Supercentenarian paradox

Vladimir Gurvich
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>National Research University Higher School of Economics, Moscow, Russia
RUTCOR, Rutgers University, Piscataway, New Jersey, United States

Mariya Naumova
Rutgers Business School, Rutgers University, Piscataway, New Jersey, United States

And the LORD said, My spirit shall not always strive with man,
for that he also is flesh: yet his days shall be one hundred and twenty years.Genesis 6:3
What should you wish for the 120th birthday? Have a nice day!A corollary

Yes, man is mortal, but that would be only half the trouble. The worst of it is that he's sometimes unexpectedly mortal—there's the trick!Mikhail Bulgakov, The Master and Margarita

§ TO BET OR NOT TO BET: THAT IS THE QUESTION

Consider the following statement:

$B(t, \Delta t)$:  a  $t$-year-old person NN will survive another $\Delta t$  years,
where $t, \Delta t \in \mathbb{R_+}$  are nonnegative real numbers.

We only know that $NN$  is  $t$  years old and nothing else, in particular, gender, origin, health conditions, etc. are not known.
Should we bet that  $B(t, \Delta t)$  will hold?
It seems that our odds are very good for any  $t$  provided
$\Delta t$ is small enough, say, $1 / 365$ (that is, one day).
However, this is not that obvious, and it depends on the life-time probability distribution.

Let  $F(t)$ be the cumulative distribution function of the lifetime of a person, where $t$ is in years, and define $\Phi(t) = 1 - F(t)$.

The following cases are possible:

* $\Phi(t)=0$ whenever $t \ge t_0$ for some
$t_0 \in \mathbb{R_+}$.

In this scenario, if $t \in [t_0 - \Delta t, t_0]$, the bet is lost with probability 1.

* $\Phi(t) \rightarrow 0$ faster than an exponential distribution.

Then, for each fixed $\Delta t > 0$, the bet is lost with probability as close to 1 as desired if $t$ is large enough.

* $\Phi(t) \rightarrow 0$ as fast as an exponential distribution.

Then for large $t$ the chances to win the bet depend only on  $\Delta t$ and not on $t$.

* $\Phi(t) \rightarrow 0$ slower than an exponential distribution.

In this scenario, in agreement with intuition, for any fixed $t$, the chances to win tend to $1$ as $\Delta t \rightarrow 0$.

The last two cases are not realistic as statistical literature suggests; see below.

For example, $normal(\mu, \sigma^2)$ distribution corresponds to case 2, as

\begin{equation}
\begin{aligned}
\displaystyle \lim_{t \rightarrow \infty} \frac{\Phi(t+\Delta t)}{\Phi(t)} &= \displaystyle \lim_{t \rightarrow \infty} \frac{1-\displaystyle \int_{- \infty}^{t+\Delta t}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}dx}{1-\displaystyle \int_{- \infty}^{t}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}dx}\\
& = \displaystyle \lim_{t \rightarrow \infty} \frac{e^{-\frac{(t+\Delta t-\mu)^2}{2\sigma^2}}}{e^{-\frac{(t-\mu)^2}{2\sigma^2}}}\\
& = \displaystyle \lim_{t \rightarrow \infty} e^{-\frac{2t\Delta t +(\Delta t)^2-2\mu\Delta t}{2\sigma^2}}\\
& = 0 \hspace{0.2in} \text{for any fixed } \Delta t>0.\nonumber
\end{aligned}
\end{equation}

Clearly, all distributions with negative excess kurtosis (platykurtic distributions), that is, the distributions with thinner tails compared to the normal distribution, satisfy this property.

In demography and actuarial sciences, the Gompertz distribution [8] is often applied to describe the distribution of adult lifespans [15], [17], [7], [9], [10]; see Section 2 for more details.

The probability density function of the Gompertz distribution is given by

\begin{equation}
\displaystyle f\left(x;\eta ,b\right)=b\eta \exp \left(\eta +bx-\eta e^{bx}\right){\text{  for }}x\geq 0,\,\nonumber
\end{equation}

where $b>0$ is the scale parameter and $\eta >0$ is the shape parameter.

The cumulative distribution function is

\begin{equation}
F\left(x;\eta ,b\right)=1-\exp \left(-\eta \left(e^{bx}-1\right)\right).\nonumber
\end{equation}

We find that

\begin{equation}
p = \frac{\Phi(t+\Delta t)}{\Phi(t)} = \frac{e^{-\eta(e^{b(t+\Delta t)}-1)}}{e^{-\eta(e^{bt}-1)}}=e^{-\eta e^{bt}(e^{b\Delta t}-1)}.\nonumber
\end{equation}

The minimal time $t$ such that the probability to win the bet is at least $p$ is

$$t = \frac{1}{b}\ln\left(\frac{-\ln p}{\eta (e^{b\Delta t}-1)}\right),$$

and the combination of parameters $\eta$ and $b$ that yields the distribution mode of 75 [For this example, we choose the mode of the distribution to be 75 years. Note that the life expectancy at birth in the US depending on state for 2019 was in statistical range of 74.4-80.9 years; see <https://www.cdc.gov/nchs/data-visualization/state-life-expectancy/index_2019.htm.>]
is $b=-\frac{1}{75}\ln \eta$, $\eta \in (0, 1)$. Statisticians fit the parameters $b$ and $\eta$ of the distribution to a series of data based on various factors (population average age, geographic location, etc). The values of $t$ for various $\Delta t$ with 50$\%$ chance of winning the bet are shown in Table <ref>.

The values of $t$ for various $\Delta t$ with 50$\%$ chance of winning the bet in case of several Gompertz distributions with mode $75$

[width=/8+2, height=1cm]$\Delta t$$\eta$   $e^{-75/3}$   $e^{-75/4}$   $e^{-75/5}$   $e^{-75/6}$   $e^{-75/7}$   $e^{-75/8}$

1 year    76.68   	78.57   	80.71   	83.04   	85.55   	88.20

1 month    84.61   	88.98   	93.60   	98.42	   103.41   	108.54

1 day   94.89   	102.68   	110.71   	118.95   	127.35	   135.90

1 hour   104.43   	115.39   	126.60   	138.02   	149.60   	161.33

1 minute   116.71   	131.77   	147.08   	162.59   	178.26   	194.08

1 second    129.00   	148.15   	167.55   	187.15   	206.92   	226.84

legend style=
legend pos=north east,
legend style=row sep=1.5pt
axis x line = bottom,
axis y line = left,
xlabel=\(\Delta t\),
xtick=1/12, 1/365,
xticklabels=$\frac{1}{12}$, $\frac{1}{365}$,
every axis x label/.style=at=(current axis.right of origin),anchor=west,
every axis y label/.style=at=(current axis.north west), above=0.5mm
[red, dotted,
very thick
$\eta = e^{-75/3}$
[blue, dotted,
very thick
$\eta = e^{-75/4}$
[brown, dotted,
very thick
$\eta = e^{-75/5}$
[black, dotted,
very thick
$\eta = e^{-75/6}$
[magenta, dotted,
very thick
$\eta = e^{-75/7}$
[green, dotted,
very thick
$\eta = e^{-75/8}$

Relation between $\Delta t$ and $t$, for $\Delta t < 1/12$ (1 month), with 50$\%$ chance of winning the bet in case of several Gompertz distributions with mode $75$

Figure <ref> demonstrates how high $t$ values are for small values of $\Delta t$ (one month or less).
For example, if $b=1/3$, $\eta = e^{-75/3}$, then $\Delta t = 1/(365\cdot 24 \cdot 60 \cdot 60)$ (one second) requires $t \approx 129$ (years), but on Earth there is no person of such age [4].

§ ACTUARY MORTALITY MODELS

Life actuaries in the US use the mortality tables published by the Society of Actuaries (SOA; see Mort.soa.org), which are based on the collected data. The Continuous Mortality Investigation (CMI) in the UK provides projections for the expected changes to those tables. Most insurance companies use these tables, adjusting them to their needs, while others use their own historical data to cover probabilities down to a personal level, including variables such as location, income, and health conditions; see <https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation>.

Meanwhile, life contingencies textbooks often have hypothetical life tables based on Gompertz and Gompertz-Makeham models [15], [13]. Despite their age, the models, also known as the law of mortality, are still widely used in the fields of demographics, biology, actuarial science and even financial engineering [12], [10]. Modern actuaries incorporate the Gompertz distribution to represent important actuarial expressions in closed form via the Gamma function, which makes representation especially convenient and popular in the annuity literature [12].

