# Critical metrics of the volume functional with pinched curvature

H. Baltazar  and  C. Queiroz  Departamento de Matemática, Universidade Federal
do Piauí
64049-550 Teresina, Piauí, Brazil<EMAIL_ADDRESS>Departamento de
Matemática, Universidade Federal do Piauí
64049-550 Teresina, Piauí, Brazil<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that a critical metric of the volume functional with
pinched Weyl curvature is isometric to a geodesic ball in $\mathbb{S}^{n}.$
Moreover, we provide a necessary and sufficient condition on the norm of the
gradient of the potential function in order to classify such critical metrics.

###### Key words and phrases:

Volume Functional; critical metrics; Weyl tensor

###### 2010 Mathematics Subject Classification:

Primary 53C25, 53C20, 53C21; Secondary 53C65

##  1\. Introduction

The investigation of critical metrics becomes a classical topic in geometric
analysis. In this sense, much attention has been given to the critical metrics
of the volume functional $V:\mathcal{M}_{\gamma}^{R}\rightarrow\mathbb{R}$,
where $\mathcal{M}_{\gamma}^{R}$ denotes the space all structures on $M$ with
constant scalar curvature and prescribed boundary metric $\gamma$. In [16],
inspired by ideas developed in [13], Miao and Tam provided sufficient
conditions for a metric $g\in\mathcal{M}_{\gamma}^{R}$ to be a critical point.
In fact, they proved that $g$ is a critical point of the functional
$V|_{\mathcal{M}_{\gamma}^{R}}$ if and only if there exist a smooth function
$f$ on $M$ such that $f|_{\partial M}=0$ and satisfies the following system of
PDE’s

(1.1) $-(\Delta f)g+Hessf-fRic=g,$

provided that the first Dirichlet eigenvalue of $(n-1)\Delta_{g}+R$ is
positive. Here, the terms $Ric$ and $Hessf$ stands for the Ricci tensor and
Hessian form associated to $g$ on $M$, respectively.

Following the terminology used in [9, 16, 17] we recall the definition of such
critical metrics investigated by Miao-Tam, which for simplicity will be called
Miao-Tam critical metrics.

######  Definition 1.

A Miao-Tam critical metric is a 3-tuple $(M^{n},\,g,\,f),$ where
$(M^{n},\,g),$ $n\geq 3,$ is a compact oriented Riemannian manifold with a
smooth boundary $\partial M$ and $f:M^{n}\rightarrow\mathbb{R}$ is a smooth
function such that $f^{-1}(0)=\partial M$ and satisfies (1.1).

There are several works about classification of Miao-Tam critical metrics
involving the Weyl curvature tensor, see for instance, [3, 4, 5, 7, 9, 17].
Hence, motivated by [12] and based in techniques developed in [1, 2], we shall
provide a classification of a such critical metrics considering a pointwise
pinching condition. More precisely, we have the following result.

######  Theorem 1.

Let $(M^{n},g,f)$ be a Miao-Tam critical metric with nonnegative scalar
curvature and satisfying

(1.2) $\left|W+\frac{\sqrt{n}}{\sqrt{2}(n-2)}\mathring{Ric}\varowedge
g\right|\leq\frac{R}{\sqrt{2(n-1)(n-2)}}.$

Then $(M^{n},g)$ is isometric to a geodesic ball in $\mathbb{S}^{n}.$

The symbol $\varowedge$, in the above expression, denotes the Kulkarni-Nomizu
product which is defined for any two symmetric $(0,2)$-tensor $S$ and $T$ as
follows

$(S\varowedge T)_{ijkl}=S_{ik}T_{jl}+S_{jl}T_{ik}-S_{il}T_{jk}-S_{jk}T_{il}.$

Since the Weyl tensor vanishes in dimension three, we obtain the following
consequence:

######  Corollary 1.

Let $(M^{3},g,f)$ be a three dimensional Miao-Tam critical metric with
nonnegative scalar curvature and satisfying

(1.3) $|\mathring{Ric}|\leq\frac{R}{\sqrt{24}}.$

Then $(M^{3},g)$ is isometric to a geodesic ball in $\mathbb{S}^{3}.$

######  Remark 1.

It is easy to verify that, the pinching (1.3) becomes in this case

$|\mathring{Ric}|\leq\frac{R}{\sqrt{24}}\leq\frac{R}{\sqrt{6}}.$

Then it suffices to apply [7, Corollary 1.5] to conclude that $(M^{3},g)$ is
isometric to a geodesic ball in $\mathbb{S}^{3}.$

In order to justify our second result will be necessary to remember the
classical simple connected examples of Miao-Tam critical metrics and a
property involving its potential function. The first one is the Euclidean ball
of radius $r_{0}$ in $\mathbb{R}^{n}$ with standard metric $g_{0}$ and
potential function

$f(x)=\frac{1}{2(n-1)}(r_{0}^{2}-|x|^{2}).$

Now, since the scalar curvature is null and $|\nabla f|_{|_{\partial
M}}=\frac{r_{0}}{n-1}$ we can deduce

$|\nabla
f|^{2}+\frac{Rf^{2}}{n(n-1)}+\frac{2f}{n-1}=\frac{r_{0}^{2}}{(n-1)^{2}}=|\nabla
f|_{|_{\partial M}}^{2}.$

At the same time, it is not hard to verify that $(\mathbb{S}^{n},g_{0}),$ the
standard sphere with canonical metric $g_{0},$ is a Miao-Tam critical metric
with potential function

$f(x)=\frac{1}{n-1}\left(\frac{cosr}{cosr_{0}}-1\right),$

where $r$ is the geodesic distance from the point $(0,\ldots,1)$ and radius
$r_{0}\neq\frac{\pi}{2}.$ Furthermore, with a straightforward computation we
obtain the same identity,

$|\nabla
f|^{2}+\frac{Rf^{2}}{n(n-1)}+\frac{2f}{n-1}=\frac{sen^{2}r_{0}}{(n-1)^{2}cos^{2}r_{0}}=|\nabla
f|_{|_{\partial M}}^{2}.$

Finally, the last interesting example is the hyperbolic space $\mathbb{H}^{n}$
embedded in the well-known Minkowski space
($\mathbb{R}^{n+1},dx_{1}^{2}+\ldots+dx_{n}^{2}-dt^{2}$),

$\mathbb{H}^{n}=\\{(x_{1},\ldots,x_{n},t)\;|\;x_{1}^{2}+\ldots
x_{n}^{2}-t^{2}=-1,\;t>0\\}.$

The geodesic ball in $\mathbb{H}^{n}$ centred at $(0,\ldots,0,1)$ is a Miao-
Tam critical metric with potential function

$f(x)=\frac{1}{n-1}\left(1-\frac{coshr}{coshr_{0}}\right).$

Similarly, the potential function must satisfy the following expression

$|\nabla
f|^{2}+\frac{Rf^{2}}{n(n-1)}+\frac{2f}{n-1}=\frac{senh^{2}r_{0}}{(n-1)^{2}cosh^{2}r_{0}}=|\nabla
f|_{|_{\partial M}}^{2}.$

It is natural to ask whether these examples are the only Miao–Tam critical
metrics that satisfy such a property. In this sense, inspired by ideas
developed by Leandro [15] we provide a complete answer to this question. More
precisely, we prove the following result.

######  Theorem 2.

Let $(M^{n},g,f),$ be a Miao-Tam critical metric satisfying

(1.4) $|\nabla f|^{2}+\frac{Rf^{2}}{n(n-1)}+\frac{2f}{n-1}=|\nabla
f|_{|_{\partial M}}^{2}.$

Then $M^{n}$ is isometric to a geodesic ball in a simply connected space form
$\mathbb{R}^{n},$ $\mathbb{H}^{n}$ or $\mathbb{S}^{n}.$

######  Remark 2.

Let us point out that, for equality (1.4) to make sense, it is necessary to
remember that a Miao-Tam critical metric has $|\nabla f|$ constant along
$\partial M,$ see [8, Section 3] for more details.

##  2\. Preliminaries

In this section we need recall some special tensors which will be important
for understanding the desired results. We start with Weyl tensor, given by the
decomposition formula

(2.1) $W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(Ric\varowedge
g)_{ijkl}+\frac{R}{2(n-1)(n-2)}(g\varowedge g)_{ijkl},$

where $R_{ijkl}$ stands for the Riemann curvature tensor. In the sequel, we
have the Cotton tensor $C$ which is given by

(2.2)
$\displaystyle{C_{ijk}=\nabla_{i}R_{jk}-\nabla_{j}R_{ik}-\frac{1}{2(n-1)}\big{(}\nabla_{i}Rg_{jk}-\nabla_{j}Rg_{ik}).}$

It is easy to check that $C_{ijk}$ is skew-symmetric in the first two indices
and trace-free in any two indices. These two tensors described above are
related as follows

(2.3) $\displaystyle{C_{ijk}=-\frac{(n-2)}{(n-3)}\nabla_{l}W_{ijkl},}$

provided $n\geq 4.$

Now, we note that the fundamental equation of a Miao-Tam critical metric can
be rewritten in the tensorial language as follows

(2.4) $-\Delta fg_{ij}+\nabla_{i}\nabla_{j}f-fR_{ij}=g_{ij}.$

Tracing (2.4) we have

(2.5) $\Delta f=-\frac{R}{n-1}f-\frac{n}{n-1}.$

Furthermore, by using (2.5) it is not difficult to check that

(2.6) $f\mathring{Ric}=\mathring{Hessf},$

where $\mathring{T}$ stands for the traceless of $T.$

Under this notation we get the following formula for a Miao-Tam critical
metric

(2.7)
$\nabla_{j}fR_{ik}-\nabla_{i}fR_{jk}=fC_{ijk}-R_{ijkl}\nabla_{l}f-\frac{R}{n-1}(\nabla_{i}fg_{jk}-\nabla_{j}fg_{ik}).$

Its proof can be found in [9].

##  3\. Key lemmas

In this section we shall deduce a couple of integral identities, which allows
us to obtain the classification of the Miao-Tam critical metrics under a
curvature estimate. Both integrals are direct consequences of the Bochner type
formulas obtained by first author and Ribeiro Jr. in [7], see Lemma 3.1 and
Lemma 3.2 for more details.

