k}y^{k},$
$\Xi=\theta^{2}-\theta_{x^{k}}y^{k}$, $\parallel
d\rho\parallel^{2}_{F}=g^{ij}\rho_{i}\rho_{j}.$ Note that $G^{i}=0$,
$L_{ijk}=0$, $Ric(y)=0$, $C_{ijn}(\tilde{\mathbf{n}})=0$. By (4.2), we get

$2C_{abc}=-\frac{(\delta_{ab}y^{c}+\delta_{ac}y^{b}+\delta_{bc}y^{a})\beta}{\lambda^{2}\alpha^{3}}+3\frac{\beta
y^{a}y^{b}y^{c}}{\lambda^{3}\alpha^{5}},$
$2C_{abn}=\frac{\delta_{ab}(\alpha\beta_{y^{n}}-\beta\alpha_{y^{n}})}{\lambda\alpha^{2}}-\frac{y^{a}y^{b}(\alpha\beta_{y^{n}}-3\beta\alpha_{y^{n}})}{\lambda^{2}\alpha^{4}},$

then $C_{abc}(\tilde{\mathbf{n}})=0$. In conclusion,
$C_{ijk}(\tilde{\mathbf{n}})=0$.

$\displaystyle 2\frac{\partial C_{abc}}{\partial y^{d}}$
$\displaystyle=-\frac{(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{bc}\delta_{ad})\beta}{\lambda^{2}\alpha^{3}}$
$\displaystyle-3\frac{(\delta_{ab}y^{c}+\delta_{ac}y^{b}+\delta_{bc}y^{a})\beta\alpha_{y^{d}}}{\lambda^{2}\alpha^{4}}+3(\frac{\beta
y^{a}y^{b}y^{c}}{\lambda^{3}\alpha^{5}})_{y^{d}},$ $\displaystyle
2\frac{\partial C_{abn}}{\partial y^{n}}$
$\displaystyle=\frac{\delta_{ab}[-\beta\alpha\alpha_{y^{n}y^{n}}-2(\alpha\beta_{y^{n}}-\beta\alpha_{y^{n}})\alpha_{y^{n}}]}{\lambda\alpha^{3}}-y^{a}y^{b}[\frac{(\alpha\beta_{y}^{n}-3\beta\alpha_{y}^{n})}{\lambda^{2}\alpha^{4}}]_{y^{n}}.$

Then $C_{abc}(\tilde{\mathbf{n}})=0,~{}2\frac{\partial C_{abc}}{\partial
y^{d}}(\tilde{\mathbf{n}})=\frac{(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{bc}\delta_{ad})be^{2\rho}}{(1+b)^{2}}$,
$\frac{\partial C_{abn}}{\partial y^{n}}(\tilde{\mathbf{n}})=0$.

$\theta(\tilde{\mathbf{n}})=0,~{}F(\tilde{\mathbf{n}})=e^{-\rho},~{}\Xi(\tilde{\mathbf{n}})=0,~{}\frac{\alpha(\tilde{\mathbf{n}})}{F(\tilde{\mathbf{n}})}=\frac{1}{1-b},~{}I_{i}(\tilde{\mathbf{n}})=0,$
$g^{ab}(\tilde{\mathbf{n}})=(1+b)\delta^{ab},~{}g^{an}(\tilde{\mathbf{n}})=0,~{}g^{nn}(\tilde{\mathbf{n}})=(1+b)^{2},$
$\parallel
d\rho\parallel^{2}_{F}(\tilde{\mathbf{n}})=g^{ij}(\tilde{\mathbf{n}})\rho_{i}\rho_{j}=(1+b)\delta^{ab}\rho_{a}\rho_{b}=(1+b)\sum\limits_{a=1}^{n-1}(\rho_{a})^{2},$
$g^{ij}(\tilde{\mathbf{n}})\rho_{i|j}=-(1+b)\delta^{ab}\frac{\pi^{2}+2\pi^{2}cos(\pi
x^{b})}{[2+cos(\pi
x^{a})]^{2}}=-(1+b)\sum\limits_{a=1}^{n-1}\frac{\pi^{2}+2\pi^{2}cos(\pi
x^{a})}{[2+cos(\pi x^{a})]^{2}},$
$[\rho^{j}I_{j}]_{|k}y^{k}(\tilde{\mathbf{n}})=(\rho^{j}_{|k}I_{j}(\tilde{\mathbf{n}})+\rho^{j}I_{j|k}(\tilde{\mathbf{n}}))\tilde{\mathbf{n}}^{k}=\rho^{j}g^{il}\frac{\partial
C_{ilj}}{\partial x^{k}}\tilde{\mathbf{n}}^{k}=0,$
$\displaystyle\rho^{j}\rho^{k}[I_{j}]_{y^{k}}(\tilde{\mathbf{n}})=\rho^{j}\rho^{k}[-2C^{il}_{k}C_{ilj}+g^{il}\frac{\partial
C_{ijl}}{\partial y^{k}})](\tilde{\mathbf{n}})$
$\displaystyle=\rho^{a}\rho^{b}[g^{cd}\frac{\partial C_{bcd}}{\partial
y^{a}}+g^{cn}\frac{\partial C_{abc}}{\partial y^{n}}+g^{nn}\frac{\partial
C_{abn}}{\partial y^{n}}](\tilde{\mathbf{n}})$
$\displaystyle=\rho^{a}\rho^{b}\frac{(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{bc}\delta_{ad})be^{2\rho}\delta^{cd}}{(1+b)}=\sum\limits_{a=1}^{n-1}(\rho^{a})^{2}\frac{(n+2)be^{2\rho}}{(1+b)}.$

So

$\displaystyle\widetilde{Ric}(\tilde{\mathbf{n}})=e^{-2\rho}(1+b)\sum\limits_{a=1}^{n-1}\left[\frac{\pi^{2}+2\pi^{2}cos(\pi
x^{a})}{[2+cos(\pi
x^{a})]^{2}}-(n-2)(\rho_{a})^{2}-(\rho^{a})^{2}\frac{(n+2)b}{(1+b)^{2}}\right].$
(4.16)

If $u_{1}=(0,0,\cdots,x_{0}^{n})\in M\cap\Omega$ and
$u_{2}=(1,1,\cdots,x_{0}^{n})\in M\cap\Omega$, we know

$\widetilde{Ric}(\tilde{\mathbf{n}})|_{\gamma_{u_{1}}}=3^{(-2n+1)}(1+b)\pi^{2}(n-1)>0,$

and

$\widetilde{Ric}(\tilde{\mathbf{n}})|_{\gamma_{u_{2}}}=-(1+b)\pi^{2}(n-1)<0.$

Meanwhile, when $t=0$,
$k_{a}^{2}|_{\gamma_{u_{1}}(0)}=k_{a}^{2}|_{\gamma_{u_{2}}(0)}=0$. Then by
(4.14), $\frac{d}{dt}\sum\limits_{i=1}^{n}k_{a}|_{\gamma_{u_{1}}(0)}>0$ and
$\frac{d}{dt}\sum\limits_{i=1}^{n}k_{a}|_{\gamma_{u_{2}}(0)}<0$. So the mean
curvatures of the parallel hypersurfaces $M_{t}$ is not constant. Note that
$\rho_{k}\tilde{\mathbf{n}}^{k}=0$ and by (4.1), we have

$\tilde{S}(\tilde{\mathbf{n}})=0.$

######  Remark 4.1.

According to Definition 2.2, $M$ with constant principal curvatures is neither
isoparametric hypersurface nor $d\mu$-isoparametric hypersurface.

