z_{0}^{2}}-1\right)-3z_{H,+}^{3}\right)\left(9z_{H,+}^{6}-8\chi^{2}z_{0}^{4}\left(e^{z_{H,+}^{3}/2\chi
z_{0}^{2}}-1\right)\right)}{384\chi z_{0}^{6}z_{H,+}^{5}}\ .$ (268)

As for the Hayward space-time, we can expand the function
$2\chi(1-e^{-z(\rho)^{3}/z_{\ast}^{3}})$ around $\rho=0$

$2\chi(1-e^{-z(\rho)^{3}/z_{\ast}^{3}})=z_{H,+}+\frac{3e^{-z_{H,+}^{3}/\chi
z_{0}^{2}}\left(2\chi z_{0}^{2}\left(e^{z_{H,+}^{3}/2\chi
z_{0}^{2}}-1\right)-3z_{H,+}^{3}\right)}{4z_{0}^{4}}\rho^{2}+\mathcal{O}(\rho^{4})$
(269)

from which we read off the coefficients $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$

$\xi_{1}=0\ ,\quad\xi_{2}=\frac{3e^{-z_{H,+}^{3}/\chi z_{0}^{2}}\left(2\chi
z_{0}^{2}\left(e^{z_{H,+}^{3}/2\chi
z_{0}^{2}}-1\right)-3z_{H,+}^{3}\right)}{4z_{0}^{4}}\mbox{\quad
and\quad}\xi_{3}=0\ .$ (270)

We observe that the conditions stated in equation (94) are satisfied in the
Dymnikova space-time as well, leading to the same conclusions as those drawn
for the Hayward black hole. These conditions ensure that the Ricci scalar and
the Hawking temperature remain well-defined and free from singularities at the
event horizons, which is indeed the case for this space-time.

##  Appendix E Conditions for regularity at inner horizons

The sufficient conditions (36) in Section 2 for regularity of the Ricci tensor
at $z_{H}$, have been derived assuming that the latter is the position of the
outer horizon: notably, we have assumed in various instances that $f(z)>0$ for
$z>z_{H}$. Generalised Schwarzschild BHs, however, may have further horizons,
which are characterised by a vanishing of the function $f(z)$ in (1), such
that the derivative of the distance (6) diverges. The latter can (in the same
way as discussed in Section 2) lead to curvature singularities that are
physically not acceptable. Assuming that the form of the metric functions
$f,h$ are still of the form (8), the presence of an inner horizon, therefore,
puts further conditions on the functions $\Phi$ and $\Psi$. We can obtain
these conditions by straightforwardly generalizing the discussion of Section
2. Here we shall briefly exhibit them, assuming a black hole with two (simple)
horizons at $z_{H,\pm}$ (with $z_{H,+}>z_{H,-}$)191919Here $z_{H,+}$ is
understood to be the position of the outer horizon, which we denote by $z_{H}$
throughout the remainder of this paper. and distances $d_{H,+}$ and $d_{H,-}$
respectively.

For simplicity, we consider the conditions that
$\mathfrak{f}_{1}:=f^{\prime}(z_{H,-})$ and
$\mathfrak{f}_{2}:=f^{\prime\prime}(z_{H,-})$ are both finite, with
$\mathfrak{f}_{1}<0$. Concretely, we write

$\displaystyle
f(z)=\mathfrak{f}_{1}\,(z-z_{H,-})+\frac{\mathfrak{f}_{2}}{2}\,(z-z_{H,-})^{2}+\mathfrak{o}\left((z-z_{H,-})^{2}\right)\,,$
for $\displaystyle z>z_{H,-}\,.$ (271)

Furthermore, we define

$\displaystyle d=:d_{H,-}+\xi(z)\,,$ with
$\displaystyle\xi(z)=\frac{2\sqrt{z-z_{H,-}}}{\sqrt{-\mathfrak{f}_{1}}}+\frac{\mathfrak{f}_{2}}{6}\,\frac{(z-z_{H,-})^{3/2}}{(-\mathfrak{f}_{1})^{3/2}}+\mathfrak{o}\left((z-z_{H,-})^{3/2}\right)\,.$

For $z>z_{H,-}$ we therefore find

$\displaystyle
z=z_{H,-}-\frac{\mathfrak{f}_{1}}{4}\,\xi^{2}+\frac{\mathfrak{f}_{1}\mathfrak{f}_{2}}{96}\,\xi^{4}+\mathfrak{o}(\xi^{4})\,,$
(272)

and thus we have the following conditions

$\displaystyle\derivative{\Phi}{y}\bigg{|}_{y=\frac{1}{d_{H,-}}}=0\,,$
$\displaystyle\derivative[3]{\Phi}{y}+6\,d_{H,-}\,\derivative[2]{\Phi}{y}\bigg{|}_{y=\frac{1}{d_{H,-}}}=0\,,$
(273)

which can be expressed in terms of the $\Phi^{(n)}$ (where we have already
taken into account (36))

$\displaystyle 0$
$\displaystyle=\left(7\,d_{H,+}\,d_{H,-}-3d_{H,+}^{2}-4d_{H,-}^{2}\right)\,\frac{\Phi^{(2)}}{d_{H,+}d_{H,-}^{2}}+\sum_{n=4}^{\infty}\frac{(d_{H,+}-d_{H,-})^{n-1}}{(n-1)!d_{H,+}^{n-1}d_{H,-}^{n-1}}\,\Phi^{(n)}\,,$
$\displaystyle 0$
$\displaystyle=42(d_{H,-}-d_{H,+})\Phi^{(2)}+\sum_{n=4}^{\infty}\frac{\left[(n+4)d_{H,+}-6d_{H,-}\right](d_{H,+}-d_{H,-})^{n-3}}{\Gamma(n-1)\,d_{H,+}^{n-2}d_{H,-}^{n-3}}\,\Phi^{(n)}\,.$
(274)

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# Optimizing RRH Placement Under a
Noise-Limited Point-to-Point Wireless Backhaul

Hussein A. Ammar1, Raviraj Adve1, Shahram Shahbazpanahi2, and Gary Boudreau3
1University of Toronto, Dep. of Elec. and Comp. Eng., Toronto, Canada
2University of Ontario Institute of Technology, Dep. of Elec. and Comp. Eng.,
Oshawa, Canada
3Ericsson Canada, Ottawa, Canada
Email: {ammarhus<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the deployment decisions and location optimization for
the remote radio heads (RRHs) in coordinated distributed networks in the
presence of a wireless backhaul. We implement a scheme where the RRHs use
zero-forcing beamforming (ZF-BF) for the access channel to jointly serve
multiple users, while on the backhaul the RRHs are connected to their central
units (CUs) through point-to-point wireless links. We investigate the effect
of this scheme on the deployment of the RRHs and on the resulting achievable
spectral efficiency over the access channel (under a backhaul outage
constraint). Our results show that even for noise-limited backhaul links, a
large bandwidth must be allocated to the backhaul to allow freely distributing
the RRHs in the network. Additionally, our results show that distributing the
available antennas on more RRHs is favored as compared to a more co-located
antenna system. This motivates further works to study the efficiency of
wireless backhaul schemes and their effect on the performance of coordinated
distributed networks with joint transmission.

###### Index Terms:

Cooperative distributed network, distributed antenna system, RRH placement,
wireless backhaul$\backslash$fronthaul.

##  I Introduction

Cooperation between the network transmitters is imperative to control
interference. In a distributed network, this can be achieved by deploying two
key modules: a central unit (CU) that executes protocols to schedule the
transmission and the reception of the signals, and remote radio heads (RRHs)
deployed throughout the network’s coverage area to cooperate and jointly serve
users [1]. However, a key component for such a scheme is the backhaul links
which connect RRHs to their CUs (also called fronthaul of the CU). Notably,
this scheme is a distributed implementation of the 5G next-generation NodeB
specified in the New Radio specification [1].

The backhaul links carry the heaviest communication load since they carry the
data for all users served by the RRHs and hence are the bottleneck to
throughput. The works in [2, 3] modeled the reliability of dedicated backhaul
links as a Bernoulli distribution to study their effect on the system outage.
The aim from these studies is to derive optimal strategies for the assignment
of the backhaul links. Important issues such as power allocation [4],
cooperation strategies [5], beamforming design [6], resource allocation [7]
and formation of serving clusters [8] have been investigated under the theme
of a limited-capacity backhaul. Additionally, the work in [9] analyzed
backhaul signal compression as a mean to minimize the impact of the limited
capacity of the backhaul.

As an alternative, several investigations have considered the problem of
transmitter placement. The work in [10] studied RRHs placement without
accounting for the backhaul. Furthermore, the study in [11] optimized the
placement of relays in in-band or out-of-band non-cooperative cellular
networks by modeling users traffic using queueing theory; their results show
that the latter provides better performance. The work in [12] also optimized
the deployment of relays, while the work in [13] studied inter-site distance
for RRHs deployment on a highway road scenario, where the outage probability
was derived. Nonetheless, these works about transmitters deployment did not
consider the effect of the backhaul on the deployment decisions in a
coordinated distributed network.

The backhaul capacity strongly affects the network performance and the
flexibility of the deployment of the RRHs. Motivated by this fact, we herein
investigate the effect of using point-to-point wireless backhaul on the
placement of the RRHs, where the aim is to enhance the access channel spectral
efficiency while meeting a backhaul outage constraint. Our work adds to the
literature on both the limited-capacity backhaul and transmitters placement
for distributed networks and provides insights into RRH deployment decisions.
The rest of the paper is organized as follows: in Section II, we present our
system model. While in Section III, we formulate the RRHs location
optimization problem. In Sections IV and V, we present our proposed solution
and results, respectively. Finally, we present our conclusions in Section VI.

##  II System Model

###  II-A Network Model

We consider a cooperative distributed network comprising $Q$ cells. Assuming
joint transmission with disjoint clustering, the RRHs found in each cell $q$
jointly serve the users inside their cell boundary $\mathcal{B}_{q}$ [14].
Each cell employs $N$ RRHs to serve $K$ users on the same time-frequency
resource block using zero-forcing beamforming (ZF-BF). Each RRH uses $M$
antennas to serve the users, where $NM>K$. In each cell, a single CU, located
at the cell center, controls the transmissions of the RRHs through point-to-
point backhaul links. The users in the cell may be concentrated in (but not
restricted) hotspots with _a priori_ known traffic pattern. The backhaul and
access links are spectrally orthogonal. To focus on network design, we assume
perfect channel state information (CSI) is available. Our aim is to deploy the
RRHs in the network so that we obtain the highest spectral efficiency over the
access channel while controlling backhaul outage.