######  Lemma 1.

Let $(M^{n},g,f)$ be a Miao-Tam critical metric. Then we have

$\displaystyle\int_{M}|\mathring{Ric}|^{2}|\nabla f|^{2}dM_{g}$
$\displaystyle=$
$\displaystyle\frac{n-3}{2(n-1)}\int_{M}f^{2}|C_{ijk}|^{2}dM_{g}+\int_{M}f^{2}|\nabla
Ric|^{2}dM_{g}$
$\displaystyle+\frac{n}{n-1}\int_{M}f|\mathring{Ric}|^{2}dM_{g}+\frac{2}{n-1}\int_{M}f^{2}R|\mathring{Ric}|^{2}dM_{g}$
$\displaystyle+\int_{M}f^{2}\left(\frac{n}{n-2}tr(\mathring{Ric}^{3})-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}\right)dM_{g}$
$\displaystyle-\frac{n-2}{n-1}\int_{M}fC_{ijk}W_{ijkl}\nabla_{l}fdM_{g},$

where $\mathring{Ric}^{3}$ is the $2$-tensor defined by
$(\mathring{Ric}^{3})_{ij}=\mathring{R}_{ik}\mathring{R}_{kl}\mathring{R}_{lj}.$

###### Proof.

Using [7, Lemma 3.1] it is immediate to obtain the following integral identity

(3.1) $\displaystyle-\int_{M}\langle\nabla f^{2},\nabla|Ric|^{2}\rangle
dM_{g}$ $\displaystyle=$
$\displaystyle-\int_{M}f^{2}|C_{ijk}|^{2}dM_{g}+2\int_{M}f^{2}|\nabla
Ric|^{2}dM_{g}$
$\displaystyle+\frac{2n}{n-2}\int_{M}f^{2}tr(\mathring{Ric}^{3})dM_{g}+\frac{2}{n-1}\int
f^{2}R|\mathring{Ric}|^{2}dM_{g}$
$\displaystyle+4\int_{M}fC_{ijk}\nabla_{j}fR_{ik}dM_{g}-2\int_{M}f^{2}W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}dM_{g},$

where we have used that

$tr(Ric^{3})=tr(\mathring{Ric}^{3})+\frac{3}{n}R|\mathring{Ric}|^{2}+\frac{1}{n^{2}}R^{3}.$

On the other hand, using (2.7) and (2.1), it is not difficult to verify that

(3.2)
$\int_{M}fC_{ijk}\nabla_{j}fR_{ik}dM_{g}=\frac{n-2}{2(n-1)}\int_{M}[f^{2}|C_{ijk}|^{2}-fC_{ijk}W_{ijkl}\nabla_{l}f]dM_{g}.$

Moreover, since $M^{n}$ has constant scalar curvature we may use (2.5) to
deduce

(3.3) $\displaystyle{\rm div}(|\mathring{Ric}|^{2}\nabla f^{2})-\langle\nabla
f^{2},\nabla|Ric|^{2}\rangle$ $\displaystyle=$ $\displaystyle 2f\Delta
f|\mathring{Ric}|^{2}+2|\mathring{Ric}|^{2}|\nabla f|^{2}$ $\displaystyle=$
$\displaystyle-\frac{2}{n-1}f^{2}R|\mathring{Ric}|^{2}-\frac{2n}{n-1}f|\mathring{Ric}|^{2}+2|\mathring{Ric}|^{2}|\nabla
f|^{2}.$

As consequence, upon integrating (3.3) over $M$ we apply the divergence
theorem to arrive at

(3.4) $\displaystyle-\int_{M}\langle\nabla f^{2},\nabla|Ric|^{2}\rangle
dM_{g}$ $\displaystyle=$
$\displaystyle-\frac{2}{n-1}\int_{M}f^{2}R|\mathring{Ric}|^{2}dM_{2}-\frac{2n}{n-1}\int_{M}f|\mathring{Ric}|^{2}dM_{g}$
$\displaystyle+2\int_{M}|\mathring{Ric}|^{2}|\nabla f|^{2}dM_{g}.$

Finally, just replace (3.2) and (3.4) into (3.1) to get the desired result. ∎

Now, we use [7, Lemma 3.2], to get another formula for
$\int_{M}|\mathring{Ric}|^{2}|\nabla f|^{2}dM_{g}.$

######  Lemma 2.

Let $(M^{n},g,f)$ be a Miao-Tam critical metric. Then we have

$\displaystyle\frac{3}{2}\int_{M}|\mathring{Ric}|^{2}|\nabla f|^{2}dM_{g}$
$\displaystyle=$
$\displaystyle-\frac{1}{n-1}\int_{M}f^{2}|C_{ijk}|^{2}dM_{g}+\int_{M}f^{2}|\nabla
Ric|^{2}dM_{g}$
$\displaystyle+\frac{n}{2(n-1)}\int_{M}f|\mathring{Ric}|^{2}dM_{g}+\frac{3}{2(n-1)}\int_{M}f^{2}R|\mathring{Ric}|^{2}dM_{g}$
$\displaystyle-\frac{n-2}{n-1}\int_{M}fC_{ijk}W_{ijkl}\nabla_{l}fdM_{g}.$

###### Proof.

By direct computation using [7, Lemma 3.2], we achieve

$\displaystyle-\frac{3}{4}\int_{M}\langle\nabla f^{2},\nabla|Ric|^{2}\rangle
dM_{g}$ $\displaystyle=$
$\displaystyle-\int_{M}f^{2}|C_{ijk}|^{2}dM_{g}+\int_{M}f^{2}|\nabla
Ric|^{2}dM_{g}$
$\displaystyle-\frac{n}{n-1}\int_{M}f|\mathring{Ric}|^{2}dM_{g}+2\int_{M}fC_{ijk}\nabla_{j}fR_{ik}dM_{g}.$

To conclude, just consider the same argument which was used in the last part
of the proof of Lemma 1. ∎

##  4\. Miao Tam critical metrics with pinched curvature

In this section we will prove the Theorem 1 announced in the introduction.

###  4.1. Proof of Theorem 1

First of all, we take the difference between the expressions obtained in Lemma
1 and Lemma 2 to infer

(4.1) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{M}|\mathring{Ric}|^{2}|\nabla
f|^{2}dM_{g}+\int_{M}f^{2}|C_{ijk}|^{2}dM_{g}+\frac{n}{n-1}\int_{M}f|\mathring{Ric}|^{2}dM_{g}$
$\displaystyle+\frac{1}{n-1}\int_{M}f^{2}R|\mathring{Ric}|^{2}dM_{g}+2\int_{M}f^{2}\left(\frac{n}{n-2}tr(\mathring{Ric}^{3})-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}\right)dM_{g}.$

Before proceeding it is important to remember the following inequality

$\displaystyle\left|\frac{n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}\right|\leq\sqrt{\frac{n-2}{2(n-1)}}\left(|W|^{2}+\frac{2n}{n-2}|\mathring{Ric}|^{2}\right)^{1/2}|\mathring{Ric}|^{2},$

which is true for an arbitrary $n$-dimensional Riemannian manifold. Its proof
can be found in [1] or [14].

Hence, using the above inequality and our curvature estimate (1.2), it is
immediate to check that

$\displaystyle\frac{1}{n-1}R|\mathring{Ric}|^{2}+2\left(\frac{n}{n-2}tr(\mathring{Ric}^{3})-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}\right)$
$\displaystyle\geq\left[\frac{1}{n-1}R-2\sqrt{\frac{n-2}{2(n-1)}}\left|W+\frac{\sqrt{n}}{\sqrt{2}(n-2)}\mathring{Ric}\varowedge
g\right|\right]|\mathring{Ric}|^{2}\geq 0.$

As consequence, returning to Identity (4.1), we force $M^{n}$ to be Einstein
manifold. Then, we are in position to use Theorem 1.1 of [17] to conclude that
$(M^{n},g)$ is isometric to a geodesic ball in $\mathbb{S}^{n}.$ So, the proof
is completed.

##  5\. Proof of Theorem 2

###  5.1. Proof of Theorem 2

To start with, taking into account that $M$ has constant scalar curvature, we
derive our hypothesis on both side to infer

$\nabla_{i}\nabla_{j}f\nabla_{j}f+\frac{Rf}{n(n-1)}\nabla_{i}f+\frac{1}{n-1}\nabla_{i}f=0.$

Hence, by $(\ref{eq:tensorial})$, we have

$fR_{ij}\nabla_{j}f+\left(\Delta
f+1+\frac{Rf}{n(n-1)}+\frac{1}{n-1}\right)\nabla_{j}f=0,$

which can be written using (2.5) as

(5.1) $Ric(\nabla f)=\frac{R}{n}\nabla f.$

Here, we have used that $f$ is positive in the interior of $M$ and the
equality immediately follows by continuity.

On the other hand, using again our hypothesis we may conclude the following
expression of the laplacian on the norm of the gradient of the potential
function $f$,

(5.2) $\displaystyle\frac{1}{2}\Delta|\nabla f|^{2}$ $\displaystyle=$
$\displaystyle-\frac{R}{2n(n-1)}\Delta(f^{2})-\frac{1}{n-1}\Delta f$
$\displaystyle=$ $\displaystyle-\frac{Rf+n}{n(n-1)}\Delta
f-\frac{R}{n(n-1)}|\nabla f|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{n}(\Delta f)^{2}-\frac{R}{n(n-1)}|\nabla f|^{2}.$

Consequently, combining the classical Bochner’s formula [11]

$\frac{1}{2}\Delta|\nabla f|^{2}=Ric(\nabla f,\nabla f)+\langle\nabla\Delta
f,\nabla f\rangle+|Hessf|^{2},$

Equalities (5.2) and (2.5), we arrive at

$\displaystyle\frac{1}{n}(\Delta f)^{2}-\frac{R}{n(n-1)}|\nabla f|^{2}$
$\displaystyle=$ $\displaystyle Ric(\nabla f,\nabla
f)+\left\langle\nabla\left(\frac{-Rf-n}{n-1}\right),\nabla
f\right\rangle+|Hessf|^{2}$ $\displaystyle=$ $\displaystyle Ric(\nabla
f,\nabla f)-\frac{R}{n-1}|\nabla
f|^{2}+|\mathring{Hessf}|^{2}+\frac{1}{n}(\Delta f)^{2},$

that is,

$|\mathring{Hessf}|^{2}+\mathring{Ric}(\nabla f,\nabla f)=0.$

Finally, using (5.1), the last equality allow us to conclude that the
$\mathring{Hessf}$ is identically zero in $M^{n}.$ This implies, from (2.6),
that the $M$ is an Einstein manifold and we are in position to use again
Theorem 1.1 of [17] to conclude that $(M^{n},g)$ is isometric to a geodesic
ball in a simply connected space form $\mathbb{R}^{n},$ $\mathbb{H}^{n}$ or
$\mathbb{S}^{n}$.

######  Acknowledgement.

The first author was partially supported by PPP/FAPEPI/MCT/CNPq, Brazil
[Grant: 007/2018] and CNPq/Brazil [Grant: 422900/2021-4].