## References

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  * [10] Rodríguez-Vázquez Alberto, _A nonisoparametric hypersurface with constant principal curvatures_ , arXiv:1903.03537.
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  * [16] Y. B. Shen and Z. M. Shen. _Introduction to Modern Finsler geometry_ , Higher Education Press, WWW.hep.com.cn, 2016.
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Peilong Dong
School of Mathematics and Statistics, Zhengzhou Normal University,
Zhengzhou Henan 450044, China.
E-mail<EMAIL_ADDRESS>

Yali Chen
School of Mathematics and Statistics, Anhui Normal University,
Wuhu, Anhui, 241000, China.
E-mail<EMAIL_ADDRESS>


11institutetext: University of Bonn

# A Dual-Source Attention Transformer for Multi-Person Pose Tracking

Andreas Doering  Juergen Gall

###### Abstract

Multi-person pose tracking is an important element for many applications and
requires to estimate the human poses of all persons in a video and to track
them over time. The association of poses across frames remains an open
research problem, in particular for online tracking methods, due to motion
blur, crowded scenes and occlusions. To tackle the association challenge, we
propose a Dual-Source Attention Transformer that incorporates three core
aspects: i) In order to re-identify persons that have been occluded, we
propose a pose-conditioned re-identification network that provides an initial
embedding and allows to match persons even if the number of visible joints
differs between the frames. ii) We incorporate edge embeddings based on
temporal pose similarity and the impact of appearance and pose similarity is
automatically adapted. iii) We propose an attention based matching layer for
pose-to-track association and duplicate removal. We evaluate our approach on
Market1501, PoseTrack 2018 and PoseTrack21.

##  1 Introduction

Multi-person pose tracking is highly relevant for a wide range of applications
such as virtual reality, autonomous driving or sports analysis and requires to
accurately estimate and track the human poses of all persons throughout a
video. Despite of the recent progress in multi-person pose tracking [37, 12,
22, 13, 42, 54, 38, 55, 49], the task remains very challenging due to camera
motion, motion blur, occlusions, and a high variety in pose and scale [11].
Consequently, a tracking approach must be robust to detection errors and
ambiguities. In particular, the assignment of highly occluded persons in
unusual poses is very difficult as shown in Fig. 1 where three persons perform
a gymnastic exercise in water. For instance, the green bounding box intersects
with three persons and the respective pose overlaps with the keypoints of the
other persons. This poses a challenge, especially for on-line methods as
assignments will be made once a new frame arrives.

Related works such as [42, 38, 55, 54, 63] try to tackle these challenges by
generating future poses from a track’s history, which are then matched with
detections based on pose similarities, _e.g_., based on Object Keypoint
Similarity (OKS) [38, 54, 63] or a pose-based matching layer [42]. Other works
such as [49] process each sequence in an off-line fashion, which is not
feasible for real-time applications. As these works mainly rely on pose-based
similarities for matching, these methods tend to fail to re-identify tracks
that have been occluded for longer periods of time or undergo high pose
deformations.

In our work, we thus focus on learning the association of detected persons to
tracks in an on-line fashion and propose an approach that leverages the
estimated poses, bounding boxes, and the appearance of the detected persons to
assign them to previous tracks or initialize new tracks. Since we can neither
rely solely on appearance-based features nor pose similarities due to multiple
instances with similar appearance, changing camera views or scene switches,
which often occur in in-the-wild sequences, we propose a dual-source
transformer that considers the similarity between detection embeddings and
previous track embeddings as well as edge embeddings that encode temporal pose
similarities. All three embeddings are updated by a dual-source decoder and a
final matching layer assigns the detections to the tracks. The matching layer
also removes duplicates, _i.e_., multiple detections for the same persons, and
initiates new tracks. Furthermore, we propose a pose-conditioned re-
identification model where the appearance features are normalized based on the
heatmaps of the detected keypoints. Our model is more robust to occlusions in
crowded scenarios compared to related works such as [45, 33] that solely
operate on a bounding-box level.

We evaluate our approach on the challenging PoseTrack 2018 [1] and PoseTrack21
dataset [11] where it achieves state-of-the-art results. Moreover, we evaluate
our pose-conditioned appearance model for the task of person re-identification
on the Market1501 dataset [62], where we achieve competitive results with the
current state-of-the-art. In summary, we propose (1) a novel pose-conditioned
re-identification model for human pose tracking, (2) a dual-source attention
transformer that combines pose and appearance similarity in a novel way for
pose-to-track association and (3) a novel matching layer for pose-to-track
association and duplicate removal.

Figure 1: Qualitative examples of our proposed method on the PoseTrack21
dataset. The first row contains visual tracking results of our method and the
second row shows visualizations of CorrTrack with ReID [11].

##  2 Related Work

We briefly discuss some related works for person re-identification and multi-
person pose tracking.

Person Re-Identification: Methods for person re-identification aim to generate
robust feature representations of a query person, which allows to re-identify
each instance of the particular person. These methods can be divided into
different categories such as re-identification based on global features [33,
20], part-based approaches [45, 25, 44], prior-based, _i.e_., pose- or mask-
guided [14, 40, 61, 53, 36] and video-based [5, 6, 26] re-identification.
Part-based models [45, 25, 44] divide each image into several parts and
extract distinct part-based features. Features for each part are either
trained individually [45] or further re-combined to obtain visibility-aware
features [44]. Pose- [14, 40, 61, 53] or mask-guided [36] approaches aim to
suppress background-noise by learning occlusion- and instance-aware features
that are more robust in crowded scenarios with occlusions. In this work, we
propose Spatially Adaptive Pose DEnormalization (SPAPDE) layers for pose-
guided re-identification and use it in the proposed dual-source tracking
transformer.