The more advanced Gompertz–Makeham model assumes that the human death rate is the sum of two components: an age-dependent term (the Gompertz function [8]), which increases exponentially with age, and an age-independent term (the Makeham function [11]). For low mortality countries the latter term is often negligible and the formula simplifies to the Gompertz model discussed in Section 1. While, based on the world data, the Makeham term has been in decline since the 1950s, the Gompertz component is surprisingly stable.

One can verify that for the above-mentioned distributions, $Pr(B(t, \Delta t)) \rightarrow 0$  as  $t \rightarrow \infty$, for any fixed  $\Delta t > 0$. Hence, for arbitrarily small positive  $\Delta t $ and $\epsilon$ there exists a sufficiently large $t$  such that  $Pr(B(t, \Delta t))  < \epsilon$, which means that we should not bet...  in theory. However, in practice we can bet safely, because for any fixed positive $\Delta t$ and $\epsilon$, a very large $t$ is required to satisfy the inequality  $Pr(B(t, \Delta t))  < \epsilon$.

For example, $\Delta t = 1/365$ and $\epsilon = 1/2$  may require  $t > 125$  years for some  typical distributions  $F$  considered in the literature. Yet, on Earth there is no person of such age. Thus, our odds are good because the chosen testee  NN is not old enough, or the bet cannot be made for technical (or, more precisely, statistical) reasons - absence of a testee.

However, there is not much statistical data about supercentenarians due to their scarceness [According to Guinness World Records, the oldest person currently living is Kane Tanaka (Japan, b. 2 January 1903) aged 119 years and 18 days, as verified on 20 January 2022; the only person verified to have lived beyond the age of 120 is Jeanne Louise Calment (France, 1875-1997).]. Therefore, the tail distribution of the human lifetime can be fitted based on an extrapolation only. Yet, it is well-known that extrapolation is not reliable. As Mark Twain noticed:

In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact. (Mark Twain, Life in the Mississippi).

The Supercentenarian Paradox is similar to the famous St.Petersburg Paradox, which we briefly review in the following section.

§ ANALOGY WITH ST. PETERSBURG'S PARADOX

The St. Petersburg paradox was named after one of the leading scientific journals of the eighteenth century Commentarii Academiae Scientiarum Imperialis Petropolitanae (Papers of the Imperial Academy of Sciences in Petersburg), in which Daniel Bernoulli published a paper entitled “Specimen Theoriae Novae de Mensura Sortis” (“Exposition of a New Theory on the Measurement of Risk”) in 1738 [1]. The problem was introduced by Nicolas Bernoulli [16] who stated it in a letter to Pierre Raymond de Montmort on September 9, 1713  [14], [6].

The paradox appears from the game that is played as follows:

A fair coin is flipped until it comes up heads the first time.
Let  $\; t = 0,1,2, \ldots$ denote the number of preceding tails.
Then,  the gambler  wins  $2^{t+1}$  dollars.
What would be a fair compensation to pay the casino for entering the game?

Assuming that the casino has unlimited resources,  the expected gain of the gambler is

$${\displaystyle {\begin{aligned}Gain&= \displaystyle \sum_{t=0}^\infty 2^{t+1} (1/2)^{t+1} =
{\frac {1}{2}}\cdot 2+{\frac {1}{4}}\cdot 4+{\frac {1}{8}}\cdot 8+{\frac {1}{16}}\cdot 16+\cdots \\&=1+1+1+1+\cdots \\&=\infty \,.\end{aligned}}}$$

Thus, in theory it is rational for the gambler to pay the casino
an arbitrary large amount for a single opportunity to play the St. Petersburg game. Yet, almost certainly the gambler will win a modest amount. Even $\$20$ compensation looks too generous.
Nicolas Bernoulli conjectured that people will neglect unlikely events [14].

Similarly, in the Supercentenarian paradox, statement
$B(t, \Delta t)$  holds for arbitrarily small  $\Delta t$ with arbitrary small probability,
provided the age $t$  is sufficiently large.
In both cases the contribution of the very large values of $t$
(that is, very long sequences of tails and very old people) is decisive, but only in theory.
In practice such values of  $t$  are too unlikely.

Interestingly, already in 1777, Buffon  [3]
mentioned certain connections between the St. Petersburg Paradox  and  $B(t, \Delta t)$.  As cited in  [5],

[Buffon] notes that a fifty-six year old man, believing his health to be good, would disregard the probability that he would die within twenty-four hours, although mortality tables indicate that the odds against his dying in this period are only 10189 to 1. Buffon thus takes a probability of 1/10,000 or less for an event as a probability which may be disregarded.

[1]
Bernoulli, D. (1738) [1954]. Specimen Theoriae Novae de Mensura Sortis,
Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5: 175–192.
English translation, 1954, Exposition of a New Theory on the Measurement of Risk,
Econometrica, 22(1): 23–36. doi:10.2307/1909829

[3]
Buffon, G. L. L. (1777). Essai d'Arithmétique Morale, Suppléments à l'Histoire Naturelle,
T. IV, Paris, 1777, 46–148. Reprinted in Oeuvres Philosophiques de Buffon, Paris, 1954.

[4]
Dong, X., Milholland, B., Vijg, J. (2016). Evidence for a limit to human lifespan. Nature. 538 (7624): 257–259.

[5]
Dutka, J. (1988). On the St. Petersburg Paradox. Archive for History of Exact Sciences, 39(1): 13–39.

[6]
Eves, H. (1990). An Introduction To The History of Mathematics (6th ed.). Thomson Brooks/Cole, Pacific Grove, CA.

[7]
Genest C., Kolev N. (2021). A law of uniform seniority for dependent lives. Scandinavian Actuarial Journal, Volume 8, 726–743.

[8]
Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies. Philosophical Transactions of the Royal Society of London. 115: 513–583.

[9]
Kulinskaya, E., Gitsels L. A., Bakbergenuly I., Wright N. R. (2021). Dynamic hazards modelling for predictive longevity risk assessment. Insurance: Mathematics and Economics, Volume 96: 222–231.

[10]
Kulinskaya, E. et al. (2021). Use of Primary Health Care Records Data in Actuarial Research. Presentation in ARC Webinar Series, Actuarial Research Centre, UK.

[11]
Makeham, W. M. (1860). On the Law of Mortality and the Construction of Annuity Tables. J. Inst. Actuaries and Assur. Mag. 8 (6): 301–310.

[12]
Milevsky M.A. (2020). Calibrating Gompertz in reverse: What is your longevity-risk-adjusted global age? Insurance: Mathematics and Economics, Volume 92: 147–161.

[13]
Missov, T. I., Lenart A., Vaupel J. W. (2012). Gompertz-Makeham life expectancies. Analytical solutions, approximations, and inferences. In: Population Association of America 2012 Annual Meeting Program. Available from: <http://paa2012.princeton.edu/abstracts/121013>.

[14]
Montmort, P. R. (1713). Essay d'analyse sur les jeux de hazard (in French) (2nd ed.), Paris: J. Quillau, 1713.

[15]
Preston, S. H., Heuveline, P., Guillot, M. (2001). Demography: Measuring and Modelling Population Processes. Oxford: Blackwell.

[16]
Plous, S. (1993). The Psychology of Judgment and Decision Making. New York: McGraw-Hill.

[17]
Spiegelhalter, D. (2016) How old are you, really? Communicating chronic risk through ‘effective age’ of your body and organs. BMC Medical Informatics and Decision Making. 16 (104).


(cvpr) Package cvpr Warning: Package ‘hyperref’ is not loaded, but highly
recommended for camera-ready version

# Knowledge Combination to Learn Rotated Detection Without Rotated Annotation

Tianyu Zhu1, 2, Bryce Ferenczi2 Pulak Purkait1 Tom Drummond3
Hamid Rezatofighi2 Anton van den Hengel1
1Amazon IML. 2Monash University 3The University of Melbourne
<EMAIL_ADDRESS>{alanzty, purkaitp<EMAIL_ADDRESS><EMAIL_ADDRESS>
{tianyu.zhu, bryce.ferenczi<EMAIL_ADDRESS>Corresponding
author.

###### Abstract

Rotated bounding boxes drastically reduce output ambiguity of elongated
objects, making it superior to axis-aligned bounding boxes. Despite the
effectiveness, rotated detectors are not widely employed. Annotating rotated
bounding boxes is such a laborious process that they are not provided in many
detection datasets where axis-aligned annotations are used instead. In this
paper, we propose a framework that allows the model to predict precise rotated
boxes only requiring cheaper axis-aligned annotation of the target dataset
111Code is available at: https://github.com/alanzty/KCR-Official.