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11institutetext: University of California, Berkeley

# The One Where They Reconstructed
3D Humans and Environments in TV Shows

Georgios Pavlakos∗ Ethan Weber∗ Matthew Tancik  Angjoo Kanazawa

###### Abstract

TV shows depict a wide variety of human behaviors and have been studied
extensively for their potential to be a rich source of data for many
applications. However, the majority of the existing work focuses on 2D
recognition tasks. In this paper, we make the observation that there is a
certain persistence in TV shows, i.e., repetition of the environments and the
humans, which makes possible the 3D reconstruction of this content. Building
on this insight, we propose an automatic approach that operates on an entire
season of a TV show and aggregates information in 3D; we build a 3D model of
the environment, compute camera information, static 3D scene structure and
body scale information. Then, we demonstrate how this information acts as rich
3D context that can guide and improve the recovery of 3D human pose and
position in these environments. Moreover, we show that reasoning about humans
and their environment in 3D enables a broad range of downstream applications:
re-identification, gaze estimation, cinematography and image editing. We apply
our approach on environments from seven iconic TV shows and perform an
extensive evaluation of the proposed system.

Figure 1: Reconstruction of humans in TV show environments. Given images
across the whole season of a TV show, we present an approach that recovers the
3D scene context, which enables accurate estimation of every actor’s 3D pose
and location. We show the input (left), the mesh reconstructions of the actors
in the camera view (center) and in a novel view (right). Human meshes are
visualized against the reconstructed scene, which is represented by a Neural
Radiance Field (NeRF). To appreciate the correct 3D localization of people,
notice the position in the novel view and the occlusions. Readers are
encouraged to watch video results in the project page:
http://ethanweber.me/sitcoms3D/. 11footnotetext: Equal contribution.

##  1 Introduction

Remember that time when you binge-watched an entire season of your favorite TV
show, e.g., “Friends”, over a weekend? After that experience, you would know
the layout of the rooms, the locations of the furniture, and even the relative
height of the characters as they interact closely on screen. As a result, for
any frame, you could tell where the room is viewed from, where the characters
are situated, and how they relate to the rest of the scene, even the parts of
the scene outside the frame. Essentially, as viewers, we aggregate all the
visual information into a dynamic 3D world where the new observations are
aligned to.

In this paper, we propose a method that can similarly aggregate 3D information
over video collections and use it to perceive accurate 3D human pose and
location of the actors. Although reconstruction of dynamic scenes is
challenging from a single video clip, our insight is that in the context of TV
shows, across many episodes, there are many video clips that depict the same
scene and people many times. The repeated observations provide a strong multi-
view signal of the underlying scene, enabling reconstruction of the camera and
the dense structure. These serve as context to accurately recover the 3D pose
and location of the people in the 3D environment. A representative result is
shown in Figure 1. Although we demonstrate our method & results on TV shows,
our insight is also applicable to other domains with repetition in the
environment and the people, e.g., sports [20, 47, 71], late night shows [15,
40] and movies [45].

We operationalize our insight by focusing on an entire season from TV shows
and collecting the sequences that correspond to a specific environment. These
sequences are organized in shots [3], which are typically captured by
different cameras. To collect a diverse set of images, we sample frames at the
shot boundaries (cuts between cameras). This ensures a wide variety of
viewpoints, while avoiding redundancy, making it practical to apply a
structure-from-motion pipeline [53] to estimate the intrinsic and extrinsic
camera parameters (calibration). Then we use a neural radiance field (NeRF-W
[36]) to disentangle the static and transient components and obtain a dense 3D
reconstruction (Figure 2, first block).

Figure 2: Overview of our workflow. First, we use a collection of videos from
a TV show environment and reconstruct the 3D scene (cameras and dense
structure). We then use this information to recover accurate 3D pose and
location of people over shot boundaries and on monocular frames. The recovered
3D information is immediately useful for various downstream applications.

The 3D scene reconstruction offers rich 3D context - cameras and scene
structure - enabling an in-depth study of humans (Figure 2, second block).
First, for frames on the shot boundaries, the viewpoints from the two
different shots act as effective multi-view (or multi-shot) information for
human reconstruction [45]. We use the calibrated cameras and propose a multi-
shot human reconstruction method, which jointly solves for body pose, body
shape, identity and location. In this calibrated multi-shot method, camera
information enables triangulation of people, which removes ambiguity and
provides significant improvement upon the equivalent uncalibrated baseline
[45]. Next, human reconstructions on the shot boundary inform us of the scale
of each person relative to the scene. This is additional 3D context that is
complementary to the cameras and scene structure. Since most frames are not on
shot boundaries, we also formulate a monocular human reconstruction method
that is explicitly guided by the extracted 3D context (camera, structure, body
scale). The successful integration of the 3D context in our contextual
monocular method leads to improvements over the state-of-the-art monocular
baselines.

Our proposed “3Dification” of TV shows opens the door to many immediate
applications (Figure 2, third block). First, our human reconstruction on the
shot boundaries associates person detections, by incorporating geometric and
anthropometric constraints. We show that this form of re-identification
consistently outperforms traditional image-based baselines [12, 22, 23]. In
parallel, from our reconstructed humans we can extract reliable gaze
information, which can outperform specialized gaze estimation [46]. Moreover,
our results provide estimates of the camera-to-person distance, which is
relevant for cinematography applications [50, 51]. Finally, we illustrate the
potential use in image editing applications, like object insertion or human
deletion.

In summary, our contributions can be summarized as follows:

  * •

We identify the significant amount of 3D context (cameras, structure and body
shape) in domains with repetition in the environment and the people, e.g., TV
shows, and propose a method to aggregate it from video sequences.

  * •

We propose a formulation that integrates this context in 3D human estimation
methods, which improves human reconstruction.

  * •

We demonstrate how the aggregated 3D information can help a wide variety of
downstream tasks: re-ID, gaze estimation, cinematography, image editing.

  * •

We perform extensive qualitative and quantitative evaluation to validate the
quality of our recovered 3D results.

##  2 Related work

###  2.1 Perceiving TV shows

The computer vision community has a long history of works on perceiving TV
shows/movies. One of the most common tasks in this setting is studying the
show characters with emphasis in face/character identification [2, 9, 39, 43,
57, 58], where different cues have also been explored, e.g., body, voice or
gaze [6, 8, 34, 35]. TV show data has been used extensively to study human
behavior. Ferrari et al. study 2D pose estimation [10] and perform pose-based
analysis [11]. Patron et al. [44] and Hoai et al. [19] focus on human
interactions, while Recasens et al. [46] and Marín-Jiménez et al. [34] use
this data to study gaze. Vondrick et al. [60] use sequences from TV shows to
learn activity forecasting, while Wang et al. [61] leverage it for affordance
learning. Despite this attention, all the above methods reason in 2D, with
only a few exceptions. Everingham and Zisserman [9] use a 3D head model for
re-identification. Here, we demonstrate how 3D location information can
significantly simplify the re-ID problem. Pavlakos et al. [45] reconstruct
humans from videos with multiple shots. However, they operate without camera
calibration, while we show the importance of recovering reliable cameras.

Figure 3: Reconstruction challenges of TV shows include transient and dynamic
objects as well as appearance changes.

###  2.2 Scene reconstruction

Reconstruction of 3D scenes is a well studied problem, e.g., [1, 21, 53, 54],
however, most methods assume static scenes. Related work focuses on dynamic
reconstruction [4, 38], but requires capture from multiple wide-baseline
synchronized cameras. Luo et al. [33] and Kopf et al. [29] present pipelines
for recovering depth in monocular videos that include humans, but they assume
that the underlying scene is static. View synthesis approaches like NeRF [37]
and follow-ups [41, 66] can be used to solve multi-view stereo, however these
also assume static scenes. Other extensions of NeRF focus on reconstructing 3D
motion in the scene [13, 30, 42], but are often limited when handling changes
in appearances and transient objects. NeRF in the wild [36] is the most
relevant approach for the type of data we use (Figure 3), since it can deal
with appearance and transient changes. In this paper, we find that when our
data is properly curated we can use NeRF-W to recover dense 3D structure. We
then show that this structure can be used to guide consistent 3D human
reconstruction.

###  2.3 Humans in 3D scenes

Most works that reconstruct humans in context with the scene assume a static,
pre-captured 3D environment. Savva et al. [52] is one of the first works that
explore 3D human-scene interactions from RGB-D video, while Hassan et al. [17]
study the recovery of 3D humans in context with their environment from
monocular images. Many works incorporate environmental constraints for motion
estimation from videos [48, 49, 55, 56, 63, 67, 69], by assuming known floor
or contact points. Recently, Guzon et al. [16] proposed a system for
localizing a person in a known environment and estimating their 3D pose.
Again, the environment is reconstructed a priori and the approach also
requires an egocentric sensor and IMUs for pose estimation. Liu et al. [31]
propose a method that reconstructs the scene and the people together using
egocentric video captured in static outdoor scenes. In this work, we
reconstruct structure from much more challenging dynamic scenes, by
aggregating 3D information over video content.

Some works [62, 68] have studied human reconstruction from single images,
while also recovering aspects of the environment. PHOSA [68] recovers humans
interacting with objects from in-the-wild images, and is followed by [62, 64]
in other settings. While they focus on visible human-object interactions, we
consider cases where the scene might not be fully visible. Knowing camera
parameters is an integral part of scene perception. SPEC [27] regresses camera
parameters from a single image. In contrast, we can recover more reliable
context for cameras by leveraging the whole collection of images from a TV
show environment.

##  3 Technical approach

For the following discussion, we use the term environment to refer to a
location, i.e., a room, kitchen, cafe, etc., that appears often in a TV show.
Figure 4 visualizes the panoramic view of the environments we reconstruct in
this paper. We use the term shot for an uninterrupted sequence captured by a
camera. Shots are organized in scenes, which are typically captured in the
same environment. Multiple scenes comprise an episode and multiple episodes
are organized into a season. In this work we collect videos across the whole
season of a TV show.

###  3.1 Camera estimation

For the first step of our workflow, we need to register the cameras in a
common coordinate frame (i.e., computing intrinsics and extrinsics) for each
environment. This amounts to hundreds of thousands of frames across the
season. To keep the number of frames at a practical scale for Structure-from-
Motion (SfM) pipelines, we sample frames at shot boundaries, which are
automatically detected [23]. This helps to increase the variety of viewpoints
- we only use two frames per shot, and inter-shot variety is typically larger
than intra-shot variety.