Multi-Person Pose Tracking: Existing works on multi-person pose tracking can
be divided into two categories: top-down methods [55, 49, 42, 38, 11, 63, 57]
and bottom-up methods [37, 22, 13, 12]. Former approaches employ a person
detector and estimate the pose for every detected person individually based on
temporal-context. Most methods employ a pose-warping [38, 11, 57] scheme that
warps tracked poses into the next frame or directly predict poses based on the
tracklet history [55, 42, 63, 24, 12], which are then matched with detected
poses using greedy or Hungarian matching. In [49], an offline approach has
been proposed that merges multiple overlapping fixed-lengths tracklets into
tracks based on bipartite matching and Dijkstra’s algorithm [10]. Bottom-up
approaches [37, 22, 24, 12], on the other hand, predict all keypoints within
an image simultaneously and generate tracks by solving spatio-temporal graphs
between detected keypoints. For instance, [37, 12, 22] generate spatio-
temporal vector fields, while various spatio-temporal embeddings for the
association of keypoints and tracks are proposed in [24]. In contrast, person
instances are tracked in [13] using a semi-supervised approach based on video
instance correspondences. In this work, we propose a Dual-Source Transformer
that considers the similarity between detection embeddings and previous track
embeddings as well as edge embeddings that encode spatial and pose
similarities.

Figure 2: The proposed multi-person pose tracking architecture entails the
following steps: a) Given a set of person crops and their respective keypoint
heatmaps at time frame $t$, we compute pose-conditioned appearance features
and feed them into N transformer encoder stages. This gives an embedding for
each detection $E^{t}_{D}$. b) We also compute temporal similarities between
tracks and detected persons by means of Intersection over Union (IoU) and
Object Keypoint Similarity (OKS), which we then encode into an edge embedding
$E^{t}_{T\rightarrow D}$. c) The dual-source decoder takes the embeddings of
the previous tracks $E^{t-1}_{T}$, the detection embeddings $E^{t}_{D}$ and
the edge embeddings $E^{t}_{T\rightarrow D}$ as input and updates the
embeddings. d) Finally, the matching stage assigns detections to tracks,
removes redundant detections and initializes new tracks.

##  3 Dual-Source Attention Transformer for Multi-Person Pose Tracking

On-line methods for multi-person pose tracking often follow the tracking-by-
detection paradigm [42, 15, 12, 37, 22, 54, 59, 38] and usually suffer from
ambiguities and occlusions. An example is shown in the second row of Fig. 1
where the blue id jumps between two persons. In this work, we thus focus on
learning robust assignments between estimated poses in a current frame and
tracked persons in previous frames. This is done by the proposed dual-source
transformer that combines appearance features and encoded temporal person
similarities. Since the importance of appearance and pose similarity varies
within a video and between videos, in particular when a person has been
occluded for a few frames, the dual-source decoder and the matching layer use
a gating mechanism to update the embeddings of the new poses and previous
tracks and to match poses to tracks. Our approach for multi-person pose
tracking is illustrated in Fig. 2.

We assume that the human poses are extracted for a new frame $t$ by a standard
multi-person pose estimator where we utilize the detector from [11] for a fair
comparison. Specifically, for a given frame at time $t$, our network takes as
input the set of estimated bounding boxes $\mathcal{B}_{D}^{t}$ and poses
$\mathcal{P}_{D}^{t}$, and additionally detected keypoint heatmaps
$\mathcal{H}^{t}_{D}$ extracted from the image crops $\mathcal{I}_{D}^{t}$ of
detected persons. From the heatmaps and image crops, an embedding $E^{t}_{D}$
for re-identification will be computed as described in Section 3.2 (Fig. 2a).
To measure pose and spatial proximity between detections and tracks,
similarities to the bounding boxes $\mathcal{B}_{T}^{t-1}$ and poses
$\mathcal{P}_{T}^{t-1}$ of the last frame of each track are predicted, which
results in the embedding $E_{T\rightarrow D}^{t}$ (Fig. 2b) that will be
described in Section 3.2 as well. Similar to [16], we call $E_{T\rightarrow
D}^{t}$ edge embedding since it is an embedding of the edges between the
tracks and detections. Given both embeddings $E^{t}_{D}$ and $E_{T\rightarrow
D}^{t}$ as well as the track embeddings $E_{T}^{t-1}$ that have been estimated
in the previous frame $t{-}1$, the proposed _Dual-Source Decoder_ (Fig. 2c)
and the _Matching Layer_ (Fig. 2d) assign the current detections to previous
tracks, update the track embeddings, and initialize new tracks. Both will be
described in the following Section 3.1.

###  3.1 Dual-Source Decoder

Figure 3: Illustration of our proposed a) Dual-Source Attention layer, b)
Dual-Source Decoder layer and c) the Matching layer. The Dual-Source Decoder
layer takes as input the detection embeddings $E_{D}^{t}$, track embeddings
$E_{T}^{t-1}$ and edge embeddings $E_{T\rightarrow D}^{t}$ and updates the
track embeddings $\hat{E}_{T}^{t}$ and edge embeddings $\hat{E}_{T\rightarrow
D}^{t}$ using the Dual-Source Attention layer. The matching layer predicts an
assignment matrix $\widehat{M}_{D\rightarrow T}^{t}$ of detections to tracks.
The $\alpha$-Gates fuse the cross-attention matrix $S_{A}$ between detections
and tracks, which measures appearance similarity, and the self-attention
matrix $S_{E}$ of the edge embeddings, which measures spatial and pose
similarity.

#### Dual-Source Attention Layer

The main components of our proposed dual-source transformer are the dual-
source attention layers and the matching layer. As illustrated in Fig. 3a, the
dual-source attention layer combines appearance-based similarities
$S_{A}=\sigma\left(O_{A}\right)$ and embedded temporal pose similarities
$S_{E}=\sigma\left(O_{E}\right)$ between detections and tracks, where $\sigma$
is the row-wise softmax layer. While the pose similarity logits
$O_{E}=E_{T\rightarrow D}^{t}W_{E}^{\intercal}$ are obtained by
$E_{T\rightarrow D}^{t}$ and the learned weight matrix $W_{E}$, the
appearance-based similarity logits $O_{A}$ are obtained by cross-attention
between the appearance features of the detections ${E}_{D}^{t}$ and tracks
${E}_{T}^{t-1}$:

$O_{A}=\frac{{E}_{T}^{t-1}W_{Q}^{\intercal}\left({E}_{D}^{t}W_{K}^{\intercal}\right)^{\intercal}}{\sqrt{d}}.$
(1)

where $W_{Q}$ and $W_{K}$ are the learned projection weights for the queries
and keys, respectively, and $d$ is the dimensionality of the embeddings $E$.

