# Building Ears for Robots: Machine Hearing in the Age of Autonomy

Xuan Zhong
Zoox, Inc.
Foster City, California, USA
<EMAIL_ADDRESS>

###### Abstract

This study explores the significance of robot hearing systems, emphasizing
their importance for robots operating in diverse and uncertain environments.
It introduces the hardware design principles using robotaxis as an example,
where exterior microphone arrays are employed to detect sound events such as
sirens. The challenges, goals, and test methods are discussed, focusing on
achieving a suitable signal-to-noise ratio (SNR). Additionally, it presents a
preliminary software framework rooted in probabilistic robotics theory,
advocating for the integration of robot hearing into the broader context of
perception and decision-making. It discusses various models, including Bayes
filters, partially observable Markov decision processes (POMDP), and
multiagent systems, highlighting the multifaceted roles that robot hearing can
play. In conclusion, as service robots continue to evolve, robot hearing
research will expand, offering new perspectives and challenges for future
development beyond simple sound event classification.

_K_ eywords machine hearing $\cdot$ autonomous vehicles $\cdot$ probabilistic
robotics

##  1 Introduction

Robot hearing system is an interesting topic of research in the fields of both
hearing science and robotics. However, compared to computer vision, robot
hearing systems received limited attention so far due to the scarcity of
robots in uncertain and open environments. The field of robotics is still in
its infancy and cannot deal with the complexity and uncertainty of the
environment. As a result, most robots could only operate in highly controlled
environments, such as automobile factories. However, in recent years, several
types of service robots, such as robotaxis, have been deployed. The field of
robotics is also quickly evolving with the advent of modern techniques like
deep reinforcement learning. In the coming decades, as robots increasingly
coexist with humans in open environments, the significance of robot hearing
systems will grow, as they uniquely contribute to robot perception.

The current body of literature on robot hearing topic comes from several
different fields. Significant progress was made in the field of sound event
localization and detection (SELD), first as summarized in [Virtanen et al.,
2018], then in several years of DCASE competitions. The work has been focused
on the application of machine learning methods in computational auditory scene
analysis. The state-of-art audio perception models can detect, locate, and
classify multiple sound sources in noisy three-dimensional (3D) environments
[Kong et al., 2020]. While many SELD methods apply to robots, the meaning of
sound only partially depends on the audio features. Robot hearing systems
should encompass capabilities beyond simple sound event classification.

Another area of study that is related to robot hearing systems is the
comparative study of human and machine hearing. Human auditory perception has
been studied through physiology and behavioral psychology experiments, and
mathematical models were established to mimic the behaviors of human auditory
systems [Pastore et al., 2020]. Such models could be potentially applied to
robots. The seminal work of [Lyon, 2017] started with an insightful review of
human hearing research and the theory of signals and systems. The focal point
of this book is the CARFAC (cascade of asymmetric resonators with fast-acting
compression), a computational model of audio features that generates a rich
representation of input auditory signals for high level processing. The
current article seeks to address unresolved questions in machine hearing
within the field of robotics.

The current study starts with a discussion on robot hearing hardware systems.
Section 2 delves into the hardware design of robot ears, using exterior
microphone arrays on self-driving cars as an example. It covers system design
principles, goals, challenges, methods, and testing procedures, with a focus
on achieving a suitable signal-to-noise ratio (SNR) amid the robot’s noise and
environmental factors.

Section 3 delves into the roles of robot hearing systems within the modern
probabilistic robotics framework, covering Bayes filters, partially observable
Markov decision processes (POMDP), and multiagent systems. Despite the
emerging nature of robotics, the study underscores the established
interactions between agents and their environments. It suggests that
integrating observation and action opens up new avenues for the design and
research of robot hearing systems. Moreover, it underscores the limitations of
current sound event localization and detection (SELD) approaches, emphasizing
the importance of considering the temporal dimension of sound events in
relation to sequential observations and decision-making. This temporal
context, or the timing of sound events relative to agent actions, is deemed
crucial for a more comprehensive sound analysis.

##  2 Robot Hearing Hardware Design

This section presents the goals, challenges, design guidelines and test
methods of the hardware of robot hearing systems. Throughout each subsection,
the discussions use the robotaxi exterior microphone arrays, one of the first
mass-produced robot hearing systems, as examples.

###  2.1 Goals

Similar to human ears, the hardware of robot hearing systems captures sound
waves and preprocesses the audio signals before sending them to the higher
levels of the perception systems.

Figure 1: Applications of machine hearing systems on autonomous vehicles

Exterior microphones on autonomous vehicles serve multiple important purposes
(Figure 1). Firstly, they are essential for reliable siren detection and
localization, ensuring that autonomous vehicles can respond appropriately to
emergency vehicles, especially in blind corners where traditional sensors may
not provide adequate information.

Secondly, in the event of a minor accident with low level impact, exterior
microphones capture impact sounds. This is essential for reporting and
assessing injuries or damages accurately, contributing to efficient post-
accident procedures.

Furthermore, autonomous vehicles also need to monitor the local weather
conditions to evaluate the possibilities of sensor degradation. Rainfalls and
many other weather conditions carry special audio features that can be used
for weather event classification.

###  2.2 Challenges

Designing effective robotic hearing systems presents a significant challenge:
converting sounds into audio signals with a sufficient SNR for high level
processing such as sound event detection. This challenge is compounded by
factors related to both the robot and the environment.

