$\mathcal{P}\left(A_{3}\right)=NP_{x}(k\geqslant 2)P^{N-1}_{x}(0)$.

  * •

$A_{4}$: Alice sends one two-photon pulse, at least one single-photon pulse,
and vacuum states, with probability
$\mathcal{P}\left(A_{4}\right)=NP_{x}(k\geqslant
2)\left(\left(P_{x}(0)+P_{x}(1)\right)^{N-1}-P^{N-1}_{x}(0)\right)$

where $\\{P_{x}(k)\\}$ are the PDS coefficients from Eq. (27) for $x=\mu$ or
the QDS coefficients from Eq. (53) for $x=\eta$.

Assuming no coherence in photon number basis, Bob’s cheating probability can
be upper-bounded individually for each of the four cases PJ+:natcomm14 . In
case $A_{1}$, Bob’s optimal strategy involves declaring a random bit
$b^{\prime}$, which will make him successfully bias the coin towards his
desired outcome with probability
$\mathcal{P}\left(b^{\prime}|A_{1}\right)=1/2$. In case $A_{2}$, Bob’s optimal
strategy consists in performing a Helstrom measurement, which is successful
with probability $\mathcal{P}\left(b^{\prime}|A_{2}\right)=y$. In case
$A_{3}$, Bob’s optimal cheating strategy also reads
$\mathcal{P}\left(b^{\prime}|A_{3}\right)=y$. In case $A_{4}$, an optimization
must be performed to find the best set of discrimination measurement operators
within the full spectrum of conclusive and inconclusive measurements. This
allows to upper bound his cheating probability as
$\mathcal{P}\left(b^{\prime}|A_{4}\right)\leqslant-2y^{2}+4y-1$ PJ+:natcomm14
.

Summing the contributions from all four cases, and assuming that Bob can cheat
with probability $1$ in all other cases, provides the final upper bound:

$\mathcal{P}_{B}^{bias}\leqslant\sum_{i=1}^{4}\mathcal{P}\left(A_{i}\right)\mathcal{P}\left(b^{\prime}|A_{i}\right)+\left[1-\sum_{i=1}^{4}\mathcal{P}\left(A_{i}\right)\right]\times
1.$ (65)

####  I.2.2 With photon number coherence

Extending the security proof to account for photon number coherence in the RE-
pumped QDS is challenging: unlike in the previous subsection, the security
analysis cannot be decomposed into independent security analyses that are
conditioned on the outcome of a photon number measurement. Dishonest Bob can
indeed exploit the presence of coherence to perform more general
discrimination attacks.

We note that Bob’s aim is to discriminate between the following two states
sent by Honest Alice with equal probabilities, in such a way that the outcome
of the flip is optimally biased towards $x=0$:

$\displaystyle\sigma_{0}^{\left(y,\eta,\phi\right)}$
$\displaystyle=\frac{1}{2}\left(\sigma_{00}^{\left(y,\eta,\phi\right)}+\sigma_{10}^{\left(y,\eta,\phi\right)}\right)$
(66) $\displaystyle\sigma_{1}^{\left(y,\eta,\phi\right)}$
$\displaystyle=\frac{1}{2}\left(\sigma_{01}^{\left(y,\eta,\phi\right)}+\sigma_{11}^{\left(y,\eta,\phi\right)}\right).$

There exists a broad spectrum of discrimination measurements, which may be
characterized by a set of parameters $\\{p_{conc},p_{corr}\\}$ Berlin et al.
(2009). The first parameter $p_{conc}$ gives the probability that implementing
the POVM will yield a conclusive outcome, i.e. that one state from the pre-
agreed set of states will be identified. The second parameter $p_{corr}$ gives
the probability that this outcome is correct, i.e. that the identified state
is indeed the one that was sent. Fully conclusive measurements have
$p_{conc}=1$, but usually display a non-unit probability of being correct
$p_{corr}<1$ (depending on the set of states). Other measurements will
increase $p_{corr}$ by allowing some probability of the measurement being
inconclusive, i.e. $p_{conc}<1$.

Of course, Bob’s optimal discrimination strategy will involve optimizing over
these parameters for all $N$ states, which is beyond the scope of this paper.
Nevertheless, we show here how one powerful attack exploiting inconclusive
measurement operators, known as unambiguous state discrimination (USD) Rudolph
et al. (2003); Lü (2021), can already provide Dishonest Bob with a cheating
advantage over states which do not exhibit photon number coherence under XX
pumping. Intuitively, a USD POVM will return an outcome that is always correct
($p_{corr}=1$), at the risk of getting an inconclusive outcome with some
probability $p_{conc}<1$. Since this attack yields inconclusive outcomes
instead of erroneous ones, Bob can repeatedly perform the same attack on each
state until the outcome is conclusive, in which case he has identified Alice’s
state without any error. If, after $(N-1)$ states, the attack is still
inconclusive, he may then perform a conclusive minimum-error discrimination
measurement on the last state, with $p_{corr}<1$ and $p_{conc}=1$, whose
maximum success probability is given by the Helstrom bound Barnett and Croke
(2009).

To derive Bob’s cheating probability in this case, we must first justify that
USD is possible for our set of states (i.e., that there exists a USD attack
which yields a non-zero probability of successfully identifying which state
was sent by Alice). For this, it suffices to note that the kernels associated
with the states $\sigma_{0}^{\left(y,\eta,\phi\right)}$ and
$\sigma_{1}^{\left(y,\eta,\phi\right)}$ living in the full photon space are
both non-zero Rudolph et al. (2003). Although this is enough to derive an
upper bound on the success of the USD measurement, i.e. on the value of
$p_{conc}$, we must find a bound that is tight in order to provide a
meaningful comparison with the over-estimated bounds from Eq. (65). For this,
we once again use the semidefinite programming techniques introduced in
Section E. We extend the pure state treatment from Eldar (2003) and the
discrimination problem from Eldar et al. (2003) to recast Dishonest Bob’s
search for the optimal USD strategy:

$\displaystyle\max$
$\displaystyle\frac{1}{2}\left[\,\mathrm{Tr}\left(M_{0}\sigma_{0}^{\left(y,\eta,\phi\right)}\right)+\,\mathrm{Tr}\left(M_{1}\sigma_{1}^{\left(y,\eta,\phi\right)}\right)\right]$
(67) s.t.
$\displaystyle\,\mathrm{Tr}\left(M_{0}\sigma_{1}^{\left(y,\eta,\phi\right)}\right)=0$
$\displaystyle\,\mathrm{Tr}\left(M_{1}\sigma_{0}^{\left(y,\eta,\phi\right)}\right)=0$
$\displaystyle M_{0}+M_{1}+M_{inc}=\mathbb{1}$ $\displaystyle
M_{0},M_{1},M_{inc}\geqslant 0,$

where $M_{0}$ and $M_{1}$ are the POVM operators which identify the states
$\sigma_{0}^{\left(y,\eta,\phi\right)}$ and
$\sigma_{1}^{\left(y,\eta,\phi\right)}$, respectively, and $M_{inc}$ is the
POVM operator yielding an inconclusive outcome. All three operators serve as
the optimization variables. The first two constraints ensure that the optimal
POVM operators identify the correct state with zero error probability.
Similarly to unforgeable quantum tokens, the tightness of this upper bound can
be shown by using strong duality (see Section E and Eldar et al. (2003)).

We label the optimal value to problem (67), i.e. Bob’s optimal USD cheating
probability, as $\mathcal{P}_{USD}$. On the other hand, the probability of a
successful Helstrom measurement is given by Barnett and Croke (2009):

$\mathcal{P}_{HEL}=\frac{1}{2}+\frac{1}{4}||\sigma_{0}^{\left(y,\eta,\phi\right)}-\sigma_{1}^{\left(y,\eta,\phi\right)}||_{1},$
(68)

where $||\odot||_{1}$ denotes the Schatten $1$-norm.