###  II-B Access Channel Transmission Scheme

The signal received at a typical user $k$ located in cell $q$ is

$\displaystyle r_{qk}=\displaystyle\underbrace{\sum_{n=1}^{N}{\bf
h}_{qn,qk}^{H}{\bf w}_{qnk}s_{qk}}_{\text{useful
signal}}+\underbrace{\sum_{\begin{subarray}{c}n=\\{1,\dots,N\\},\\\
k^{\prime}=\\{1,\dots,K\\},k^{\prime}\neq k\end{subarray}}{\bf
h}_{qn,qk}^{H}{\bf
w}_{qnk^{\prime}}s_{qk^{\prime}}}_{\begin{subarray}{c}\text{intra-
cluster}\text{ interference}\end{subarray}}$
$\displaystyle\quad+\underbrace{\sum_{\begin{subarray}{c}q^{\prime}=\\{1,\dots,Q\\},q^{\prime}\neq
q,\\\ n^{\prime}=\\{1,\dots,N\\},k^{\prime}=\\{1,\dots,K\\}\end{subarray}}{\bf
h}_{q^{\prime}n^{\prime},qk}^{H}{\bf
w}_{q^{\prime}n^{\prime}k^{\prime}}s_{q^{\prime}k^{\prime}}}_{\begin{subarray}{c}\text{inter-
cluster}\text{ interference}\end{subarray}}+z_{qk}$ (1)

where $s_{qk}\in\mathbb{C}$ is the power-limited signal sent to user $k$ by
the serving RRHs in cell $q$, that is $\mathbb{E}\\{{\bf s}_{q}{\bf
s}_{q}^{H}\\}=p{\bf I}_{K}$ for ${\bf s}_{q}=[s_{q1},\dots,s_{qK}]$, with $p$
being the total power budget for the RRHs in the cell, which is referred as
vector normalization [15].

The vector ${\bf h}_{qn,qk}\in\mathbb{C}^{M}$ denotes the channel between RRH
$n$ in cell $q$ and user $k$ in cell $q$ and accounts for the small-scale and
large-scale fading. We have ${\bf
h}_{qn,qk}=\sqrt{\ell_{k}(x_{qn},y_{qn})}{\bf g}_{qn,qk}$, where ${\bf
g}_{qn,qk}\sim\mathcal{CN}(0,{\bf I}_{M})$ is a white complex Gaussian random
vector representing the Rayleigh fading, while
$\ell_{k}(x_{qn},y_{qn})=\left(1+d_{qn,qk}/d_{0}\right)^{-\alpha}$ is the path
loss of the signal, with
$d_{qn,qk}=\sqrt{(x_{qn}-\widetilde{x}_{k})^{2}+(y_{qn}-\widetilde{y}_{k})^{2}}$
being the distance between RRH $n$ and user $k$, $d_{0}$ is the reference
distance, and $\alpha$ is the path loss exponent. The terms $x_{qn}$ and
$y_{qn}$ are the $x,y$-coordinates of RRH $n$ and will be our optimization
variables. Furthermore, ${\bf w}_{qnk}\in\mathbb{C}^{M}$ is the linear
precoding vector used by RRH $n$ to serve user $k$ in its cell $q$, and
$z_{qk}\sim\mathcal{CN}(0,\sigma_{z}^{2})$ is the independent additive white
Gaussian noise (AWGN), with $\sigma^{2}_{z}$ being the noise power.

The RRHs use ZF-BF to serve the $K$ users in each cell. To construct the
beamformer, we denote the concatenation of the channel matrix for all the
users in cell $q$ as ${\bf H}_{q}=[{\bf h}_{q1}\dots\bf
h_{qK}]\in\mathbb{C}^{NM\times K}$ with presumably linearly independent rows,
with ${\bf h}_{qk}=[{\bf h}_{qk,qk}^{T}\dots\bf
h_{qN,qk}^{T}]^{T}\in\mathbb{C}^{NM}$ being the concatenation of the access
channels for user $k$. We can write ${\bf h}_{qk}={\bf D}_{qk}^{1/2}{\bf
g}_{qk}$, where ${\bf D}_{qk}\in\mathbb{C}^{NM\times NM}$ is a diagonal matrix
representing the large-scale fading between all the RRHs in the network and
user $k$. Hence, $[{\bf D}_{qk}]_{mm}=\ell_{k}(x_{qn},y_{qn})$ where
$n=\lceil{m/M}\rceil$ for $m=\\{1,\dots,NM\\}$. Furthermore, ${\bf
g}_{qk}\in\mathbb{C}^{NM}$ represents the vector of the small-scale fading
coefficients between the antennas of the RRHs and user $k$.

The precoding matrix ${\bf W}_{q}$ is designed for each cell $q$ assuming CSI
at the RRHs is available. We have ${\bf W}_{q}=[{\bf v}_{q1}\dots\bf
v_{qK}]\in\mathbb{C}^{NM\times K}$ with ${\bf v}_{qk}=[{\bf
w}_{q1k}^{T}\dots\bf w_{qNk}^{T}]^{T}\in\mathbb{C}^{NM}$, where ${\bf
w}_{qnk}\in\mathbb{C}^{N}$ is the precoding vector used at RRH $n$, and is
given by

$\displaystyle{\bf W}_{q}={\bf\widetilde{W}}_{q}\bm{\mu}_{q}=\left({\bf
H}_{q}\right)^{\dagger}\bm{\mu}_{q}={\bf H}_{q}\left({\bf H}_{q}^{H}{\bf
H}_{q}\right)^{-1}\bm{\mu}_{q}$ (2)

Here, $\bm{\mu}_{q}\in\mathbb{C}^{K\times K}$ is a diagonal matrix which
satisfies the power budget $\mathbb{E}\left\\{\text{tr}\left\\{{\bf W}_{q}{\bf
W}_{q}^{H}\right\\}\right\\}=p$, i.e., using average power normalization [16],
and its $k^{th}$ entry is

$\displaystyle\mu_{qk}=\left[\bm{\mu}_{q}\right]_{k,k}=\sqrt{pK^{-1}\mathbb{E}\\{\|{\bf\widetilde{v}}_{qk}\|^{2}\\}^{-1}}$
(3)

with ${\bf\widetilde{v}}_{qk}=[{\bf H}_{q}\left({\bf H}_{q}^{H}{\bf
H}_{q}\right)^{-1}]_{.k}$ is $k^{th}$ column of the matrix.

###  II-C Backhaul Transmission Scheme

We consider point-to-point single-input single-output (SISO) noise-limited
wireless transmission between the CUs and their RRHs (representing an upper-
bound for the performance of a SISO solution). We assume that the CU
implements directional antennas, providing parallel _Rician channels_ between
the CU and the $N$ RRHs.

The downlink average achievable rate between CU $q$ and its served RRHs is
defined as

$\displaystyle R_{qn}^{(b)}\left(x_{qn},y_{qn}\right)$
$\displaystyle=\mathbb{E}\left\\{\log\left(1+\rho_{c}\left|\bar{g}_{qn}\right|^{2}\bar{\ell}_{q}(x_{qn},y_{qn})\right)\right\\}$
(4)

where $\rho_{c}=\frac{p_{c}}{\sigma^{2}}$,
$\bar{\ell}_{q}(x_{qn},y_{qn})=\left(1+\bar{d}_{qn}/d_{0}\right)^{-\alpha}$ is
the path loss on the backhaul link, and it depends on the distance between CU
$q$ and RRH $n$ denoted as
$\bar{d}_{qn}=\sqrt{(x_{qn}-\bar{x}_{q})^{2}+(y_{qn}-\bar{y}_{q})^{2}}$, where
$\left(\bar{x}_{q},\bar{y}_{q}\right)$ are the coordinates of the CU. The term
$\bar{g}_{qn}$ is the small-scale fading of the channel, which follows a
Rician fading model. This Rician channel is characterized by the Rician
parameter (the Rician K-factor) $\mathcal{K}=\eta_{1}^{2}/\eta_{2}^{2}$, where
$\eta_{1}^{2}$ and $\eta_{2}^{2}$ represent the power of the line-of-sight
(LoS) and NLoS components respectively.

##  III Problem Formulation

We aim to find the optimal deployment for the RRH locations in the presence of
specific traffic distributions, inter-cell interference, and most importantly,
under limited capacity wireless backhaul. These important factors will be
characterized in the next subsections.

###  III-A Traffic Distribution

We use a traffic probability density function (PDF) that is a combination of
both a uniform distribution, chosen with probability $P_{0}$, and a number
$N_{h}$ of bivariate normal distributions, representing hotspots (a choice of
small $P_{0}$ means more dense hotspots). The number of these hotspots is
uniformly distributed, i.e., $N_{h}\sim
U\left(N_{h}^{\text{min}},N_{h}^{\text{max}}\right)$. This number can be
generated for the whole network or for each cell separately. These hotspots
are centered at locations
$\left[{\bf\widetilde{x}}_{h},{\bf\widetilde{y}}_{h}\right]=\left[\left[{\widetilde{x}}_{h1},\dots,{\widetilde{x}}_{hN_{h}}\right]^{T},\left[{\widetilde{y}}_{h1},\dots,{\widetilde{y}}_{hN_{h}}\right]^{T}\right]\in\mathbb{R}^{N_{h}\times
2}$, and they have equal variances $\sigma_{h}^{2}$ on both the $x$ and $y$
axis, without any correlation between the two axis. Therefore, the cell
traffic distribution PDF bounded by the cell boundary $\mathcal{B}_{q}$ is
defined as

$\displaystyle
f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})=f_{0}\left(P_{0}\left(\frac{1}{B_{q}}\right)+\left(1-P_{0}\right)\right.$
(5)

where $B_{q}$ (different from cell boundary $\mathcal{B}_{q}$) is the area of
cell $q$, $f_{0}$ is a normalizing factor that can be calculated numerically
to normalize the PDF. This traffic model is flexible in the sense that it can
be constructed from a traffic survey that identifies the locations of hotspots
in the network.