In autonomous driving scenarios, robotaxis generate internal noise sources,
such as engine and air-conditioner sounds, which can obscure acoustic
features, especially for low-level signals such as distant sirens.
Additionally, the robot’s physical size can create sound reflections and
diffractions, leading to substantial sound attenuation in directions not
directly facing incoming sound waves. To address this, multiple microphone
arrays can be placed on different sides of the vehicle to enhance coverage.
Furthermore, accommodating the robotic hearing system alongside other modules
and subsystems within the vehicle necessitates thoughtful packaging and
harnessing considerations.

Environmental factors further complicate the hardware design of robot hearing
systems. For exterior microphones on robotaxis, wind noise increases with
higher vehicle speeds, and traffic noise can diminish the SNR. These
microphones are also exposed to environmental elements like water and dust,
making it challenging to strike a balance between wind noise reduction and
protecting against ingression. Moreover, for sound event detection at long
distances, sounds may not propagate along a straight line due to local
atmospheric conditions like temperature gradients and local winds [Hannah,
2003].

In summary, calculating the SNR of exterior microphones on robots must account
for the variability of both the signal and the noise. The noise level is
determined by the most dominant noise sources within the environment and
originating from the robot itself.

###  2.3 Design Methods

Given the complexity of both the robot itself and its operating environment,
the hardware design of robot hearing systems should follow a system design
approach [Hamming, 1998]. According to Hamming, system design involves keeping
the high level objectives in mind and translating local design efforts into
global outcomes. Several key principles can guide this approach:

  1. 1.

Understand the Whole as a Whole: Comprehend the system as a holistic entity,
not merely the sum of its individual components.

  2. 2.

Flexibility and Adaptability: Prepare for potential changes and adapt to
evolving circumstances.

  3. 3.

Subsystem Specifications with Redundancy: Define subsystem specifications with
built-in redundancies to enhance reliability.

In the case of designing robotaxi exterior microphones, using a system design
approach requires the following:

  1. 1.

Top-down Design Approach: Define the vehicle system level performance before
specifying subsystem and module performance.

  2. 2.

General Purpose Design: Design a general purpose microphone for a wide range
of tasks instead of, for instance, building a resonator that only picks up a
narrow range of target sounds.

  3. 3.

Hierarchical Design with Redundancy: break the top level design requirements
to low level design targets while keeping some redundancies.

Figure 2: Placement of multiple exterior microphones on a Zoox autonomous
vehicle

The Zoox robotaxi’s exterior microphone system embodies the design principles
discussed earlier. It comprises six pairs of microphones strategically
positioned around the vehicle’s perimeter (Figure 2). This placement is
informed by extensive field and lab tests focused on vehicular and wind noise.
At the microphone module level, the detailed design is inspired by the feather
structure of barn owl [Koch and Wagner, 2002], which finds a balance among
acoustics, wind noise reduction and ingression protection performance. The
design delivers wide-band audio signals at high speeds. The design passes the
IP69k test, which means that the microphones are fully dust-proof and can
withstand water jet streams of 1500 PSI at a temperature of 80 °C.

###  2.4 Test Methods

Two types of tests are critical to the success of product development. For the
design team, a series of validation tests are required to ensure that the
design could fulfill the required tasks. At the factory, a battery of quality
control tests is implemented, starting from the component level and
progressing upward to ensure consistent adherence to specifications.

For example, frequency responses of microphones are tested at various levels.
The MEMS microphone components are tested before they are soldered onto the
printed circuit board assembly (PCBA). Further testing occurs at the PCBA
level. Once the PCBA is integrated into enclosures, module-level testing is
performed, with each test requiring specialized fixtures.

After the module level tests, the fully assembled modules are mounted onto a
limited number of vehicles for validation tests. The frequency response is
tested in two settings. In one test, the reference microphone is mounted on
the vehicle, right next to the device-under-test. This “local” frequency
response represents the capability of the microphone to capture nearby sounds,
such as human speech or low-level impact. In another test, the reference
microphone is used to calibrate a test loudspeaker without the vehicle. This
"global" frequency response represents how good the microphone is at capturing
distant sounds, such as the sirens. The results from multiple loudspeaker
locations with different incident angles can also be named vehicle-related
transfer functions, akin to the head-related transfer functions for human
listeners.

Figure 3: Simulated rain test in a semi-anechoic wind tunnel

The wind noise tests also follow the “bottom-up” sequence. Airflow is
generated at various scales to provide test environments at different levels.
A small wind box with a 30-cm test section is used to study the wind noise
reduction effect of different surface conditions. A medium-sized wind tunnel
with a 5-m test section offers the capability of wind-blown rain testing at
the module and sub-assembly levels. At the vehicle level, the lab tests
leverage a big wind tunnel with a 20-m test section and weather simulation
capabilities (Figure 3). Moreover, vehicle-level field tests are done at
multiple test tracks to provide data of microphone performance with real
winds.

##  3 Robot Hearing Software Framework

This section presents a preliminary software framework of robot hearing
systems based on probabilistic robotics theory. This is an attempt to answer
the question “what does a hearing system mean to a robot?” rather than the
specifics of system implementation.