Since Bob performs the same attack on each of the $(N-1)$ states sent by
Alice, followed by a Helstrom measurement on the $N$th state, we can finally
upper-bound his cheating probability as:

$\mathcal{P}_{B}^{bias}\leqslant\left[1-\left(1-\mathcal{P}_{USD}\right)^{N-1}\right]\times
1+\left(1-\mathcal{P}_{USD}\right)^{N-1}\times\mathcal{P}_{HEL}.$ (69)

##  Appendix J Security of quantum bit commitment

We focus on the quantum protocol from NJC:NatComms12 , under the bounded
storage assumption. This assumption circumvents the no-go theorem for two-
party computations Mayers (1997); Lo and Chau (1997) by placing restrictions
on Dishonest Bob’s storage capabilities: he may store only $S$ of the $N$
quantum states sent by Honest Alice, over a time duration no longer than
$\Delta t$. Similarly to BB84 QKD in Section F, the practical security
analysis assumes that states emitted by Honest Alice bear no coherence in the
photon number basis. We therefore consider only phase-randomized PDS, and
LA/TPE QDS. We adapt here the conditions provided in the Appendix of
NJC:NatComms12  for a $3\epsilon$-secure quantum bit commitment
implementation. Here, $\epsilon$ is a fixed parameter which upper-bounds the
occurrence of bad events, governed by the Hoeffding inequality.

In a nutshell, the practical bit commitment protocol is constructed from a
weak string erasure sub-routine with errors (WSEE) Konig et al. (2012). An
$\left(N,\lambda,\epsilon,e\right)$-WSEE provides Alice with a string $X_{N}$
and Bob with a randomly chosen subset $I\in[N]$, as well as a substring
$\widetilde{X}_{I}$. This substring is given by the substring $X_{I}$ (the
bits of $X_{N}$ corresponding to the indices in $I$) passed through a binary
symmetric channel that flips each bit of $X_{I}$ with probability $e$. The
security statements then read as follows:

  * •

If Alice is honest, then the amount of information a Dishonest Bob holds about
$X_{N}$ is limited, i.e. the $\epsilon$-smooth min entropy of $X_{N}$
conditioned on a Dishonest Bob’s information is lower bounded by a value
$\lambda$.

  * •

If Bob is honest, then Alice does not have any information $I$. That is, Alice
does not learn which bits of $X_{N}$ are known to Bob.

Essentially, losses allow Dishonest Bob to discard a fraction of single-photon
detection events, and keep more multiphoton events so that his chance of
guessing $X_{N}$ correctly is increased. The resulting min-entropy rate
$\lambda$ can thereby be calculated as a function of experimental parameters
$\\{P_{x}(k)\\}$ and $\\{P_{\left(x\eta_{c}\right)}(k)\\}$, where
$\\{P_{x}(k)\\}$ are the PDS coefficients from Eq. (27) for $x=\mu$ or the QDS
coefficients from Eq. (53) for $x=\eta$, and
$\\{P_{\left(x\eta_{c}\right)}(k)\\}$ are defined similarly, only with an
extra channel transmission factor $\eta_{c}$ multiplying $x$ to account for
honest losses.

Following Lemma $12$ from the Supplementary Information of NJC:NatComms12 ,
given an experimental error rate $e$, channel losses $\eta_{c}$, fixed
parameters $\epsilon$ and $\left(\beta,\gamma\right)\in\left(0,0.01\right]$,
and considering a Dishonest Bob with bounded storage size $S$, we define the
following parameters:

$\displaystyle m_{2}=P_{x}(1)-P_{\left(x\eta_{c}\right)}(0)+P_{x}(0)-3\gamma$
(70) $\displaystyle m_{3}=1-P_{\left(x\eta_{c}\right)}(k)$ $\displaystyle
L^{\prime}=\max_{s\in\left(0,1\right]}-\frac{1}{s}\left[\log\left(1+2^{s}\right)-1-s\right]-\frac{3\epsilon}{s}$
$\displaystyle\delta=2\>\frac{e+\frac{\beta}{\sqrt{1-2\beta}}}{1-4\sqrt{5}\beta}$
$\displaystyle\lambda=H_{2}(\delta)+3\beta^{2}$ $\displaystyle
M_{1}=\frac{1}{2\gamma^{2}}\log\frac{2}{\epsilon}$ $\displaystyle
M_{2}=\frac{\log\frac{1}{\epsilon}}{\epsilon\>m_{2}}$ $\displaystyle
M_{3}=\frac{\log\frac{2}{\epsilon}}{\left(m_{3}-\gamma\right)\beta^{2}}$
$\displaystyle M_{4}=\frac{S}{m_{2}L^{\prime}-m_{3}\lambda}$

where $H_{2}(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ is the binary entropy
function for $0<x\leqslant 1$. For security to hold in a practical
implementation, the following condition must hold:

$m_{2}L^{\prime}-m_{3}\lambda>0.$ (71)

If Eq. (71) is true, then quantum bit commitment under the bounded storage
model can be implemented $3\epsilon$-securely by using a randomly constructed
error-correcting code, whenever the number of states sent by Alice:

$N>\max\\{M_{1},M_{2},M_{3},M_{4}\\}.$ (72)

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# High-definition event frame generation using SoC FPGA devices

Krzysztof Blachut, Tomasz Kryjak<EMAIL_ADDRESS><EMAIL_ADDRESS>The work presented in this paper was supported by the programme “Excellence
initiative –- research university” for the AGH University of Krakow. The
authors would like to thank Piotr Wzorek and Mateusz Wąsala for their help
with the project. Embedded Vision Systems Group, Department of Automatic
Control and Robotics,AGH University of Krakow, Poland

###### Abstract

In this paper we have addressed the implementation of the accumulation and
projection of high-resolution event data stream (HD – $1280\times 720$ pixels)
onto the image plane in FPGA devices. The results confirm the feasibility of
this approach, but there are a number of challenges, limitations and trade-
offs to be considered. The required hardware resources of selected data
representations, such as binary frame, event frame, exponentially decaying
time surface and event frequency, were compared with those available on
several popular platforms from AMD Xilinx. The resulting event frames can be
used for typical vision algorithms, such as object classification and
detection, using both classical and deep neural network methods.

##  I Introduction

Event cameras (DVS – Dynamic Vision Sensors) are state-of-the-art vision
sensors inspired by biology. They only detect changes in brightness in an
image and can operate with microsecond resolution. For this reason, they are
able to register changes in the environment very quickly, i.e. they have low
latency. In addition, they operate correctly under difficult lighting
conditions, e.g. limited brightness or high dynamic range. The aforementioned
properties make DVS an interesting sensor for currently developing autonomous
vehicles, including drones, where both speed of operation and robustness to a
variety of lighting conditions are important.

However, processing event data, which is in the form of a sparse spatio-
temporal cloud, is challenging. Several approaches are encountered in the
literature: from the direct analysis of the point cloud, to the projection of
events onto a plane (representations) and reconstruction [1]. Particularly
popular is the use of a representation in the form of a so-called event
frames, i.e. events collected over a certain time interval, projected onto the
image plane. This allows the use of typical, well-known vision algorithms –
both classical and those based on deep learning.

In order to process event data quickly enough, a powerful hardware platform is
needed on which relevant information can be extracted in real-time, e.g. the
position and direction of potential objects in the drone’s environment. This
prerequisite is met by SoC FPGA (System-on-Chip Field Programmable Gate Array)
or embedded GPU (embedded Graphics Processing Unit) platforms, which enable
parallel computing and, in addition, are small and lightweight and therefore
well suited to be placed on a drone.