###  III-B Access Channel Spectral Efficiency

On the access channel, the ZF beamformer is formed per cell $q$. Hence, the
intra-cluster interference found in (II-B) will be completely canceled. As a
result, the mean achievable spectral efficiency over the access channel for
user $k$ in cell $q$ is

$\displaystyle R$ ${}_{qk}^{(a)}\left({\bf x},{\bf y}\right)=$ (6)

The concatenated form of the RRHs’ locations in the network is written as
$\left[{\bf x},{\bf y}\right]$, where the the x-coordinates are ${\bf x}=[{\bf
x}_{1}^{T},\dots,{\bf x}_{q}^{T},\dots,\dots,{\bf
x}_{Q}^{T}]^{T}\in\mathbb{C}^{NQ}$, for $q\in\\{1,\dots,Q\\}$ (similarly for
${\bf y}$). These will form our optimization variables. We use $[{\bf
x}_{q},{\bf y}_{q}]$ to refer to the locations of the RRHs in cell $q$ and
$[{\bf x}_{-q},{\bf y}_{-q}]$ to denote the locations of all the RRHs in the
network except those in cell $q$.

The spectral efficiency over the access channels depends on the distances
between all the RRHs found in the network and the user $k$, which in its turn
depends on the locations of all the RRHs. Given the distances to the serving
and interfering RRHs, the lower-bound of spectral efficiency can be written as
[10]

$\displaystyle R_{qk}^{(a)}\left({\bf x},{\bf
y}\right)=\leavevmode\resizebox{160.43727pt}{}{$\displaystyle\log\left(1+\displaystyle\gamma_{k}\left({\bf
x}_{-q},{\bf
y}_{-q}\right)^{-1}\sum_{n=1}^{N}\ell_{k}(x_{qn},y_{qn})\right)$}$ (7)

with

$\displaystyle\gamma_{k}\left({\bf x}_{-q},{\bf
y}_{-q}\right)=\leavevmode\resizebox{151.76964pt}{}{$\displaystyle\frac{NK}{(NM-K)\rho}\left(\displaystyle\frac{M\rho}{K}\sum_{q^{\prime}=1,q^{\prime}\neq
q}^{Q}\text{ICI}_{q^{\prime}k}({\bf x}_{q^{\prime}},{\bf
y}_{q^{\prime}})+1\right)$}$ (8) $\displaystyle\text{ICI}_{q^{\prime}k}({\bf
x}_{q^{\prime}},{\bf
y}_{q^{\prime}})=\sum_{l=1}^{N}\ell_{k}(x_{q^{\prime}l},y_{q^{\prime}l})\sum_{j=1}^{K}\frac{\ell_{j}(x_{q^{\prime}l},y_{q^{\prime}l})}{\xi(q^{\prime},j)}$
(9)

where $\rho=\frac{p}{\sigma_{z}^{2}}$, and $\text{ICI}_{q^{\prime}k}({\bf
x}_{q^{\prime}},{\bf y}_{q^{\prime}})$ denotes the inter-cell interference
(ICI) contributed by cell $q^{\prime}$ to the user under test and
$\xi(q^{\prime},j)=\text{tr}\\{{\bf
D}_{q^{\prime}j}\\}=M\sum_{m=1}^{N}\ell_{j}(x_{q^{\prime}m},y_{q^{\prime}m})$.
As can be seen, this interference depends on the locations of the users in the
interfering cells because the beamformers in each cell is designed based on
the channels of the users in these cells.

Using the traffic distribution PDF,
$f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})$, in (III-A), we further average
this achievable rate over
$f_{q^{\prime}}(\widetilde{x}_{k},\widetilde{y}_{k})$ in the interfering cells
by writing the ICI term as

$\displaystyle\text{ICI}_{q^{\prime}k}({\bf x}_{q^{\prime}},{\bf
y}_{q^{\prime}})=\leavevmode\resizebox{143.09538pt}{}{$\displaystyle\sum_{l=1}^{N}\ell_{k}(x_{q^{\prime}l},y_{q^{\prime}l})K\mathbb{E}_{\widetilde{x}_{j},\widetilde{y}_{j}}\left\\{\frac{\ell_{j}(x_{q^{\prime}l},y_{q^{\prime}l})}{\xi(q^{\prime},j)}\right\\}$}$
$\displaystyle\
=\leavevmode\resizebox{190.79385pt}{}{$\displaystyle\sum_{l=1}^{N}\ell_{k}(x_{q^{\prime}l},y_{q^{\prime}l})K\int\\!\\!\int_{\widetilde{x}_{j},\widetilde{y}_{j}\in\mathcal{B}_{q^{\prime}}}\\!\\!\frac{\ell_{j}(x_{q^{\prime}l},y_{q^{\prime}l})}{\xi(q^{\prime},j)}f_{q^{\prime}}(\widetilde{x}_{j},\widetilde{y}_{j})\mathop{}\\!\mathrm{d}\widetilde{x}_{j}\mathop{}\\!\mathrm{d}\widetilde{y}_{j}$}$
(10)

where, as indicated earlier, $\mathcal{B}_{q^{\prime}}$ is the boundary of
cell $q^{\prime}$.

###  III-C Problem Definition

We define our problem of optimizing the locations of the RRHs in the network
as

$\displaystyle\underset{{\bf x},{\bf
y}}{\text{max}}\quad\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\},\ \forall q$ (11a) s.t.

$\displaystyle\displaystyle\mathbb{P}\left\\{\omega_{c}R_{qn}^{(b)}(x_{qn},y_{qn})\leq
K\omega\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right\\}\leq\epsilon,$

$\displaystyle\quad\quad\quad n=1,\dots,N$ (11b)

where

$\displaystyle\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf
y})\right\\}=\leavevmode\resizebox{125.74689pt}{}{$\displaystyle\int\\!\\!\int_{\widetilde{x}_{k},\widetilde{y}_{k}\in\mathcal{B}_{q}}R_{qk}^{(a)}({\bf
x},{\bf
y})f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})\mathop{}\\!\mathrm{d}\widetilde{x}_{k}\mathop{}\\!\mathrm{d}\widetilde{y}_{k}$}$
(12)

The terms $\omega$, $\omega_{c}$ are the bandwidth allocated for the access
channel and the backhaul, respectively, and $\epsilon$ is the allowed backhaul
outage probability for each CU-RRH link.

The formulation in (11) maximizes the average spectral efficiency of typical
user $k$ in the network, and this spectral efficiency is averaged over the
traffic distribution in the user’s cell as shown in (11a), which means it
maximizes the spectral efficiency for the system. Additionally, the $N$
constraints in (11b) place an upper bound on the sum of the rates over the
access channel with respect to the backhaul achieved capacity. If this
constraint is not respected the backhaul will experience an outage.

_Proposition:_ The probability of outage, $P_{qn}\left({\bf x},{\bf y}\right)$
is given by

$\displaystyle P_{qn}\left({\bf x},{\bf y}\right)$
$\displaystyle=\leavevmode\resizebox{164.77771pt}{}{$\displaystyle\mathbb{P}\left\\{\omega_{c}R_{qn}^{(b)}(x_{qn},y_{qn})\leq
K\omega\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right\\}$}$
$\displaystyle=1-Q_{1}\left(\frac{\sqrt{2}\eta_{1}}{\eta_{2}},\frac{\sqrt{2\zeta_{qn}({\bf
x},{\bf y})}}{\eta_{2}}\right)$ (13)

where $Q_{1}(.)$ is the Marcum $Q$-function, and

(14)

###### Proof.

Please see Appendix -A. ∎

##  IV Optimizing RRH Placement Under Backhaul Constraints

In the next subsections, we solve the optimization problem in (11) using two
different approaches.

###  IV-A Direct Approach

For notational simplicity we define the following term.

$\displaystyle J_{1_{qn}}({\bf x},{\bf y})=\frac{\sqrt{2\zeta_{qn}({\bf
x},{\bf y})}}{\eta_{2}}=J_{2_{qn}}(x_{qn},y_{qn})\times J_{3_{q}}({\bf x},{\bf
y})$
$\displaystyle=\leavevmode\resizebox{195.12767pt}{}{$\displaystyle\frac{1}{\eta_{2}}\sqrt{\frac{2}{\rho_{c}\bar{\ell}_{q}(x_{qn},y_{qn})}}\times\sqrt{\left(\exp\left(K\frac{\omega}{\omega_{c}}\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right)-1\right)}$}$ (15)

We can write the Lagrangian formulation of our problem as

$\displaystyle\mathcal{L}({\bf x},{\bf
y},\bm{\lambda})=\leavevmode\resizebox{160.43727pt}{}{$\displaystyle-\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}+\sum_{n=1}^{N}\lambda_{n}\left(P_{qn}\left({\bf x},{\bf
y}\right)-\epsilon\right)$}$ (16)

where the $\bm{\lambda}\succcurlyeq 0$ denotes the vector of Lagrange
multipliers. For a specific RRH $m\in\\{1,\dots,N\\}$ in cell $q$, we
differentiate the Lagrangian formulation with respect to the x-coordinate
$x_{qm}$ of RRH $m$, set it to zero, and obtain an iterative formula for
$x_{qm}$ that can be written as

$\displaystyle
x_{qm}^{(i+1)}=\leavevmode\resizebox{177.78578pt}{}{$\displaystyle\frac{\displaystyle\left(1-A_{2}-A_{4}\right)\int\\!\\!\int_{\widetilde{x}_{k},\widetilde{y}_{k}\in\mathcal{B}_{q}}\widetilde{x}_{k}A_{1}(\widetilde{x}_{k},\widetilde{y}_{k})f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})\mathop{}\\!\mathrm{d}\widetilde{x}_{k}\mathop{}\\!\mathrm{d}\widetilde{y}_{k}+A_{3}\bar{x}_{q}}{\displaystyle\left(1-A_{2}-A_{4}\right)\int\\!\\!\int_{\widetilde{x}_{k},\widetilde{y}_{k}\in\mathcal{B}_{q}}A_{1}(\widetilde{x}_{k},\widetilde{y}_{k})f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})\mathop{}\\!\mathrm{d}\widetilde{x}_{k}\mathop{}\\!\mathrm{d}\widetilde{y}_{k}+A_{3}}$}$
(17)

The same formulation applies by differentiating the Lagrangian with respect to
the y-coordinate.