On this reduced set of frames, we use DISK [59] to find correspondences. Since
our data includes dynamic actors, we run Mask R-CNN [18] to detect human
masks, and we reject correspondences on these regions. We use COLMAP [53] on
the remaining feature matches and estimate the sparse 3D reconstruction and
camera registration. We use a simple pinhole camera model, and allow each
camera to have different focal length. For each frame $t$ we get estimates of
camera intrinsics $K_{t}\in\mathbb{R}^{3\times 3}$ and extrinsics
$R^{{\scaleto{CW}{3.7pt}}}_{t}\in\mathbb{R}^{3\times
3},T^{{\scaleto{CW}{3.7pt}}}_{t}\in\mathbb{R}^{3}$, where $CW$ denotes camera
to world transformation. This sparse reconstruction is used to register other
frames (non shot-boundary images). Since we do not have access to 3D ground
truth for TV show environments, the quality of our cameras is evaluated
implicitly by the effect it has on the human reconstruction (see also Sup.
Mat.).

Figure 4: Panoramic views of the reconstructed TV show environments. We obtain
and render the static structure using NeRF-W [36]. The environments represent
seven TV shows: “The Big Bang Theory”, “Frasier”, “Everybody Loves Raymond”,
“Friends”, “Two And A Half Men”, “Seinfeld” and “How I Met Your Mother”.

###  3.2 Dense structure

Besides the camera registration returned from SfM, we also estimate the dense
structure of the environment to help with human position estimation.
Traditional dense reconstruction methods assume static scenes, but these
assumptions are not satisfied in TV show environments which contain many
images of extreme diversity (Figure 3). Instead, we use a NeRF-W network [36]
for dense structure estimation. NeRF-W extends NeRF [37], to account for
varying appearances and transient occluders. For efficiency, instead of
training NeRF-W with all images, we use an automatic selection method to
maximize viewpoint variety. We cluster the images based on camera location and
viewing direction. For each cluster we select the image with least percent of
Mask R-CNN human pixels to use for training (i.e., maximum number of scene
rays). After training, NeRF-W returns a volumetric 3D representation of the
static structure of the scene.

Figure 5: Calibrated cameras for scale estimation and identity association.
Given calibrated cameras, we can use frames at a shot change to solve for the
actors’ pose, location, relative scale and association. The four overlapping
regions (left) indicate possible locations triangulated by the cameras.
Circles indicate correct matches after Hungarian matching. Reconstructed
humans are visualized in a NeRF (right).

###  3.3 Calibrated multi-shot human reconstruction

In movies and TV shows, scenes are filmed in consecutive shots. The shot
changes within a scene correspond to consecutive time frames seen by different
viewpoints. This serves as effective multi-view information, providing signal
to recover the 3D location and pose of the actors [45]. However, doing so
requires knowledge of the identity of the actors across the shot changes.
Prior work utilizes a pre-trained recognition-based re-ID model to establish
these correspondences, but this is not always reliable, for example when only
the back of the character is visible. We make an observation that when camera
information is available, the association can be solved jointly with the 3D
human pose, shape, and location. We refer to this approach as calibrated
multi-shot optimization.

Let us assume there are $M$ actors in frame $t$, $N$ actors in frame $t+1$,
and a shot change happens from frame $t$ to $t+1$. We need to solve a matching
problem to associate the two sets of actors. We propose to use the objective
of SMPLify fitting [5] to model the cost for this matching.

Formally, let us consider a detection of a person at time instance $t$, with
detected 2D keypoints $J_{est,t}$ [7]. We denote with $\theta_{t}$ the pose
parameters and with $\beta_{t}$ the shape parameters of the person in the SMPL
format [32]. We use $J_{t}$ for the joints and $T^{{\scaleto{C}{3.7pt}}}_{t}$
for the translation of the body in the camera frame. Moreover, from SfM, we
have access to the transformations
$R^{{\scaleto{CW}{3.7pt}}}_{t},T^{{\scaleto{CW}{3.7pt}}}_{t}$ from the camera
frame to the world frame at time $t$. Given all of the above, we minimize the
objective function with respect to
$\\{\theta_{t},\theta_{t+1},\beta_{t},\beta_{t+1},T^{{\scaleto{C}{3.7pt}}}_{t},T^{{\scaleto{C}{3.7pt}}}_{t+1}\\}$:

$E=\underbrace{E_{J_{t}}+E_{J_{t+1}}}_{\textrm{2D
reprojection}}+\underbrace{E_{{\textrm{priors}}_{t}}+E_{{\textrm{priors}}_{t+1}}}_{\textrm{anthropometric
constraints}}+\underbrace{E_{{\textrm{glob}}_{t,t+1}}}_{\textrm{3D
consistency}}$ (1)

Here, $E_{J_{t}}=E_{J_{t}}(\beta_{t},\theta_{t},K_{t},J_{est,t})$ is the
joints reprojection term and $E_{\textrm{priors}}$ are anthropometric priors
similar to [5]. The key constraint is multi-shot consistency, which encourages
the estimated bodies to be similar in the global frame:

$E_{\textrm{glob}_{t,t+1}}=\|(R^{{\scaleto{CW}{3.7pt}}}_{t}J^{{\scaleto{C}{3.7pt}}}_{t}+T^{{\scaleto{CW}{3.7pt}}}_{t})-(R^{{\scaleto{CW}{3.7pt}}}_{t+1}J^{{\scaleto{C}{3.7pt}}}_{t+1}+T^{{\scaleto{CW}{3.7pt}}}_{t+1})\|^{2}.$
(2)

In contrast to prior work, we do not need to solve for the camera as we have
access to reliable extrinsics and intrinsics (prior works [25, 45, 68] use a
heuristic for focal lengths). This leads to more accurate human placement and
constraints that allow for solving associations. Using this fitting cost $E$,
we solve association by Hungarian matching. See Figure 5 for an illustration
of this optimization.

Figure 6: Contextual monocular human reconstruction. For an input frame, we
can leverage (a) the body shape (scale) of the person from a neighboring shot
change, (b) the camera registration, and (c) the static structure of the
environment. This enables monocular reconstruction of the person in context
with their environment.

###  3.4 Contextual monocular human reconstruction

Although shot changes provide effective multi-view information for free, the
majority of the frames in the video only have monocular observations.
Monocular human reconstruction is challenging, particularly so for TV shows
with many close-up shots; however, in our case, we can capitalize on the
contextual information we have recovered. In this subsection, we explain how
we can make use of this 3D context in an effective way. We demonstrate this
using a single-frame optimization approach, SMPLify [5], but other methods
could also benefit from our context, e.g., we show representative results for
HuMoR [48] in the Sup. Mat.

A high-level overview of this step is presented in Figure 6. First, given the
sparse reconstruction of the environment, we can register the camera for a new
frame. This gives us both extrinsics $R^{{\scaleto{CW}{3.7pt}}}_{t}$,
$T^{{\scaleto{CW}{3.7pt}}}_{t}$ and intrinsics $K_{t}$ for the camera via
solving PnP with COLMAP [53]. We leverage these parameters for accurate
projection. Moreover, we can employ the structure captured by our NeRF-W
network. In general, it is not trivial to extract the structure from NeRF [41,
66]. The native representation used by NeRF is in the form of densities for
each point. Here, we propose to use this density as a proxy for occupancy of
the 3D space. With this in mind, we formulate an objective to discourage the
human body vertices $V$ from occupying areas with high density values:

$E_{{{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\textbf{{structure}}}}}=\rho\Bigl{(}\sum_{v\in
V}{{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\textbf{$\tilde{\sigma}$}}}(v)\Bigr{)},$
(3)

where $\tilde{\sigma}$ samples values from the density field $\sigma$ using
trilinear interpolation, while $\rho$ is the Geman-McClure robust error
function [14]. Finally, we leverage the shape parameters $\hat{\beta}$ that
capture the relative scale of the person with respect to the environment, and
are recovered from the nearest shot change with the calibrated multi-shot
reconstruction. This value can be used explicitly in the optimization to
resolve the scale ambiguity.

Eventually, our monocular fitting objective minimizes:

$E_{J}(\beta={{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\textbf{$\hat{\beta}$}}},\theta,K={{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{$\hat{K}$}}},J_{est})+E_{\textrm{priors}}+E_{{{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\textbf{{structure}}}}},$
(4)

with respect to ${\theta_{t},T_{t}^{{\scaleto{C}{3.7pt}}}}$, where we employ
the camera information and the body shape parameters of the person during the
fitting, while also discouraging the body mesh from penetrating the static
structure of the scene.

###  3.5 Applications

An important argument in favor of 3D reconstruction for people in TV show
environments is that it can simplify many reasoning tasks in this domain. For
example, the calibrated multi-shot optimization explicitly reasons about the
identity of the detected humans, as part of the Hungarian matching. This
enables reliable re-identification in the challenging case of shot changes
where the viewpoint can change significantly (Section 4.3). Moreover, one can
extract gaze information from our 3D humans by considering the 3D pose of the
face/head. With knowledge of camera pose, we can easily estimate the gaze
direction in the global space, and thus compute gaze across the shot change
(Section 4.4). Finally, we perform an analysis of our data which could be
useful for cinematography applications, and highlight the potential of image
editing using our results (Section 4.5).

##  4 Experiments

In this section, we present the quantitative and qualitative evaluation of our
approach. We use seven popular TV shows (Figure 4) and one season from each.
We follow the procedure described in Section 3 to collect the images we use.
Each environment has 1k-5k frames from shot changes. For evaluation, we select
per TV show a set of 50 person identities present on these shot changes. We
use these frames as a test set to evaluate our method qualitatively with a
crowd-sourced perceptual evaluation on AMT and curate it with the information
we require for quantitative evaluation, i.e., human-human associations, body
keypoints, top-down location of the pelvis in the scene and gaze target across
the shot change.

Figure 7: Calibrated multi-shot and re-ID results. Using input shot changes
(left), we perform our calibrated multi-shot optimization which jointly solves
for pose, shape, location and association (middle). Note that identity,
illustrated with colors, is not available a priori, but is estimated jointly
with the 3D reconstruction. The recovered humans can be rendered in novel
views using the NeRF of the environment (right).

###  4.1 Calibrated multi-shot human reconstruction

For a proof of concept, we first evaluate our proposed calibrated multi-shot
optimization in a controlled setting, with the Human3.6M dataset [24], where
we have accurate 3D ground truth for pose. Since our focus is on the effect of
having access to camera parameters, we compare with the equivalent
uncalibrated baseline, which is similar to [45]. The results are presented in
Table 1. The significant improvement when having access to camera information
further motivates the importance of our calibrated multi-shot algorithm.