###  3.1 Space and Hearing

Mostly of the previous literature on human spatial hearing has emphasized
"hearing" rather than "space". Typically, a stationary listener listens to
stationary sound sources, so that spatial hearing cues, such as interaural
time difference (ITD), interaural level difference (ILD) and human-related
transfer function (HRTF), could be studied [Blauert, 1997]. The result is a
comprehensive but at best partial understanding of the relationship between
space and hearing. For instance, a recent study pointed out the limitations of
head-centric spatial hearing cues in world-centric sound source localization
tasks [Yost et al., 2021].

In the field of robotics, intelligent agents, including humans and robots,
navigate their environments with at least six degrees of freedom, sometimes
sharing space with other agents. For example, a human listener is searching
for a lost phone at home. If the lost phone is set to vibration mode, low-
frequency sound sources are challenging to locate accurately. Hence it is
unlikely that the listener can locate the phone at once. However, by moving
around, the listener perceives variations in sound intensity. Continuously
moving to maximize sound intensity eventually leads the listener to discover
the phone on the couch.

Figure 4: The interactions between an intelligent agent and its environment

This example illustrates the use of a hearing system in a setting of
probabilistic robotics [Burgard et al., 2005], in which the problem of finding
a sound source is framed as a long-term state-estimation problem based on
sequential observations and actions (Figure 4). In this context, the concept
of "space" discussed earlier is referred to as the "environment".

Theoretically, a state in probabilistic robotics encompasses all information
pertaining to the future in the environment and the agent itself. In practice,
a selective subset of these variables is calculated to optimize time and
computational resources.

Intelligent agents engage with their environments in two ways: through actions
performed by actuators and through observations made using sensors, including
microphones. It’s important to note that this study doesn’t assume that an
agent can accomplish its tasks solely with audio inputs.

Many existing robot perception systems use an open-loop design, in which the
role of the system is to generate a percept regarding nearby objects and
events without considering how these perceptions inform decision-making. The
majority of the current SELD research and development falls in this category
and they can fulfill a wide range of tasks.

However, there are distinct advantages to integrating perception with planning
within the framework of probabilistic robotics. Hence the remainder of this
section focuses on the discussion of the closed-loop design of sequential
decision-making processes. The taxonomy of robot perception and decision-
making is introduced in the order of Bayesian filters, partially-observable
Markov decision processes (POMDP) and multiagent systems, with increasing
complexity and capabilities. The exploration of the potential roles of robot
hearing within these systems follows.

###  3.2 Robot hearing in Bayes filters

The agent interacts with its environment through a sequence of observations
and actions that changes the state. The sequence of these state changes is
illustrated in Figure 5.

Figure 5: The probabilistic model of state changes over time

In probabilistic robotics, the state is described by a probabilistic density
function rather than a deterministic function. The sequence of interactions
consists of measurement updates and action updates. Action updates typically
add to the uncertainty due to the error of the actuator and environment
uncertainties (similar to walking in darkness), whereas the measurement update
usually reduces the uncertainty by providing added information [Burgard et
al., 2005].

The agent does not have direct access to the true state but must infer the
state through a series of actions and observations. Observations can come from
a range of sensors, including cameras, lidars, radars and microphones. The
agent’s internal knowledge of the state is referred to as belief. The general
approach of deriving belief from actions and observations is the Bayes method,
in which the belief converges to the true state after a series of actions and
observations with sensors and actuators with Gaussian noise. The errors reduce
significantly as more observations and actions are incorporated.

The transition model describes how the dynamics of the environment changes
based on the previous state and action: $s_{t}\sim p(s_{t}|s_{t-1},a_{t-1})$

The measurement model describes the partial sensor observation of the state:
$o_{t}\sim p(o_{t}|s_{t})$

The goal is to estimate the state from sensor data and controls.

The most studied form of Bayes filters is the Kalman filter and its nonlinear
version, the extended Kalman filter (EKF). In a previous study [Zhong et al.,
2016], robot head rotation and interaural time differences (ITD) are combined
in an EKF model for sound source localization. Here the state represents the
location of the sound sources, the observations consist of the binaural audio
inputs and ITDs, and the actions correspond to the head rotations.

When sequential observations are allowed during head rotations, the amount of
ITD change becomes a reliable new cue for sound source localization, allowing
the robot to estimate not only the azimuth of the sound source, but the
elevation as well, although at any given moment, the ITD alone does not
support sound source localization due to the problem of the cone of confusion.

As another example, in the case of “finding the phone”, the listener can
potentially locate the phone even with only one ear. The monaural sound level
changes over a trajectory of motion are another spatial hearing cue. In fact,
for the simplest model of binaural hearing, which is two microphones attached
to the two ends of a rod, all the degrees of freedom contribute to spatial
hearing except the rotation along the axis of the rod.

###  3.3 Robot hearing in POMDP

In the Bayes filter model, a control of action is given for each step in the
sequence of state changes. However, the Bayes model itself does not prescribe
how the control is generated. In the realm of probabilistic robotics, the
basic framework of planning and control is the Markov decision process (MDP).
A more general form of MDP is the POMDP (Figure 6), where the sensors only
carry partial information about the states [Burgard et al., 2005].

The primary objective of robot software is to select the right actions.
However, when a robot interacts with the environment, there are two types of
uncertainty that a robot model must consider. The first type of uncertainty
pertains to the sensors. At best, each type of sensor only generates signals
that reflect a part of the true state with some noise. The second type of
uncertainty concerns the actuators. Due to the actuator noise and the
complexity of real-world dynamics, the effects of actions are not
deterministic but probabilistic in nature.