To achieve this, we leverage the fact that neural networks are capable of
learning richer representation of the target domain than what is utilized by
the task. The under-utilized representation can be exploited to address a more
detailed task. Our framework combines task knowledge of an out-of-domain
source dataset with stronger annotation and domain knowledge of the target
dataset with weaker annotation. A novel assignment process and projection loss
are used to enable the co-training on the source and target datasets. As a
result, the model is able to solve the more detailed task in the target
domain, without additional computation overhead during inference. We
extensively evaluate the method on various target datasets including fresh-
produce dataset, HRSC2016 and SSDD. Results show that the proposed method
consistently performs on par with the fully supervised approach.

Acknowledgement This paper is inspired by a computer vision project conducted
at Amazon. We would like to express our sincere gratitude to the following
individuals for their contributions to this research project: Gil Avraham,
Hisham Husain, Chenchen Xu, Ravi Garg, Shatanjay Khandelwal and Philip Schulz,
who all work at Amazon. Their support, insights, and feedback were invaluable
throughout the research process, and we are truly grateful for their help.

##  1 Introduction

Rotated detectors introduced in recent works [32, 17, 20] have received
attention due to their outstanding performance for top view images [33, 34,
15]. They reduce the output ambiguity of elongated objects for downstream
tasks making them superior to axis-aligned detectors in dense scenes with
severe occlusions [18]. However, the rotated annotation is more expensive
compared to axis-aligned annotation. Furthermore, popular 2D annotation tools
such as Sagemaker Groundtruth 222https://aws.amazon.com/sagemaker/data-
labeling/ and VGG app 333https://www.robots.ox.ac.uk/vgg/software/via/ do not
support rotated bounding box annotations. As a result, many popular detection
datasets only have axis-aligned annotations [11, 3, 4]. These problems reduces
the potential scope of the implementation of rotated detectors. In this work,
we introduce Knowledge Combination to learn Rotated object detection, a
training scheme that only requires cheaper axis-aligned annotation for the
target dataset in order to predict rotated boxes.

Figure 1: KCR combines the task knowledge of a source dataset with stronger
rotated annotation and the domain knowledge of the target dataset with weaker
axis-aligned annotation, which enables the model to predict rotated detection
on the target domain.

Neural networks encode data into a latent space, which is then decoded to
optimize the given task. The latent embedding is an abstract representation of
the data, containing much richer information than the output [29]. Early works
in deep learning show that the model implicitly learns to detect image
features such as edges and corners [10, 12], which can be used for more
detailed tasks if decoded properly. We believe decoding to a more precise task
on the target domain can be learnt via co-optimizing with a strongly labelled
source dataset. We design a framework that combines task knowledge of rotated
detection from a source dataset, and the domain knowledge of a disjoint class
of objects in the target dataset with only axis-aligned annotation, as shown
in Figure 1. This approach combines the advantage of both weakly-supervised
learning and transfer learning.

We follow a design principal that the framework should maximize the target
domain knowledge learnt by the model while minimizing the negative impact
caused by weaker labels. This is achieved by co-training the source and target
dataset with projection losses and a novel assignment process. The design
choices are validated through ablation studies. We conduct extensive
experiments to demonstrate that our framework is robust to a large domain gap
between source and target dataset. Therefore, box orientation can practically
be learnt for free with KCR, due to the availability of free public source
datasets such as DOTA [32] with rotated annotations. We show the efficacy of
this method on a fresh-produce dataset with high density of objects and severe
occlusions. The performance (AP50) gap between the proposed method, learning
from weak axis-aligned boxes, and the fully-supervised model learning from
strong rotated annotation, reduces to only $3.2\%$ for the challenging
cucumber dataset. We apply the same framework to HRSC2016 [16] and SSDD [27]
datasets to show that our method consistently performs on par with fully
supervised models. The performance gap reduces to $1.0\%$ for SSDD. We believe
our approach can greatly increase the usage and impact of rotated object
detectors. The source code will be publicly available for the community to
save future annotation cost. In summary, our main contributions are as
follows:

  1. 1)

We introduce a framework that combines task knowledge of a strongly labelled
source dataset and domain knowledge of a weakly labelled target dataset.

  2. 2)

We apply this method in 2D rotated detection task, enabling the model to
predict rotated bounding box with only axis-aligned annotation and verify the
generality of the method with several datasets.

  3. 3)

We demonstrate robustness of the framework to various domain gaps between
source and target datasets. Hence, box orientation can be learnt with no
additional annotation cost in practical applications.

##  2 Related Work

Rotated Detection Task. Rotated object detection requires the model to predict
minimum area rectangles with five degrees of freedom, namely rotated bounding
boxes, enclosing objects of interests [32]. In axis-aligned object detection,
the output rectangles have four degrees of freedom, which are aligned with the
image axes [13]. The rotated boxes occupy a much smaller area when estimating
the location status of diagonally positioned elongated objects as shown in
Figure 1. Rotated detection is strictly superior to axis-aligned detection as
there is less background within the box, and the orientation can potentially
convey object pose information. However, there are only a handful of datasets
with rotated annotations [32, 16, 27] comparing to large number of readily
available large-scale axis-aligned datasets [11, 19, 4, 13]. A potential
contributor to this phenomenon is the ease of annotating axis-aligned boxes by
simple click-and-drag with current labelling tools. Rotated boxes require much
more effort to tightly enclose the objects with an extra degree of freedom.
Popular annotation tools such as AWS Sagemaker and VGG app do not support
rotated boxes. In order to acquire tighter rotated annotations, users must pay
for instance segmentation, which is significantly more expensive and
unnecessary for the final task. Therefore, we propose this work to address the
shortcomings of the rotated object detection task pipeline by making
orientation free to learn.

Weakly-Supervised Learning. Weakly supervised learning sits between fully
supervised learning and unsupervised learning, in the sense that only weak
labels are available. The labels are weak either because they are incomplete,
inexact or noisy [6, 23, 31, 9, 1, 30]. In computer vision, popular weakly-
supervised learning tasks include object detection with only image level
annotation [5, 35] and instance segmentation with only box annotation [21, 2].
Due to the difficulty of pixel-wise prediction, weakly-supervised instance
segmentation still falls significantly behind a fully-supervised model [28] on
novel objects. In this paper, we focus on learning rotated detection requiring
five parameters with only axis-aligned annotation during training which
provides four parameters. The one parameter difference makes our approach
weakly-supervised. To the best our knowledge, such problem has only been
attempted on specific category of objects [8] but not approached generally.
For elongated objects, solving this problem is more appropriate than solving
weakly-supervised instance segmentation directly.

Figure 2: Overall framework of KCR, which learns the knowledge of rotated
detection from source dataset combined with domain knowledge of axis-aligned
target dataset to infer rotated bounding boxes of target objects. The rotated
detector [33] takes in an image encoded by a CNN and generates first-stage
proposals $(x,y,w,h,\alpha,\beta,p)$ and second-stage refinements
$(x^{*},y^{*},w^{*},h^{*},\theta^{*},p^{*})$. We use a function $P$ to either
project or enlarge the box representation during the assignment processes and
loss functions.

Transfer Learning. Computer vision models are frequently initialised with
backbones[7, 22, 14] pretrained on ImageNet or COCO [13, 26], which is a basic
form of transfer learning and a common practice. Transfer learning is highly
effective when the target dataset has small sample size such as medical
imaging [24]. Despite the fact that conventional transfer learning reduces the
number of data samples required for a specific task, strong annotation of the
target domain is still required for the model to learn the detailed task. In
this paper, we utilize co-training strategy to transfer the ability to make
more detailed predictions from the source to the target dataset with weaker
annotations.

##  3 KCR

The goal of our work is to learn to predict rotated bounding boxes on a target
dataset of which we only have axis-aligned bounding boxes in training
examples. We develop a co-training scheme that utilizes an out-of-domain but
strongly labelled source dataset to learn accurate rotations of elongated
objects. An example of a potential source and target dataset pair are a
satellite imagery dataset, DOTA [32], and a fresh-produce dataset, as shown in
Figure 1. In this section, we first briefly describe the workflow of the
detector following by our training scheme which enables weakly-supervised
learning and knowledge transfer.