In order to assign detections to tracks, appearance-based similarities $S_{A}$
and embedded temporal pose similarities $S_{E}$ are fused by the
$\alpha{-}Gate$ that weights the contribution of appearance-based attention
weights and the pose-based attention weights by a hyperparameter $\alpha$,
which we evaluate in our experiments:

$A=\alpha\cdot S_{A}+(1-\alpha)\cdot S_{E}.$ (2)

Intuitively, pose and spatial similarities $S_{E}$ provide a strong matching
prior, but are less reliable in crowded scenarios as shown in Fig. 1.
Appearance similarities $S_{A}$ allow to re-identify an occluded person after
some frames, but are unreliable in case of motion blur or person instances
with similar appearance as it is common in team sport videos. While none of
them can resolve all ambiguities, fusing the normalized similarities $S_{A}$
and $S_{E}$ automatically gives higher weight to the similarity measure where
the matching confidence of a detection to a track is higher. This, however,
only works if also we allow that none of the detections is assigned to a
track, _e.g_., if a person has not been detected or is occluded. Prior to
applying the softmax to the attention matrices, we thus add a column of zeros.
In other words, the last column of $S_{A}$ and $S_{E}$ indicates if a track
does not match with any detection.

Finally, the dual-source attention layer calculates a proposed track embedding
update as follows:

$\Delta E_{T}^{t}=(A_{:,:-1}E_{D}^{t})W_{A}^{\intercal},$ (3)

where $A_{:,:-1}$ denotes the attention weights without the last column and
$W_{A}$ are the weights of a linear layer.

#### Dual-Source Decoder Layer

The dual-source decoder layer shown in Fig. 3b then updates the track
$E_{T}^{t-1}$ and edge $E_{T\rightarrow D}^{t}$ embeddings based on the output
of the dual-source attention layer. As common for transformer blocks [47], we
use a residual feed-forward network (FNN) as shown in Fig. 3b). Specifically,
we compute

$\widehat{E}^{t}_{T}=LN(\widetilde{E}^{t}_{T}+FFN(\widetilde{E}^{t}_{T})),\;\;\;\;\;\widetilde{E}^{t}_{T}=LN(E_{T}^{t-1}+\Delta
E_{T}^{t}),$ (4)

where LN denotes layer normalization. As shown in Fig. 2c, we employ an
additional _Track Embedding Head_ after the last decoder layer that predicts
the final track embeddings. The _Track Embedding Head_ consists of two linear
layers, where the first layer comprises LayerNorm [2] and GELU [19] non-
linearities.

To update the edge embeddings, we employ an $\alpha{-}Gate$ to weight the
contribution of appearance-based and edge-based attention logits $O_{A}$ and
$O_{E}$ similar to the dual-source attention layer. We then apply a feed-
forward network $FFN_{E}$ on the weighted sum of attention logits,

$\widehat{E}_{T\rightarrow D}^{t}=FFN_{E}\left(\alpha\cdot
O_{A}+(1-\alpha)\cdot O_{E}\right),$ (5)

where $FFN_{E}$ is a feed-forward network as $FFN$.

#### Confidence-Guided Track Update

As shown in Fig. 2c, we perform the final update of the tracks embedding
before the final assignment of detections to tracks is done by the matching
layer (Fig. 2d). This is done for two reasons. Firstly, the final assignment
might be wrong and, secondly, we do not want to update the track embedding
when the detection embedding is noisy due to occlusion or motion blur. To
prevent track embeddings from being updated by noisy or low-confident
detections, we employ a confidence-guided update of the track embeddings. Let
$\mathcal{A}=\\{A_{:,:-1}^{n}\\}_{n=1}^{N}$ be the set of dual-source
attention weights of all $N$ decoder stages, without the last column. As the
number of detections per frame is dynamic, we apply max pooling on the rows of
each attention matrix to obtain the maximum attention score for each track and
estimate an importance weight $w_{j}$ with a linear layer as

$\displaystyle\widehat{\mathcal{A}}_{j}$ $\displaystyle=$
$\displaystyle\text{concat}\\{\max_{i}A_{ji}^{n}|n\in[1,N]\\}$ (6)
$\displaystyle w_{j}$ $\displaystyle=$
$\displaystyle\sigma\left(\sum_{n}w_{n}\cdot\widehat{\mathcal{A}}_{j}^{n}+b_{n}\right),$
(7)

where $\sigma$ is the sigmoid function and $\widehat{A}_{j}^{n}$ the maximum
attention for track $j$ at layer $n$. Ultimately, we update the embedding for
each track $j$ as a confidence-guided moving average by using the importance
weight $w_{j}$ following $E_{T_{j}}^{t}=(1-w_{j})\cdot
E_{T_{j}}^{t-1}+w_{j}\cdot\widehat{E}_{T_{j}}^{t}$. As we will show in our
experiments, the confidence-guided embedding update improves the performance
compared to solely relying on the _Track Embedding Head_ (Fig. 2c) while only
adding a negligible overhead.

#### Matching Layer

The matching layer (Fig. 3c)) comprises a structure similar to the dual-source
attention layer, but differs in two aspects. 1) Unlike the dual-source
attention layer, the matching layer does not use the attention weights to
predict a track embedding update and therefore does not consists of a linear
layer after the $\alpha{-}Gate$. 2) The attention weights (2) are used as
assignment matrix $\widehat{M}_{D\rightarrow T}^{t}$ to assign detections to
tracks, as we show in Fig. 2d). In the presence of duplicate detections, this
allows to match multiple detections to a single track. Given the matching
matrix $\widehat{M}_{D\rightarrow T}^{t}$, we utilize Hungarian matching to
assign detections and tracks. All the remaining detections $i$ that have a
matching probability $\hat{m}_{i\rightarrow j}>\tau_{dup}$ for any track $j$
are considered as duplicate detections and are removed. Any other detection
will be passed to the _new track embedding head_ and initialize a new track
embedding (Fig. 2d). The new track embedding head has the same structure as
the _track embedding head_. All tracks that have not been tracked for
$\tau_{age}$ frames will be removed. We evaluate the impact of $\tau_{dup}$
and $\tau_{age}$ in the experiments.

###  3.2 Encoder Stage

#### Pose-Conditioned Appearance Features

Multi-person pose tracking in crowded real-world scenarios is particularly
challenging due to many occlusions, especially for appearance-based
associations as two persons in close proximity are likely to share similar
features. Towards this end, we employ a ResNet50 [17] as our backbone network
(Fig. 2a) that we pre-trained for person re-identification [33]. Since we need
to deal with diverse poses and occlusions, we propose a modification to the
ResNet and condition the backbone on the keypoint heatmaps
$\mathcal{H}_{D}^{t}$ of the pose estimator. Specifically, we propose a
Spatially Adaptive Pose DEnormalization (SPAPDE) layer, which is inspired by
[35]. Each SPAPDE layer operates on a set of keypoint heatmaps
$\mathcal{H}_{D}^{t}\in\mathbb{R}^{N\times
K\times\frac{256}{s}\times\frac{128}{s}}$ along with the current ResNet
features $f\in\mathbb{R}^{N\times C\times\frac{256}{s}\times\frac{128}{s}}$
extracted from the corresponding image crops
$\mathcal{I}_{D}^{t}\in\mathbb{R}^{Nx3x256x128}$, where $N$ is the number of
persons in frame $t$, $s$ denotes the respective scaling factor, $K$ denotes
the number of keypoints and $C$ denotes the number of feature channels.