Figure 6: Partially-observable Markov Decision Process (POMDP)

The POMDP model formulates the problem as the following:

$s$: set of states

$o$: set of observations

$a$: set of actions

$r$: rewards

The decision process is modeled as follows:

Transition model: $s_{t}\sim p(s_{t}|s_{t-1},a_{t-1})$

Measurement model: $o_{t}\sim p(o_{t}|s_{t})$

Reward function: $r_{t}\sim p(r_{t}|s_{t})$

Policy: $a_{t}\sim p(a_{t}|o_{t-1},a_{t-1})$

The primary aim of the POMDP model is to determine a policy of actions that
maximizes the cumulative reward in the process. Solving POMDPs can be
challenging, and various methods are employed, including traditional
techniques such as value iterations [Burgard et al., 2005] and modern
approaches like deep reinforcement learning networks [Wu et al., 2023]. A
comprehensive overview of reinforcement learning methods is available in [Dong
et al., 2020].

A properly solved POMDP model automatically regulates the behavior of
observations and actions. The decisions of the policy at a given time step can
be taking an action or “doing nothing and observing”. When the uncertainty is
high, the agent keeps observing and collecting more information. On the
contrary, when the uncertainty is low, the agent takes actions according to
the policy. There could also be a process of exploration, in which the only
goal is to collect more information about the environment through active
sensing [Burgard et al., 2005].

The roles of robot hearing system within the POMDP framework is multi-faceted:

  1. 1.

Providing Information about the States: Sensors should provide information to
generate a proper belief state for POMDP. Each type of sensor reflects a
different aspect of the state. Hence robot hearing reduces belief variance by
reflecting some information of the state. Hearing does not require significant
computing power and is a rapid mode of perception. Hearing also does not
require direct field of the view of the sound sources, so it is beneficial for
state estimation when the field of view is obstructed for other sensors such
as cameras. In many cases, what blocks the field of view is the enclosure of
objects under observation. Sounds of diesel engines and hair blowers provide
information about the states which may be difficult to obtain with other
methods.

  2. 2.

Assessing Effects of Actions: Robot hearing also plays a role in evaluating
the consequences of previous actions. For example, when a car is stared, the
sound of the engine can indicate whether it started normally or remained
silent due to a dead battery. Similarly, when a door is closed, the sound of
the lock can reveal whether it engaged as expected. In the context of POMDP,
silence can hold meaning: it signifies an anticipated state change that did
not occur after the actions were taken. Consequently, the meaning of sound
events does not only depend on the audio inputs, but also the context of
interactions between the robot and its environment in the decision process.

###  3.4 Multiagent systems

The concept of multiagent systems introduces several complexities and
challenges into the realm of robotics and artificial intelligence. In
multiagent systems, two or more agents coexist and interact within the same
environment, each striving to maximize their individual rewards or objectives.
This introduces a level of complexity beyond what is typically addressed in
traditional POMDP settings, where a single agent operates in a more controlled
environment. [Shoham and Leyton-Brown, 2008].

Multiagent systems can be fully cooperative, fully competitive, or a mix of
both. They can be modeled as Markov games, where all agents decide
simultaneously, or extensive-form games, with turn-based decisions. Agents
have limited information about the state and lack access to other agents’
policies, making it challenging to predict their actions and effects.

In the realm of multiagent systems, robot hearing systems fulfill several
critical roles in addition to those in the case of POMDP. Firstly, they serve
as detectors for identifying the presence of other agents in the environment.
In dynamic scenarios, the number of agents can fluctuate as they enter or exit
the immediate area of interest. Consequently, agents equipped with robot
hearing have an enhanced capacity to detect the presence of new agents. This
capability is essential for tasks such as recognizing potential mates,
predators, or prey.

Additionally, robot hearing systems can monitor the actions of other agents.
When other agents take actions, the environment may undergo changes that are
independent of the primary agent’s actions. In contrast, in moments of
silence, the likelihood of external actions occurring is low, and the primary
agent can reasonably assume that its existing policies remain applicable. The
primary agent may also gain insights into the intentions, rewards, and
policies of other agents by analyzing a series of their actions.

Robot hearing systems must take into account the broader context of multiagent
decision-making to accurately interpret sound events. For instance, hearing
the front door open may have very different implications depending on whether
one expects a roommate or is home alone at night. Neglecting the influence of
multiagent interactions on sound event interpretation can limit the
applicability of existing methods in complex scenarios. This underscores the
idea that sound events should be studied in conjunction with the context in
which they occur, as their meaning is only partially determined by the audio
inputs.

##  4 Discussions

This study first introduces the hardware design of robot hearing systems with
the example of robotaxi. The design challenges, principles and test methods
were discussed. Looking ahead, service robots may take a myriad of sizes and
forms. There may be large robots for mining and very small drones for
pollination, but the design and validation principles of the hearing systems
should still apply.

The following section delved into the significance of a hearing system for a
robot. Drawing from probabilistic robotics models, a hearing system
contributes to improved belief certainty through multiple observations (in
EKF), provides accurate and timely information about the state and action
effects (in POMDP), and tracks environmental changes caused by other agents
(in multiagent systems).

Indeed, this preliminary framework emphasizes the importance of not isolating
the study of hearing systems but integrating them with actions and sequential
decision-making within the broader field of robotics. This approach opens up
several future research directions that extend beyond the current paradigms in
both SELD and hearing research. In the future, as the functions, capabilities
and complexity of service robots quickly evolves, so will their hearing
systems.