###  3.1 Rotated Detection Overview

In this work, we employ the same architectural choices as [33] and the flow is
briefly demonstrated in Figure 2. The forward propagation comes in two stages:
the oriented RPN followed by oriented R-CNN, where both methods contain a
classification head and a regression head. The RPN takes an image and
generates $N$ oriented region proposals $\left\\{{R_{i}}\right\\}_{i=1}^{N}$
which each take the form $R_{i}=(x_{i},y_{i},w_{i},h_{i},\alpha,\beta,p_{i})$,
where $(x_{i},y_{i})$ denotes the center, $w_{i}$ and $h_{i}$ are the width
and height of the tightest axis-aligned external box. $\alpha$ and $\beta$ are
the are the offsets relative to the midpoints of the top and right sides of
the external rectangle. $p_{i}$ is an object score. The proposed rotated
region will then be cropped by rotated roi [33] in feature space. The proposal
is then fed to Oriented R-CNN which is a CNN followed by another
classification head and a bounding box regression head rectifying the spatial
location. We denote the output of of second stage is
$\left\\{{R_{i}^{*}}\right\\}_{i=1}^{N}$ and
$R_{i}^{*}=(x_{i}^{*},y_{i}^{*},w_{i}^{*},h_{i}^{*},\theta_{i}^{*},p_{i}^{*},c_{i}^{*})$
where $(x_{i}^{*},y_{i}^{*})$ denotes the center, $w_{i}^{*}$ and $h_{i}^{*}$
are the width and height and $\theta_{i}^{*}$ the rotation angle of the final
predicted rotated box. $p_{i}^{*}$ is the second stage object score and
$c_{i}^{*}$ represents the classification score.

###  3.2 Learning Rotated Region Proposal

We develop a co-training scheme that utilizes both source and target datasets.
We denote $\left\\{{B_{j}^{s}}\right\\}_{j=1}^{m}$ to be the $m$ rotated
bounding boxes of a source image where
$B_{j}^{s}=(x_{j}^{s},y_{j}^{s},w_{j}^{s},h_{j}^{s},\theta_{j}^{s})$ is a
rotated box. For the target dataset, we have $n$ axis-aligned boxes
$\left\\{{B_{j}^{t}}\right\\}_{j=1}^{n}$ on a target image where
$B_{j}^{t}=(x_{j}^{t},y_{j}^{t},w_{j}^{t},h_{j}^{t})$ is a axis-aligned box.

In the first stage, a set of $N$ oriented regions are proposed where each
proposal $R_{i}=(x_{i},y_{i},w_{i},h_{i},\alpha,\beta,p_{i})$ is assigned a
ground truth label $B_{\sigma(i)}$ based on intersection over union (IoU)
matching, where the assignment $\sigma(i)$ and matching score $\tau(i)$ are

$\displaystyle\sigma(i)=\operatorname*{arg\,max}_{j\in\left\\{{1,\ldots,m}\right\\}}iou((x_{i},y_{i},w_{i},h_{i}),P(B_{j})),$
(1)
$\displaystyle\tau(i)=\max_{j\in\left\\{{1,\ldots,m}\right\\}}iou((x_{i},y_{i},w_{i},h_{i}),P(B_{j})).$
(2)

Since in first stage, $(x_{i},y_{i},w_{i},h_{i})$ represents the tightest
external axis-aligned box instead of the rotated region itself, we need a
transformation function $P$ to project the ground truth to axis-aligned box
for source rotated dataset. It is important to notice that $P(B_{j})$ is
strictly equal or larger than canonical axis-aligned box of the object.
Therefore, for target axis-aligned dataset, we need to enlarge $B_{j}^{t}$.
Here we formally define the transformation function

$\displaystyle P(B)=\begin{cases}(x_{min},y_{min},x_{max},y_{max})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{ for $B=(x,y,w,h,\theta)$,
}\\\ (x_{min}-\gamma w_{i},y_{min}-\gamma h_{i},x_{max}+\gamma w_{i},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}y_{max}+\gamma h_{i})\text{ for
$B=(x,y,w,h)$, }\end{cases}$ (3)

where $\gamma$ is simply an enlargement factor that we can tune. The loss for
this RPN is defined as follows

$\displaystyle L_{\mathcal{S}}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}-\mathbf{1}_{{\tau(i)}\geq
0.5}\log(p_{i})+l_{1}(R_{i},B_{\sigma(i)}^{s})$ (4)
$~{}~{}L_{\mathcal{T}}=\frac{1}{N}\sum_{i=1}^{N}-\mathbf{1}_{{\tau(i)}\geq
0.5}\log(p_{i})\\\ +l_{1}((x_{i},y_{i},w_{i},h_{i}),P(B_{\sigma(i)}^{t})),$
$\displaystyle L$
$\displaystyle=L_{\mathcal{S}}+L_{\mathcal{T}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(5)

where the loss for both target and source examples are composed by a binary
cross entropy (BCE) loss and an $l_{1}$ regression loss on spatial outputs.
The label for the BCE is determined by whether the proposal has an overlap
larger than 0.5 with any projected ground truth. For the source dataset, we
can compute ground truth $(\alpha^{s},\beta^{s})$ from $\theta^{s}$ which is
the regression for the rotation representation. However, for the target
dataset, we only compute regression loss for $(x_{i},y_{i},w_{i},h_{i})$. The
model must learn the rotation knowledge from the source dataset. To simplify
the mathematical notation, we omit classification loss as it is less relevant
to our contribution than object score.

###  3.3 Learning Rotated R-CNN

In the second stage, the Oriented R-CNN takes in a subset of proposals
generated in first stage and make final prediction
$R_{i}^{*}=(x_{i}^{*},y_{i}^{*},w_{i}^{*},h_{i}^{*},\theta_{i}^{*},p_{i}^{*},c_{i}^{*})$.
The bounding box regression is less important because of two reasons. Firstly,
the cropped region proposal is already an approximate detection. The goal of
regression of this stage is to fine-tune. Secondly, we choose to use class-
agnostic rotated bounding box regression here as the model is able to learn
general bounding box regression from the source dataset. The classification of
the second stage is also straight forward to train because canonical axis-
aligned detector fundamentally identical in that respect. The most important
and challenging aspect is to produce an accurate object score $p_{i}^{*}$,
which is a direct result of ground truth assignment process. We first
formulate the assignment process for the source dataset as

$\displaystyle\sigma_{s}^{*}(i)=\operatorname*{arg\,max}_{j\in\left\\{{1,\ldots,m}\right\\}}iou((x_{i},y_{i},w_{i},h_{i},\theta_{i}),B_{j}^{s}),$
(6)
$\displaystyle\tau_{s}^{*}(i)=\max_{j\in\left\\{{1,\ldots,m}\right\\}}iou((x_{i},y_{i},w_{i},h_{i},\theta_{i}),B_{j}^{s}).$
(7)

Note that the assignment process is based on the proposal instead of the final
refined prediction. The difference from the first stage is that here we use
accurate rotated ground truth $B_{j}^{s}$ against $R_{i}$ instead of the
external enclosing axis-aligned box. We compute $\theta$ of the first stage
with $(\alpha,\beta)$ inline with [33]. This works appropriately with the
source dataset. However, it creates a challenge for the target dataset which
lacks rotation information. If we follow the identical assignment process as
the source dataset, then a close-to-correct proposal might be classified as
negative example if the overlap between the rotated bounding box and axis-
aligned bounding box is low. Consequently, false negative rate will be
increased, reducing the recall of such training scheme. Therefore, we propose
two additional strategies, projection assignment and heuristic selection.

Projection Assignment. To avoid the aforementioned increased false negative
problem, we can project the predicted $(x_{i},y_{i},w_{i},h_{i},\theta_{i})$
as an axis-aligned box and then compute the overlap between projected box and
axis-aligned ground truth. Then we have,

$\displaystyle\sigma_{t}^{*}(i)=\operatorname*{arg\,max}_{j\in\left\\{{1,\ldots,m}\right\\}}iou(P(x_{i},y_{i},w_{i},h_{i},\theta_{i}),P(B_{j}^{t})),$
(8)
$\displaystyle\tau_{t}^{*}(i)=\max_{j\in\left\\{{1,\ldots,m}\right\\}}iou(P(x_{i},y_{i},w_{i},h_{i},\theta_{i}),P(B_{j}^{t})).$
(9)

where $P$ is the transformation defined in previous section. By doing this, we
reduce the false negative rate as rotated boxes can be correctly assigned to
the axis-aligned counterpart and increase the recall of the model. However,
consider a rotated box that has a rotation angle of $\pi-\theta$ or $-\theta$.
It will have the same projected axis-aligned box as the one with $\theta$ and
these two are completely different boxes. Therefore, this process might
misclassify negative examples as positive, which results in reduction in
precision. The losses are:

$\displaystyle L_{\mathcal{S}}^{*}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}-\mathbf{1}_{{\tau_{s}^{*}(i)}\geq
0.5}\log(p_{i}^{*})+l_{1}(R_{i}^{*},B_{\sigma(i)}^{s}),$ (10) $\displaystyle
L_{\mathcal{T}}^{*}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}-\mathbf{1}_{{\tau_{t}^{*}(i)}\geq
0.5}\log(p_{i}^{*})$ (11) $\displaystyle L^{*}$
$\displaystyle=L_{\mathcal{S}}^{*}+L_{\mathcal{T}}^{*}.$ (12)