In this paper, we address the subject of event frame generation in SoC FPGAs
for data in HD (High Definition – $1280\times 720$ pixels) resolution or
higher. To the best of our knowledge, this issue has not yet been considered
in the scientific literature for such high resolutions (Section II-B) and it
presents many challenges, both scientific (hardware architecture development)
and technical (implementation).

The remainder of this paper is organised as follows. Section II provides
background information on event cameras and the approaches available in the
scientific literature to the problem considered. Section III is devoted to the
presentation and comparison of different event data representations. Details
of the hardware implementation of the algorithm, together with its potential
enhancements, are collected in Section IV. The final Section V summarises the
work done and indicates further directions for development.

##  II Event-frame generation

###  II-A Event camera

Dynamic Vision Sensors differ from traditional frame cameras in both design
and operation. Firstly, they only record brightness changes independently
(asynchronously) for each pixel. Each such change is called an event and
consists of four pieces of information: the exact time of occurrence
(timestamp), the $x$ and $y$ positions in the image, and the polarity, i.e.
information about the increase or decrease in brightness (positive and
negative polarity respectively), which can be written as $e=\\{t,x,y,p\\}$.

When monitoring a static scene, event cameras do not transmit any data (apart
from noise), which significantly reduces data transfer and energy
requirements. Secondly, changes are recorded at a very high rate – more than a
thousand times per second, whereas frame cameras typically operate at lower
frequencies (e.g. 30, 60, 120 or up to 240 frames per second). Another
difference resulting from this is the significant reduction in blur effect
associated with object movement on event data, which enables correct
registration of a highly dynamic scene. A fourth advantage over traditional
cameras is the high dynamic range, allowing very bright and very dark areas to
be recorded in a single image, which is usually not possible with frame
cameras.

Event cameras are used, among others, in unmanned aerial vehicles [2], for
increasing the frequency of video sequences [3], for traffic sign detection
[4], or for fast objects counting [5]. A comprehensive overview of event
camera applications can be found in the article [6].

###  II-B Previous works

A number of solutions can be found in the scientific literature where frames
were generated to perform certain tasks based on event data. The paper [4]
realised traffic sign detection using neural networks and proposed a multi-
channel approach (different simple representations on three image channels).
The work [7] proposed a system for object detection using neural networks and
a representation based on the frequency of event occurrence. The paper [8]
presented a system for pedestrian detection using a neural network and a
representation called neighbourhood suppression time surface. For the gesture
recognition task, a temporal binary representation method was proposed in [9],
where a sequence of 8 binary frames was generated, aggregated to a single
greyscale image. The only work where frame generation was done for HD data is
[5], which proposed a system to control the flow of elements in time (falling
corn grains).

One of a few similar hardware implementations on an FPGA platform is [10],
whose authors performed object classification using a NullHop accelerator
based on a convolutional neural network. In this work, frames were generated
for a specific number of events instead of a time interval. Two memory blocks
for data of only $64\times 64$ elements were used, so the authors did not
encounter a number of challenges in implementing the algorithm for high-
resolution data.

The generation of frames without distinguishing events’ polarity was part of
the optical flow determination algorithm on an FPGA platform [11], in which
data was accumulated in 3 blocks from consecutive time intervals. In the
project an event camera with a resolution of $240\times 180$ pixels was used.

The work [12] developed a gesture recognition system using an FPGA platform
and a hierarchy of time surfaces representation, inspired by an approach from
multi-layer neural networks, and a single-layer linear decay kernel
representation, i.e. linear decay of events in time. Each single event was
specified using the time context concept, a quadratic matrix of timestamp
differences in a small neighbourhood. In the project two event sensors (with
$304\times 240$ and $128\times 128$ pixels resolutions) were used, but for the
purposes of the algorithm, the data was stored in an accumulator scaled to
$128\times 128$.

In summary, in a number of research papers projecting event data onto the
image plane using different representations was proposed, and further
processing of the frames was rather performed using neural networks than
classical methods. In some of these papers, an FPGA platform was used for the
implementation of the frame generation algorithm, but for very low resolution
data. The implementation of such an algorithm for high-resolution data has not
yet been addressed as it raises several problems, mainly related to the
handling of incoming events and real-time frame generation, which at low
resolutions – due to much less data per time interval – simply do not occur.
This paper therefore attempts to fill this gap, which becomes particularly
relevant as the resolution of commercially available event cameras increases –
from $128\times 128$ pixels in 2008 up to $1280\times 960$ in 2020 [6].

##  III Event-frame representations

There are several approaches to processing event data, e.g. ‘straightforward’,
which requires the development of new algorithms due to the different data
format. Therefore, a more ‘natural’ approach is the generation of so-called
event frames, which consist of events from certain time intervals projected
onto a plane, resulting in pseudo-images similar to these of traditional frame
cameras, to which well-known algorithms can be applied. However, the frame
generation poses a number of challenges: selecting the event accumulation
period, taking into account time information when projecting onto the plane or
ensuring real-time frame generation and further processing.

To date, several basic data representations for event frame generation have
been proposed in the literature, with additional modifications for specific
applications. According to [13], a function $\Sigma_{e}$ can be defined that
assigns the time of occurrence of an event $t$ to each coordinate pair
$\textbf{u}(x,y)$, and a function $P_{e}$ that assigns a polarity from the set
$\\{1,-1\\}$ or, using another convention, $\\{1,0\\}$. In addition, a
parameter $\tau$ can be defined, indicating the event accumulation time for
the generation of one frame. The representations listed in this section are
defined (for simplicity) for times $t\in(0,\tau)$.

###  III-A Binary frame

The first representation – for maximum simplicity – can be a binary
information about the occurrence of an event in given coordinates, as in [11]:
$f(\textbf{u},t)=1$. A binary frame can also be generated from events of only
one polarity.

###  III-B Event frame

The second approach is a slight modification of the previous one, as the
polarity of events is taken into account, according to the work [14]:
$f(\textbf{u},t)=P_{e}(\textbf{u})$. This is one of the most popular
representations due to its simplicity, but also the considerable amount of
information it contains.

###  III-C Exponentially decaying time surface

In this representation the temporal information is taken into account,
precisely by specifying the time from the event occurrence to the end of the
accumulation period, as in [14]. This value is then used for determining the
brightness of a pixel in the resulting image:
$f(\textbf{u},t)=P_{e}(\textbf{u})\cdot e^{\frac{\Sigma_{e}(u)-t}{\tau}}$.
This is motivated by the observation that the weight of information
contributed by an event decreases with time to 0.

###  III-D Event frequency

The next approach uses information about the frequency of a given pixel in the
image, as proposed in [7]: $f(x)=\frac{255}{1+e^{-x/2}}$, where the sum of
polarisations for a pixel is denoted as $x$.

###  III-E Comparison

A comparison of generated event frames using the representations described is
shown in Figure 1. An exemplary sequence shows a fast-moving ball flying
through a room recorded by a moving event camera. Apart from aforementioned
representations, more complex arithmetic operations can be proposed to
determine the values of individual pixels, as mentioned in Section II-B. There
are also solutions in which an image with multiple channels of particular
representations [4] or aggregated into one-channel [9] is generated.

(a) Binary frame

(b) Event frame

(c) Exp. decaying time surface

(d) Event frequency

Figure 1: Comparison of event frames generated using different representations
over a 10 ms interval. The fast-moving object generates a big amount of
events. In case of images (c) and (d), it is blurred as the influence of
particular events on the resulting image differs depending on exact time or
frequency.

##  IV The proposed frame generation module

In this work, we have developed several versions of a hardware module for
event frame generation on an FPGA platform. Their architectures were prepared
in SystemVerilog language using the Vivado environment and tested for AMD
Xilinx’s FPGA platforms. The input of each module was an event
$e=\\{t,x,y,p\\}$ and the output was an 8-bit pixel brightness value. For
testing purposes, sample sequences with a thrown ball were recorded with a
moving (rotating) Prophesee EVK1 camera in HD resolution. The events were
saved to a text file, from which they were read in the Vivado simulation.