In contrast to batch normalization [23], the normalized input features are
scaled and shifted with respect to the keypoint heatmaps. In particular,
SPAPDE computes the modulation parameters $\beta$ and $\gamma$ following

$\gamma=\mathrm{conv}(a),\;\;\;\beta=\mathrm{conv}(a),\;\;a=\mathrm{ReLU}(\mathrm{conv}(\mathcal{H}_{D}^{t})),$
(8)

where $\mathrm{conv}$ denotes a 3x3-convolution and
$\beta,\gamma\in\mathbb{R}^{N\times C\times\frac{256}{s}\times\frac{128}{s}}$.
The image features $f$ are then conditioned on the heatmaps
$\mathcal{H}_{D}^{t}$ as follows. Let $f_{n,c,y,x}$ be the feature value for
the detected person $n$ and feature channel $c$ at pixel location $(x,y)$. The
SPADPE layer first calculates the mean $\mu_{c}$ and standard deviation
$\sigma_{c}$ over all persons and pixels of $f$ and then adaptively de-
normalizes the image features by

$\widehat{f}_{n,c,y,x}=\gamma_{n,c,y,x}\cdot\frac{f_{n,c,y,x}-\mu_{c}}{\sigma_{c}}+\beta_{n,c,y,x}.$
(9)

We replace every batch normalization layer within ResNet50 by SPAPDE layers
and train the network following [33].

Moreover, we use a vanilla N-stage transformer encoder [47] without positional
encoding to further disentangle the backbone features of severely occluded
persons and noisy pose predictions and generate a set of encoded person
features $E_{D}^{T}\in\mathcal{R}^{N\times 256}$ as shown in Fig. 2a.

#### Temporal Edge Features

The edge features $E_{T\rightarrow D}^{t}$ (Fig. 2b) used in our dual-source
architecture are based on similarities between bounding boxes and poses using
Intersection over Union (IoU) and Object Keypoint Similarity (OKS),
respectively. In this work, we rely on three variants of OKS: the first
variant considers keypoints which are present in both poses. While the first
variant provides a good measure of keypoint alignment, its expressiveness
suffers if two poses only share a small subset of keypoints. For that reason,
the remaining two variants consider all keypoints present in one of the two
poses, respectively. OKS denotes thus a 3d vector.

We leverage a warping function $\phi_{W}$ to warp all tracks into the current
frame $t$ and calculate the temporal edge features $E_{T\rightarrow D}$
between tracks and detections as follows:

$E_{T\rightarrow
D}=\phi_{E}\left(\left[IOU(\widehat{\mathcal{B}}_{T}^{t-1},\mathcal{B}_{D}^{t})\;||\;OKS(\widehat{\mathcal{P}}_{T}^{t-1},\mathcal{P}_{D}^{t})\right]\right).$
(10)

Here, $\widehat{\mathcal{P}}_{T}^{t-1}=\phi_{W}(\mathcal{P}_{T}^{t-1})$ and
$\widehat{\mathcal{B}}_{T}^{t-1}=\phi_{W}(\mathcal{B}_{T}^{t-1})$ represent
the set of warped track poses and track bounding boxes, respectively. The
operator $\left[\cdot\;||\;\cdot\right]$ represents concatenation, and
$\phi_{E}$ denotes the edge embedding head, which consists of three linear
layers with LayerNorm [2] and GELU [19].

###  3.3 Training Objective

In a first step, we train the re-identification network following [33]: We
apply the triplet loss [20] and center loss [53] after the last pooling layer
of ResNet50 and we employ the cross-entropy loss with label smoothing [46] on
the classification layer. Subsequently, we freeze the re-identification
network and proceed to train our dual-source transformer.

In our approach, the matching layer is trained using a cross-entropy loss,
which is defined as follows:

$\mathcal{L}_{match}=-\frac{1}{N_{D}}\sum_{i}y_{i}\cdot
log(p^{m}_{ij})+(1-y_{i})\cdot p^{m}_{i0},$ (11)

where $N_{D}$ is the total number of detections, $p^{m}_{i,j}$ represents the
probability of matching the $i$-th detection to its corresponding ground truth
track $j$, and $p^{m}_{i0}$ is the probability of not matching the $i$-th
detection to any track. The variable $y_{i}$ takes the value 1 if the $i$-th
detection is assigned to a ground truth track and 0 otherwise.

To optimize the performance of our approach, we provide direct feedback to
both the encoder and decoder, thereby enabling maximum guidance during the
training process. Since we discard duplicates after the final matching layer,
we intentionally integrate duplications into the training process and allow
multiple detection assignments to a single track. Specifically, we leverage
both, detected poses and ground truth poses in the training process that can
share the same person identity. To assign identities to the detected poses we
employ OKS-based greedy matching to the ground truth poses. We then utilize a
duplicates-aware cross-entropy loss function that operates on the attention
weights of the encoder and decoder layers. The loss function is defined as
follows:

$\mathcal{L}_{attn}=-\frac{1}{N_{T}}\sum_{j}\log(p_{j}),\;\;\;p_{j}=\left(\sum_{i}A_{ji}\mathbb{I}_{i}(j)\right)+\mathbb{I}_{\nexists
i}(j)A_{j0}.$ (12)

$N_{T}$ represents the total number of tracks and $p_{j}$ denotes the
accumulated matching probability for track $j$, where $A_{ji}$ is the
attention weight of the respective encoder/decoder layer and
$\mathbb{I}_{i}(j)$ is 1 if the identity of the current track $j$ and the
detection $i$ are the same, and 0 otherwise. If none of the detections
matches, _i.e_., $\mathbb{I}_{\nexists i}(j)$, we maximize the no-match
probability $A_{j0}$, which is the last column of the attention matrix as
discussed in Section 3.1. In other words, we want that $A_{ji}$ is large for
the correct assignment if and only if a match exists.

The final objective function is a combination of the cross-entropy loss
function for the matching layer ($\mathcal{L}_{match}$) and the duplicate-
aware cross-entropy loss functions for each encoder and decoder layer:

$\mathcal{L}=\mathcal{L}_{match}+\sum_{k}\mathcal{L}_{attn}^{enc_{k}}+\sum_{k}\mathcal{L}_{attn}^{dec_{k}},$
(13)

where $\mathcal{L}_{attn}^{enc_{k}}$ and $\mathcal{L}_{attn}^{dec_{k}}$
represent the duplicate-aware cross-entropy loss functions for the $k$-th
encoder and decoder layer, respectively.