For our data from TV shows, we do not have access to 3D ground truth for
humans, so we perform two evaluations for the human reconstructions. First, we
perform a system evaluation by Amazon Mechanical Turk (AMT) workers. For each
3D human reconstruction, we task the annotators to select the rendered result
video (our method vs. a baseline) where the human reconstruction is more
accurate and consistent with the scene and shot boundary images. Each result
video is 10 seconds and provides multiple viewpoints of the person in the
scene. We test on our test set, resulting in 2100 human labels from 48
participants who went through quality control (please see Sup. Mat. for more
details). We report the percent of choices where our method is preferred over
the baselines in Table 1. The uncalibrated baseline (first row; without
intrinsics or extrinsics) is very rarely preferred over our calibrated
baseline (last row; with estimated intrinsics and extrinsics). Having access
to estimated intrinsics can help with localization (middle row), but it is
still preferred only 35% of the time. Besides the crowd-sourced evaluation, we
also evaluate the location of each person quantitatively. In this case, we
compute the mean metric distance error for the pelvis joint in the top-down
projection, which is reported in Table 1. The conclusions are consistent with
the AMT evaluation, highlighting the importance of camera information. Some
representative results of our calibrated multi-shot optimization are presented
in Figure 7, where we also indicate the estimated identity association (which
we evaluate in more detail in Section 4.3).

Figure 8: Results for the contextual monocular reconstruction. We ablate the basic components of the contextual reconstruction to demonstrate their effects. Our method uses all three forms of context. Without our estimated camera intrinsics (left) and without body shape (middle), the person is incorrectly placed in the scene due to scale ambiguity. Using structure (right) avoids interpenetration with the environment. Method | Camera information | Human3.6M | TV shows
---|---|---|---
Multi-shot | Intrinsics | Extrinsics | MPJPE | PA-MPJPE | % preferred | Distance
optimization | vs. Ours $\uparrow$ | error $\downarrow$
Uncalibrated [45] | ✗ | ✗ | 131.9 | 56.9 | 4% | 889cm
Partial Calibration | ✓ | ✗ | 123.8 | 56.3 | 35% | 59cm
Calibrated | ✓ | ✓ | 65.8 | 47.1 | — | 38cm
Table 1: Evaluation of the proposed calibrated multi-shot optimization. We
ablate the effect of camera information in multi-shot optimization. On
Human3.6M, we report results on the standard 3D pose metrics in mm [70]. On
our TV show data, we perform a system evaluation on AMT and provide
quantitative results based on the spatial localization of the reconstructed
person in the scene.

###  4.2 Monocular contextual human reconstruction

Next, we investigate our proposed contextual monocular reconstruction. For
this evaluation, we study the effect of each component separately – knowledge
of the cameras, access to the person’s body shape, and finally scene structure
information. We present the results of this ablation in Table 2, where we
report cross-shot PCK @ $\alpha=0.5$ [45]. Effectively, we project the person
to the view across the shot boundary and measure localization accuracy for the
joints in that space (more details in the Sup. Mat.). First, we see that
state-of-the-art monocular methods without context [5, 26, 28] perform
similarly on this data. Then, we examine the effect of context, using the
optimization baseline [5] as our starting point (third row). Access to camera
intrinsics is important to estimate a rough location of the person, and
without it the method performs as the baseline without context. Knowledge of
the body scale of the person, can make our estimate even more accurate.
Finally, structure gives a smaller quantitative improvement but has a more
pronounced qualitative effect by placing the person coherently in the
environment. See Figure 8 for qualitative results.

Method | cross-shot PCK
---|---
No context: ProHMR [28] | 14.7%
No context: PARE [26] | 14.2%
No context: SMPLify [5] | 16.5%
Context w/o camera (intrinsics) | 16.0%
Context w/o body shape (scale) | 65.9%
Context w/o structure | 87.5%
Context (full) | 88.7%
Table 2: Ablation of the main components of our contextual reconstruction. Cross-shot PCK @ $\alpha=0.5$ is reported. Knowledge of the camera focal length is very important to get a good 3D location for the human. Information about body shape can have significant improvements, as it resolves the scale ambiguity. Structure helps to avoid the incoherent interpenetrations with the scene. Matching costs | Re-ID F1 $\uparrow$
---|---
Fu et al. [12] (Appearance) | 0.78
Huang et al. [22] (Appearance) | 0.79
Huang et al. [23] (Appearance) | 0.80
Keypoint triangulation (Geometry) | 0.86
Ours (Geometry + Anthropometric) | 0.91
Table 3: Re-ID results for actors in shot boundary frames. We use different
methods to estimate matching costs for detections and we run Hungarian
matching to establish associations. A geometric baseline using the
reprojection error from person keypoint triangulation improves upon SOTA
image-based baselines [12, 22, 23], but using our multi-shot fitting cost
performs better because it also includes anthropometric constraints, i.e., the
triangulated points should respect the human body priors.

###  4.3 Re-identification

For the re-ID evaluation, we examine the challenging case of person
association after a shot change. For our case, re-ID is directly estimated
from our calibrated multi-shot optimization. We compare this result with two
types of baselines for computing affinities/costs between instances for
Hungarian matching. The first type is image-based re-ID networks for affinity
estimation, where [12] achieves SOTA on standard re-ID benchmarks, while [22,
23] are trained on movies, a source of data similar to TV shows. The second
type is a geometric baseline that uses our recovered cameras and is based on
human keypoint triangulation, where the reprojection error is used as the cost
for the Hungarian algorithm. Notice, that unlike SMPL fitting, this does not
incorporate anthropometric constraints, i.e., it considers every keypoint
match independently, without using human body shape priors or measuring the
holistic result. We report re-ID F1 scores in Table 3 using the visible pairs
of actors before/after the shot change. Based on the results, our re-
identification can consistently outperform these baselines.

Method | PCGD ($\alpha=20^{o}$) $\uparrow$
---|---
all | w/ face
Recasens et al. [46] | 16% | 32%
Ours | 62% | 67%
Table 4: Gaze following results. We report the Percentage of Correct Gaze
Directions (see text for description). Our approach outperforms the baseline
of [46].

###  4.4 Gaze estimation

For gaze estimation, we compare with the method of [46] that estimates the
gaze target after the shot change. We evaluate the angular error in the gaze
direction projected on the image plane. We report the Percentage of Correct
Gaze Directions (similar to PCK [65]), using $\alpha=20^{o}$ as threshold.
Please see Sup. Mat. for details.

Results are reported in Table 4. Since [46] relies on face detection, we
report results on our whole test set (column “all”) and on the subset where
face detection is successful (column “w/ face”). Our approach outperforms [46]
in both cases. Since we rely on body detection, we are more robust even when
the face is occluded. Moreover, our extracted camera poses allow us to follow
gaze across shots in a more accurate way. Further improvements are expected by
modeling eye pose and saliency estimation to detect the gaze target, similarly
to [46]

###  4.5 Cinematography/Image editing applications

We provide an initial analysis of our results in Figure 9, and present an
extended study in the Sup. Mat. First, we visualize the distribution of the
estimated field of view for the cameras. Here we can see the long-tail
distribution for the views with a large field of view, i.e., more informative
viewpoints for 3D reconstruction. This justifies our insight to process data
across the whole season, since the large majority of the views is typically
close-ups. Moreover, we visualize the locations of the cameras and the human
actors. The camera data could be useful for cinematography analysis [51], and
the person data for behavior or affordance analysis [61]. Finally, we
illustrate potential editing applications enabled by the reconstruction of
humans and the environment: person removal and object insertion. More editing
options are possible, given the 3D nature of our processing.

Figure 9: Cinematography applications/Image editing. We present analysis of
our processed data, including distribution of field of view, camera pose
distribution and person location distribution for Friends (top). Moreover, we
present editing options after our processing, including person removal and
object insertion (bottom).

##  5 Discussion

Conclusion: To the best of our knowledge, we are the first to reconstruct the
people and the environment in TV shows and reason about them in 3D. We start
with multi-shot video sequences associated with a specific environment and
recover the camera, structure and relative human scale. We use this
information as context to reconstruct humans even from a single frame, in a
way that is consistent with their environment. We demonstrate our approach on
seven different TV shows and present qualitative and quantitative results, as
well as a wide variety of applications and analysis of the reconstructed data.

Future Directions: Our work has only scratched the surface of this extremely
challenging and in-depth problem. Currently, we do not reconstruct the
transient objects or dynamic objects that humans interact with (e.g., chairs
that move around, fridge opening). Also, the recovered pose of the humans is
completely dependent on the quality of the 2D keypoint detections. It would be
an interesting direction to incorporate appearance models for pose fitting.

Acknowledgements: This research was supported by the DARPA Machine Common
Sense program as well as BAIR/BDD sponsors. Matthew Tancik is supported by the
NSF GRFP.

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# Counting ideals in numerical semigroups

Parth Chavan  ID 1Department of Mathematics, Massachusetts Institute of
Technology, Cambridge, MA, USA<EMAIL_ADDRESS>
###### Abstract.

We calculate the ideal generating function counting ideals in a numerical
semigroup with respect to their codimension for numerical semigroups with
embedding dimension $2$ and certain families of numerical semigroups with
embedding dimension $3.$ We also show that the degree of the ideal generating
function of a numerical semigroup is its Frobenius number.

###### Key words and phrases:

Numerical semigroup, Frobenius number, Ideal generating function

###### 1991 Mathematics Subject Classification:

20M14, 05A19.

The author was supported by The 2022 Spirit of Ramanujan Fellowship and The
2022 Mehta Fellowship

##  1\. Introduction

Let $\mathbb{N}$ denote the set of non-negative integers and let
$\mathbb{N}_{\geq a}=\\{i\in\mathbb{N},i\geq a\\}$. A _numerical semigroup_
$S$ is an additive submonoid of $(\mathbb{N},+)$ such that
$|\mathbb{N}\setminus S|$ is finite. Given a nonempty finite subset $A$ of
$\mathbb{N}$, denote by $\langle A\rangle$ the submonoid of $(\mathbb{N},+)$
generated by elements in $A$, that is,

$\langle A\rangle=\\{a_{1}x_{1}+\ldots+a_{n}x_{n}\mid a_{1},\ldots,a_{n}\in
A,x_{1},\ldots,x_{n}\in\mathbb{N}\\}.$

If $\langle A\rangle=S$, we call $A$ the _generating set_ and elements of $A$
the _generators_ of $S$. Note that $\langle A\rangle$ is a numerical semigroup
if and only if the elements in $A$ are coprime. Moreover, it is well known
that any numerical semigroup $S$ has a unique minimal set of generators with
respect to containment [9]. If $a_{1},a_{2},\ldots,a_{k}$ is a minimal set of
generators, we say that $S$ has _embedding dimension_ $e(S)=k$.