###  IV-A Basic version

The architecture of the basic version of the algorithm consists of three
elements: a memory for storing accumulated events, logic controlling the
reading and writing of data, and a temporary buffer for incoming data during
frame reading.

The first of these elements can be realised in two ways – using the FPGA
chip’s internal block memory (BRAM) or external RAM (usually dynamic). In the
first case, the memory resources are quite limited, so the appropriate
representation and resolution of the image must be selected. In the second
case, the resources are much larger, but this approach increases the
complexity of the architecture, introduces additional latency in data
processing and is more energy consuming. Therefore, the first solution is used
in this work due to its greater simplicity and operational efficiency. The
number of elements in memory was set to 921600 (the number of pixels in an HD
image). The number of bits allocated for one element depends on the chosen
representation, but in the simplest case just 1 bit can be assumed (Section
III-A). Other variants are described as algorithm extensions in Section IV-B1.

The second element is the logic that controls the communication with the two-
port block memory, of which one port is used for writing data and the other
for reading. As long as events belonging to a certain time interval $\tau$
(e.g. 10 ms) occur at the input, the algorithm operates in write mode. Based
on the coordinates of the incoming event, the address in memory is determined
as: $address=address\\_Y\cdot image\\_width+address\\_X$, while the value
stored equals 1.

When the condition to stop the write mode is met (e.g. $\tau$ value exceeded),
the algorithm enters read mode, resetting the pixel position counter in the
image. Its value is given as the address of the memory cell from under which
the content of the accumulator is read before being incremented by 1. The
value read is further decoded to the range 0-255 to display the output event
frame in greyscale. At the same time, the address used for reading data is
delayed by one clock cycle and fed to the input of the second port used for
writing, together with the value 0, to reset the memory cell. When the counter
reaches the last pixel in the image, the algorithm returns to write mode. A
schematic visualisation of the operation of the algorithm in write and read
mode is shown in Figure 2.

Figure 2: Operation of the frame generation algorithm in write and read mode.
For simplicity, an image of $12\times 8$ pixels was assumed. Writing was
visualised using an example event $e=\\{t,8,5,p\\}$. Filled fields indicate
earlier events. In read mode, the counter and the reset signal go through the
entire image to the last pixel. Based on the representation, the values
written and read from memory (and after decoding) may differ.

Due to event camera properties, just after entering the read mode, new events
may appear at the input of the module, which should be used for the generation
of the next frame. So the last necessary element is to handle these events. In
order to separate the ‘older’ events (in the accumulator) from the ‘newer’
ones (at the input), a temporary buffer must be prepared to store them until
the read mode finishes. It was realised as a FIFO (First In First Out) queue
using the FPGA’s block memory, in which all event components were stored.

Regardless of the operation mode, each event was firstly written to the FIFO.
Once in write mode, data was read from the queue and, in the meantime, the
latest events from the input were written to the FIFO. A difficult issue is to
determine the maximum size of the queue in order to store all incoming events
but with low memory usage, as it depends on the dynamics of the scene and the
$\tau$ value. In our solution, the queue size was set to 32768 elements. In
addition, a mechanism of removing ‘the oldest’ events from the queue to add
‘the newest’ was implemented, if the FIFO was full in read mode. This was
motivated by the fact that ‘newer’ events are more important, as they
represent the most recent changes.

###  IV-B Extensions

The architecture described in Section IV-A is sufficient to correctly generate
event frames in real-time on an FPGA platform. However, a number of
enhancements and extensions can be applied to achieve better performance or
results.

####  IV-B1 Other representations

The first element is another representation of event data. The simplest and
most common modification is to add polarisation, creating an event frame. In
this case, an event with a polarity of 1 generates an output pixel value of
255, the one with $-$1 polarity generates 0, while pixels without events
receive a value of 128. To save hardware resources, a 2-bit value is stored in
the accumulator, denoting 1 (positive event), $-$1 (negative event) or 0 (no
event).

For other representations described in Section III, Look-Up-Table (LUT) can be
used to store the approximate values of the exponential function. In the case
of the method in Section III-C, knowing the value of $\tau$ (e.g. 10 ms), the
difference between the current timestamp and the maximum value for the
interval can be calculated. The rounded result can be stored in an 8-bit
accumulator, thus obtaining the output pixel value.

For the representation from Section III-D, it was necessary to determine the
maximum number of events for the given coordinates, which depends on dynamics
of the scene and $\tau$ value. After tests on an example sequence, a
limitation to the interval $(-16,15)$ turned out to be sufficient, as larger
polarity sums had no apparent effect on the final pixel value. Therefore, 5
bits were stored in the accumulator and then were decoded to an 8-bit output.
The polarity counting itself was done in a way that with the arrival of a new
event with the given coordinates, the current value was read from the
accumulator using one port of the block memory, after which, depending on the
polarity, the number 1 was added or subtracted and then written back to the
same address using the other port.

####  IV-B2 Multiple Block RAMs

Figure 3: Use of multiple block memories for frame generation. For clarity,
only 2 blocks are used in the diagram. To determine block and cell indexes,
the actual number of blocks (2) is used during division. This results in a
2ppc (2 pixels per clock cycle) stream at the output of the module.

It is possible to use a set of smaller block memories, working in parallel, so
that the reading of several pixels can be performed in the same clock cycle.
This approach is inspired by processing a traditional video stream on an FPGA
platform in a vector format, the so-called Xppc (X pixels per clock cycle) as
in [15]. Apart from calculating the address based on the event coordinates,
the horizontal index determines the memory block into which it is written, as
in Figure 3. In read mode, pixels with consecutive indices (from 0 onwards)
are read from all X blocks simultaneously and fed to the module output in
successive clock cycles. With this approach, the output image is identical to
that of a single memory block, but it allows to reduce X times the latency of
reading the image or the clock frequency and thus the energy consumed. It is
also possible to read multiple values from one block using memory properties
in Vivado, but this solution has limited versatility in terms of possible X
values and the way of resetting memory (all cells at once).

####  IV-B3 Rolling window

A so-called rolling window, used in typical contextual operations on the
image, can be proposed for event frame generation, which is partly inspired by
[11]. The idea is to accumulate event data from $N$ ms, with an image
generated every $K$ ms and covering the last $M$ ms, where $K\leq M\leq N$.
For the test, the values chosen were $N$ = 8, $M$ = 4, $K$ = 1, so a new frame
was generated every 1 ms, covering the last 4 ms, while accumulating events
from the last 8 ms. In this way, in images generated at high frequency, the
oldest events are removed and the newest are added.

Figure 4: Operation of the rolling window method. For simplicity of the
scheme, the values set are $N$ = 4, $M$ = 2, $K$ = 1. When writing to memory,
apart from the address, the index of the ‘sub-window’ is determined, in this
case from 1 to 4. On reading, the indexes of the stored events are compared
with the recent one and some pixels are selected and passed to the output,
creating an image. In the example, only pixels from ‘sub-windows’ 3 and 4
(green and yellow) form the output image and pixels with index 1 are removed
from memory. In the next cycle, the pixels with indexes 4 and 1 will be read,
and those with index 2 will be reset and so on.

On the hardware implementation side, it required additional memory due to
‘adding’ of $\log_{2}N=3$ bits to the data representation, informing about the
index of the ‘sub-window’. Its resetting included only the oldest index, i.e.
events before $N-$1 = 7 ms relative to the current timestamp. A schematic of
how this method works is presented in Figure 4.

It is worth mentioning that this approach preserves more accurate information
about the timestamp of events and can be applied to any data representation.
Other variants of this method may include the use of several frames, offset by
$K$ ms, as components of a multi-channel image or aggregated into a single one
as in [9], but also the use of dynamically changing $M$ value from which the
output images are generated.