## Acknowledgments

The author would like to thank Shaminda Subasingha, William Yost, Aurora
Cramer and Liang Sun for their comments and suggestions. The materials covered
in this article were presented and discussed at the Hearing Seminar at
Stanford University, which was organized by Malcolm Slaney in October, 2023.

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# An elementary proof of an isoperimetric inequality for paths with finite
$p$-variation.

George Galvin

(August 2016)

###### Abstract

In this article we will prove that if the continuous closed curve
$\gamma:[0,1]\rightarrow\mathbb{R}^{2}$ has finite $p$-variation with $p<2$,
then

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma,(x,y))|^{q}\,dx\,dy)^{1/q}\leq(\frac{1}{2})^{\frac{1}{q}}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p,[0,1]})^{\frac{2}{q}}$

for all $q\in[1,\frac{2}{p})$, where $\eta(\gamma,(x,y))$ is the winding
number of $\gamma$ at $(x,y),\zeta$ is the Reimann zeta function, and
$||\gamma||_{p,[0,1]}$ is the $p$-variation of $\gamma$ on the interval
$[0,1]$.

Our main contribution is that we have explicitly given a bound by known
constants, and we have found this by an elementary proof. We are going to be
using a method introduced by L.C. Young [8] in 1936.

##  1 Introduction

Isoperimetric problems have been studied since the time of Ancient Greece. The
simplest isoperimetric problem was solved in the second century BC, when the
Greek mathematician Zenodorus proved that a circle has greater area than any
polygon with the same perimeter . This was later generalised to produce the
classical isoperimetric inequality, which states that for any shape with
perimeter $L$ and area $A$,

$L^{2}-4\pi A\geq 0,$

with equality when the shape is a circle. [2]

This inequality was generalised by Banchoff and Pohl [3] to include curves
that self-intersect. The result of their paper was that if $\gamma$ is a two-
dimensional closed path with finite length, and $L$ is the length of $\gamma$,
and $\eta(\gamma,(x,y))$ is the winding number (which will be defined below)
of $\gamma$ at $(x,y)$, then

$L^{2}-4\pi\iint\limits_{\mathbb{R}^{2}}\eta(\gamma,(x,y))^{2}\,dx\,dy\geq 0.$

More recently it has been shown [4] that for $1\leq p<2$, and for all
$q<\frac{2}{p}$, there exists $C_{p,q}>0$ such that for all paths
$\gamma:[0,1]\rightarrow\mathbb{R}^{2}$ with finite $p$-variation,

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma,(x,y))|^{q}\,dx\,dy)^{1/q}\leq
C_{p,q}\max(||\gamma||_{p,[0,1]},||\gamma||^{p}_{p,[0,1]}),$ (1)

where $||\gamma||_{p,[0,1]}$ is the $p$-variation (which will also be defined
below) of $\gamma$ on the interval $[0,1]$.

In this article, our main contribution is that we have bounded this integral
using known constants, and we have found this result with an elementary proof.

More precisely, we will prove the following:

######  Theorem 1.1.

If the continuous closed curve $\gamma:[0,1]\rightarrow\mathbb{R}^{2}$ has
finite $p$-variation with $p<2$, then

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma,(x,y))|^{q}\,dx\,dy)^{1/q}\leq(\frac{1}{2})^{\frac{1}{q}}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p,[0,1]})^{\frac{2}{q}}$

for all $q\in[1,\frac{2}{p})$, where $\eta(\gamma,(x,y))$ is the winding
number of $\gamma$ at $(x,y),\zeta$ is the Riemann zeta function, and
$||\gamma||_{p,[0,1]}$ is the $p$-variation (which will be defined below) of
$\gamma$ on the interval $[0,1].$

In section 2 (Preliminary), we are going to state and prove some important
theorems and lemmas we are going to use to prove Theorem 1.1.

In section 3 (Main proof), we are going to use this knowledge to complete this
proof. We will use Young’s method [8] of successively removing partition
points.

This idea has also been used in rough path theory. The interested reader is
referred to the texts by Friz and Victoir [10], Lyons [9], and Lyons, Caruana
and Lévy [5].

##  2 Preliminary

We start by defining the concepts of the partition and $p$-variation, which we
will use heavily throughout this article. Throughout this section we will let
$\gamma$ be a continuous closed path in $\mathbb{R}^{2}$.

######  Definition 2.1.

We say $\mathcal{P}=(t_{0},...,t_{r})$ is a partition of $[0,T]$ if
$t_{0}=0,t_{r}=T$ and $t_{0}<t_{1}<...<t_{r-1}<t_{r}$.

######  Definition 2.2.

Given a partition $\mathcal{P}=(t_{0}<t_{1}<...<t_{r})$, define

$\gamma^{\mathcal{P}}_{t}=\gamma_{t_{i}}+\frac{t-t_{i}}{t_{i+1}-t_{i}}(\gamma_{t_{i+1}}-\gamma_{t_{i}}).$

This means that if we model $\gamma$ as a path, $\gamma^{\mathcal{P}}$ will be
a polygonal interpolation of $\gamma$.