$\displaystyle
y_{qm}^{(i+1)}=\leavevmode\resizebox{177.78578pt}{}{$\displaystyle\frac{\displaystyle\left(1-A_{2}-A_{4}\right)\int\\!\\!\int_{\widetilde{x}_{k},\widetilde{y}_{k}\in\mathcal{B}_{q}}\widetilde{y}_{k}A_{1}(\widetilde{x}_{k},\widetilde{y}_{k})f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})\mathop{}\\!\mathrm{d}\widetilde{x}_{k}\mathop{}\\!\mathrm{d}\widetilde{y}_{k}+A_{3}\bar{y}_{q}}{\displaystyle\left(1-A_{2}-A_{4}\right)\int\\!\\!\int_{\widetilde{x}_{k},\widetilde{y}_{k}\in\mathcal{B}_{q}}A_{1}(\widetilde{x}_{k},\widetilde{y}_{k})f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})\mathop{}\\!\mathrm{d}\widetilde{x}_{k}\mathop{}\\!\mathrm{d}\widetilde{y}_{k}+A_{3}}$}$
(18)

where

$\displaystyle
A_{1}(\widetilde{x}_{k},\widetilde{y}_{k})=\leavevmode\resizebox{160.43727pt}{}{$\displaystyle\frac{\gamma_{k}\left({\bf
x}_{-q},{\bf
y}_{-q}\right)^{-1}\left(1+d_{qm,qk}/d_{0}\right)^{-1-\alpha}}{d_{qm,qk}\left(1+\gamma_{k}\left({\bf
x}_{-q},{\bf
y}_{-q}\right)^{-1}\displaystyle\sum_{n=1}^{N}\left(1+d_{qn,qk}/d_{0}\right)^{-\alpha}\right)}$}$
(19) $\displaystyle
A_{2}=\frac{K\frac{\omega}{\omega_{c}}\lambda_{m}J_{1_{qm}}({\bf x},{\bf
y})\exp\left(K\frac{\omega}{\omega_{c}}\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf
y})\right\\}\right)}{\eta_{2}\sqrt{2\rho_{c}\left(\exp\left(K\frac{\omega}{\omega_{c}}\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right)-1\right)}}$
$\displaystyle\times\leavevmode\resizebox{201.6317pt}{}{$\displaystyle\exp\left(-\left(\frac{\eta_{1}^{2}}{\eta_{2}^{2}}+\frac{\left(J_{1_{qm}}({\bf
x},{\bf
y})\right)^{2}}{2}\right)\right){}_{0}F_{1}\left(;1;\frac{\eta_{1}^{2}}{2\eta_{2}^{2}}\left(J_{1_{qm}}({\bf
x},{\bf
y})\right)^{2}\right)\left(1+\frac{\bar{d}_{qm}}{d_{0}}\right)^{\frac{\alpha}{2}}$}$
(20) $\displaystyle
A_{3}=\leavevmode\resizebox{186.45341pt}{}{$\displaystyle\frac{\lambda_{m}J_{1_{qm}}({\bf
x},{\bf
y})\sqrt{\left(\exp\left(K\frac{\omega}{\omega_{c}}\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right)-1\right)}}{\eta_{2}\sqrt{2\rho_{c}}}$}$
$\displaystyle\times\leavevmode\resizebox{199.4681pt}{}{$\displaystyle\exp\left(-\left(\frac{\eta_{1}^{2}}{\eta_{2}^{2}}+\frac{\left(J_{1_{qm}}({\bf
x},{\bf
y})\right)^{2}}{2}\right)\right){}_{0}F_{1}\left(;1;\frac{\eta_{1}^{2}}{2\eta_{2}^{2}}\left(J_{1_{qm}}({\bf
x},{\bf
y})\right)^{2}\right)\left(\frac{\left(1+\bar{d}_{qm}/d_{0}\right)^{\frac{\alpha}{2}-1}}{\bar{d}_{qm}}\right)$}$
(21) $\displaystyle
A_{4}=\leavevmode\resizebox{186.45341pt}{}{$\displaystyle\frac{K\frac{\omega}{\omega_{c}}\exp\left(K\frac{\omega}{\omega_{c}}\exp\left(-\left(\frac{\eta_{1}^{2}}{\eta_{2}^{2}}+\frac{\left(J_{1_{qm}}({\bf
x},{\bf
y})\right)^{2}}{2}\right)\right)\right)}{\eta_{2}\sqrt{2\rho_{c}\left(\exp\left(K\frac{\omega}{\omega_{c}}\exp\left(-\left(\frac{\eta_{1}^{2}}{\eta_{2}^{2}}+\frac{\left(J_{1_{qm}}({\bf
x},{\bf y})\right)^{2}}{2}\right)\right)\right)-1\right)}}$}$
$\displaystyle\times\\!\\!\sum_{n\in\mathcal{D}_{q},n\neq
m}\lambda_{n}J_{1_{qn}}({\bf x},{\bf
y})\exp\left(-\left(\frac{\eta_{1}^{2}}{\eta_{2}^{2}}+\frac{\left(J_{1_{qn}}({\bf
x},{\bf y})\right)^{2}}{2}\right)\right)$
$\displaystyle\quad\quad\times{}_{0}F_{1}\left(;1;\frac{\eta_{1}^{2}}{2\eta_{2}^{2}}\left(J_{1_{qn}}({\bf
x},{\bf
y})\right)^{2}\right)\left(1+\frac{\bar{d}_{qn}}{d_{0}}\right)^{\frac{\alpha}{2}}$
(22)

###### Proof.

Follows from the derivative chain rule and has been skipped due to lack of
space. ∎

The terms $A_{1}(\widetilde{x}_{k},\widetilde{y}_{k}),A_{2},A_{3}$ and $A_{4}$
depend on the RRHs locations, but we do not write the RRHs ${\bf x}$ and ${\bf
y}$ coordinates as parameters to minimize the notation. Here
${}_{0}F_{1}(;.;.)$ is the regularized confluent Hypergeometric function, and
we can write it in an alternate form as a function of the modified Bessel
function of first kind as
${}_{0}F_{1}\left(;1;\frac{\eta_{1}^{2}}{2\eta_{2}^{2}}\left(J_{1_{qm}}({\bf
x},{\bf
y})\right)^{2}\right)=I_{0}\left(\frac{\sqrt{2}\eta_{1}}{\eta_{2}}J_{1_{qm}}({\bf
x},{\bf y})\right)$, i.e.,
${}_{0}F_{1}\left(;1;z\right)=I_{0}\left(2\sqrt{z}\right)$.

The derivative of $\mathcal{L}({\bf x},{\bf y},\bm{\lambda})$ in (16) with
respect to $\lambda_{m}$ is

$\displaystyle\mathcal{L}_{\lambda_{m}}({\bf x},{\bf
y},\bm{\lambda})=\frac{\partial\mathcal{L}({\bf x},{\bf
y},\bm{\lambda})}{\partial\lambda_{m}}$
$\displaystyle\quad\quad=\left(1-Q_{1}\left(\frac{\sqrt{2}\eta_{1}}{\eta_{2}},\frac{\sqrt{2\zeta_{qm}({\bf
x},{\bf y})}}{\eta_{2}}\right)-\epsilon\right)$ (23)

Using the batch gradient descent, we can obtain an iterative formula for
$\lambda^{(i+1)}_{m}$ as

$\displaystyle\lambda^{(i+1)}_{m}=\left[\lambda^{(i)}_{m}+\nu\mathcal{L}_{\lambda_{m}}({\bf
x}^{(i)},{\bf y}^{(i)},\bm{\lambda}^{(i)})\right]^{+}$ (24)

where $\left[\cdot\right]^{+}=\max\left(\cdot,0\right)$, and
$\nu\in\mathbb{R}_{+}$ is a step size chosen small enough to guarantee
convergence. Based on this analysis, we can construct Algorithm 1 to obtain
the optimal locations of the RRHs in the network as described below.

1 Generate random locations for the RRHs in all cells

2 Define $d_{\text{max}}$ big enough

3 while  _$d_{\text{max}} >d_{\text{cvg}}$_ do

4 for  _ $q\in\\{1,\dots,Q\\}$_ do

5 Wrap-around cells to make cell $q$ at center

6 for  _ $m\in\mathcal{D}_{q}$_ do

7 Update $x_{qm}$, $y_{qm}$, $\lambda_{m}$ using eq. (17), (18), (24)

8

9 end for

10
$d_{q}=\underset{n}{\max}\left\\{\underset{x,y}{\max}\left\\{|x_{qn}^{(i+1)}-x_{qn}^{(i)}|,|y_{qn}^{(i+1)}-y_{qn}^{(i)}|\right\\}\right\\}$

11 end for

12 $d_{\text{max}}=\underset{q}{\max}\ d_{q}$

13 end while

Algorithm 1 RRHs Locations Optimization

The algorithm starts by choosing random locations for the RRHs (Step 1). Then,
we define a $d_{\text{max}}$ as a distance large enough to start the locations
update. For the RRHs location update in each algorithm iteration, we perform a
cell wrap-around (Step 1) to place the cell of these RRHs at the center, hence
eliminating network border effect. After that, we update the locations of the
RRHs in this cell using the indicated equations in Step 1, and we calculate
the maximum RRHs location change. We do one iteration for each cell at a time
until we iterate through all the network cells. Hence, the total number of
iterations will be the same for all the cells, and the updated RRHs locations
in the interfering cells will be used, which is very reliable. At last, the
algorithm convergence is determined when the maximum change in the RRHs
locations ($d_{\text{max}}$ in Step 1) is lower than a small distance
$d_{\text{cvg}}$. We note that our scheme is sub-optimal mainly because of the
complex traffic distribution shown in (III-A).