##  4 Experiments

###  4.1 Datasets and Evaluation

We evaluate our work on the PoseTrack datasets [1, 11]. Both datasets are
large-scale benchmarks for multi-person pose tracking and contain 593 videos
for training and 170 for evaluation. The videos contain various activities and
include highly diverse poses and severe occlusions as shown in Fig. 1. Since
for PoseTrack 2018 the evaluation server is not anymore available, we only
report results on the validation set. Compared to PoseTrack 2018, PoseTrack21
[11] provides more annotations and additional benchmarks for multi-object
tracking (MOT) and person search. We thus primarily focus on PoseTrack21 [11]
in our experiments. For evaluation, we use keypoint HOTA [11]. Keypoint HOTA
consists of sub-metrics that measure the detection accuracy (DetA), the
association accuracy (AssA) and the fragmentation accuracy (FragA). In
addition, we report results for the keypoint-based MOTA metric [1]. Both
metrics are evaluated on a keypoint level and then averaged. For completeness,
we report the keypoint detection performances in terms of mean average
precision (mAP).

We follow common practice [42, 54, 38, 55, 11, 49] and utilize a multi-frame
pose estimation approach to compensate for missed detections due to motion
blur and occlusions during inference. In particular, we utilize keypoint
correspondences as in [38, 11]. In the following, we compare our approach to
the state-of-the-art. Implementation details and detailed ablation studies are
provided as supplementary material.

###  4.2 Comparison with State-of-the-Art

Approach Online AssA FragA DetA HOTA MOTA mAP CorrTrack [11] ✓ 58.02 57.75
45.48 51.13 63.0 72.3 CorrTrack w. ReID [11] ✓ 60.21 59.66 46.56 52.71 63.8
72.7 Tracktor++ [3] w. Poses [11] ✓ 59.41 58.61 46.30 52.21 63.3 71.4
Tracktor++ [3] w. Correspondences [11] ✓ 54.05 52.02 44.67 48.90 61.6 73.6
Ours ✓ 62.20 60.93 47.20 53.94 64.1 73.6 CorrTrack [11] ✗ 60.93 60.37 45.48
52.42 63.9 72.3

Table 1: Comparison to state-of-the-art methods on the PoseTrack21 dataset.

Method AssA DetA LocA HOTA MOTA TRMOT [52] 54.98 40.91 79.92 46.85 47.2
FairMOT [60] 61.45 47.43 83.16 53.53 56.3 Tracktor++ [3] 65.43 52.71 83.09
58.29 59.5 CorrTrack + ReID [11] 64.19 51.33 82.80 56.95 52.0 Ours 66.89 51.81
82.71 58.42 55.3

Table 2: Comparison to state-of-the-art methods on the PoseTrack21-MOT
dataset.

PoseTrack21: We first evaluate our model on the PoseTrack21 validation set and
compare the performance to methods proposed in [11] using the keypoint HOTA
[11] and the MOTA metrics [1]. The results are shown in Table 1. Our proposed
pose tracker consistently outperforms existing methods, achieving a HOTA score
of 53.94 and a MOTA score of 64.1. Compared to CorrTrack W. ReID, our approach
boosts the association accuracy (AssA) and fragmentation accuracy (FragA) by
$+1.99\%$ and $+1.27\%$ to $62.20$ and $60.93$, respectively. Additionally,
the detection accuracy (DetA) and the mAP increase to $47.20$ and $73.6$,
respectively. While our approach performs online multi-person pose tracking,
it also outperforms the offline approach CorrTrack.

We further evaluate our approach on the PoseTrack21-MOT benchmark and compare
the performance to the methods in [11]. As we show in Table 2, our approach
consistently outperforms existing methods in terms of AssA (66.89) and HOTA
(58.42). Tracktor++ [3] achieves a slightly higher DetA and localization
accuracy (LocA), which also results in higher MOTA. While Tracktor++ has been
trained on the annotated bounding boxes for MOT, our approach has been trained
for pose tracking and we simply generate the bounding boxes from the estimated
poses. Consequently, MOT methods achieve a better MOTA score due to higher
bounding box detection and localization accuracy. As discussed in [32], HOTA
is a better metric than MOTA for MOT.

In the supplementary material, we provide detailed ablation studies on
PoseTrack21, which include the impact of the loss terms, encoder, confidence-
guided track updates, new track embedding head and the hyperparameters
$\alpha$, $\tau_{dup}$ and $\tau_{age}$.

PoseTrack 2018: The comparison with related works on PoseTrack 2018 is
difficult for two reasons: i) The PoseTrack 2018 dataset is not available
anymore and it is no longer possible to submit results to the official test
server; ii) the validation set was released in two different versions. The
first version (v1) contains 74 whereas the second version (v2) contains 170
sequences, respectively. For a better comparability, we grouped related works
by the version of the used validation set in Table 3. The results for
CorrTrack [38] show that the version v2 is much more difficult. Compared to
CorrTrack, our method improves the MOTA score by $0.9\%$ to $64.5$. The pose
estimation performance increases from $75.9$ to $76.4$ in terms of mAP. Our
works performs similar to [13] in terms of MOTA while achieving a higher mAP.
CombDet [49] achieves a higher accuracy, but it uses a stronger multi-frame
person detector and is an offline approach, whereas our approach is an online
approach. LDGNN [55] also uses a better multi-frame pose estimator, but the
code is not publicly available.

Market1501: For a comparison with related works on person re-identification,
we train our proposed person re-identification model SPADPE (Section 3.2) on
the Market1501 dataset [62]. Market1501 is a small dataset for person re-
identification and contains 32,688 annotated bounding boxes with a total of
1501 identities. The training and testing sets consist of 751 and 750
identities, respectively. As this dataset does not provide annotated
keypoints, we utilize our pose estimator trained on PoseTrack21 to estimate
the human pose of all person crops of Market1501. For the evaluation of the
person re-identification model, we follow [62] and report the mean average
precision (mAP) and rank-1 metrics. Since Market1501 contains pedestrians with
very limited pose variations, it is not expected that our approach achieves
state-of-the-art results on Market1501. Nevertheless, the results are on par
with the state-of-the-art as shown in Table 4. We evaluate SPADPE on
PoseTrack21 in the supplementary material.