####  IV-B4 Ultra RAM

Another option could be to use Ultra RAM resources available in some SoC
FPGAs. This additional internal memory is larger than BRAM and can be used to
store the contents of an accumulator or a FIFO queue. This can allow the use
of more complex representations, but also support more dynamic scenes due to
possible larger queue size. However, only a subset of AMD Xilinx’s chips are
equipped with this memory (UltraScale and UltraScale+ series), so this
enhancement is dependent on the chosen hardware platform.

####  IV-B5 Temporary buffer

Apart from described buffering (Section IV-A), other ways of handling input
events can be used.

One is to duplicate the accumulator module and use a ‘ping-pong buffering’
method, as in the work of [10], and swap the roles of two accumulators in a
way that one is in read mode and the other in write mode. However, this
solution is not very efficient, as the entire accumulator memory must be
doubled, which may not be feasible for more complex representations due to the
limited memory resources of the FPGA device.

Second idea is to use external RAM, located on the FPGA board. Its size is
considerably larger than block memory, so a large FIFO queue can be generated.
However, this approach also has several disadvantages, among which are:
greater system complexity, higher resource utilisation, additional latency
(due to reading and transferring data from memory), and the need to use the
processor for communication and thus higher power consumption. An example
comparison of the available memory types in several popular FPGA platforms
from various AMD Xilinx board families is provided in Table I.

TABLE I: Comparison of available memory resources for exemplary AMD Xilinx platforms. Values in parentheses specify the number of ‘units’ of a memory type. SoC FPGA platform | Block RAM [Kb] | Ultra RAM [Mb] | External RAM [GB]
---|---|---|---
ZCU 104 | 11 (312) | 27 (96) | 4.5
Kria KV260 | 5 (144) | 18 (64) | 4
Zybo Z7-20 | 5 (140) | - | 1

####  IV-B6 Accumulation time

Event accumulation time $\tau$ is one of the most important parameters in the
event frame generation algorithm. A smaller value provides more frames per
second and better information about the dynamics of the scene, while a larger
value allows a better understanding of the overall motion and reduces the
impact of noise. Therefore, a potential improvement of the algorithm in this
case could be an adaptive approach in which the accumulation time $\tau$ can
be increased or decreased depending on the dynamics of the scene.

####  IV-B7 Accumulation number

Another enhancement could be the generation of frames every fixed number of
events $Z$, as in the work of [10]. Event data is recorded asynchronously for
each pixel and the dynamics of the scene determines the exact number of events
per time interval. In this approach, it may possible (depending on the
representation) to omit the timestamps and reduce the amount of data
processed. However, the generated frames will appear at the output in
different time intervals, which may raise problems especially in cases with
high dynamics (and thus high fps value). A potential solution to this problem
could be an adaptive version, in which the length of the time interval is
additionally analysed and the number of $Z$ events is reduced or increased
based on it.

###  IV-C Performance

TABLE II: Comparison of hardware resource utilisation on an FPGA platform for different data representations and variants (for a 2-bit event frame only) for a 10 ms accumulation interval. The size of the FIFO queue was set to the minimum (512 elements) not to disturb the comparisons. Power estimation for the FPGA chip only was performed by the Vivado software. Due to the nature of BRAM memory, the smallest indivisible element has 0.5 size of a single block (meant as a ‘piece’). Representation/Algorithm variant | No. of bits | Block RAM | LUT | Flip-Flop | Power est. [W]
---|---|---|---|---|---
Binary frame (Sec. IV-A) | 1 | 29.5 | 191 | 142 | 5.9
Event frame (Sec. IV-B1) | 2 | 57.5 | 240 | 157 | 9.0
Exp. decaying time surface (Sec. IV-B1) | 8 | 226 | 1289 | 168 | 29.1
Event frequency (Sec. IV-B1) | 5 | 142 | 363 | 158 | 27.3
Basic (Sec. IV-A) | 2 | 57.5 | 240 | 157 | 9.0
Multiple BRAMs (Sec. IV-B2) | 2 | 61 | 204 | 258 | 12.0
Rolling window (Sec. IV-B3) | 5 | 142 | 8369 | 150 | 18.2

A comparison of the most important parameters of several data representations
and variants for event frame generation algorithm is provided in Table II. The
use of memory resources increases significantly with the number of bits per
pixel due to larger accumulator size. Due to the amount of available memory
resources, some options described may not be feasible to implement on smaller
FPGA chips – therefore simpler representations, additional approximations or
lower data resolution may be needed. The use of the remaining resources is
very low, as the logic itself controlling the writing and reading of data from
the accumulator is relatively simple.

The described variants concern only one most popular data representation
(event frame). In case of the rolling window method, it is necessary to
allocate more memory to store the ‘sub-window’ indices (3 extra bits – 8
indices). The use of multiple memory blocks generates a slight increase in
resource consumption, because with suboptimal parameters, a part of each
allocated block remains unused. However, this solution allows faster reading
of pixels, vector data processing (Section IV-B2) and reduced clock frequency
and power consumption. Therefore, using one block configured in Vivado with a
specified set of parameters and a global reset can also be considered, if
smaller versatility of the module is not a problem.

##  V Conclusion

In this paper we have proposed and compared various ways of event frame
generation in SoC FPGA devices for an HD event stream. For a pipelined
hardware implementation on an FPGA platform, this operation brings a number of
challenges and trade-offs depending on the available resources, mainly memory:
the choice of resolution of the generated frames, the representation used, the
realisation of a temporary buffer and additional hardware enhancements.

The use of SoC FPGA chips from AMD Xilinx can yield real-time and energy-
efficient data processing, but the resources available on a chosen platform
condition the details of the implementation. In case of Zybo Z7-20, only the
simplest representations (binary and event frame) can be realised due to the
small size of the available memory. For Kria KV260, any of the presented
variants can be used, but Ultra RAM or external memory must be used, while for
ZCU 104, all of them can be implemented using only block memory.

As part of future work, several applications of the proposed event frame
generation module are planned, as the frames themselves can be used in typical
vision-based systems with both classical methods and deep neural networks.
Running on an exemplary hardware platform together with the loading of data
from an SD card (or later receiving directly from the camera), it can be
possible to use this module in a larger vision system, e.g. for the detection
of fast-moving objects. Ultimately, the developed system is planned to be used
on an unmanned autonomous drone to enable it, among other things, to fly
through a dynamic environment with obstacles.

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because of the strong forward diffraction introduced by the large particles
(e.g., Kolokolova & Kimura 2010; Min et al. 2016; Tazaki & Tanaka 2018). Model
calculations for disks with such models are presented by Tazaki et al. (2019),
who investigate in detail the wavelength dependence and the fractional
polarization of the disk radiation parameters $(I/I_{\star})_{\lambda}$ and
$(\langle p_{\varphi}\rangle)_{\lambda}$ on the properties of dust aggregates.
Further calculations of this kind using detailed dust models for the
derivations of the scattering parameters $\sigma(\lambda,a)$,
$\omega(\lambda,a)$, $g(\lambda,a)$, and $p_{\rm max}(\lambda,a)$, or even the
radiation parameters of disks considering the effects of the radiative
transfer and the disk geometry are required in addition to accurate
measurements in order to derive the nature of the dust particles in
protoplanetary disks.