######  Definition 2.3.

Let $p\geq 1$. Let $\gamma:[0,T]\rightarrow\mathbb{R}^{2}$ be a continuous
function. The $p$-variation of $\gamma$ on $[0,T]$ is defined by

$||\gamma||_{p,[0,T]}=(\sup\\{\sum\limits_{j=0}^{r-1}|\gamma_{t_{j}}-\gamma_{t_{j+1}}|^{p}:(t_{0},t_{1},...,t_{r})\mbox{
is a partition of }[0,T]\\})^{\frac{1}{p}}.$

Sometimes we will omit writing the interval $[0,T]$ \- in this case the
interval is taken to be $[0,1]$.

######  Definition 2.4.

Suppose that $(x,y)\in\mathbb{R}^{2}\backslash\gamma[0,1]$. Then the function

$S^{\gamma}:t\rightarrow\frac{\gamma_{t}-(x,y)}{|\gamma_{t}-(x,y)|}$

maps $[0,1]$ to the unit circle centred at $0$.

Let $\theta^{\gamma}:[0,1]\rightarrow\mathbb{R}$ be a lift of $S^{\gamma}$
such that

$S^{\gamma}_{t}=(\cos\theta^{\gamma}_{t},\sin\theta^{\gamma}_{t})\ \forall
t\in[0,1].$

Then the winding number of $\gamma$ at the point $(x,y)$ is defined as

$\eta(\gamma,(x,y))=\frac{\theta^{\gamma}(1)-\theta^{\gamma}(0)}{2\pi}.$

The function $\eta(\gamma,(x,y))$ is independent of the lift $\theta^{\gamma}$
(see chapter 3, Lemma 1 and 2, in [7]).

To prove Theorem 1.1 we need to also reference an important lemma on winding
numbers:

######  Lemma 2.1.

(Theorem 7.2 [6]) Let $\beta_{1}:[0,1]\rightarrow\mathbb{R}^{2}$ and
$\beta_{2}:[0,1]\rightarrow\mathbb{R}^{2}$ be continuous paths such that
$\beta_{1}(1)=\beta_{1}(0)$ and $\beta_{2}(1)=\beta_{2}(0).$ Also let

$\beta_{1}\star\beta_{2}(t)=\begin{cases}\beta_{1}(2t),&0\leq
t\leq\frac{1}{2},\\\
\beta_{2}(2t-1)-\beta_{2}(0)+\beta_{1}(1),&\frac{1}{2}\leq t\leq
1.\end{cases}$ (2)

Then,

$\eta(\beta_{1}\star\beta_{2},(x,y))=\eta(\beta_{1},(x,y))+\eta(\beta_{2},(x,y)).$
(3)

We will also use the Minkowski inequality as displayed below.

######  Lemma 2.2.

Let $f$ and $g$ be measurable functions. Then

$(\iint\limits_{\mathbb{R}^{2}}|f(x,y)+g(x,y)|^{q}\,dx\,dy)^{1/q}\leq(\iint\limits_{\mathbb{R}^{2}}|f(x,y)|^{q}\,dx\,dy)^{1/q}+(\iint\limits_{\mathbb{R}^{2}}|g(x,y)|^{q}\,dx\,dy)^{1/q}.$
(4)

######  Lemma 2.3.

Let $\gamma$ be a continuous closed path in $\mathbb{R}^{2}$ with finite
$p$-variation with $p<2$. Let $\mathcal{P}=(t_{1},...,t_{r})$. Then for any
$j_{1},...,j_{n}$ and for all $n\in\\{1,2,...,r-2\\}$,

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},...,t_{j_{n}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}},(x,y))|^{q}\,dx\,dy)\\\
\leq\frac{1}{2\times
2^{\frac{2}{p}}}||\gamma||^{2}_{p,[t_{j_{n+1}-1},t_{j_{n+1}+1}]}.$

###### Proof.

Figure 1:  This figure illustrates the removal of a partition point from a
pentagonal path. The diagram on the left represents
$\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},...,t_{j_{n}}\\}}$, and the diagram
on the right represents
$\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}}$.

Let $\Phi$ denote

$[\gamma_{t_{j_{n+1}-1}},\gamma_{t_{j_{n+1}}}]\star[\gamma_{t_{j_{n+1}}},\gamma_{t_{j_{n+1}+1}}]\star[\gamma_{t_{j_{n+1}+1}},\gamma_{t_{j_{n+1}-1}}],$
(5)

with $[a,b]$ denoting the straight line between $a$ and $b$.

By the additivity of the winding numbers, Lemma 2.1, the functions

$(x,y)\rightarrow\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n}}\\}},(x,y))$

and

$(x,y)\rightarrow\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}},(x,y))$

coincide except inside $\Phi$, where the two functions differ by
$\eta(\Phi,(x,y))$. Figure 1 is an illustration of this.

Therefore,

$\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}},(x,y))=\eta(\Phi,(x,y)).$
(6)

It follows that

$\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}},(x,y))|^{q}\,dx\,dy\\\
=\iint\limits_{\mathbb{R}^{2}}|\eta(\Phi,(x,y))|^{q}\,dx\,dy.$

Since $\eta(\Phi,(x,y))$ is $1$ or $-1$ inside $\Phi$ and $0$ outside,

$\iint\limits_{\mathbb{R}^{2}}|\eta(\Phi,(x,y))|^{q}\,dx\,dy=\mbox{Area}(\triangle_{t_{j_{n+1}-1},t_{j_{n+1}},t_{j_{n+1}+1}}),$
(7)

where $\triangle_{t_{j_{n+1}-1},t_{j_{n+1}},t_{j_{n+1}+1}}$ is the triangle
with vertices $\gamma_{t_{j_{n+1}-1}},\gamma_{t_{j_{n+1}}},$ and
$\gamma_{t_{j_{n+1}+1}}$.