Approach Online Val. Set Detector MOTA mAP STAF [37] ✓ v1 - 60.9 70.4 T CPN++
[57] ✓ v1 Cascade R-CNN [4] 64.0 80.9 MIPAL [22] ✓ v1 - 65.7 74.6 KeyTrack
[42] ✓ v1 HTC [7] 66.6 81.6 CorrTrack [38] ✓ v1 Cascade R-CNN [4] 68.8 79.2
TKMRNet [63] ✓ v1 Faster R-CNN FPN DCN [64] 68.9 76.7 CorrTrack [38] ✗ v1
Cascade R-CNN [4] 69.1 79.2 CorrTrack [38] ✓ v2 Cascade R-CNN [4] 63.6 75.9
LTIVA [13] ✓ v2 - 64.7 71.4 CombDet [49] ✗ v2 ResNet-101 SNIPER [41] 68.7 81.5
LDGNN [55] ✓ v2 Faster R-CNN FPN DCN [64] 69.2 77.9 Ours ✓ v2 Cascade R-CNN
[4] 64.5 76.4

Table 3: Comparison to the state-of-the-art on PoseTrack 2018 [1]. Two
versions of the validation set have been released containing 74 (v1) and 170
(v2) sequences, respectively.

Metric PSE [40] GLAD [53] PGFA[34] PCB[45] OAMN[8] HOReID[48] FED[51]
BPBreIDRN [43] PAT[28] r = 1 87.7 89.9 91.2 92.3 92.3 94.2 95.0 95.1 95.4 mAP
69.0 73.9 76.8 77.4 79.8 84.9 86.3 87.0 88.0 NGFR [58] TransReID[18] RFCNet
[21] PFD[50] FPC [56] DIP [25] BoT [33] DC-Former [27] Ours r = 1 95.5 95.0
95.2 95.5 95.1 95.6 95.4 96.0 95.8 mAP 88.1 88.2 89.2 89.7 91.4 90.3 94.2 90.4
92.8

Table 4: Comparison of SPADPE (ours) to state-of-the-art methods on Market1501
[62]. The best results are bold and the second best results are underlined.

##  5 Conclusion

We presented a novel approach for multi-person pose tracking that can be
combined with any off-the-shelf multi-person human pose estimator. Our method
employs a duplicate-aware association and dynamically weights the impact of
temporal pose similarities and appearance-based similarities based on the
attention probabilities of each similarity measure. We evaluated our approach
on the challenging PoseTrack21 dataset where our approach outperforms previous
works for multi-person pose tracking. On PoseTrack 2018, the approach is only
outperformed by methods that use a more expensive human pose estimator. We
also evaluated the proposed Spatially Adaptive Pose DEnormalization (SPAPDE)
for the task of person re-identification on PoseTrack21 and the less
challenging Market1501 dataset where our approach performs on par with the
state-of-the-art.

Appendix

##  Appendix 0.A Implementation Details

#### Track Embedding and Edge Embedding Heads:

Fig. 4 visualizes the structure of the proposed track embedding head Fig. 4a)
and the edge embedding head Fig. 4b). The new track embedding head has the
same structure as the track embedding head.

#### Re-Identification Model:

We train our re-identification model for 244 epochs on the PoseTrack21 person
search dataset with a batch size of 256 and a learning rate of $0.00035$ that
we decay to $0.000035$ after 75 epochs. During the first 10 epochs, we apply a
linear learning rate warm-up. Additionally, we apply data augmentation such as
random scaling, random rotation and horizontal flipping.

#### Dual-Source Attention Transformer:

Our proposed dual-source transformer employs two encoder and two decoder
stages. On the PoseTrack21 dataset, we train our transformer for 14 epochs
with a learning rate of $0.0001$, which is decayed by a factor of 10 after 13
epochs. We further incorporate a linear learning rate warm-up over the course
of 16k iterations and optimize the network with the AdamW [31] optimizer. Each
training sequence is split into sub-sequences of length three with an overlap
of one frame. We follow the same settings on the PoseTrack 2018 dataset and
train the dual-source transformer for 11 epochs.

#### Person Detector:

For a fair comparison to related works, we utilize the same person detector
and pose estimation model for all our experiments. In particular, we use the
person detector from [11], which consists of a FasterRCNN [39] with a
ResNet50-FPN [29] backbone. The detector was first pre-trained on MSCOCO [30]
and further fine-tuned on PoseTrack21 for 30 more epochs. As pose estimator,
we employ the released model [11], that was originally proposed in [38]. The
pose estimation model was trained on MSCOCO and PoseTrack21 for 215 and 16
epochs, respectively. On PoseTrack 2018, we use Cascade R-CNN [4] as object
detector and the pose estimator from CorrTrack [38].

Figure 4: Illustration of the a) track embedding head and b) edge embedding
head.

$\mathcal{L}_{attn}^{dec}$ Encoder $\mathcal{L}_{attn}^{enc}$ CG-Update HOTA
53.04 ✓ 53.40 ✓ ✓ 53.56 ✓ ✓ ✓ 53.61 ✓ ✓ ✓ ✓ 53.94

Table 5: Impact of several components in our pose tracking network on the
tracking performance. CG-Update denotes the Confidence-Guided Track Update as
discussed in Section 3.1.

##  Appendix 0.B Ablation Studies

We perform ablation experiments to examine the influence of each building
block of our proposed method. All experiments are conducted on the PoseTrack21
dataset.

###  0.B.1 Evaluation of the Network Architecture

#### Loss terms, Encoder and Confidence-Guided Track Update:

We first evaluate the impact of each component in our tracking model on the
keypoint HOTA score [11] in Table 5. As our base model, we start with the
dual-source decoder with matching layer that we trained with the matching loss
$\mathcal{L}_{match}$, _i.e_. (11). Subsequently, we incrementally activate
several components and evaluate their impact on the overall performance.
Direct supervision on the dual-source attention layers using
$\mathcal{L}_{attn}^{dec}$ ((12)) increases the overall performance from
$53.04$ to $53.40$ in terms of HOTA. Using additional transformer encoder
layers (Fig. 2a) ) boosts the performance to $53.56$. Using the loss
$\mathcal{L}_{attn}^{enc}$ (12) also for the encoder increases HOTA to
$53.61$. The impact of the Confidence-Guided Track Update (Section 3.1) is
shown in the last row. It further increases the tracking performance by $0.33$
to a HOTA score of $53.94$.

Re-ID Backbone SPAPDE Tepmoral Person Similarity HOTA ResNet50 & BoT IoU + OKS
53.47 ResNet50 & BoT ✓ IoU 53.59 ResNet50 & BoT ✓ OKS 53.81 ResNet50 & BoT ✓
IoU + OKS 53.94

Table 6: We evaluate the impact of different re-identification backbones and
different temporal person similarities used for the calculation of edge
embeddings.