###  IV-B Distance-based Approach

Choosing an appropriate step size $\nu$ for ${\bf\lambda}$ update in (24) can
be tricky especially when the constraint (11b) is tight. To address this
issue, we define an equivalent approach that is distance-based to solve
problem (11) by formulating it as

$\displaystyle\underset{{\bf x},{\bf
y}}{\text{max}}\quad\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\},\forall q$ (25a)
$\displaystyle\text{s.t.}\quad\frac{\bar{d}_{qn}}{\bar{d}_{\text{out}}}\leq
1,\quad n=1,\dots,N$ (25b)

where $\bar{d}_{\text{out}}$ is the maximum allowed backhaul distance to
guarantee that the outage is lower than $\epsilon$. When the achievable
spectral efficiency over the access channel is fixed, $\bar{d}_{\text{out}}$
can be easily obtained using a bisection search to obtain $P_{qn}\left({\bf
x},{\bf y}\right)=\epsilon$ defined in (III-C). Once $\bar{d}_{\text{out}}$ is
found, we can write an iterative formula for $x_{qm}$ and $y_{qm}$ as

(26)

Similarly for the y-coordinates:

(27)

To update $\lambda_{m}^{i}$, we perform another bisection search, such that
$\bar{d}_{qm}=\bar{d}_{\text{out}}$. This method eliminates the need for a
step size $\nu$ to update $\bm{\lambda}$, and at the same time (26) and (27)
guarantee that the RRHs will be placed in the locations that maximize (25a) as
we will see in the results section. Consequently, we can obtain the optimal
locations of the RRHs in our network using Algorithm 1, but with replacing
Step 1 with two steps; one that obtains $\bar{d}_{\text{out}}$ using bisection
search and the other updates $x_{qm}$ and $y_{qm}$ using (26) and (27)
respectively.

##  V Numerical Results

We consider a network of $Q=9$ cells with wrap-around and cell dimension of
$1000\times 1000$ meters. The cells have square shapes, but any other
preferred shape can be used if needed, e.g., circular. Moreover, we consider a
system of $25$ resource blocks (RBs), where each RB has a bandwidth (BW) of
$180$ KHz. We use the traffic distribution in (III-A) to model the locations
of users. We summarize the rest of the simulation parameters in Table V, where
the available system bandwidth is divided between the backhaul (of BW
$\omega_{c}$) and the access channel (of BW $\omega$).

| Parameter | Value
---|---|---
Cell config. |  $Q$, $N$, $M$, $K$ |  $9$, $10$, $8$, $10$
Power |  $p$, $p_{c}$ |  $30$ dBm, $45$ dBm
Bandwidth |  RB, $\omega$, $\omega_{c}$ |  $180$ KHz, $5$ RBs, $20$ RBs
Noise |  spectral density $S_{z}$, noise figure $F_{z}$ |  $-174$ dBm/Hz, $8$ dBm
Hotspots |  20cm$P_{0}$, $\sigma_{h}$; |
$N_{h}^{\text{min}}$, $N_{h}^{\text{max}}$ |  20cm$0.1$, $100$ meters; |
Per network: $2Q$, $4Q$ |  |
Path loss, Fading |  $d_{0}$, $\alpha$, $\mathcal{K}$, $\eta_{1}$, $\eta_{2}$ |  $0.392$ meters, $3.76$, $15$ dB, $8$, $\sqrt{2}$
Algorithm |  $\epsilon$, $d_{\text{cvg}}$, $\nu$ |  $0.2$, $1$ meter, $1$

TABLE I: Simulation parameters.

Figure 1: Typical generated network hotspots showing the optimal RRHs location
when $\omega=6$ RBs, and at no constraint.

In Figure 1, we show a typical generated traffic distribution (equation
(III-A)), and we present the resulted optimized locations of the RRHs when we
have $\omega=6$ RBs (i.e., $\omega_{c}=19$ RBs). We note that the users exist
also in locations outside the hotspots (hotspots are represented with yellow
areas) with a probability $P_{0}$ shown in Table V. We include the no
constraint case for comparison purpose. The results show that even at this
high bandwidth allocation for the backhaul compared to the access channel, the
backhaul constraint is still the limiting factor in the deployment of RRHs in
each cell.

In Figure 2, we plot the spectral efficiency on the access channel as a
function of different RBs allocations between the access channel and the
backhaul. A ratio of $\frac{K\omega}{\omega_{c}}=1.36$ corresponding to
$\omega\leq 3$ RBs for the access channel allows deploying the RRHs freely in
the network for the typical parameters specified in Table V. In such an
allocation, the wireless backhaul is not a bottleneck and the RRHs can be
freely deployed as if the backhaul has unlimited bandwidth. This constraint
becomes even more relaxed if we use only $M=2$ antennas at each RRH, which
leads to a lower spectral efficiency over the access channel and hence an
$\omega\leq 5$ RBs would be enough for freely deploying the RRHs.

In Figure 3, we plot the spectral efficiency as a function of the number of
the RRHs per cell, $N$. We show the results when the number of antennas $M$
per RRH is fixed ($M=8$), and when the total number of antennas per cell is
fixed ($NM=80$), i.e., as $N$ is increased we get more distributed network.
Interestingly, the results show that distributing available antennas on more
RRHs per cell is a good strategy to increase the spectral efficiency even when
we have a backhaul constraint. On the other hand, we plot the obtained
efficiencies when the backhaul bandwidth is further divided among the RRHs to
provide frequency division among the RRHs links, i.e., the $w_{c}/N$ plots.
For these plots, the backhaul constraint cannot be satisfied even when the
RRHs are co-located at the CU, which means distributing the RRHs in the cell
will not be possible, and other solutions should be taken to make such
approach successful, e.g., decreasing $K$ or provide more bandwidth for the
backhaul.

Figure 2: Achievable spectral efficiency as a function of different RBs
allocation, $M=8$ (solid line) and $M=2$ (dashed).

Figure 3: Access channel spectral efficiency at $\omega=5$ RBs.

##  VI Conclusion

We analyzed the effect of a limited capacity backhaul on the achievable rate
in a coordinated distributed network. We used point-to-point noise-limited
wireless backhaul links between the CUs and the RRHs and analyzed its effect
on the deployment decisions of the RRHs. We used ZF-BF on the access channel
to allow the RRHs to jointly serve the users that are distributed according to
some traffic distribution. Our results show that we need to allocate a very
large bandwidth for the backhaul compared to the access channel to allow
serving a large number of users and to allow free deployment of RRHs in the
network. Our work underlines the fundamental role the backhaul plays in the
design of distributed networks.

###  -A Proof of outage on the Noise-limited SISO Backhaul

The backhaul outage probability can be written as

$\displaystyle\displaystyle P_{qn}\left({\bf x},{\bf
y}\right)=\leavevmode\resizebox{164.77771pt}{}{$\displaystyle\mathbb{P}\left\\{R_{qn}^{(b)}\left(x_{qn},y_{qn}\right)\leq
K\frac{\omega}{\omega_{c}}\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right\\}$}$ $\displaystyle\
=\leavevmode\resizebox{195.12767pt}{}{$\displaystyle\mathbb{P}\left\\{\left|\bar{g}_{qn}\right|^{2}\leq\frac{1}{\rho_{c}\bar{\ell}_{q}(x_{qn},y_{qn})}\left(\exp\left(K\frac{\omega}{\omega_{c}}\mathbb{E}_{\widetilde{x}_{k},\widetilde{y}_{k}}\left\\{R_{qk}^{(a)}({\bf
x},{\bf y})\right\\}\right)-1\right)\right\\}$}$ (28)

where $\bar{g}_{qn}$ is the small-fading parameter for both the LoS and NLoS
components between CU $q$ and RRH $n$, which is assumed to be Rician fading,
hence the probability density function (PDF) of the fading power
$\delta_{qn}=\left|\bar{g}_{qn}\right|^{2}$ is
$f(\delta_{qn})=\frac{1}{\eta_{2}^{2}}\exp\left(-\frac{\eta_{1}^{2}+\delta_{qn}}{\eta_{2}^{2}}\right)I_{0}\left(\frac{2\eta_{1}}{\eta_{2}^{2}}\sqrt{\delta_{qn}}\right)$,
where
$I_{0}(z)=\frac{1}{\pi}\int_{0}^{\pi}\exp\left(z\cos(\theta)\right)\mathop{}\\!\mathrm{d}\theta$
is the modified Bessel function of first kind. Let us denote the right side of
the inequality in (-A) as $\zeta_{qn}({\bf x},{\bf y})$, hence

$\displaystyle\displaystyle P_{qn}({\bf x},{\bf
y})=\leavevmode\resizebox{169.11153pt}{}{$\displaystyle\int_{0}^{\zeta_{qn}({\bf
x},{\bf
y})}f(\delta_{qn})\mathop{}\\!\mathrm{d}\delta_{qn}=1-Q_{1}\left(\frac{\sqrt{2}\eta_{1}}{\eta_{2}},\frac{\sqrt{2\zeta_{qn}({\bf
x},{\bf y})}}{\eta_{2}}\right)$}$ (29)

where $Q_{1}(\cdot)$ is the Marcum Q-function.

## Acknowledgment

This work was supported in part by Ericsson Canada and in part by the Natural
Sciences and Engineering Research Council (NSERC) of Canada.