There will most likely always exist different (competing) dust models that can
explain the observationally derived parameters for the dust scattering of a
circumstellar disk. Therefore, the characterization of the dust scattering
phase matrix, or a simple description thereof with the three parameters
$\omega$, $g$, and $p_{\rm max}$, seems to be a useful interface between dust
models and disk observations to constrain the dust properties. The model
results presented in this work show that there exist well-defined and
relatively universal dependencies between these dust scattering parameters and
the reflected radiation from disk that are measurable with high-resolution
observations. It is important to collect quantitative measurements for more
disks because additional constraints on the dust can be derived from
circumstantial evidence based on systematic effects found for disks with
different physical conditions, like dust temperatures, stellar irradiation,
mass accretion rates, dust-to-gas ratios, and different complementary dust
characteristics like silicon oxide(SiO) or polycyclic aromatic hydrocarbons
(PAHs) emission features in the mid-IR. This will provide a more comprehensive
picture of the particle properties and clarify the evolution of the dust in
protoplanetary disks.

###### Acknowledgements.

We thank the anonymous referee for the valuable comments on the manuscript.
This work has been carried out within the frame of the National Centre for
Competence in Research PlanetS supported by the Swiss National Science
Foundation (SNSF).

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##  Appendix A Results for plane parallel surfaces

Our models calculate the reflected photon intensity $I$, the linearly
polarized intensity $P$ (or Stokes $Q$ and $U$), and the fractional
polarization $p$ from a plane parallel surface as two-dimensional functions of
the polar and azimuthal angles $\theta$ and $\phi$ (see Fig. 3). All our
models assume that the incident radiation is unpolarized. Model parameters are
the polar incidence angle $\theta_{0}$, and the dust parameters scattering
albedo $\omega$, scattering asymmetry $g$, and fractional scattering
polarization $p_{\rm max}$. The main text of this paper describes in Sect. 3.1
results for the principal plane ($\phi=0^{\circ}$ or $180^{\circ}$) for the
0.5-model with parameters $\omega=0.5$, $g=0.5$, $p_{\rm max}=0.5$, and
$\theta_{0}=57.5^{\circ}$. These results are plotted in this Appendix with red
color as references. We present in Appendix A.1 additional results for the
0.5-model for the radiation escaping in other directions than in the principal
plane. Appendix A.2 describes how the reflected radiation in the principal
plane depends on the model parameters $\omega$, $g$, $p_{\rm max}$, and
$\theta_{0}$.

Figure 17: Reflected intensity $I$, Stokes parameters $q,u$ and fractional
polarization $p$ for the 0.5-model for the principal plane (red) and the two
vertical planes for the $\phi$-angles $40^{\circ}\colon 220^{\circ}$ (blue)
and $90^{\circ}\colon 270^{\circ}$ (yellow)

###  A.1 Dependencies on the azimuth angle

##### Intensity.

The principal plane picks only a special case $\phi=0^{\circ}\colon
180^{\circ}$ of the reflected intensity $I(\theta,\phi)$ of a plane parallel
surface. The first panel in Fig. 17 also plots the reflected intensities for
planes perpendicular to the surface with $\phi$-angles $40^{\circ}\colon
220^{\circ}$ and $90^{\circ}\colon 270^{\circ}$ using the same
$\pm\theta$-convention as for $\theta_{\rm pp}$ for the principal plane (Sect.
3.1. The scattering problem is symmetric with respect to the principal plane,
meaning that the reflected intensity for $I(\pm\theta,40^{\circ}\colon
220^{\circ})$ is equal to $I(\mp\theta,140^{\circ}\colon 320^{\circ})$ and
$I(\theta,90^{\circ})=I(-\theta,270^{\circ})$. All intensity curves go through
the same value $I(\theta=0^{\circ})$ for the photon escape direction
perpendicular to the surface.

Figure 17(a) shows that the forward peaked scattering phase function creates
the strongest $I(\pm\theta)$-asymmetry in the principal plane, with a gradual
evolution for other $\phi$-angles towards the symmetric case
$I(\pm\theta,90^{\circ}\colon 270^{\circ})$. Therefore the reflected intensity
in the principal plane $\phi=0^{\circ}\colon 180^{\circ}$ can be used for
guessing the intensity at other $\phi$-angles.

##### Polarization.

The fractional polarization of the scattered radiation given in Fig. 17(b) and
(c) needs to be described by $q=Q/I$ and $u=U/I$ for arbitrary $\phi$ angles,
because there are not only perpendicular and parallel Stokes $q$ components
with respect to the vertical plane but also nonzero Stokes $u$ polarization
components in the $45^{\circ}$ and $135^{\circ}$ direction. Only in the
principal plane the $u$-component is zero for all angles $\theta$, and the
deviations of the red curve from zero in panel (c) illustrate very well the
statistical uncertainties of the Monte Carlo simulations. For the principal
plane there is also $p=(q^{2}+u^{2})^{1/2}=|q|$ as can be inferred from panel
(b) and (d).

For all other azimuth angles the polarization signals are more complex. For
example the yellow curves in Fig. 17(b) and (c) for the scattered polarization
for the $\phi=90^{\circ},270^{\circ}$ plane has for all $\theta$ a negative
$q$-component and a $u$-component which changes from strongly positive to
strongly negative. This corresponds to a rotation of the polarization angle
from $57.5^{\circ}$ at $\theta=-90^{\circ}$ for photons escaping to the one
side just along the surface to an angle $122.5^{\circ}$ at
$\theta=+90^{\circ}$ for photons escaping to the opposite side along the
surface, both with scattering angles of $90^{\circ}$. The polarization angle
is $90^{\circ}$ or parallel to the vertical plane for photons escaping
vertically at $\theta=0^{\circ}$ where the scattering angle is about
$32.5^{\circ}$. All these polarization position angles are roughly
perpendicular to the propagation direction of the incoming photon and such a
perpendicular polarization is also expected for single scattering. Multiple
scattering events add small contributions which lead to small second-order
deviations from the perpendicular polarization angles.

The perpendicular scattering polarization would be more obvious, if we would
show the polarization relative to the scattering plane defined by the
propagation directions of the incoming $\theta_{0},0^{\circ}$ and escaping
photon $\theta,\phi$, instead of the polarization relative to the
perpendicular planes as used in Fig. 17. However, the geometry of the
scattering plane depends on the incidence angle $\theta_{0}$ and this
complicates comparisons with the red curve for scattering in the principal
plane.

The yellow curve for the fractional polarization $p$ in Fig. 17(d) for the
$\phi=90^{\circ},270^{\circ}$ plane has maxima for the right angle scattering
at $\theta=\pm 90^{\circ}$ and a minimum for $\theta=0^{\circ}$. For
intermediate $\phi$-angles the fractional polarization $p$ is (very) roughly
between the red and yellow curve.

###  A.2 Dependencies on model parameters

##### Scattering albedo.

Figure 18: Reflected intensity $I(\theta_{\rm pp})$, fractional polarization
$p(\theta_{\rm pp})$, and polarized intensity $P(\theta_{\rm pp})$ in the
principal plane for a scattering surface with a HGpol phase function. The red
curves is the standard model with $\theta_{0}=57.5^{\circ}$, $\omega=0.5$,
$g=0.5$, and $p_{\rm max}=0.5$ and in each column one of these parameters is
varied as indicated in the top panel.

The single scattering albedo $\omega$ has a very important impact on the
amount of reflected intensity as can be seen in Fig. 18 for the principal
plane $I(\theta_{\rm pp})$. For our standard model parameters
$\theta_{0}=57.5^{\circ}$, $g=0.5$ and $p_{\rm max}=0.5$, the peak
reflectivity max$(I(\theta_{\rm pp}))$ decreases for standard model parameters
from 0.074, to 0.031 and 0.010, if the single scattering albedo is reduced
from 0.8 to 0.5 and 0.2. These characteristic values are also given in Table 5
together with the angle $\theta_{\rm pp}^{\rm maxI}$ where this maximum
occurs.