Heuristic Selection. Aside from the projection assignment, we also propose a
parallel strategy. We use the same assignment rule as the source dataset 7.
However, we evaluate how reliable each assignment is based on the aspect ratio
and area size of axis-aligned ground truth and try to learn more from the
reliable ground truth. For a particular axis-aligned ground truth box, we can
compute its aspect ratio $r_{j}$ defined by
$r_{j}=\max(w_{j},h_{j})/\min(w_{j},h_{j})$. Larger aspect ratio implies the
ground truth box is more reliable. In addition, we can find an area threshold
$a_{threshold}$ a minimum un-occluded stereotypical object area on a target
dataset. For $w_{j}*h_{j}<a_{threshold}$, it is likely an occluded box.
Following these heuristics, we define a binary reliability switch and loss for
target dataset as

$\displaystyle g_{i}$ $\displaystyle=\begin{cases}1\text{ if
($r_{\sigma^{*}(i)}>3$ or
$w_{\sigma^{*}(i)}*h_{\sigma^{*}(i)}<a_{threshold}$})\\\ 0\text{
otherwise}\end{cases}$ (13) $\displaystyle L_{\mathcal{T}}^{*}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}-g_{i}\mathbf{1}_{{\tau_{t}^{*}(i)}\geq
0.5}log(p_{i}^{*}).$ (14)

and we choose to mask unreliable examples. Despite heuristic selection is less
general than projection assignment, this can be beneficial for confined
industry applications where the aspect ratio of the target object is known
such as detecting a bottle on the conveyor belt.

##  4 Experiments and Discussion

In this section, we first outline our implementation details and experimental
setup including the datasets. Then we show the effectiveness of our method
through ablation studies followed by the comparison with different target and
source datasets. Finally, we include some qualitative results.

###  4.1 Experimental setup

Source Datasets. Source datasets, in this work, are datasets strongly labelled
with box orientations. The model has access to the source datasets to learn
the task of rotated box detection. We choose a variety of source datasets
including DOTA [32], COCO [13] and Fresh-produce dataset 1 where DOTA [32] is
a popular satellite imagery dataset with rotated annotations and COCO [13] is
a popular general object detection dataset. In COCO, rotated bounding box
ground truth can be generated by finding the minimum enclosing rectangle on
each instance segmentation mask ground truth. Fresh-produce dataset 1 is a
challenging dataset with high object density and heavy occlusion. The number
of images and instances are shown in table 1. There are three long-shape
subsets including banana, cucumber and carrot.

Target Datasets. The training subset of the target datasets only contains
axis-aligned bounding box ground truth which is used for the model to learn.
Alternatively, the validation and test subsets contain rotated ground truth to
evaluate the performance of the model. We select HRSC2016 [16], SSDD [27],
cucumber and carrot datasets 1 as our target datasets to cover a variety of
domain permutations including satellite, single channel, natural and various
object density.

Table 1: Fresh-produce datasets.  Class | Number of images | Number of instances
---|---|---
banana | 158 | 7391
cucumber | 48 | 2036
carrot | 47 | 4647
Table 2: Ablation Studies of the proposed methods. We use DOTA as source and fresh cucumber as target dataset. This table decoupled the effectiveness of rotated cotraining, rpn projection loss, single class, heuristic selection, projection assignment and axis-aligned pretraining using other objects. Rotated Cotraining | RPN Proj. | Single Class | H. Select. | P. Assign. | Pretraining |  AP50$\uparrow$
---|---|---|---|---|---|---
- | - | - | - | - | - | 0.491
✓ | - | - | - | - | - | 0.542
✓ | ✓ | - | - | - | - | 0.581
✓ | ✓ | ✓ | - | - | - | 0.633
✓ | ✓ | ✓ | ✓ | - | - | 0.666
✓ | ✓ | ✓ | - | ✓ | - | 0.664
✓ | ✓ | ✓ | - | ✓ | ✓ | 0.683
Table 3: Analysis for enlargement factor $\gamma$. We use DOTA as source and HRSC as target dataset for this analysis. $\gamma$ | 1.00 | 1.05 | 1.10 | 1.15 | 1.20 | 1.25
---|---|---|---|---|---|---
AP50$\uparrow$ | 0.791 | 0.763 | 0.756 | 0.733 | 0.718 | 0.650

Implementation Details. We pretrain our detector, Oriented-RCNN [33] using
DOTA [32]. The ship class and the images with the ships are removed from DOTA
for this paper for pretraining and its role as a source dataset, due to class
overlap with some target datasets. The main statistic we use in the paper to
evaluate our method is AP50. It calculates the average precision with IOU
threshold of $0.5$. Average precision (AP) is the area under precision and
recall curve. We utilise the mmrotate framework [36] and [33] for training. We
use a batch of $2$ images from target dataset and a batch of $2$ images from
the source dataset for a combined mini-batch of $4$. The losses and forward
propagation for two batches are computed independently, then the losses are
added and backpropagated through the network. We train the models up to $50$
epochs of target dataset. During inference, we set the non-maximum-suppression
threshold to $0.5$ instead of $0.1$ used generally for aerial datasets, as the
higher object density fresh-produce dataset contains far more object
overlapping cases. We conduct the experiments using one 2080ti GPU with 11GB
of memory. The training time depends on the dataset size, ranging from 30
minutes to 2 hours to train. The test can be done within 5 minutes for each
dataset with speed of 15 FPS, which is the same as the original detector.

###  4.2 Ablation Studies

Table 4: We evaluate KCR with dense cucumber and carrot datasets with severe occlusion as target datasets. Tight rotated ground truth is used to evaluate AP50 performance of the model in the test time. We compare KCR method with original Oriented-RCNN [33] with weak axis-aligned annotation for training and fully-supervised models trained with strong rotated ground truth. Results show that KCR enables the model to perform on par with fully-supervised model and works reasonably well under large domain gap. | Method | Source | Target Train |  AP50$\uparrow$
---|---|---|---|---
Cucumber |  original [33] | - | axis-aligned | 0.491
KCR(ours) |  DOTA [32] | axis-aligned | 0.683
KCR(ours) | banana | axis-aligned | 0.788
fully-supervised | - | rotated | 0.820
Carrot |  original [33] | - | axis-aligned | 0.560
KCR(ours) |  DOTA [32] | axis-aligned | 0.614
KCR(ours) | banana | axis-aligned | 0.723
fully-supervised | - | rotated | 0.765
Table 5: Evaluation of KCR method with HRSC2016 [16] and SSDD [27] as target datasets against tight rotated ground truth in the test. Influence of source dataset is investigated using COCO [13], COCO with rotation augmentation and DOTA [32]. We also compare KCR against original Oriented-RCNN [33], Grabcut [25] with weak axis-aligned annotation for training and fully-supervised model trained with strong rotated ground truth. We use projection assignment methods for KCR except for the ones with HS, which stands for heuristic selection. | Method | Source | Target Train |  AP50$\uparrow$
---|---|---|---|---
HRSC2016 [16] |  original [33] | - | axis-aligned | 0.175
Grabcut[25] | - | axis-aligned | 0.240
Grabcut Train | - | axis-aligned | 0.629
KCR(ours) |  COCO [13] | axis-aligned | 0.579
KCR(ours) | COCO Aug | axis-aligned | 0.783
KCR(ours) |  DOTA [32] | axis-aligned | 0.791
KCR-HS(ours) |  DOTA [32] | axis-aligned | 0.778
| fully-supervised | - | rotated | 0.903
SSDD [27] |  original [33] | - | axis-aligned | 0.432
Grabcut Train | - | axis-aligned | 0.585
KCR(Ours) |  COCO [13] | axis-aligned | 0.888
KCR(Ours) | COCO Aug | axis-aligned | 0.874
KCR(Ours) |  DOTA [32] | axis-aligned | 0.881
KCR-HS(Ours) |  DOTA [32] | axis-aligned | 0.874
| fully-supervised | - | rotated | 0.898

To tackle the weakly-supervised learning of rotated bounding box given only
axis-aligned ground truth, we build our approach progressively. In this
section, we show how the approach evolves using cucumber as our axis-aligned
training target dataset and DOTA [32] as our source dataset. We choose this
particular pair because their domain gap is large in terms object appearance,
density and occlusion severity. It is more convincing if the model is able to
learn rotation with such a large domain gap between source and target.

We first establish a baseline by training a rotated detector using our axis-
aligned cucumber training dataset and test it on rotated cucumber test set.
This baseline is the first row of table 2 with AP50 of 0.491. Then we
introduce the cotraining strategy with the source dataset without any
modification of the training scheme. That means we simply treat both source
and target datasets equally and completely follow the training scheme of [33].
This gives us AP50 of 0.542 which is an incremental improvement over the
baseline.