## References

  * [1] 3GPP.  Release 15, Technical Specification Group Services and System Aspects, 2019.  www.3gpp.org/ftp/Specs/archive/21_series/21.915/.
  * [2] T. A. Khan, P. Orlik, K. J. Kim, and R. W. Heath.  Performance analysis of cooperative wireless networks with unreliable backhaul links.  IEEE Communications Letters, 19(8):1386–1389, Aug 2015.
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  * [5] Q. Zhang, C. Yang, and A. F. Molisch.  Downlink base station cooperative transmission under limited-capacity backhaul.  IEEE Transactions on Wireless Communications, 12(8):3746–3759, August 2013.
  * [6] K. Nguyen, Q. Vu, M. Juntti, and L. Tran.  Globally optimal beamforming design for downlink comp transmission with limited backhaul capacity.  In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 3649–3653, March 2017.
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  * [9] M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, and M. Debbah.  Cell-free massive mimo with limited backhaul.  In 2018 IEEE International Conference on Communications (ICC), pages 1–7, May 2018.
  * [10] A. Minasian, R. S. Adve, S. Shahbazpanahi, and G. Boudreau.  On RRH placement for multi-user distributed massive MIMO systems.  IEEE Access, 6:70597–70614, 2018.
  * [11] M. Minelli, M. Ma, M. Coupechoux, J. Kelif, M. Sigelle, and P. Godlewski.  Optimal relay placement in cellular networks.  IEEE Transactions on Wireless Comm., 13(2):998–1009, February 2014\.
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Robotic systems often require precise scene analysis capabilities, especially
in unstructured, cluttered situations, as occurring in human-made environments.
While current deep-learning based methods yield good estimates of object poses,
they often struggle with large amounts of occlusion and do not take inter-object
effects into account.
Vision as inverse graphics is a promising concept for detailed scene
analysis. A key element for this idea is a method for inferring scene parameter
updates from the rasterized 2D scene. However, the rasterization process
is notoriously difficult to invert, both due to the projection and occlusion
process, but also due to secondary effects such as lighting or reflections.
We propose to remove the latter from the process by mapping the rasterized image
into an abstract feature space learned in a self-supervised way from pixel
Using only a light-weight inverse rendering module, this allows us to refine
6D object pose estimations in highly cluttered scenes by optimizing a simple
pixel-wise difference in the abstract image representation.
We evaluate our approach on the challenging YCB-Video dataset, where it
yields large improvements and demonstrates
a large basin of attraction towards the correct object poses.

§ INTRODUCTION

Robust robotic interaction in environments made for humans is an open research
field. An important prerequisite in this context is scene perception, yielding
the necessary information such as detected objects and their poses or affordances
for later manipulation actions.
While there are various high-accuracy methods for scene understanding,
the problem becomes significantly harder in the presence of clutter and
inter-object effects.
As such, current works in humanoid manipulation that require precise grasping
are often limited to non-cluttered
or even isolated scenes (e.g. <cit.>).
While the manipulation action itself and planning for it is certainly more difficult
in cluttered scenes, robust 6D object pose estimation is a necessary prerequisite.

An interesting approach in this context is the idea of viewing computer vision
as an inverse graphics process <cit.>.
It promises to perform
scene analysis by inverting the rasterization process, which sounds highly
promising—today's rendering techniques are capable of producing convincing
photo-realistic renderings of highly complicated scenes, so inversion of the
process should yield high-quality scene analysis.

However, the problem plaguing the inverse graphics field is that the rendering
process is largely unidirectional, with complex physical effects such as lighting,
surface scattering, transparency, and so on. Furthermore, scene analysis is
especially in demand for cluttered scenes, e.g. in warehouse automation contexts,
but occlusion effects caused by clutter are among the most difficult to invert or differentiate.

To take a step towards a solution of this problem, we propose to first remove
most secondary rendering effects from the scene using abstract
surface features learned in an unsupervised manner. This way, only the
primary effects remain—occlusion and projection.
These effects can then be explained and analyzed by a simpler differentiable
rendering component.

We apply our render-and-compare framework to the task of monocular 6D pose
estimation, specifically pose refinement, where initial pose guesses are available.
In our approach, 6D pose predictions from state-of-the-art pose estimation methods are refined by
minimizing the pixelwise difference between the rendered image and the observed image
in the proposed abstract descriptor space invariant to secondary rendering effects.
We further make the assumption that meshes of the objects are available, as
is the case in many industrial and robotic applications. For example,
service robots operating in human environments working with tools designed for human usage and
industrial part handling robots can greatly benefit from having precise 6D pose estimation.

label/.style=anchor=north,align=center, text depth=0,
axislabel/.style=font=,inner sep=1pt,

[inner sep=0] (input) at (-4,0) ;

[label] at (-7.5, -0.3) Mesh Database;
[label] at (-4.5, 1.9) Feature-Annotated
Mesh Database;
[label] at (-4.5, -1.35) Observed Scene $I_O$ ;

[label] at (-.9, 1.45) Abstract Scene $A_R$;
[label] at (-.9, -1.15) Abstract Scene $A_O$;

[label] at (-6.05, .8) Abstraction;
[label] at (-6.05, .57)  Module;

[label] at (-2.9, 0.8) Differentiable;
[label] at (-2.9, 0.58) Renderer;

[label] at (-2.8, -.38) Abstraction ;
[label] at (-2.8, -.6)  Module ;

[label] at (-2.9, 1.4) Pose $P$;

[label] at (0.2, -0.1) Loss $L$  ;

Scene analysis using differential rendering and learned abstraction
module. An abstract representation is extracted both for the input scene
and the mesh database.
The differential rendering module then tries to match the abstract scene
representation with the feature-annotated meshes.
The loss gradient (red dotted line) only needs to
be backpropagated through the differentiable renderer.

In short, our contributions proposed in this work include:

* A scene abstraction method that removes secondary render effects, so
that scene analysis by render-and-compare becomes feasible,

* fusion of surface features onto object meshes for direct rendering
of scenes in the abstract feature space,

* a fast and light-weight differentiable rendering component, and

* the integration of these components into a pose refinement pipeline,
which is evaluated on the YCB Video Dataset <cit.>.
To the best of our knowledge, this is the first differentiable rendering
pipeline capable of optimizing object poses in cluttered real-world scenes.

This paper is structured as follows. After discussing related works in the section II, we
describe the descriptor learning process in section III and introduce our differentiable renderer, LightDR in section IV.
In section V, we apply our pipeline to 6D object pose refinement,
and further analyze the robustness of the proposed refinement process.

§ RELATED WORK

Vision as inverse graphics aims at inferring object parameters like shape, illumination, reflectance, and pose, scene parameters like camera parameters, lighting, and secondary reflections by inverting the rendering process. The process of rendering 3D scene to discrete 2D pixels involves discretization steps that are not differentiable. However, several approximation methods have been to proposed to realize a differentiable renderer.  <cit.> proposed OpenDR, a generic differentiable renderer that can compute gradients with respect to object and scene parameters. <cit.> introduced a differentiable renderer that is suited for neural networks. <cit.> treat forward rendering as a black-box and used REINFORCE <cit.> to compute gradients. <cit.> proposed edge sampling algorithm to differentiate ray tracing that can handle secondary effects such as shadows or global illumination. <cit.> proposed a differentiable probabilistic formulation instead of discrete rasterization. The differentiable renderer used in this work is closely modeled after OpenDR but tailored for object pose refinement with a strong focus on speed.

Vision as inverse graphics is most often formulated as a render-and-compare approach, where model parameters are optimized by minimizing the difference between rendered and observed images. <cit.> used render-and-compare for real-time height mapping fusion. Several recent works used render-and-compare for solving a wide range of vision problems: <cit.> learned unsupervised monocular face reconstruction; <cit.> introduced a framework for instance-level 3D scene understanding; <cit.> estimated 6D object pose in cluttered synthetic scenes.
More closely related is the DeepIM method by <cit.>, who
formulated 6D object pose estimation as an iterative pose refinement process
that refines the initial pose by trying to match the rendered image with the observed image.
In contrast to our approach, they avoid the need for backpropagating gradients
through the renderer by training a neural network to output pose updates.
While the method yields very promising results, it is not directly clear how to
apply this method to symmetric objects without specifying symmetry axes,
whereas our method inherently optimizes to a suitable pose.
We also note that DeepIM is object-centric, refining each object's pose
separately. In contrast, our method retains the entire scene, refining all
object poses simultaneously and thus is able to account for inter-object

In this work, we use render-and-compare to refine 6D object poses in cluttered real-world scenes. Instead of comparing the rendered and observed RGB images, we propose to use an abstraction network to deal with the difficulties in comparing images from two different modalities.

§ LEARNED DESCRIPTORS FOR SCENE ABSTRACTION

Real scenes exhibit a large variety of secondary effects such as lighting,
camera noise, reflections, and so on. All of these effects are very difficult
to model and severely constrain the applicability of differentiable rendering
We propose an additional abstraction module $f: \mathbb{I} \rightarrow \mathbb{A}$,
mapping the RGB image space $I$ to an abstract feature space $A$. Ideally,
the mentioned secondary effects lie in the null space of $f$.
For convenience, we require that $A$ is also image-like so that pixels in
$I$ correspond to feature vectors in $A$.

The very difficult problem of decomposing an image into its different intrinsic components, such as shading, reflectance, and shape,
has been studied extensively <cit.>.
However, in this application such a complex and physically accurate decomposition
is not required.
In order to be usable for differential rendering, we only require that features
are similar for corresponding points on the same object (under varying
lighting conditions etc), and dissimilar for non-corresponding objects.
Many traditional feature extractors exhibit this property (e.g. SIFT).
Of particular interest, however, are feature extractors designed for
dense output.
Recently, <cit.> showed that highly precise
feature extractors can be trained in a self-supervised way from ground truth
correspondences. The learned descriptors outperform sparse feature extractors
by a large margin.

§.§ Real-Synthetic Correspondences for Descriptor Learning

Learning dense descriptors from real-synthetic correspondences.
Top: Real and synthetic input frames $A$ and $B$. Positive correspondences
matches are shown in green for one object.
Bottom: Learned dense abstract representation of the scene.

In order to leverage this idea in the differential rendering setting, we propose
to learn descriptors from the object meshes in conjunction with a training
dataset for pose estimation. For a pose-annotated real dataset frame $A$,
we render a synthetic frame $B$ with the same object set, but using
different poses (see <ref>).
Corresponding points in both RGB frames can be easily determined
from the object poses through projective geometry.

With a probability of 0.5, the synthetic object is drawn in the same orientation
as the real object (see <ref>). This results in a large
number of positive correspondences. In the other case, the orientation is drawn
uniformly random—resulting in a larger number of negative examples and ensuring
that the learned descriptors stay globally unique.
The object translation is always sampled uniformly at random.