A famous problem in additive number theory is to determine the largest number
$G(S)\notin S$ in terms of the minimal generating set of $S$. $G(S)$ is called
the _Frobenius number_ of $S$. The problem of computing $G(S)$ is known as the
Frobenius problem. This problem has been also known as the _Coin Exchange
Problem_ , _Postage Stamp Problem_ or _Chicken McNugget Problem_ , and has a
long history.

For $|S|=2$, Sylvester [11] obtained $G(a_{1},a_{2})=a_{1}a_{2}-a_{1}-a_{2}$
in 1882. For $|S|=3$, the problem has been studied and efficient algorithms
have been given by G. Denham [4] and A. Tripathi [12]. Moreover, many formulas
for special cases have been determined; see [10, 7] for more details. In the
general case $|S|\geqslant 3$, Curtis showed that $G(S)$ can not be given by
polynomial formulae in terms of generators [3]. In fact, computing the
Frobenius number of numerical semigroups with more than three generators is
NP-hard [8]. For a detailed overview and developments related to the Frobenius
problem, we refer the reader to [7].

In this article, we are interested in enumerating subsets of $S$ which have an
additive structure over $S$: a non-empty set $I\subset S$ is called an _ideal_
of $S$ if $I+S\subseteq I$. For an ideal $I\subseteq S$, the _codimension_ of
$I$ is defined as $\operatorname{codim}(I)=|S\setminus I|$. Denote by
$I(a_{1},\ldots,a_{r})$ the set of all ideals of $\langle
a_{1},\ldots,a_{r}\rangle$.

######  Definition 1.1.

We define the _ideal generating function_ corresponding to the numerical
semigroup $\langle a_{1},\ldots,a_{r}\rangle$ by

$I(a_{1},\ldots,a_{r};q)=\sum_{I\,\in\,I(\langle
a_{1},\ldots,a_{r}\rangle)}q^{\operatorname{codim}(I)}.$

Let $R=k[t^{a_{1}},\ldots,t^{a_{r}}]$ be the _semigroup ring_ over a field $k$
associated with the semigroup $\langle a_{1},\ldots,a_{r}\rangle$ and let
$J(a_{1},\ldots,a_{r})$ be the set of all monomial ideals in this ring. The
ideal generating function can alternatively be defined as

(1.1) $I(a_{1},\ldots,a_{r})=\sum_{J\in
J(a_{1},\ldots,a_{r})}q^{\dim_{k}(R/J)}$

For a rational function $\frac{p(x)}{q(x)}\in\mathbb{R}(x)$, we define its
degree as $\deg(p(x))-\deg(q(x))$. It turns out that the ideal generating
function is related to the Frobenius number as described below.

######  Theorem 1.2.

The ideal generating function of a numerical semigroup $S=\langle
a_{1},\ldots,a_{r}\rangle$ can be written as

$I(a_{1},\ldots,a_{r};q)=\frac{f(q)}{1-q},$

where $f(q)\in\mathbb{Z}[q]$. Moreover, we also have

$G(S)=\deg\left(I(a_{1},\ldots,a_{r};q)\right)=\deg\left(f(q)\right)-1.$

##  2\. Ideal generating function and Frobenius number

In this section we prove Theorem 1.2. Given a numerical semigroup $S$, its
_gap-invariant_ is defined to be $\Delta(S)=|\mathbb{N}\setminus S|$. Denote
by $s(I)$ the set $I\setminus\mathbb{N}_{\geq G(S)+1}$.

######  Proposition 2.1.

Given any ideal $I\subset S$, the set $I+1=\\{i+1\mid i\in I\\}$ is an ideal
if and only if $|s(I)|=0$.

###### Proof.

Suppose that the set $I+1$ is an ideal and $s(I)\neq\phi$. Let $m(s(I))$ be
the maximum element in $s(I)$. From the definition of an ideal,
$m(s(I))=m(s(S))$. Since $I+1$ is an ideal, $m(S)+1\in I+1\subseteq S$. We now
have two cases, $m(s(S))<G(S)-1$ and $m(s(S))=G(S)-1$. In the first case, we
get that $m(s(I))+1\in s(I)$. However, this contradicts the maximality of
$m(s(I))$. In the second case we get $G(S)\in S$ since $I+1$ is an ideal.
However, this contradicts the definition of $G(S)$. Both the cases contradict
our assumption and thus $s(I)=\phi$.

Now suppose for an ideal $I$, $s(I)=\phi$. We clearly have $I+1\subset S$. For
any $s\in S$ and $i+1\in I+1$, we have $i+1+s\in I+1$ since $i+s\in I$. Thus
$I+1$ is an ideal. ∎

Denote by $m\left(\langle a_{1}.\ldots,a_{n}\rangle,k\right)=\left|\\{I\in
I(a_{1},\ldots,a_{n})\mid\operatorname{\operatorname{codim}}(I)=k\\}\right|$.
Then we have the following result.

######  Proposition 2.2.

Given an integer $k\geq G(S)$, any ideal $I\in m(S,k)$ satisfies $s(I)=\phi$.

###### Proof.

Suppose for an ideal $I\in m(S,k)$, $s(I)\neq\phi$. We now prove that
$\operatorname{codim}(I)\leq G(S)-1$. Note that $|s(S)\setminus
s(I)|\leq|s(S)|-1=G(S)-\Delta(S)-1$. Since $s(I)\neq\phi$, $G(S)+1\in I$ which
implies that $(s(S)+G(S)+1)\cup\mathbb{N}_{\geq 2G(S)+2}\subset I$. Thus we
have $|\mathbb{N}_{\geq G(S)+1}\setminus(I\setminus
s(I))|\leq|\mathbb{N}_{\geq G(S)+1}\setminus((s(S)+G(S)+1)\cup\mathbb{N}_{\geq
2G(S)+2})|\leq\Delta(S)$ which implies
$\operatorname{\operatorname{codim}}(I)\leq
G(S)-\Delta(S)-1+\Delta(S)=G(S)-1$. This contradicts our assumption. Thus
$s(I)=\phi$. ∎

######  Proposition 2.3.

For any ideal $I\subset S$ with $\operatorname{\operatorname{codim}}(I)>G(S)$
we have $G(S)+1\notin I$.

###### Proof.

Suppose $G(S)+1\in I$. From proposition 2.2 we get $s(I)=\phi$. We also have
$(s(S)+G(S)+1)\cup\mathbb{N}_{\geq 2G(S)+2}\subseteq I$. Clearly
$I^{\prime}=(s(S)+G(S)+1)\cup\mathbb{N}_{\geq 2G(S)+2}$ is an ideal. Thus we
have $\operatorname{\operatorname{codim}}(I)=s(S)+|\mathbb{N}_{\geq
G(S)+1}\setminus I|\leq s(S)+|\mathbb{N}_{\geq G(S)+1}\setminus
I^{\prime}|=(G(S)-\Delta(S))+\Delta(S)=G(S)$. ∎

######  Proposition 2.4.

Given a numerical semigroup $S$, for all integers $j>G(S)$ we have

$m\left(S,G(S)\right)=m\left(S,j\right).$

###### Proof.

Let $S_{i}=\\{I\in I(S)\mid\operatorname{\operatorname{codim}}(I)=i\\}$.
Consider the maps

$F:S_{G(S)}\mapsto S_{j}\,\,\textrm{and}\,\,G:S_{j}\mapsto S_{G(S)},$

defined by $F(I)=I+1$ and $G(I)=I-1$ respectively. We first prove that $F,G$
are well defined. Proposition 2.1 and proposition 2.2 imply that $F$ is well
defined. Proposition 2.2 and proposition 2.3 imply that $G$ is well defined.
Clearly, both the maps and injective as well, which gives us a bijection
between $S_{G(S)}$ and $S_{j}$. ∎

###  2.1. Proof of Theorem 1.2

Using proposition 2.4, the ideal generating function can be written as

$I(a_{1},\ldots,a_{r};q)=\sum_{\operatorname{\operatorname{codim}}(I)<G(S)}q^{\operatorname{\operatorname{codim}}(I)}+\frac{m(S,G(S))q^{G(S)}}{1-q}.$

Thus we get $I(a_{1},\ldots,a_{r};q)=\frac{f(q)}{1-q}$ for some polynomial
$f(q)\in\mathbb{Z}[q]$. Since $m(S,G(S))=m(S,j)$ for all $j>G(S)$ we get
$\deg(f(q))-1=G(a_{1},\ldots,a_{r})$.

##  3\. Computing $I(a,b;q)$

In this section we compute the ideal generating function of the numerical
semigroups $S$ with embedding dimension two. We assume that $S=\langle
a,b\rangle$ where $\gcd(a,b)=1$. Let $R(a)$ be the rectangle having $a$ rows
and an infinite number of columns. Label the bottom left corner of $R(a)$ as
$0$. We now label the remaining boxes of $R(a)$ with integers so that, shift
by $1$ up adds $b$, and shift by $1$ to the right adds $a$. As a result, a box
is labelled by $ax+by$ where $(x,y)$ are the coordinates of the bottom left
vertex of the box. Let $R(a,b)$ denote the grid we obtain after labelling
$R(a)$ in this way. For instance, Figure 1 shows $R(3,4)$. Notice that each
element $s\in\langle a,b\rangle$ occurs exactly once as a label in $R(a,b)$
since each element $s$ can be uniquely written as $ax+by$ where
$x\in\mathbb{N},b\in\\{0,1,\ldots,a-1\\}$.

$0$$3$$6$$9$$4$$8$$7$$10$$13$$11$$14$$17$$12$$16$$20$$15$$19$$23$$\ldots$$\ldots$$\ldots$
Figure 1. Labelling of $R(3)$.

Let $P(a,b)$ be the set of lattice paths $P$ that uses the steps $(1,0)$ and
$(0,-1)$ and satisfy the following conditions

  1. (1)

$P$ starts at $(0,a)$ and ends at $(x,0)$ for some integer $x\in\mathbb{N}$.

  2. (2)

$P$ ends with a step of the form $(0,-1)$

  3. (3)

The number of steps of the form $(1,0)$ between the first and last step of the
form $(0,-1)$ in $P$ is at most $b$.

Given a lattice path $P\in P(a,b)$, let $L(P)$ denote the set of labels of the
boxes that lie below $P$ and let $I(P)=\langle a,b\rangle\setminus L(P)$. Let
$r(P,k)$ denote the least label in row $k$ of $R(a)$ that is not below the
path $P$.