RPN projection 5 improves the performance to 0.581 (Table 2). Instead of
learning the wrong rotation supervision from the axis-aligned dataset, it
chooses not to learn. After that, we try the single class strategy: only use
one class label for all classes in source dataset. This effectively improves
the performance to 0.633 because the model focuses on learning the object
score and rotation angle instead of the classification problem of the source
dataset. If the target dataset is multi-class, we can apply the single class
strategy to the source dataset because the classification is not important.

The projection assignment and heuristic selection methods are implemented
separately. The projection assignment strategy as described in section 3.3
improves the performance to 0.664 since it reduces the false negative rate. It
cannot further improve performance due to the false positive assignment
problem. Heuristic selection strategy improves the model performance to 0.666.
Finally, we pretrain this model on other axis-aligned objects and improve this
performance to 0.683 which is 0.192 higher than the baseline. We take the last
row of table 2 as our final approach and apply it to permutations of target
and source datasets in section 5.

We also provide an ablation of the enlargement factor $\gamma\geq 1$. As we
can see from Table 3, $\gamma=1.00$ is the best, which is what we have used
throughout all the experiments.

Figure 3: Visualization of KCR performance against original Oriented-Rcnn
trained with weak axis-aligned annotation. The images are from test sets of
HRSC2016 [16] and SSDD [27]. We use the COCO as our source dataset which has
large domain gap from the target. The model trained with KCR methods learnt to
predict accurate rotated bounding box, which is much more precise than the
original model.  Figure 4: Distribution of Aspect ratios on 4 datasets. Figure
5: Visualization of KCR performance trained with axis-aligned only dataset.
The images are chosen from an unlabeled set to ensure generality. The model is
clearly capable of predicting tight rotated bounding box even with high object
density and occlusion.

###  4.3 Main results

In this section, we apply KCR framework to various combinations of source and
target datasets to investigate the generality of the method under different
domain gaps. We also compare KCR with a popular foreground segmentor [25].

Fresh-produce dataset. In Table 4 we can see results progressively improve
when information that is closer to desired result is introduced. We initially
evaluate our method where no knowledge combination is used, that is, for the
cucumber and carrot, we use the axis-aligned bounding boxes and allow the
network to learn incorrect $\alpha,\beta$ and $\theta$ values. In the next
iteration, we use DOTA[32] aerial satellite images for knowledge combination.
The domain gap between these images and the cucumber and carrots is large
because of the sparsity of objects in DOTA. Nonetheless, introducing rotated-
bounding box information, even with dissimilar dataset, results in a large
performance improvement for both cucumber and carrot as seen in the second row
of Table 4. The next progression combines knowledge with bananas annotated
with rotated bounding-box. This further improves the results and provides an
insight into task similarity strengthening knowledge combination. The banana
dataset offers the opportunity for the model to learn rotated detection with
severe occlusion. For reference, the last row in Table 4 is the result when
training the model directly on ground truth rotated bounding box, i.e. fully
supervised training. We note that the performance gap between our approach and
fully supervised training is small.

HRSC2016. HRSC2016 [16] is a rotated satellite imagery dataset that focuses on
ships. The objects in HRSC2016 typically have a larger aspect ratio and
occupies larger area of the image in comparison DOTA [32] targets. To prevent
class overlap, images with ships have been removed from DOTA when using as a
source dataset. We establish our baseline by using original training regime
and only axis-aligned ground truth of HRSC2016 yielding AP50 of 0.175, which
is 72.8% lower than a fully supervised result as shown in Table 5. We
investigate the performance of KCR with COCO [13], containing only natural
images, as source dataset because the domain gap between COCO and HRSC2016 is
visually large. We generate rotated bounding box ground truth from instance
segmentation masks. The result significantly improves to 0.579 when we use KCR
framework. Although the rotated bounding box is used in source dataset,
objects in COCO are typically axis-aligned, hence produce a weak rotation
training signal. We therefore rotate images by $0-180$ degrees followed by
horizontal and diagonal flip augmentation to increase the number of rotated
examples in the source dataset. As a result, AP50 rises to 0.783 with strongly
augmented COCO. Thus we show that KCR enables the transfer learning of extra
parameter under large domain gap. At last, we use DOTA dataset as source and
achieves AP50 of 0.791, which is only 11.2% lower than the fully supervised
model. The performance gap is potentially due to lack of equally long objects
with large aspect ratio in the source dataset.

SSDD. SSDD [27] is SAR dataset which also focuses on detection of ships.
Images in SSDD are single channel and vary in frequency depending on the
sensor used for acquisition. These images are commonly low resolution and
contain high frequency noise, which results in a visually large domain gap
from any source datasets. We follow the same knowledge combination strategy as
before. Using KCR, a trained rotated box model performs almost equally well
from three different source datasets (DOTA, coco and coco augmented). The
final AP50 of 0.888 is 45.6% better than the baseline and only 1% lower than a
fully supervised model.

Rotated Boxes with Grabcut. GrabCut [25] is a computer vision algorithm that
predicts a foreground initialised with a region-of-interest. This foreground
segmentation can then be used for predicting a rotated box with a variety of
heuristic algorithms. This predicted rotation can be used at inference time as
a post-processing step, transforming the axis aligned boundary box to a
rotated boundary box. At test time, this improves the baseline by a minimal
6.5%, however reduces inference throughput to 0.3 FPS. Alternatively, this
algorithm can be performed offline on the axis-aligned ground truth to produce
a noisy rotated boundary box ground truth for training. This method is
significantly more effective, improving test time performance on rotated
ground truth to 0.629 AP50 for HRSC and 0.585 for SSDD. However, KCR
outperforms training on noisy ground truth by 16.2% on HRSC and 30.3% on SSDD.

Analysis on aspect ratios. We show histograms of aspect ratio in Fig 4. The
biggest difference between an axis-aligned and rotated box happens when the
instance has high aspect ratio and rotated. Most COCO objects are neither long
nor rotated while DOTA has a fair distribution of long and rotated objects.
Performance gap between original COCO and DOTA as source dataset is bigger for
HRSC than SSDD because HRSC has a more tail heavy aspect ratio distribution.

###  4.4 Qualitative Results

We visualize performance of KCR on HRSC2016 [16] and SSDD [27] in Figure 3.
The model has successfully learnt to predict accurate rotated bounding box
with weak axis-aligned annotation. The source dataset we use to gain rotation
knowledge is COCO [13], which has a large domain gap distant from the target
dataset. The rotated prediction from our framework produces boxes much tighter
than a model which was trained on and predicts axis-aligned boxes.

Fresh-produce dataset is more challenging due to higher density of objects
with severe occlusion. We visualize the performance of KCR with cucumber and
carrot as target datasets and a banana dataset as the source dataset supplying
knowledge of rotation. The depicted images are from a separate unlabeled set
to ensure generality. As shown in Figure 5, the model is clearly capable of
predicting tight rotated bounding boxes in a challenging scenario which is
core contribution of this paper. The model is able to complete the task with
high precision and recall in a scene with frequent and extremely occlusions,
various lighting conditions and different object sizes. For the two images in
the left column, the model detects almost every object.

##  5 Conclusion

Rotated detection improves the performance of downstream tasks by reducing the
overall area of the enclosing box, improving the foreground to background
ratio. This is particularly important for scenes with high density of targets
and complex occlusions. However, most existing datasets only provide axis-
aligned annotation, with the lack of the capability to annotate rotated boxes.
In this paper, we address this problem by proposing KCR, which is a novel
knowledge combination training scheme that only requires axis-aligned
annotation for the target object class to train the model. At inference time,
the model predicts accurate rotated bounding boxes on par with fully-
supervised approach. This approach will enable the detector to predict an
extra but crucial parameter. We believe this work can greatly extend the use
case of rotated object detection by reducing annotation costs.