§.§ Network Details & Training

Since our application depends on high spatial resolution of the computed
features, we follow the architecture of
Light-Weight RefineNet <cit.>, a
state-of-the-art semantic segmentation method, which successively upsamples
lower-resolution feature maps of higher abstraction levels and combines them
with higher-resolution feature maps of lower abstraction levels.
In our network, we use ResNet-34 <cit.> with ImageNet pretraining as
the RefineNet backbone network.
The final convolutional layer is adapted to not only output semantic segmentation
($C{\times}H{\times}W$) but also the dense features ($D{\times}H{\times}W$).
Here, semantic segmentation is included as an auxiliary task, as it is not
used in the following pipeline stages (but could be in the future).
In our experiments, we use $D=3$ for easier visualization, as in

Following <cit.> and <cit.> we minimize a
pixel-wise contrastive loss function $\mathcal{L}_C(A,B)$:

\begin{align}
\mathcal{L}_+(A,B) &= \frac{1}{|M_+|} \sum_{u \in M_+} || f_A(u_A) - f_B(u_B) ||_2^2 \\
\mathcal{L}_-(A,B) &= \frac{1}{H_-} \sum_{u \in M_-} \max (0, M - || f_A(u_A) - f_B(u_B) ||_2)^2 \\
\mathcal{L}_C(A,B)   &= \mathcal{L}_+(A,B) + \mathcal{L}_-(A,B),
\end{align}

where $f_P(u)$ is the descriptor value in image $P$ at location $u$,
$M_+$ is the set of correspondent pixel pairs $(u_A, u_B)$, $M_-$ is a set
of randomly sampled negative correspondences (also limited to the object mask),
and $H_- = \sum_{u \in M_-} \mathds{1}[M - || f_A(u_A) - f_B(u_B) ||_2 > 0]$ the number of
hard negatives.

To encourage the network to disregard clutter in the background, we also
introduce a loss on background pixels for a single frame $F$:

\begin{align}
\mathcal{L}_\text{bg}(F) &= \lambda \frac{1}{G} \sum_{u \in G} || f_F(u) ||_2^2,
\end{align}

where $G$ is the set of background pixels. The loss balancing factor $\lambda=0.1$
is chosen rather small in order not to hurt descriptor learning in the foreground

The combined loss function is simply

\begin{align}
\mathcal{L}(A,B) &= \mathcal{L}_C(A,B) + \sum_{F \in \{A,B\}} \mathcal{L}_S(F) + \mathcal{L}_\text{bg}(F),
\end{align}

where $\mathcal{L}_S$ is the cross-entropy loss for pixel-wise segmentation.

The network is trained using the Adam optimizer with learning rate 1e-4 and
parameters $\beta_1 = 0.9, \beta_2 = 0.999$ on 350k image pairs. Note that
the synthetic image $B$ of each image pair is generated on the fly, i.e. the
network never is presented with the same pair twice.

§.§ Mesh Representation for Surface Features

Learned surface features, projected and fused onto the mesh.
The 3D feature vectors are visualized directly as RGB colors.

Since the learned features should be constant for each local surface patch,
independent of the viewing pose, we can fuse the descriptor information
onto the mesh representation. To this end, we render $N$ views (50 in our experiments)
of the object
from randomly sampled viewing directions and viewing distances. After feature
extraction using the learned network, the resulting point-feature pairs are
aggregated in the object frame. A voxel grid downsampling is applied, where
descriptors and positions are each averaged inside each voxel.
The voxel size is determined heuristically from the object bounding box s.t.
the voxel count is constant—this results in constant-size output.
This method is robust and easy to tune in case more complex geometry needs to
be supported. In our experiments, we use 5000 voxels.
Finally, each vertex of the object mesh is assigned an interpolated descriptor
from the four nearest voxels using inverse-distance weighting.
<Ref> shows exemplary meshes and corresponding feature

§ DIFFERENTIABLE RENDERER

The pose refinement problem is an optimization problem. In our case, we assume
that we start with a pose initialization of reasonable quality, such that
local optimization methods can find the optimum solution.
In this context, it is very favorable to be able to compute derivatives
of the rendering process, since the number of parameters grows linearly with
the number of objects in the scene (at least six parameters per object).
Optimization without gradient information quickly becomes infeasibly slow.

We base our differential rendering module on the method of
OpenDR <cit.>, which is able to approximate gradients with respect
to lighting parameters, camera parameters, object poses, etc.
We note that in our setting, only pose parameters need to be optimized, because
lighting and other surface effects are removed by the abstraction network
and camera parameters are assumed to be fixed in monocular pose estimation.
Encouraged by this simplification, we implemented a lightweight differentiable
renderer, which we call LightDR.

The OpenDR method is built around the screen-space approximation of the derivative
of the rendering process. Gradients due to occlusion effects during this 3D-2D
reduction are approximated from the local intensity gradient.
In essence, this idea assumes that occluded pixels are similar to their visible

In order to simplify gradient computation, we locally linearize the pose:

\begin{align}
T(\alpha,\beta,\gamma,a,b,c) &= T_0 \left( \begin{matrix}
1 & -\gamma & \beta & a \\
\gamma & 1 & -\alpha & b \\
-\beta & \alpha & 1 & c \\
0 & 0 & 0 & 1 \\
\end{matrix} \right)
\end{align}

The rotation part of $T$ is orthonormalized after each optimization step.

The scenes we are interested in feature high levels of occlusion between the
individual objects. <cit.> makes several assumptions in computation
of the screen-space gradients.  While these assumptions help in simplifying the computation,
in real world scenarios, they are often violated.
We discuss the assumptions involved, scenarios where these assumptions are violated, and propose solutions
for better approximation of the pose gradients.

§.§ Gradients on Occlusion Boundaries

(a)   (b)   (c)   (d)

Corner cases for render gradient estimation.
(a) and (b): Front and top view of an exemplary scene with occlusion.
(c) Top:  Occlusion-free scene; Bottom: Scene with mug occluded.
(d) Magnitude of pixel-wise loss corresponding to scenarios depicted in (c)
when the mug is translated. Black depicts: high loss magnitude.
When the rendered mug is moved behind the occluder, all pixels with high
loss (and thus gradient information) lie outside of the rendered object mask.

Occlusion boundaries are highly important for pose refinement, since they
offer much information about the scene layout.
On the occlusion boundary pixels, OpenDR uses Sobel kernel ($\frac{1}{2} [-1, 0, 1 ]$) and its transpose to compute the gradient along the horizontal axis and the vertical axis respectively, with the underlying assumption that a shift in the occlusion boundary can be approximated by the replacement of the current pixel by the neighboring pixel (of the other object).
However, this assumption is valid only if the occlusion boundary pixel belongs to the object in the foreground. <ref> (a), (b) depict the front view and top view of an example scene where the mug is occluding the can. Translating the mug results in covering or uncovering of can pixels, which is well approximated using the local Sobel gradient.
Conversely, translating the can in the background does not result in covering/uncovering mug pixels, rather, more can pixels will become visible or become covered. Thus using the Sobel derivative is incorrect in this case.
To address this issue, we detect such cases using the Z-buffer during rendering
suppress the Sobel gradient on these pixels.
We note that the occluded pixel belongs to the same object in this case, so that
zero gradient should be a good approximation.

§.§ Propagating Image-space Gradients to Object Coordinates

While propagating the image-space gradients to the object coordinates, only gradient from the pixels belonging to the object needs to be propagated. The naïve way to do this is to mask the image-space gradient with rendered object mask. However, in certain situations this means we are ignoring exactly the pixels where a pixel-wise loss function generates high gradients, namely just outside of the rendered object boundary. <Ref>(c-d) illustrates this point.

To address this issue, we propose a dilation of the rendered object mask by one
pixel, in order to include gradient information directly outside of the object

§.§ Implementation

LightDR uses OpenGL via the Magnum engine[<https://magnum.graphics/>]
for forward rendering. The gradient backpropagation is built on top of PyTorch
to facilitate faster computations on GPU. LightDR needs a few milliseconds for
forward rendering and around 50 ms for backpropagation of gradients to object

§ POSE REFINEMENT

label/.style=anchor=north,align=center, text depth=0,
axislabel/.style=font=,inner sep=1pt,

[inner sep=0] (input) at (-4,0) ;

[label] at (-7.4, -.55) Mesh Database;

[label] at (-4.1, 1.4) Rendered Scene $I_R$;
[label] at (-4.1, -1.35) Observed Scene $I_O$;

[label] at (-0.85, 1.4) Abstract Scene $A_R$;
[label] at (-0.85, -1.35) Abstract Scene $A_O$;

[label] at (-2.63, .775) Abstraction ;
[label] at (-2.6, .53) Module ;
[label] at (-2.63, -.5) Abstraction ;
[label] at (-2.63, -.73) Module ;

[label] at (-6., 0.66) Differentiable;
[label] at (-6., 0.44) Renderer;

[label] at (-5.96, 1.25) Pose$P$;

[label] at (0.3, -0.25) Loss $L$;

Render-Abstraction pipeline. The renderer produces an RGB image of
the scene, which is then mapped into the abstract feature space.
The loss gradient (red dotted line) is propagated back through the
abstraction module and the differentiable renderer.

Armed with the abstraction module and our differentiable renderer, we can tackle
the 6D pose refinement problem in cluttered real-world scenes.
We first experimented with the architecture depicted in <ref>.
The 3D scene with the objects in the current estimated pose $P$ is rendered to generate image $I_R$.
The abstraction module (see <ref>) is used to generate abstract representations $A_R$ and $A_O$ from the images $I_R$ and $I_O$.
The loss $L$ is computed as the pixel-wise loss between $A_R$ and $A_O$.
Finally, we can derive the gradient of $L$ with respect to the poses $P$:

\begin{align}
\frac{\partial L}{ \partial P} &=  \frac{\partial I_R}{ \partial P} \cdot \frac{\partial A_R}{ \partial I_R} \cdot \frac{\partial L}{ \partial A_R}
\end{align}

(a)   (b)   (c)   (d)

Gradient of pixel-wise loss w.r.t. the rendered scene in the two pipeline variants.
(a) Observed scene. (b) Rendered scene.
(c) Render-Abstraction pipeline (see <ref>).
(d) Abstraction-Render pipeline (see <ref>).
The actual gradient magnitudes are scaled for better visualization.
Gray corresponds to zero gradient.