Now, using the inequality $ab\leq\frac{a^{2}+b^{2}}{2}$,

$\begin{aligned}
\mbox{Area}(\triangle&{}_{t_{j_{n+1}-1},t_{j_{n+1}},t_{j_{n+1}+1}})\\\
&\leq\frac{|\gamma_{t_{j_{n+1}-1}}-\gamma_{t_{j_{n+1}}}||\gamma_{t_{j_{n+1}+1}}-\gamma_{t_{j_{n+1}}}|}{2}\\\
&=\frac{(|\gamma_{t_{j_{n+1}-1}}-\gamma_{t_{j_{n+1}}}|^{\frac{p}{2}}|\gamma_{t_{j_{n+1}+1}}-\gamma_{t_{j_{n+1}}}|^{\frac{p}{2}})^{\frac{2}{p}}}{2}\\\
&\leq\frac{(|\gamma_{t_{j_{n+1}-1}}-\gamma_{t_{j_{n+1}}}|^{p}+|\gamma_{t_{j_{n+1}+1}}-\gamma_{t_{j_{n+1}}}|^{p})^{\frac{2}{p}}}{2\times
2^{\frac{2}{p}}}\\\
&\leq\frac{||\gamma||^{2}_{p,[t_{j_{n+1}-1},t_{j_{n+1}+1}]}}{2\times
2^{\frac{2}{p}}}.\end{aligned}$ (8)

∎

##  3 Main proof

In this section we will let $\gamma:[0,1]\rightarrow\mathbb{R}^{2}$ be a
continuous closed path in $\mathbb{R}^{2}$ with finite $p$-variation, $p<2$,
and let $\mathcal{P}$ be a partition of $[0,1]$.

We begin by stating an important theorem to be used:

######  Theorem 3.1.

([5]) For any partition $\mathcal{P}=(t_{0}<t_{1}<...<t_{k})$ there exists
$j\in\\{1,...,k-1\\}$ such that

$||\gamma||_{p,[t_{j-1},t_{j+1}]}\leq\frac{2}{k-1}||\gamma||_{p,[0,1]}.$ (9)

We now use Young’s method of successively removing points to form an upper
bound. Let $\mathcal{P}=(t_{0}<...<t_{r})$; by (9) with partition
$\mathcal{P}$ there exists $j_{1}\in\\{1,...,r-1\\}$ such that

$||\gamma||^{p}_{p,[t_{j_{1}-1},t_{j_{1}+1}]}\leq\frac{2}{r-1}||\gamma||^{p}_{p,[0,1]}.$

We now remove $t_{j_{1}}$ from $\mathcal{P}$ \- therefore we have $r-2$ points
left. By abuse of notation, we relabel these $r-2$ points as
$t_{1},...,t_{r-2}.$

Now with partition $\mathcal{P}\backslash\\{t_{j_{1}}\\}$ there exists
$j_{2}\in\\{1,...,r-2\\}$ such that

$||\gamma||^{p}_{p,[t_{{j_{2}}-1},t_{{j_{2}}+1}]}\leq\frac{2}{r-2}||\gamma||^{p}_{p,[0,1]}.$

We then generalise this for removing any number of points. For all
$n\in\\{1,2,...,r-2\\}$, we can remove $n-1$ points from $\mathcal{P}$, using
abuse of notation to label the $r-n$ points left as $t_{1},...,t_{r-n}$. Then
there exists $j_{n}\in\\{1,...,r-n\\}$ such that

$||\gamma||^{p}_{p,[t_{{j_{n}}-1},t_{j_{n}+1}]}\leq\frac{2}{r-n}||\gamma||^{p}_{p,[0,1]}.$
(10)

Removing $r-2$ points from $\mathcal{P}$, we are left with the path
$\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}}$. As
the partition only has three points left, and

$\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}}(t_{0})=\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}}(t_{r}),$

the path only has two points left. Therefore,

$\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}},(x,y)))=0\
$ (11)

for all $(x,y)$ outside
$\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}}$. It
is then clear that for all $(x,y)$ outside $\gamma^{\mathcal{P}}$, we have

$\begin{split}\eta(\gamma^{\mathcal{P}},(x,y))&=(\eta(\gamma^{\mathcal{P}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}}\\}}),(x,y))\\\
+&(\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}}\\}},(x,y)))\\\
+&...\\\
+&(\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-3}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}},(x,y))).\end{split}$
(12)

Therefore, by an extension of the Minkowski inequality (4) to $r-2$ functions,

$\begin{aligned}
&(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}},(x,y))|^{q}\,dx\,dy)^{1/q}\\\
&\leq(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}}\\}},(x,y))|^{q}\,dx\,dy)^{1/q}\\\
&+(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}}\\}},(x,y))|^{q}\,dx\,dy)^{1/q}\\\
&+...\\\
&+(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-3}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{r-2}}\\}},(x,y))|^{q}\,dx\,dy)^{1/q}.\end{aligned}$
(13)

We can now create the inequality which forms the base of the final proof.