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# Modulator-free coherent-one-way quantum key distribution

George L. Roberts  Toshiba Research Europe Ltd, 208 Cambridge Science Park,
Milton Road,
Cambridge CB4 0GZ, United Kingdom  Cambridge University Engineering
Department, 9 J J Thomson Avenue,
Cambridge, CB3 0FA, United Kingdom  Marco Lucamarini  Toshiba Research Europe
Ltd, 208 Cambridge Science Park, Milton Road,
Cambridge CB4 0GZ, United Kingdom  James F. Dynes  Toshiba Research Europe
Ltd, 208 Cambridge Science Park, Milton Road,
Cambridge CB4 0GZ, United Kingdom  Seb J. Savory  Cambridge University
Engineering Department, 9 J J Thomson Avenue,
Cambridge, CB3 0FA, United Kingdom
Zhiliang Yuan<EMAIL_ADDRESS>Toshiba Research Europe Ltd,
208 Cambridge Science Park, Milton Road,
Cambridge CB4 0GZ, United Kingdom  Andrew J. Shields  Toshiba Research Europe
Ltd, 208 Cambridge Science Park, Milton Road,
Cambridge CB4 0GZ, United Kingdom

###### Abstract

Time-bin encoding is an attractive method for transmitting photonic qubits
over long distances with minimal decoherence. It allows a simple receiver for
quantum key distribution (QKD) that extracts a key by measuring time of
arrival of photons and detects eavesdropping by measuring interference of
pulses in different time bins. In the past, coherent pulses have been
generated using a CW laser and an intensity modulator. A greatly simplified
transmitter is proposed and demonstrated here that works by directly
modulating the laser diode. Coherence between pulses is maintained by a weak
seed laser. The modulator-free source creates time-bin encoded pulses with a
high extinction ratio (29.4 dB) and an interference visibility above 97 %. The
resulting QKD transmitter gives estimated secure key rates up to 4.57 Mbit/s,
the highest yet reported for coherent-one-way QKD, and can be programmed for
all protocols using weak coherent pulses.

Quantum key distribution (QKD) uniquely allows two parties to exchange secure
keys with secrecy guaranteed by the fundamental laws of physics
gisin_quantum_2002 . Its potential for real-world applications has stimulated
a large amount of progress in developing implementation technologies
Diamanti_practical_2016 . Over optical fiber links, QKD has been demonstrated
to distribute quantum keys with rates exceeding 1 Mbit/s dixon_continuous_2010
, over a hundred kilometers of distance Korzh_provably_2015 ; frohlich_long-
distance_2017  and/or in the presence of strong classical signals in the same
fiber patel_quantum_2014 . The technology is being extensively tested in
installed fiber network environments Qiu_quantum_2014 . Moreover, satellite
QKD and quantum repeater technologies are also being pursued to extend
communication to the global scale Diamanti_practical_2016 .

Since the inception of QKD in 1984 with the Bennett-Brassard (BB84) protocol
Bennett_Quantum_1984 , a number of diverse implementations have been proposed.
Two broad classes of protocol that share popularity are discrete variable and
distributed phase reference protocols Scarani_security_2009 . The protocols
within these classes have a variety of requirements for transmitters, for
example phase modulation and/or intensity modulation. Each provides different
benefits, whether it is high bit rate, long achievable distance, a rigorous
security proof or simplicity in experimental implementation. It is therefore
highly beneficial to develop a versatile QKD transmitter that can operate
different QKD protocols. Promising work has recently been demonstrated by
combining a laser with a number of external phase and intensity modulating
elements sibson_integrated_2017 .

The modulator-free transmitter proposed by Yuan et al yuan_directly_2016  has
a number of attractive properties for phase modulation. The light source uses
a pair of laser diodes in an optical injection configuration. A master laser
provides phase modulation, and randomization when required, while the slave
laser is responsible for generating short optical pulses. The light source has
successfully been demonstrated for two important phase-encoded QKD protocols,
BB84 and differential phase shift (DPS) Inoue_differential_2002 . However, the
suitability of a modulator-free design to offer high extinction ratio
intensity modulation in quantum communications is yet to be explored.

Here, we tackle this issue by implementing the coherent-one-way (COW) QKD
protocol with a modulator-free transmitter. This transmitter allows us to
achieve record-breaking key rates with quantum bit error rates (QBERs) below 1
% and visibilities over 97 %. Using a realistic finite key-size scenario, we
can distribute keys from Alice to Bob at losses between 1.5 dB and 30 dB,
equivalent to 7.5 km and 150 km of standard single mode optical fiber.

The COW protocol Korzh_provably_2015  uses time-bin encoding to share a key
between two parties. Security of the key is ensured by Alice maintaining a
fixed coherence between pulses. Bob can measure the interference visibility
between adjacent pulses using an interferometer and infer the presence of an
eavesdropper, Eve, by a break in the coherence. It has been used in a real-
fiber system to transmit a secure key between two parties separated by 307 km
- the longest distance for any two party quantum protocol Korzh_provably_2015
. This is possible because the time-bin encoding produces a lower QBER than
other protocols, for example BB84 dixon_continuous_2010 . The main downside to
the protocol is that whilst security against a number of attacks has been
demonstrated, no comprehensive security proof exists for all families of
attack Scarani_security_2009 , which could mean the protocol is vulnerable to
an all-powerful Eve.

In the COW protocol, Alice prepares two values of a logical bit using empty or
full time bins:
$\rvert$$\beta_{0}$$\rangle$=$\rvert$$\alpha$$\rangle$$\rvert$0$\rangle$ and
$\rvert$$\beta_{1}$$\rangle$=$\rvert$0$\rangle$$\rvert$$\alpha$$\rangle$,
where $\rvert$0$\rangle$ is the vacuum state and $\rvert\alpha\rangle$
represents a coherent state of light loudon_quantum_2000  with intensity
$\mu$=$|\alpha|^{2}$. For these requirements, the transmitter used in this
protocol must be able to modulate intensity whilst maintaining coherence. Bob
decodes the signals by measuring their arrival times with a single photon
detector (SPD). He also takes a portion of the received photons in order to
measure the coherence between adjacent time bins, allowing him to test the
channel against unauthorized external intrusions. This measurement is
performed by overlapping two consecutive optical pulses on a beam splitter and
measuring the resultant interference visibility. This is possible for the
pulse sequence $\rvert$$\beta_{1}$$\rangle$$\rvert$$\beta_{0}$$\rangle$. To
increase the number of consecutive non-empty pulses, Alice also prepares a
decoy sequence
$\rvert$$\beta_{2}$$\rangle$=$\rvert$$\alpha$$\rangle$$\rvert$$\alpha$$\rangle$.
This reduces the amount of bits Bob needs to collect before he can accurately
measure the visibility, which is important in the finite key-size scenario.
Moreover, the decoy sequence is used as a security feature, as Eve does not
know whether she is attacking a logical bit or a decoy sequence. During
sifting, Alice informs Bob when she sent decoy pulses and Bob tells Alice
whether he measured the arrival time or the visibility of the optical pulses.

Figure 1 shows the experimental setup. The quantum transmitter (Alice) is
configured for time-bin encoding using a slave laser that is optically seeded
by a master laser. An optical attenuator attenuates the non-empty time bins to
around 0.1 photons per pulse before transmitting them through the quantum
channel, implemented by an optical attenuator, to the quantum receiver (Bob).
Bob uses a 90:10 beamsplitter to passively route most of the photons to a
superconducting nanowire detector (SPD1) for arrival time measurements. The
remaining 10 % are fed into an unbalanced Mach-Zehnder interferometer (UMZI)
for measuring the phase coherence with a second superconducting nanowire
detector (SPD2). The UMZI is based on a planar lightwave circuit with a
differential delay of 500 ps and has a loss of 3 dB. A built-in heater allows
direct control of the phase delay across one arm. The superconducting nanowire
SPDs used feature a dark count rate (DCR) of 10 Hz, alongside an efficiency of
34 % at a wavelength of 1550 nm, allowing us to reach long distances.

Intensity-modulated gain-switched pulses are produced through electrical
modulation of the slave laser at 3.3 V liu_Modulator-free_2014 ;
He_amplification_2016 . A DC bias above the lasing threshold is applied to the
master laser to ensure the phase is coherent when it is injected into the
slave laser. We use a wavelength-tunable, continuous-wave fiber laser as the
master laser and a semiconductor distributed feedback (DFB) laser diode as the
slave. The slave laser is kept at room temperature with a free-running
wavelength of 1550.1 nm. The master laser is wavelength-tuned to give a
maximum coherence transfer, which occurs when both lasers have the same free-
running wavelength. The output pulses from this system have a pulse width of
70 ps, which is much smaller than the inverse modulation frequency, and a
spectral width of 0.10 nm. For time-bin encoding, a pseudo-random number
generator creates a repeated 512-bit sequence, generating decoy sequences with
a probability of 1 % and signal sequences for the remaining time bins equally
distributed between the bit values 0 and 1. The pattern generated is applied
to gain-switch the slave laser at a clock rate of 2 GHz, therefore
implementing a COW transmitter at an effective bit-rate of 1 GHz. A high
intensity extinction ratio of 29.4 dB can be achieved between non-empty and
empty pulses, thus ensuring a low encoding error in the time basis.