It is known that lattice paths in $P(a,b)$ are in one-to-one correspondence
with ideals in $I(\langle a,b\rangle)$ [5]. However, we provide a proof for
the convenience of the reader.

######  Proposition 3.1.

Lattice paths in $P(a,b)$ are in one-to-one correspondence with ideals in
$I(\langle a,b\rangle)$.

###### Proof.

Consider the map $F:P(a,b)\mapsto I(\langle a,b\rangle)$, defined by
$F(P)=I(P)$. We first prove that $F$ is well-defined and that $I(P)$ is an
ideal. Since a shift by $1$ to the right adds $a$, it suffices to prove that
$w(P,k)+b\in I(P)$ for all $k\in\\{1,2,\ldots,a\\}$. This clearly follows from
the definition of paths in $P(a,b)$.

Now, consider the map $G:I(\langle a,b\rangle)\mapsto P(a,b)$, defined as
follows. First, we associate the elements in an ideal $I$ to the corresponding
labels in $R(a,b)$. Let $s(I,k)$ denote the least label in row $k$ of $R(a,b)$
such that it is also contained in $I$. Now build the path corresponding to an
ideal $I$ by placing steps of the form $(0,-1)$ on the left edges of the boxes
containing labels $s(I,k)$ and adding steps of the form $(1,0)$ where needed
to complete it. Since, $s(I,a)+b\in I$, the number of horizontal steps between
the first and the last step of the form $(0,-1)$ is at most $b$. Thus the map
$G$ is well-defined and injective as well.

∎

######  Example.

In the following figure of $R(3,8)$, let the dark lines denote path $P$.
Clearly $P\in P(3,8)$. $I(P)$ corresponds to the ideal generated by
$11,13,12$. Also, note that $r(P,1)=12,r(P,2)=13,r(P,3)=11$ and
$L(P)=\\{0,3,4,6,7,8,9,10\\}$.

$0$$3$$6$$9$$4$$8$$7$$10$$13$$11$$14$$17$$12$$16$$20$$15$$19$$23$$\ldots$$\ldots$$\ldots$

######  Example.

In the following figure of $R(3,8)$, let the dark lines denote path $P$.
Clearly $P\notin P(3,8)$ since the number of steps of the form $(1,0)$ between
the first and last step of the form $(0,-1)$ is greater than $3$. We also have
that $I(P)\notin I(3,8)$ since $11\in I(P)$ but $4+11=15\notin I(P)$.

$0$$3$$6$$9$$4$$8$$7$$10$$13$$11$$14$$17$$12$$16$$20$$15$$19$$23$$\ldots$$\ldots$$\ldots$

We can map any path $P\in P(a,b)$ to a word over the alphabet $\\{D,R\\}$ by
replacing the steps $(0,-1),(1,0)$ by $D,R$ respectively. Let $W(a,b)$ be the
set of corresponding words. Given a path $P\in P(a,b)$, let $w(P)\in W(a,b)$
be the corresponding word. Given a word $w$, let $w_{i}$ denote the letter at
its $i^{\text{th}}$ place.

######  Definition 3.2.

Given a word $w$ over the alphabet $\\{D,R\\}$, an inversion is a pair $(i,j)$
where $i<j$ and $w_{i}=R,w_{j}=D$.

Notice that the area under a lattice path $P$ equals the number of inversions
in the word $w(P)$. Given a word $w$ over the alphabet $\\{D,R\\}$, let
$\operatorname{inv}(w)$ denote the number inversions in $w$. Let $Q(n,k)$
denote the set of all words over the alphabet $\\{D,R\\}$ have $n,k$
occurrences of the letters $D,R$ respectively. The following result holds [1]

(3.1) $\binom{n}{k}_{q}=\sum_{w\in Q(n,k)}q^{\operatorname{inv}(w)},$

where

$\binom{n}{k}_{q}=\frac{\prod_{i=1}^{n}(1-q^{i})}{\prod_{i=1}^{k}(1-q^{i})\prod_{i=1}^{n-k}(1-q^{i})}.$

Now we are ready to explicitly evaluate $I(a,b;q)$.

######  Theorem 3.3.

The ideal generating function of the numerical semigroup $\langle a,b\rangle$
is given by

$I(a,b;q)=\frac{1}{1-q^{a}}\binom{a+b-1}{a-1}_{q}$

###### Proof.

From proposition 3.1, any ideal $I\in I(a,b)$ can be mapped to a word in
$W(a,b)$. It is easy to see that the codimension of an ideal equals the area
under its corresponding path. Let $S_{i}(a,b)$ be the set of paths in $P(a,b)$
such that the first $i$ steps are $(1,0)$ and the $(i+1)^{\text{th}}$ step is
$(0,-1)$. Let

$S_{i}(q)=\sum_{P\in S_{i}(a,b)}q^{\operatorname{area}(P)}.$

Clearly we have $S_{i}(q)=q^{a(i-1)}S_{1}(q)$. Since the codimension of an
ideal equals the area under its corresponding lattice path, we have

$I(a,b;q)=\frac{S_{1}(q)}{1-q^{a}}.$

From equation (3.1) we have,

$S_{1}(q)=\binom{a+b-1}{a-1}_{q},$

proving the desired result. ∎

######  Remark 3.4.

It can be calculated that $\deg(I(a,b;q))=ab-a-b$. Thus using Theorem 1.2, we
rederive Sylvester’s theorem. Surprisingly, it also turns out that the number
of ideals with codimension greater than $ab-a-b-1$ is given by the rational
catalan number

$c_{a,b}=\frac{1}{a+b}\binom{a+b}{a}.$

##  4\. Computing $I(a,b,c;q)$

In this section, we compute the ideal generating function of certain families
of numerical semigroups with embedding dimension three. In particular, we
compute $I(a,b,c;q)$ when $a,b,c$ are mutually coprime, $a\mid b+c$ and
$a<b<c$ by using a tabular representation like the one used in section $2.$ In
what follows, let $[n]=\\{1,2,\ldots,n\\}$ for some positive integer $n$. To
provide a tabular representation of $\langle a,b,c\rangle$, we need the
following result.

######  Proposition 4.1.

Given a semigroup of the form $\langle a,b,c\rangle$, there exists a positive
integer $k_{1}$ such that any $s\in\langle a,b,c\rangle$ can be uniquely
written as either $ax+by$ or $ax+cz$ where
$x\in\mathbb{N},y\in[k_{1}-1]\cup\\{0\\}$ and, $z\in[a-k_{1}]$.

###### Proof.

Let $k_{1}$ and $k_{2}$ be the least positive integers such that
$bk_{1}\in\langle a,c\rangle,ck_{2}\in\langle a,b\rangle$. Note that
$k_{1},k_{2}\in[a-1]$. We now prove that any element $s\in\langle
a,b,c\rangle$ can be uniquely written as $ax+by$ or $ax+cz$ where
$y\in[k_{1}-1],z\in[k_{2}-1]$. Let $ck_{2}=aj+bk$ for some $j,k\in\mathbb{N}$.
Clearly, any $s=ax_{1}+bx_{2}+cx_{3}$ can be written in the form $s=ax+bv+cw$
for some $v,w\in\mathbb{N}_{\leq a-1}$. Since $a\mid b+c$, this can be reduced
to a representation of the form $s=ax+bu$ or $s=ax+cu$ for some $u\in[a-1]$.
For the sake of brevity, assume $s=ax+cu$ where $u>k_{2}-1$. Let $i$ be such
that $y-ik_{2}<ki$. We now have the representation

$s=a(x+ij)+b(ik)+c(y-ik_{2}).$

Since $a\mid b+c$, this representation can be rewritten as $s=ax(1)+bu(1)$ for
some $u(1)\in[a-1]$. If $u(1)>k_{1}-1$, we can repeat this process to get a
representation of the form $s=ax(2)+cu(2)$. Since the size of the coefficient
of $a$ strictly increases at every step, $u(2)<u$. Clearly, we can continue
this process until we have a representation of the form $ax+by$ or $ax+cz$
where $y\in[k_{1}-1],z\in[k_{2}-1]$. Now suppose $ax+by=ax^{\prime}+cz$ for
some $y\in[k_{1}-1],z\in[k_{2}-1]$. Since $\gcd(b,c)=1$, we get $x\neq
x^{\prime}$. For the sake of brevity, assume $x<x^{\prime}$. We now get
$by=a(x^{\prime}-x)+cz\in\langle a,c\rangle$ which contradicts the minimality
of $k_{1}$. Thus the representation is unique. Since $\langle a,b,c\rangle$
contains all residue classes modulo $a$, we get $k_{1}+k_{2}=a-1$. ∎

Using proposition 4.1, any element $s\in\langle a,b,c\rangle$ occurs exactly
once in one of the sequences $\\{d_{i,j}\\}_{j=1}^{\infty}$ defined by

$d_{i,j}=\begin{cases}c(a+1-k_{1}-i)+a(j-1)\,\,\,\,\,\,\,\,\,\,\,i\in[a-k_{1}]\\\
a(j-1)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\quad\,\,\,\,i=a-k_{1}+1\\\
b(i+k_{1}-a-1)+a(j-1)\,\,\,\,\,\,\,\,\,\,\,i\in\\{a-k_{1}+2,a-k_{1}+2,\ldots,a\\}\end{cases}$

Thus, similar to section $2$, we can give a tabular representation of $\langle
a,b,c\rangle$ by labelling a box in row $i$, column $j$ in $R(a)$ by
$d_{i,j}$. Let $R(a,b,c)$ denote the grid we obtain after labelling $R(a)$ in
this way.

For an ideal $I\subseteq I(a,b,c)$, let $d_{k}(I)$ denote the number of
elements in the $k^{\text{th}}$ row of $R(a,b,c)$ that are not in $I$, i.e.
$d_{k}(I)=|\\{d_{k,i}\\}_{i\geq 1}\setminus I|$.

######  Proposition 4.2.

Given any ideal $I\in I(a,b,c)$, the following inequalities hold

$d_{1}(I)\leq d_{2}(I)\leq\ldots\leq
d_{a-k_{1}+1}(I)\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,d_{a}(I)\leq
d_{a-1}(I)\leq\ldots\leq d_{a-k_{1}+1}(I).$

###### Proof.

We first prove that $d_{i}(I)\leq d_{i+1}(I)$ for $i\in[a-k_{1}-1]$. Let
$d_{i}(I)=m-1$. Thus, $d_{i,m}\in I$ which implies $d_{i,m}+c=d_{i-1,m}\in I$.
Therefore, $d_{i}(I)\leq d_{i+1}(I)$. The remaining set of inequalities can be
proven analogously. ∎

For some $k\in\mathbb{N}_{\geq 1}$, denote by $R_{k}(a,b,c)$ the set of ideals
$I\subseteq I(a,b,c)$ such that $d_{j}(I)=k-1$ for all $j\in[a]$ and
$d_{1,k}\in I$ or $d_{a,k}\in I$. Using proposition 4.2, we have the set
partition $I(a,b,c)=\bigcup_{i\in\mathbb{N}_{\geq 1}}R_{i}(a,b,c)$.