######  Lemma 3.2.

For $q<\frac{2}{p}$,

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}},(x,y))|^{q}\,dx\,dy)^{1/q}\leq(\frac{1}{2})^{\frac{1}{q}}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p,[0,1]})^{\frac{2}{q}}.$
(14)

###### Proof.

From Lemma 2.3,

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}},(x,y))|^{q}\,dx\,dy)^{\frac{p}{2}}\\\
\leq\frac{1}{2\times
2^{\frac{p}{2}}}||\gamma||^{p}_{p,[t_{j_{n+1}-1},t_{j_{n+1}+1}]}\\\
\leq\frac{1}{2^{\frac{p}{2}}}(\frac{1}{r-(n+1)})||\gamma||^{p}_{p,[0,1]}\
\mbox{(applying (\ref{eq:rempoints}))}\\\
\Rightarrow(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n}}\\}},(x,y))-\eta(\gamma^{\mathcal{P}\backslash\\{t_{j_{1}},t_{j_{2}},...,t_{j_{n+1}}\\}},(x,y))|^{q}\,dx\,dy)^{\frac{1}{q}}\\\
\leq\frac{1}{2^{\frac{1}{q}}}(\frac{1}{(r-(n+1))}||\gamma||^{p}_{p,[0,1]})^{\frac{2}{pq}}.$
(15)

We then use this inequality to simplify each integral in (13), as follows:

$\begin{split}(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}},(x,y))|^{q}\,dx\,dy)^{1/q}&\\\
\leq&\frac{1}{2^{\frac{1}{q}}}(\frac{1}{r-1}||\gamma||^{p}_{p,[0,1]})^{\frac{2}{pq}}\\\
&+\frac{1}{2^{\frac{1}{q}}}(\frac{1}{r-2}||\gamma||^{p}_{p,[0,1]})^{\frac{2}{pq}}\\\
&+...\\\
&+\frac{1}{2^{\frac{1}{q}}}(\frac{1}{2}||\gamma||^{p}_{p,[0,1]})^{\frac{2}{pq}}.\end{split}$
(16)

Then, taking the sum,

$\begin{split}(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}},(x,y))|^{q}\,dx\,dy)^{1/q}&\\\
\leq&\frac{1}{2^{\frac{1}{q}}}(\sum\limits_{n=2}^{r-1}(\frac{1}{n})^{\frac{2}{pq}})||\gamma||^{\frac{2}{q}}_{p,[0,1]}\\\
\leq&\frac{1}{2^{\frac{1}{q}}}(\zeta(\frac{2}{pq})-1)||\gamma||^{\frac{2}{q}}_{p,[0,1]}.\end{split}$
(17)

To bound the above integral from above, we thus require $\zeta(\frac{2}{pq})$
to be finite. For this we need $1<\frac{2}{pq}$ so we need $q<\frac{2}{p}$.

∎

Now to complete the proof we create a new partition $\mathcal{P}_{n}$.

######  Definition 3.1.

Let $\mathcal{P}_{n}=\\{0,\frac{1}{n},\frac{2}{n},...,\frac{n-1}{n},1\\}$.

We also need one more lemma; it is known that

######  Lemma 3.3.

([1]) For all $(x,y)$ outside the image of
$\cup^{\infty}_{n=1}\gamma^{\mathcal{P}_{n}}\cup\gamma^{n}$,

$\liminf_{n\to\infty}\eta(\gamma^{\mathcal{P}_{n}},(x,y))=\eta(\gamma,(x,y)).$
(18)

We are now in a position to complete the proof of Theorem 1.1.

###### Proof.

We can take the limit infinum of both sides of (14) as the number of partition
points $n$ goes to infinity:

$\liminf_{n\to\infty}(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}_{n}},(x,y))|^{q}\,dx\,dy)^{1/q}\leq\liminf_{n\to\infty}(\frac{1}{2^{\frac{1}{q}}}(\zeta(\frac{2}{pq})-1)||\gamma||_{p,[0,1]}^{\frac{2}{q}}).$

As the right hand side is constant, its limit infinum is itself, so we can
rewrite the equation as follows:

$\liminf_{n\to\infty}(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma^{\mathcal{P}_{n}},(x,y))|^{q}\,dx\,dy)^{1/q}\leq\frac{1}{2^{\frac{1}{q}}}(\zeta(\frac{2}{pq})-1)||\gamma||_{p,[0,1]}^{\frac{2}{q}}.$
(19)

Due to Fatou’s Lemma,

$(\iint\limits_{\mathbb{R}^{2}}\liminf_{n\to\infty}|\eta(\gamma^{\mathcal{P}_{n}},(x,y))|^{q}\,dx\,dy)^{1/q}\leq\frac{1}{2^{\frac{1}{q}}}(\zeta(\frac{2}{pq})-1)||\gamma||_{p,[0,1]}^{\frac{2}{q}}.$
(20)

Therefore due to (18),

$(\iint\limits_{\mathbb{R}^{2}}|\eta(\gamma,(x,y))|^{q}\,dx\,dy)^{1/q}\leq\frac{1}{2^{\frac{1}{q}}}(\zeta(\frac{2}{pq})-1)||\gamma||_{p,[0,1]}^{\frac{2}{q}}.$

∎

##  4 Acknowledgements

I would like to acknowledge the support of the EPSRC Vacation Bursary. I would
like to thank my supervisor Horatio Boedihardjo for his support and insights.
I also thank the referee for their useful suggestions and comments.

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