Figure 1: System schematics. The master laser in Alice injects CW light into
the slave laser, which produces low-jitter gain-switched pulses. These are
then attenuated to the single photon level before being transmitted through
the quantum channel to Bob. SPD1 detects the arrival time of the photons, from
which the key is generated, and SPD2 is used to measure the phase coherence.
BS=beamsplitter; UMZI=unbalanced Mach-Zehnder interferometer; att=attenuator.

Optical injection ensures there is coherence among the gain-switched slave
pulses, as they all inherit the phase of the CW master laser. The injected
light transfers the coherence, thereby suppressing the randomness of the phase
that would occur if pulses were triggered by spontaneous emission
Comandar_quantum_2016 . To illustrate the physical principle, we gain-switch
the slave laser to produce a 2 GHz pulse train and replace the SPD2 in Fig. 1
with an optical power meter to measure the interference fringe visibility,
defined as

$V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$ (1)

where Imax and Imin are the average pulse intensities for constructive and
destructive interference respectively. The attenuator in Fig. 1 is set to
maximum transmission, while a second attenuator (not shown) is used to vary
the seed optical power into the slave laser. The interference visibility
increases monotonically with seed power, as a result of the increasing
dominance of the injected light over spontaneous emission in the slave laser
cavity. The visibility saturates at 99.78 % with a seed power of 216 $\mu$W.
In order to achieve a visibility of 99 %, a modest 12 $\mu$W of seed power is
sufficient. In the subsequent QKD experiment, we use a seed power of 50 $\mu$W
to ensure the pulses are sub-100 ps.

The quantum transmitter and receiver are linked via a short optical fiber and
extra attenuation is applied to simulate the loss of the quantum channel. The
transmitting photon flux is set to 0.1 photons per non-empty pulse at the
output of the transmitter. At the lowest attenuation we decrease the photon
flux to 0.07 photons per pulse to minimize time-jitter effects, as described
later. In the QKD experiment, two channels of a digitizer with 100 ps time
resolution simultaneously record the arrival times of single photons at the
photon detectors (SPD1 and SPD2). Bob uses a 90:10 beamsplitter to passively
direct most of the photons to SPD1, where he sifts the key by measuring time-
bins. The sifting loss here is minimal, caused only by the small portion of
photons routed through the phase coherence measurement path. An example
histogram measured by SPD1 is shown in Fig. 2(a), giving a QBER of less than 1
%. SPD2 is placed at the destructive output port of the interferometer to
enable an accurate measurement of the visibility. To highlight the
interference effect, we show in Fig. 2(c) an example measurement from the
constructive output port of the interferometer. The height of constructive
peaks is approximately four times that of the non-interfering peaks, as
expected from first-order optical interference.

Figure 2: Detected signals. Complementary signals received by Bob in a) the
time basis; b) the destructive arm of his interferometer; c) the constructive
arm of his interferometer, measured using the SPDs. The transmitted key for
these patterns is $\rvert$$\beta_{0}$$\rangle$, $\rvert$$\beta_{0}$$\rangle$,
$\rvert$$\beta_{1}$$\rangle$, $\rvert$$\beta_{2}$$\rangle$,
$\rvert$$\beta_{1}$$\rangle$, $\rvert$$\beta_{1}$$\rangle$,
$\rvert$$\beta_{0}$$\rangle$, $\rvert$$\beta_{1}$$\rangle$, where each logical
bit is separated by a vertical grey line. Data is acquired for 60 s in a
quantum channel with 15 dB loss, with Alice transmitting 0.1 photons per
pulse.

Each QKD session is continued until over 2$\times$107 counts are collected in
the time basis. The QBER and visibility are collected alongside the number of
counts in each arm. The key rates are calculated using the finite key size
analysis derived by Korzh et al Korzh_provably_2015  with a total security
parameter of $\varepsilon_{QKD}=10^{-10}$. The key rate dependence on
visibility and number of photons per pulse, $\mu$, is given by

$\begin{split}\zeta&=(2V-1)\times\exp(-\mu)\\\
&-2\left\\{\left[1-\exp(-2\mu)\right]V(1-V)\right\\}^{1/2}.\end{split}$ (2)

The extracted key length is then calculated using

$\begin{split}l&=n\left[1-Q-(1-Q)\textrm{h}\left(\frac{1-\zeta}{2}\right)\right]-7\left[n\log_{2}(\beta^{-1})\right]^{1/2}\\\
&-f_{IR}\times\textrm{h}(Q)\times
n-\log_{2}\left(\frac{1}{2\varepsilon_{cor}\beta^{2}}\right),\end{split}$ (3)

where n is the block size used for post processing, Q is the QBER, h is the
binary entropy function truncated to unity at input values over 0.5, $\beta$
is optimised at $\varepsilon_{QKD}$/4, fIR is the efficiency of information
reconciliation and $\varepsilon_{cor}$ is the probability with which the key
is incorrect. The total measurement time increases with channel attenuation,
although only 600 s are required at 30 dB channel loss to collect the required
number of counts.

Figure 3: Experimental (symbols) and simulated (lines) key rates and
associated QBERs and visibilities. COW protocol with a finite key-size
analysis Korzh_provably_2015 .

Fig. 3(a) shows the estimated secure key rate as a function of channel
attenuation. This is the first time that megabit per second estimated key
rates have been shown using the COW protocol. These key rates are extracted in
a finite key-size scenario, attaining 4.57 Mbit/s at 1.5 dB of attenuation. As
the channel attenuation increases, the secure key rate decreases
exponentially. At 20 dB of optical attenuation, equivalent to 100 km of
ordinary optical fiber (0.2 dB/km loss), a secure key rate of 127.8 kbps is
delivered. This rate is ten times higher than that measured by Korzh et al
Korzh_provably_2015  using the COW protocol at similar attenuations. We
attribute this enhanced performance to the lower QBER enabled by our source,
alongside high efficiency detectors.

We also plot the QBER and the interference visibility in Fig. 3(b). These
parameters give a direct evaluation of the performance of our light source as
a quantum transmitter. Because the single photon detectors have negligible
DCRs, we do not expect a strong variation of QBER across the entire range of
the channel attenuation. This is indeed the case for attenuations equal to and
above 20 dB, where the QBER is measured at below 0.15 %. This is low relative
to other QKD protocols, which achieve QBERs of around 4 % at similar distances
dixon_continuous_2010 ; Lucamarini_efficient_2013 . At the lowest channel
attenuation of 1.5 dB, the QBER increases to 0.78 %. We attribute this QBER
increase to the deterioration of the timing jitter performance of the
superconducting nanowire detectors at high count rates, where the jitter
increases from 40 ps to 90 ps. We also increase the time-bin width on the
digitizer at short distances to ensure all counts are measured. This
deterioration causes an overlap between the detected time-bins, as shown in
Fig. 2(a), creating an ambiguity in the bit value of a photon.

The interference visibility does not suffer from the time jitter deterioration
because the count rate of SPD2 is 30 times lower than SPD1. As shown in Fig.
2(b) and (c), the detection peaks are well separated from each other. We
measure a visibility of 97.81 % at 10 dB attenuation, illustrating high
quality coherence transfer to the intensity-modulated pulses of the slave
laser. This value is lower than the master laser visibility because the direct
intensity modulation slightly weakens the indistinguishability among optical
pulses due to the limited bandwidth of the slave laser. While our simulations
show that improvement of the visibility would only entail a relatively small
increase in the secure key rates, there is potential to reach far higher
modulation rates using different slave laser diodes. Transmission at 10 Gbit/s
has been shown in classical communications by using a gain-switched vertical-
cavity surface-emitting laser with optical injection locking Lin_beyond-
bandwidth_2010 .

In summary, we have successfully demonstrated the suitability of a modulator-
free QKD transmitter for the COW protocol. This system has produced estimated
secure key rates between 4.57 Mbit/s to 6.38 kbit/s over equivalent distances
of 7.5 km to 150 km. The lack of external modulators reduces both the system
size and complexity. An exciting prospect opened up by this work is the
potential for implementation in a multi-protocol network. Current work towards
this has used bulky systems with a number of active components. The system
presented in this work would enable a single transmitter to quickly switch
between protocols depending on a client’s requirement on bit-rate, distance or
security. The wide range of functionalities offered by this transmitter,
namely amplitude and phase modulation with on-demand phase randomization, mean
that newly developed protocols could be easily adopted with firmware updates.
An example of this would be an extension to the COW protocol that incorporates
block-wise phase randomization to offer unconditional security, similar to
work done for the DPS protocol Tamaki_unconditional_2012 .

## Acknowledgment

G. L. R. Acknowledges financial support via the EPSRC funded CDT in Integrated
Photonic and Electronic Systems and Toshiba Research Europe Limited.

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