#### Re-Identification Model and Edge Features:

To evaluate the impact of the re-identification network, we trained our
tracking model with two different re-identification networks. We used the
proposed SPAPDE network (Section 3.2) and the re-identification network [33],
which we term ResNet50 & BoT. The first and last row in Table 6 show that
adding Spatially Adaptive Pose DE-normalization layers (SPAPDE) to the network
increases HOTA from $53.47$ to $53.94$. Furthermore, Table 6 shows the impact
of different embedded pose similarities. IoU only provides coarse information
and does not allow to distinguish between spatially closely located person
instances. Even though a temporal similarity based on IoU achieves a HOTA
score of $53.59$ and outperforms all related works (Table 1), OKS-based
temporal similarity performs better and the combination performs best.

Track Embedding Initialization AssA FragA DetA HOTA New Track Embedding Head
62.20 60.93 47.20 53.94 Detection Embedding 61.87 60.73 47.15 53.76

Table 7: Impact of the New Track Embedding Head on the overall tracking
performance on PoseTrack21.

Edge Embedding Update Learnable $\alpha$ AssA FragA DetA HOTA $\alpha$-gated
attention logits 62.20 60.93 47.20 53.94 $\alpha$-gated attention logits ✓
61.80 60.59 47.15 53.74 $\alpha$-gated attention weights 54.74 50.77 47.14
50.53

Table 8: Impact of updating edge embeddings based on $\alpha$-gated attentions
weights or attention logits on the overall tracking performance on
PoseTrack21.

#### New Track Embedding Head:

Given detections in frame $t$ that can not be matched to any existing track,
we use the New Track Embedding Head (Figure 2d) to generate new track
embeddings from the unmatched detection embeddings. In Table 7, we evaluate
the impact of the _New Track Embedding Head_ compared to initializing new
tracks directly from the detection embeddings. The _New Track Embedding Head_
provides a better initialization of the track embeddings.

#### Edge Embedding Update:

To update the edge embeddings, we utilize an $\alpha$-Gate to weight the
contributions of the appearance-based and edge-based attention logits (eq.
(5)). While we evaluate the impact of $\alpha$ in Figure 5, Table 8 shows that
learning $\alpha$ does not improve the results. We further evaluate the
tracking performance when the fusion is not done at the logits, _i.e_., before
the softmax, but after the softmax as in (2), _i.e_.,

$\widehat{E}_{T\rightarrow D}^{t}=FFN_{E}\left(A_{:,:-1}\right),$ (14)

where $A_{:,:-1}$ denotes the attention weights without the last column
(Section 3.1). The last row in Table 8 shows that the performance largely
decreases from a HOTA score of $53.94$ to $50.53$ in this case.

###  0.B.2 Hyperparameter Evaluation on PoseTrack21

Figure 5: Impact of the parameters $\alpha$ and maximum track ages
$\tau_{age}$ on the tracking performance. a) visualizes the impact of $\alpha$
with $\tau_{age}=60$ and $\tau_{dup}=0.4$. b) shows the performance evaluation
with respect to the maximum track age $\tau_{age}$ with $\alpha=0.3$ and
$\tau_{dup}=0.4$.

#### Impact of $\alpha$:

We evaluate the impact of $\alpha$ in the $\alpha$-Gate ((2) and (5)) in Fig.
5a. $\alpha$ weights the contribution of the appearance-based attention
weights and the pose-based attention weights, where $\alpha=1.0$ only
considers appearance-based attention weights and, vice versa, $\alpha=0.0$
only considers edge-based attention weights. As we can observe in Fig. 5a, the
tracking performance drastically decreases for $\alpha>0.4$. Appearance-based
person features are very sensitive to persons with similar appearance as it is
common in sports videos, resulting in false associations and a high degree of
identity switches. For $0\leq\alpha\leq 0.3$, we can observe a linear increase
in the overall performance, peaking at $\alpha=0.3$, which we used in all
other experiments. This shows that pose-based and appearance-based
similarities complement each other. Pose similarities provide strong guidance
between consecutive frames, while appearance-based features allow to recover
inactive tracks, _e.g_., due to occlusion.

#### Impact of $\tau_{age}$:

We close tracks that have not been tracked for more than $\tau_{age}$ frames
and do not include them for the detection-to-track matching anymore. Fig. 5b
shows that the accuracy saturates at $\tau_{age}=40$. We used $\tau_{age}=60$
in all other experiments.

#### Impact of $\tau_{dup}$:

During tracking, we remove unmatched detections if they have a matching
confidence $m_{ij}>\tau_{dup}$ with an already matched track (Section 3.1).
Fig. 6 shows that the accuracy drops without such a threshold since duplicates
generate new tracks in this case. We use $\tau_{dup}=0.4$.

Figure 6: Evaluation of the duplicate confidence threshold $\tau_{dup}$ with
$\alpha=0.3$ and $\tau_{age}=60$.

##  Appendix 0.C Person Re-Identification

We finally evaluate the person re-identification model on PoseTrack21 and
measure the performance in terms of mean average precision (mAP), _i.e_., we
calculate the area under the Precision-Recall curve for each query and average
over all queries. As we show in Table 9, training a ResNet50 [17] with proper
data sampling results in a significant performance gain and results in a mAP
score of $68.65$. In more detail, we sample $K=6$ different instances of the
same person identity for every batch. Surprisingly, training the ResNet50 from
scratch results in a better mAP performance ($68.65$) compared to training a
ResNet50 pre-trained on ImageNet [9] ($68.31$). As discussed in Section 3.3,
we follow [33], which further increases the performance to $73.78$. To
incorporate pose information, we replace each batch normalization layer by
SPAPDE layers (eq. (9)), which incorporates pose information by the keypoint
heatmaps. The best performance is achieved by using keypoint heatmaps with a
kernel size of 10. Specifically, we denote the kernel size as the standard
deviation of a Gaussian distribution. For each keypoint we calculate a
Gaussian distribution with the mean set to the respective keypoint location
and a standard deviation of 10. Further following [33], we achieve a total
performance of $78.0$ and $74.42$ with ground truth and estimated poses,
respectively. By adding SPAPDE, mAP thus increases from $73.78$ to $74.42$.

Model GT Poses Kernel Width Sampling pre-trained mAP loss ResNet50 ✓ 64.24
triplet ResNet50 ✓ ✓ 68.31 triplet ResNet50 ✓ 68.65 triplet ResNet50 & back-
of-tricks [33] ✓ 73.38 triplet + ce + center SPAPDE ResNet50 ✓ 2 ✓ 66.77
triplet SPAPDE ResNet50 ✓ 5 ✓ 70.39 triplet SPAPDE ResNet50 ✓ 10 ✓ 71.17
triplet SPAPDE ResNet50 ✓ 15 ✓ 69.62 triplet SPAPDE ResNet50 & bag-of-tricks
[33] ✓ 10 ✓ 78.00 triplet + ce + center SPAPDE ResNet50 & bag-of-tricks [33]
10 ✓ 74.42 triplet + ce + center

Table 9: We evaluate the person re-identification performance (mAP) on the
PoseTrack21 dataset for various model settings.

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