The reflected polarized flux $P(\theta_{\rm pp})$ shows also a correlation
with $\omega$, but with a less steep gradient when compared to the intensity.
A low single scattering albedo reduces the multiple-scattered photons—which
contribute more ”randomized” or unpolarized reflected light— more
significantly, while the strongly polarizing single-scattered photons are less
suppressed. Therefore, the maximum fractional polarization of the reflected
light max($p(\theta_{\rm pp})$) is high for low albedo and increases from
about 19 % to 31 % and 43 % for $\omega$ decreasing from 0.8, to 0.5 and 0.2,
respectively (see Table 5).

##### Scattering asymmetry parameter.

The scattering asymmetry parameter $g$ can significantly change the strength
and the angular distribution of the reflected intensity $I(\theta_{\rm pp})$
as shown in Fig. 18 for $\theta_{0}=57.5^{\circ}$, $\omega=0.5,$ and $p_{\rm
max}=0.5$. For isotropic scattering $g\approx 0,$ about half of the incident
photons propagate after the first scattering upwards and have a high
probability of escaping, while for strongly forward scattering particles
$g\approx 0.5$ or 0.75, the photons propagate after the scattering
predominantly further into the dusty disk and they have a high probability of
absorption and the reflectivity decreases strongly with the $g$-parameter.
Also, the peak of the $I(\theta_{\rm pp})$-curves shifts from perpendicular
escape directions $\theta_{\rm pp}^{\rm maxI}\approx 0^{\circ}$ for isotropic
scattering to large scattering angles $\theta_{\rm pp}^{\rm maxI}$ for larger
$g$ according to Table. 5. Interestingly, the fractional polarization of the
scattered radiation $p(\theta_{\rm pp})$ in Fig. 18 depends only slightly on
the scattering asymmetry parameter. The polarization $p$ follows from the
ratio between escaping single scattered and multiple scattered photons, and
this ratio clearly does not significantly change for different $g$-values.
Independently of $g$, the maximum $p$-values occurs near $\theta_{\rm pp}^{\rm
maxp}\approx-35^{\circ}$ or approximately near right-angle scattering
$\theta_{\rm pp}^{\rm maxp}-\theta_{0}\approx 90^{\circ}$.

##### Scattering polarization.

The maximum fractional polarization $p_{\rm max}$ of the polarization phase
function has a strong effect on $p(\theta,\phi)$ of the reflected radiation.
For the models with $\theta_{0}=57.5^{\circ}$, $\omega=0.5,$ and $g=0.5$ shown
in Fig. 18, there exists a well-defined relation for the maximum fractional
polarization of the reflected radiation max$(p(\theta_{\rm pp}))\approx
0.62\cdot p_{\rm max}$ (see Table. 5). This tight relation is helpful for
deriving the dust parameter $p_{\rm max}$ from observations. However,
$p(\theta,\phi)$ depends also on $\omega$ and this must be considered for the
derivation of $p_{\rm max}$ (see Sect. 5.4).

The impact of different $p_{\rm max}$-parameter on the reflected intensity
$I(\theta_{\rm pp})$ is hardly visible in Fig. 18. There are small differences
because the higher order scatterings depend on the polarization of the
scattered photons as described in Chandrasekhar (1960) for Rayleigh
scattering, but this effect is not important when compared to the strong
dependencies caused by the parameters $\omega$ and $g$.

##### Incidence angle.

The incidence angle $\theta_{0}$ defines for a given viewing direction the
scattering angle. Forward scattering dust $g>0$ and an incidence close to
perpendicular $\theta_{0}\approx 0^{\circ}$ will scatter the photons
predominantly into deeper layers and the surface reflectivity is low. For
grazing incidence $\theta_{0}\rightarrow 90^{\circ}$ the forward scattering
dust will produce a high reflectivity in the forward direction $\theta_{\rm
pp}\approx-70^{\circ}$. This produces the frequently observed intensity
maximum for the front side surface of inclined disks. This effect is further
amplified by the fact that the first interaction for photons with grazing
incidence takes place often close to the surface, and if scattering occurs,
then the probability for photon escape is relatively high.

The fractional polarization $p(\theta_{\rm pp})$ has a strong maximum near
angles $\theta_{\rm pp}^{\rm maxp}\approx\theta_{0}-90^{\circ}$ corresponding
to right-angle scattering. For the polarized intensity $P$, the angle for the
maximum reflectivity is intermediate between the peaks for the fractional
polarization $p$ and the intensity $I$.

Table 5: Key values for the reflected radiation in the plane parallel surface
models

Parm. | Ref. |  $\omega$-dep. |  $g$-dep. |  $p_{\rm max}$-dep. |  $\theta_{0}$-dep.
---|---|---|---|---|---
$\omega$ |  $\omega_{r}$=$0.5$ | 0.20 | 0.80 | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$
$g$ | $g_{r}=0.5$ | $g_{r}$ | $g_{r}$ | 0.00 | 0.25 | 0.75 | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$
$p_{\rm max}$ | $p_{r}=0.5$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | 0.20 | 0.80 | $p_{r}$ | $p_{r}$ | $p_{r}$
$\theta_{0}$ |  $\theta_{0,r}$=57.5∘ | $\theta_{0,r}$ | $\theta_{0,r}$ | $\theta_{0,r}$ | $\theta_{0,r}$ | $\theta_{0,r}$ | $\theta_{0,r}$ | $\theta_{0,r}$ | $7.5^{\circ}$ | $32.5^{\circ}$ | $82.5^{\circ}$
${\cal{R}}$ | 0.091 | 0.025 | 0.253 | 0.155 | 0.126 | 0.047 | 0.091 | 0.091 | 0.048 | 0.058 | 0.194
${\rm max}(I(\theta_{\rm pp}))$ | 0.031 | 0.010 | 0.074 | 0.039 | 0.033 | 0.023 | 0.032 | 0.032 | 0.011 | 0.015 | 0.120
$\theta_{\rm pp}^{\rm maxI}[^{\circ}]$ | -62.5 | -67.5 | -57.5 | 7.5. | -52.5 | -72.5 | -67.5 | -67.5 | -2.5 | -47.5 | -77.5
${\rm max}(p(\theta_{\rm pp}))$ | 0.31 | 0.43 | 0.19 | 0.35 | 0.33 | 0.30 | 0.12 | 0.50 | 0.37 | 0.30 | 0.39
$\theta_{\rm pp}^{\rm maxp}[^{\circ}]$ | -32.5 | -32.5 | -37.5 | -37.5 | -37.5 | -27.5 | -37.5 | -37.5 | -87.5 | -67.5 | -12.5

555Given values include integrated reflectivity ${\cal{R}}$, peak reflectivity
${\rm max}(I(\theta_{\rm pp}))$ and peak fractional polarization ${\rm
max}(p(\theta_{\rm pp}))$ for the principal plane and the corresponding angle
bin $\theta_{\rm pp}^{\rm maxI}$ and $\theta_{\rm pp}^{\rm maxp}$
respectively, as discussed in Fig. 18. Numerical uncertainties for the
calculated model results are at a level of about one or two units for the last
digit of the given values.

##  Appendix B Additional results for the disk models

The main text of this paper provides quantitative results for the radiation
parameters for the reference disk model in Fig. 7 and lists in Table 2
corresponding values for four inclinations and for many different model
parameters. This appendix provides for most disk models in Table 2 the plots
of radiation parameters as functions of the disk inclination which should be
helpful for a detailed assessment of the model results.

###  B.1 Disk integrated and disk averaged radiation parameters

Figure 19: Integrated disk intensity $\overline{I}(i)/I_{\star}$, azimuthal
polarization $\overline{Q}_{\varphi}(i)/I_{\star}$, Stokes polarization
$\overline{Q}(i)/I_{\star}$ and the disk averaged fractional polarizations for
the azimuthal polarization $\langle
p_{\varphi}\rangle=\overline{Q_{\varphi}}/\overline{I}$ and the Stokes $Q$
polarization $\langle p_{Q}\rangle=\overline{Q}/\overline{I}$ as functions of
disk inclination $i$. Model parameters are $\omega=0.5$, $g=0.5$ and
$\chi=32.5^{\circ}$ unless other values are given in the leftmost panel. Fixed
model parameters are $\alpha=10^{\circ}$, and $p_{\rm max}=0.5$. The red curve
in each panel is the reference model.