We now prove the following result:

######  Theorem 4.3.

The ideal generating function for the semigroup $\langle a,b,c\rangle$ is
given by

$I(a,b,c;q)=\dfrac{1}{1-q^{a}}\left(\sum_{I\in
R_{1}(a,b,c)}{q^{\operatorname{\operatorname{codim}}(I)}}\right).$

###### Proof.

For some positive integer $m\in\mathbb{N}_{\geq 1}$, let
$F:R_{1}(a,b,c)\mapsto R_{m}(a,b,c)$, $G:R_{m}(a,b,c)\mapsto R_{1}(a,b,c)$ be
maps defined by $F(I)=I+a(m-1)$ and $G(I)=I-a(m-1)$. We first prove that $F$
is well defined. Clearly, when $I\in I(a,b,c)$ we have $I+a\in I(a,b,c)$. Thus
$I+a(m-1)$ is an ideal. Since $d_{1,1}\in I$ or $d_{a,1}\in I$ we have
$d_{1,1}+a(m-1)=d_{1,m}\in I$ or $d_{a,1}+a(m-1)=d_{a,m}\in I$. Thus
$I+a(m-1)\in R_{m}(a,b,c)$. One can analogously prove that $G$ is well
defined. Clearly $F,G$ are injective implying that
$|R_{1}(a,b,c)|=|R_{m}(a,b,c)|$. Now let

$S_{i}(q)=\sum_{I\in R_{i}(a,b,c)}q^{\operatorname{\operatorname{codim}}(I)}.$

From the bijection between $R_{1}(a,b,c)$ and $R_{m}(a,b,c)$ we clearly have

$S_{m}(q)=q^{a(m-1)}S_{1}(q).$

Thus we have

$I(a,b,c;q)=\sum_{m=1}^{\infty}S_{m}(q)=\frac{S_{1}(q)}{1-q^{a}},$

as desired. ∎

We now compute the ideal generating function for particular examples.

###  4.1. Computing $I(3,n+2,2n+1)$

We enumerate ideals in $I(3,n+2,2n+1)$ by removing certain elements from the
tabular representation of $\langle 3,n+2,2n+1\rangle$. Since
$2(n+2)=3+(2n+1)$, we get that $k_{1}=2$. Thus the associated sequences are
$\\{d_{i,j}\\}_{j=0}^{\infty},\\{d_{2,j}\\}_{j=1}^{\infty},\\{d_{3,j}\\}_{j=1}^{\infty}$
with initial terms $d_{1,1}=2n+1,d_{2,1}=3,d_{3,1}=n+2$. Figure 2 shows the
tabular representation of $\langle 3,n+2,2n+1\rangle$.

We can partition the set $R_{1}$ as $\langle 3,n+2,2n+1\rangle\cup P_{1}\cup
P_{2}\cup P_{3}$ where $P_{1}=\\{I\in R_{1}(3,n+2,2n+1)\mid 0\notin I,n+2\in
I,2n+1\in I\\},P_{2}=\\{I\in R_{1}(3,n+2,2n+1)\mid 0\notin I,n+2\in
I,2n+1\notin I\\},P_{3}=\\{I\in R_{1}(3,n+2,2n+1)\mid 0\notin I,n+2\notin
I,2n+1\in I\\}$. Denote by $S(i_{1},i_{2},i_{3})=\\{d_{1,i}\\}_{i\geq
i_{1}}\cup\\{d_{2,i}\\}_{i\geq i_{1}}\cup\\{d_{3,i}\\}_{i\geq i_{3}}$. For
$i\in\\{1,2,3\\}$, let

$F(P_{i},q)=\sum_{I\in P_{i}}q^{\operatorname{\operatorname{codim}}(I)}.$

Since for any ideal $I\in R_{1}(3,n+2,2n+1)$, either $n+2\in I$ or $2n+1\in I$
we get that $n+2+2n+1=3n+3\in I$. Thus for any ideal $I\in R_{1}$, we have
$d_{2}(I)\in[n+1]$. Clearly, any set of the form $S(0,i,0)$ where $i\in[n+1]$
is an ideal. From proposition 4.2 we have $P_{1}=\\{S(0,i,0)\mid
i\in[n+1]\\}$. Thus we have

(4.1) $F(P_{1},q)=\sum_{i=1}^{n+1}q^{i}.$

We now calculate $F(P_{2},q)$. Since for an ideal $I\in P_{2}$, $n+2\in I$ we
have that $(n+2)+(n+2)=2n+4\in I$. This implies that $d_{1}(I)=1$. Clearly,
any set of the form $S(1,i,0)$ where $i\in[n+1]$ is an ideal. From proposition
4.2 we have $P_{2}=\\{S(1,i,0)\mid i\in[n+1]\\}$. Thus we have

(4.2) $F(P_{2},q)=\sum_{i=1}^{n+1}q^{i+1}.$

Finally we calculate $F(P_{3},q)$. Since for any ideal $I\in P_{2}$, $2n+1\in
I$ we have that $(2n+1)+(2n+1)=4n+2\in I$. This implies that $d_{3}(I)\in[n]$.
From proposition 4.2, we conclude that any set of the form
$S(0,i,j),S(0,n+1,k)$ where $1<j<i$, $i\in[n]$ , and $k\in[n]$ is an ideal.
Thus we have $P_{3}=\\{S(0,i,j)\mid
i,j\in\mathbb{N},i\in[n],0<i<j+1\\}\cup\\{S(0,n+1,i)\mid
i\in\mathbb{N},i\in[n]\\}$. We now have

(4.3)
$F(P_{3},q)=\sum_{i=1}^{n}q^{i}\sum_{j=1}^{i}q^{j}+q^{n+1}\sum_{i=1}^{n}q^{i}.$

Combining equations (4.1),(4.2),(4.3) we get the following result.

$2n+1$

$0$

$n+2$

$2n+4$

$3$

$n+5$

$\ldots$$\ldots$$\ldots$

$5n+3$

$3n$

$4n+2$

$5n+6$

$3n+3$

$4n+5$

$\ldots$

$\ldots$

$\ldots$

Figure 2.  Tabular representation of $\langle 3,n+2,2n+1\rangle$.

######  Theorem 4.4.

Let $n$ be a positive integer such that $n\not\equiv 1\pmod{3}$. Then, the
ideal generating function for the numerical semigroup $\langle
3,n+2,2n+1\rangle$ is given by

$\displaystyle
I(3,n+2,2n+1;q)=\frac{1}{1-q^{3}}\left(1+\frac{q\left(q^{n+1}-1\right)\left(2q^{n+3}+q^{2}-2q-1\right)-q^{4}\left(q^{2n+1}-q^{n+1}-q^{n}+1\right)}{\left(1-q\right)^{2}\left(1+q\right)}-q^{2n+2}\right).$

Using Theorems 1.2, 4.4 we find that $G(3,n+2,2n+1)=2n-2$ for $n\not\equiv
1\pmod{3}$

##  5\. Concluding Remarks and Further Research

Let $P(L)$ denote the HOMFLY polynomial of an oriented link $L\in S^{3}$. It
is known that $P(L)$ is an element of $\mathbb{Z}[a^{\pm 1},(q+q^{-1})^{\pm
1}]$. Let $T_{k,n}$ denote the $(k,n)$ Torus knot. Given a pair of coprime
integers $(k,n)$ and the curve $y^{k}=x^{n}$, it is known that $T_{k,n}$ is
the link of its singularity at the origin. Oblomkov and Shende give the
following result [6]

######  Theorem 5.1.

Let $C$ be the curve cut out by $y^{k}=x^{n}$ and let $p$ be the origin.We
have

(5.1)
$P(T_{k,n})=\left(\frac{a}{q}\right)^{(k-1)(n-1)}(1-q^{2})\int_{C_{p}^{[*]}}q^{2l}(1-a^{2})^{m-1}d\chi.$

Note that the complete local ring associated with the curve $y^{k}=x^{n}$ is
$\mathcal{O}=\mathbb{C}[[t^{k},t^{n}]]$. Let $J(k,n)$ denote the set of all
monomial ideals in $\mathbb{C}[[t^{k},t^{n}]]$. The integral in equation (5.1)
can alternatively be written as [6]

(5.2) $\int_{C_{p}^{[*]}}q^{2l}(1-a^{2})^{m-1}d\chi=\sum_{J\in
J(k,n)}q^{2\dim_{\mathbb{C}}(\mathcal{O}/J)}(1-a^{2})^{m(J)}.$

From equations (1.1),(5.2) and Theorem 5.1 we have

(5.3) $P(T_{k,n})\mid_{a=0}=I(k,n;q^{2}).$

It can be asked whether there exist relations of the form (5.3) for the ideal
generating function of numerical semigroup with more than two generators. We
formulate this for numerical semigroup $\langle 3,n+2,2n+1\rangle$.

It can be calculated that $\mathbb{C}[[t^{3},t^{n+2},t^{2n+1}]]$ is the
complete local ring associated with curve $x^{n+1}=yz,y^{2}=xz$. Let
$L(3,n+2,2n+1)$ be the link associated with these curves. Note that
$L(3,n+2,2n+1)$ is a link in Lens space. Cattibriga et.al. [2] extend the
HOMFLY polynomial to links in lens space. It can now be asked whether
$I(3,n+2,2n+1;q)$ is related to the HOMFLY polynomial of $L(3,n+2,2n+1)$.

Another possible direction is to study the numbers $m(S,k)$ for $k\geq G(S)$.
As noted before in Remark 3.4, $m(\langle a,b\rangle,k)=c_{a,b}$ for $k\geq
ab-a-b$ are the rational Catalan numbers which have a variety of combinatorial
interpretations. Thus it will be interesting from a combinatorial point of
view to study the numbers $m(S,k)$ for numerical semigroups with more than two
generators.

##  6\. Acknowledgements

The majority of this research was done during the Research Science Institute
(RSI) at MIT in the summer of 2022. First, I would like to thank my mentor
Jeffery Yu for his invaluable guidance throughout the completion of this work,
and Minh-Tam Quang Trinh for proposing the project. I am grateful to the CEE &
MIT for their hospitality and support during the preparation of this work.

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