Figure 19 gives an overview of the model parameter dependence for the
integrated reflected intensity $\overline{I}(i)/I_{\star}$, the azimuthal
polarization $\overline{Q}_{\varphi}(i)/I_{\star}$, the Stokes parameter
$\overline{Q}(i)/I_{\star}$, and the disk averaged fractional polarizations
$\langle p_{\varphi}\rangle=\overline{Q}_{\varphi}/\overline{I}$ and $\langle
p_{Q}\rangle=\overline{Q}/\overline{I}$. Each panel includes the result for
the reference model plotted in red.

A general feature of the transition disk model is that they are bright in
intensity $\overline{I}(i)$ and polarized intensity $\overline{Q}_{\phi}(i)$
for pole-on viewing angles ($i=0^{\circ}$), when the illuminated inner wall is
well visible. The signal decrease for larger inclination and become zero for
edge-on disks ($i=90^{\circ}$) when the illuminated regions are hidden.

The four model parameters $\omega$, $g$, $\chi$, and $\alpha$ investigated in
Fig. 19 have a strong influence on the overall level of the $\overline{I}(i)$,
$\overline{Q}_{\varphi}(i),$ and $\overline{Q}(i)$ curves. The parameter
$p_{\rm max}$ is not included because it only changes —to first order— the
polarization level, as discussed for the plane-parallel case and as
illustrated below.

The relative inclination dependence for $\overline{I}(i)$,
$\overline{Q}_{\varphi}(i)$, and $\overline{Q}(i)$ or the shape of the curves
are not strongly changed by $\omega$, $g,$ or $\alpha$. According to the
numerical values in Table 1 for $i=7.5^{\circ},\,32.5^{\circ},\,57.5^{\circ},$
and $77.5^{\circ}$, the reflected intensities $\overline{I}(i)/I_{\star}$ for
$\omega=0.2$ are simply scaled by a factor of $0.28\pm 0.01$, and for
$\omega=0.8$ by a factor of $2.7\pm 0.2$ relative to the reference case
$\omega=0.5$. As described for the plane parallel case, a low $\omega$
produces scattered light with less multiple scattering and therefore a higher
fractional polarization. For the asymmetry parameter $g$, both the intensity
and polarized intensity are enhanced for low $g$, while the fractional
polarization signal stays roughly the same. The angular wall height $\alpha$
scales the integrated radiation parameters approximately by a factor
$\alpha/\alpha_{r}$ except for large $i$ when the front wall can hide
illuminated disk regions on the back side and therefore introduce more complex
$\alpha$-dependencies. The wall slope $\chi$ has important effects on both the
level and the shape of the $\overline{I}(i)$ and $\overline{Q}_{\varphi}(i)$
curves. Disks with flat walls, equivalent to small $\chi$, are significantly
brighter when compared to disks with steep walls. For flat walls the incidence
angle $\theta_{0}$ of the incoming light is large and close to grazing
incidence and therefore the photons have a high probability to escape from the
disk surface. Contrary to this, photons penetrate deeply into the disk for
steep incidence angles (small $\theta_{0}$) and for strongly forward
scattering dust, the probability for photon absorption is high. The shape of
the curves for the fractional polarizations $\langle p_{\varphi}(i)\rangle$
and $\langle p_{Q}(i)\rangle$ show the same behaviour as the reference case
for many disk parameters.

Most model parameters hardly change the ”normal” $i$-dependence for
$p_{\varphi}(i)$ and $p_{Q}(i)$ except for the wall slope $\chi$, which has a
strong impact. The curves show for small $i\ll 90^{\circ}-\chi$ the expected
decrease with $i$ for $p_{\varphi}(i)$ or increase with $i$ for $p_{Q}(i)$ as
long as the illuminated inner wall on the front side is well visible. However,
when $i$ approaches $\chi$ the typically lower $p$, negative $p_{Q}$ front
side component disappears and the overall $p_{Q}$ shows a bump around
$i\approx 90^{\circ}-\chi$. This bump is strong and narrow for small $\chi$
and weak and broad for large $\chi$ and for $\chi=82.5^{\circ}$ it is around
$i\approx 5^{\circ}$ and therefore hardly visible. For $i>90^{\circ}-\chi$
various effects play a role which depend on details of the scattering
geometry, like hiding of parts of the back wall, grazing photon emergence from
the lower or upper back side, negative polarization because of multiple
scattering under $\theta_{s}\gtrsim 150^{\circ}$ back-scattering conditions.
These special effects, which become important for cases $i\gtrsim
90^{\circ}-\chi$, are beyond the scope of our general description of model
parameter dependencies for transition disks.

###  B.2 Quadrant polarization parameters

The azimuthal polarization depends significantly on the scattering asymmetry
parameter $g$ and the wall slope $\chi$ as illustrated in Figs. 8(a) and 8(b).
These dependencies can be quantified by the polarization quadrant parameters
as illustrated in Figs. 20 and 21. In general, high reflectivity and therefore
also strong signals for the quadrant parameters $Q_{xxx}/I_{\star}$ and
$U_{xxx}/I_{\star}$ are obtained for low $g=0$ and flat wall slopes.

The quadrant polarization parameters $Q_{xxx}/I_{\star}$ and
$U_{xxx}/I_{\star}$ also depend strongly on the model parameters $\omega$,
$p_{\rm max}$, and $\alpha$ which enhance or lower the overall level of the
scattering polarization for the entire disk, almost independently of the
azimuthal angle. Therefore, the values $Q_{xxx}/I_{\star}$ and
$U_{xxx}/I_{\star}$ have a limited diagnostic value. More useful for the
investigation of the scattering asymmetry $g$ or the wall slope $\chi$ are
relative quadrant values $Q_{xxx}/\overline{Q}_{\varphi}$ and
$U_{xxx}/\overline{Q}_{\varphi}$ as given in Table 1 or ratios between
quadrant polarization intensities like $Q_{180}/Q_{000}$ or $U_{135}/U_{045}$,
because these ratios depend mainly on $g$ and $\chi$ but are almost
independent of $\omega$, $p_{\rm max}$ and $\alpha$.

The interpretation of the front-to-back disk brightness ratio
$Q_{180}/Q_{000}$ should also consider the corresponding intensity effect.
Therefore, we also list in Table 1 the corresponding front-to-back intensity
ratio $I_{180}/I_{000}$ derived in exactly the same way as $Q_{180}/Q_{000}$,
but from the $I(x,y)$-image instead of the $Q(x,y)$-image. The diagnostic
potential of these ratios for the interpretation of data is briefly discussed
in Sect. 5.5.

Figure 20: Quadrant polarization values $Q_{xxx}(i)/I_{\star}$ (left) and
$U_{xxx}(i)/I_{\star}$ (right) for the scattering asymmetry parameters
$g=0.0,\,0.25,\,0.5$ and $0.75$, and fixed parameter $\omega=0.5$, $p_{\rm
max}=0.5$, $\chi=32.5^{\circ}$ and $\alpha=10^{\circ}$. The red curves is the
reference model.  Figure 21: $Q_{xxx}(i)/I_{\star}$ (left) and
$U_{xxx}(i)/I_{\star}$ (right) for the wall slopes
$\chi=7.5^{\circ},\,32.5^{\circ},\,57.5^{\circ}$ and $82.5^{\circ}$ and fixed
parameter $\omega=0.5$, $g=0.5$, $p_{\rm max}=0.5$ and $\alpha=10^{\circ}$.
The red curves is the reference model.