As depicted in <ref>,  $ \frac{\partial I_R}{ \partial P}$ is approximated by the differentiable renderer
and $\frac{\partial A_R}{\partial I_R}\cdot\frac{\partial L}{\partial A_R}$ is computed by standard backpropagation.
During experiments, we noticed that the latter gradient is rather sparse
and focuses most of its magnitude on few spatial locations in the image (see <ref>).
This effect is well-known, for example in the field of adversarial example
generation for CNNs <cit.>, <cit.>. Here, it is highly undesirable,
since the differential rendering process works best with smooth, uniform

Learned scene abstraction, removing secondary effects such as
background, lighting, shadows, and camera noise.
Top row: RGB input scene, bottom row: RGB visualization of the output feature
The leftmost column shows a real test scene from the YCB dataset, the other
columns contain rendered scenes with random lighting and camera noise.

(a)     (b)     (c)     (d)     (e)

Qualitative examples from the YCB Video Dataset.
(a): Observed scene.
(b) and (c): Renderings (blue) with initial and optimized pose parameters, respectively.
(d) and (e): Renderings of the feature-annotated meshes in initial and
optimized pose, respectively.

To mitigate this issue, we investigated a second pipeline shown in <ref>.
We use the method described in <ref> to create meshes with fused
surface descriptors.
Rendering these meshes directly results in the abstract rendered $A_R$.
In this case, $\frac{\partial L}{ \partial P} $ is simpler:

\begin{align}
\frac{\partial L}{ \partial P} &=  \frac{\partial L}{ \partial A_R} \cdot \frac{\partial A_R}{ \partial P}.
\end{align}

Here, $\frac{\partial A_R}{ \partial P}$ is approximated directly by the differentiable renderer.
An additional benefit of this variant is that only one forward pass of the abstraction module
for $A_O$ is required. This directly translates to significant reduction in the optimization process runtime.

We optimize the object poses with the AdaGrad optimization scheme. This avoids
manual tuning of learning rates for translation and rotation parameters—which
differ largely in scale. In our experiments, we use a learning rate $\lambda = 1e-2$
with a decay of $0.99$. The optimization runs for 50 iterations, which corresponds
to roughly 2 s per frame.

§ EXPERIMENTS

We perform our experiments on the YCB Video dataset <cit.>,
which consists of 133,936 images extracted from 92 videos, showing 21 different
objects in cluttered arrangements.
Importantly, the dataset comes with high-quality meshes, which are also used for
synthetic data generation by most of the pose estimation methods applied to the
dataset <cit.>.

We first qualitatively show the result of the learned mapping into the abstract
feature space in <ref>. To be able to control
primary and secondary effects separately, we rely on our rendering pipeline
and generate multiple scenes with random secondary parameters as detailed in
Our trained model is able to
effectively suppress the background pixels and produces robust, consistent
output under changing lighting conditions and camera model parameters.

For a quantitative analysis, we measure the
the ADD and ADD-S metrics as in <cit.> for each
object occurrence, which measure average point-wise distances between transformed
objects and the ground truth, for non-symmetric and symmetric objects, respectively:

\begin{align}
\text{ADD} &= \frac{1}{m} \sum_{x\in \mathbb{M}} || (Rx + T) - (\tilde{R}x + \tilde{T})||, \\
\text{ADD-S} &= \frac{1}{m} \sum_{x_1\in \mathbb{M}}\min_{x_2\in\mathbb{M}} || (Rx_1 + T) - (\tilde{R}x_2 + \tilde{T})||,
\end{align}

where $R$ and $T$ are the ground truth rotation and translation,
$\tilde{R}$ and $\tilde{T}$ denote the estimated pose, and $\mathbb{M}$
is the set of model points as included in the YCB Video dataset.
We aggregate all results and measure the area under the
threshold-accuracy curve for distance thresholds from zero to 0.1 m, which
is the same procedure as in <cit.>.

Pose refinement results on the YCB Video Dataset

every column/.style=
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before row=    2cPoseCNN <cit.>     4cPoseCNN refined (ours)     2cHeatMaps <cit.>     4@c@HeatMaps refined (ours)     2cDeepIM <cit.>

(lr) 2-3 (lr) 2-3 (lr.5cm) 4-7 (lr) 8-9 (lr.5cm) 10-13 (lr) 14-15
after row=,
every last row/.style=after row=,
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We report the area
under the accuracy curve (AUC) for varying error thresholds on the ADD and ADD-S metrics.

We demonstrate pose refinement from the initialization of
PoseCNN <cit.> and the newer method by <cit.>.
<Ref> displays qualitative refinement examples, while
<Ref> gives quantitative results.
In our experiments, we assume that objects were correctly detected so that
we can focus on the problem of refining poses rather than correcting detections.
Our pipeline gives consistent improvements across nearly all objects of the
dataset for the PoseCNN initialization.
Note that we do not compare against the PoseCNN variant with ICP
post-refinement, since our pipeline works with RGB only and ICP requires
depth measurement.
The improvement is especially significant for large and textured objects.
On the initializations of <cit.>, which are already of
very high quality, our gains are smaller.
We hypothesize that our approach is currently limited by the spatial resolution
of the computed feature representation.

Finally, compared to DeepIM <cit.>, our method almost reaches
the same overall performance. We note that the experiments performed in
<cit.> apparently started from a better PoseCNN initialization
than what was available to us, though the difference seems small.
Interestingly, our method obtains significantly better results on a few object
classes—suggesting that a combination of the techniques (e.g. by
making the abstract representation and computed pose updates accessible to the
DeepIM network) could yield further improvements.

ymin=0, ymax=110,
bar width=6pt,

tickwidth         = 0pt,
enlarge x limits  = 0.05,

xtick=(0-10], (10-20],(20-30], (30-40], (40-50], (50-60], (60-70], (70-80], (80-90], (90-100] ,
xtick style=draw=none,
symbolic x coords = (0-10], (10-20],(20-30], (30-40], (40-50], (50-60], (60-70], (70-80], (80-90], (90-100] ,
nodes near coords,
legend pos=south east,
ylabel=Area under Curve (AUC),
xlabel=Pixel overlap [%],
nodes near coords style=anchor=south east,inner sep=0,shift=(4pt, 5pt),font=,/pgf/number format/.cd,fixed,fixed zerofill,precision=1,
ylabel near ticks, ylabel shift=-5pt,
((0-10],     42.9)
((10-20],    52.3)
((20-30],    64.0)
((30-40],    67.9)
((40-50],    72.1)
((50-60],    74.4)
((60-70],    78.6)
((70-80],    76.7)
((80-90],    76.7)
((90-100],   74.5);

((0-10],     61.4)
((10-20],    70.7)
((20-30],    79.3)
((30-40],    82.7)
((40-50],    85.1)
((50-60],    86.5)
((60-70],    89.0)
((70-80],    88.2)
((80-90],    88.6)
((90-100],   86.8);
ADD, ADD-S

Basin of attraction in translation dimensions. We show resulting
metrics for varying initial 2D overlap of ground truth pose and initial

ymin=0, ymax=110,
bar width=6pt,

tickwidth         = 0pt,
enlarge x limits  = 0.05,

xtick=(0-5], (5-10],(10-15], (15-20], (20-25], (25-30], (30-35], (35-40], (40-45] ,
xtick style=draw=none,
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nodes near coords,
legend pos=south west,
ylabel=Area under Curve (AUC),
xlabel=Angular perturbation [$^\circ$],
nodes near coords style=anchor=south east,inner sep=0,shift=(4pt, 5pt),font=,/pgf/number format/.cd,fixed,fixed zerofill,precision=1,
ylabel near ticks, ylabel shift=-5pt,
((0-5],      90.3)
((5-10],     89.1)
((10-15],    83.7)
((15-20],    82.7)
((20-25],    77.5)
((25-30],    74.3)
((30-35],    69.2)
((35-40],    62.7)
((40-45],    58.1);

((0-5],      94.5)
((5-10],     94.4)
((10-15],    92.0)
((15-20],    91.8)
((20-25],    89.6)
((25-30],    88.4)
((30-35],    86.1)
((35-40],    83.1)
((40-45],    80.8);
ADD, ADD-S

Basin of attraction in rotation dimensions. We show resulting
metrics for varying initial angular perturpations from the ground truth pose.

To quantify the robustness of our render-and-compare pipeline to the quality of the initialization, we analyzed
the basin of attraction of the refinement process. We experimented with 295 scenes from the validation set of
YCB-Video dataset ($\sim$10 % of the total validation scenes) by randomly perturbing the translation and
rotation components of the ground truth poses to varying degrees and optimizing the perturbed poses.
The translation perturbations were uniformly sampled in a range of $ \pm 5$ centimeters.
Since the impact the translation perturbations has for an object depends of the size of the object,
we compute the percentage of pixel overlap between the observed image and the rendered image for an object.

Similarly, we uniformly sample an axis of rotation and an rotation angle in the range $ \pm 45$ degrees.
The AUC of the optimized pose with respect to different overlaps is shown in <ref> and the rotation angle
is shown in <ref>.
Our method is able to robustly handle translation perturpations with almost no
loss in accuracy down to $30\%$ remaining overlap. In the rotation experiment,
the ADD-S metric is almost unaffected by rotations of up to $45^\circ$.
The ADD metric drops off more steeply—this is caused by the entirely symmetric
objects, where the system has no chance of correcting the perturbation around
the symmetry axis.

§ CONCLUSION

We introduced a technique for scene abstraction by dense object features
learned in a self-supervised way and scene analysis by render-and-compare
utilizing a fast differential renderer implementation.
Our proposed method yields good results on the challenging YCB Video dataset,
where it robustly refines rough initial pose estimates to precise localizations.
We further demonstrated its large basin of attraction from perturbed initializations.
We see our result as a proof-of-concept for differential rendering in the
context of scene analysis.
In future work, we will increase performance further
and investigate other applications, for example non-rigid registration of
category-level models.
Due to its formulation the method could also be combined with other
iterative refinement procedures, contributing its holistic scene understanding.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

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Robots (Humanoids)
titleAutonomous dual-arm manipulation of familiar objects

booktitle2018 IEEE/RSJ International Conference on Intelligent
Robots and Systems (IROS)
titleSupervised autonomous locomotion and manipulation for
disaster response with a centaur-like robot

titleLectures in Pattern Theory-Volume 1: Pattern Synthesis
journaltitleApplied Mathematical Sciences, Berlin: Springer, 1976

titleLectures in Pattern Theory: Volume II: Pattern Analysis

titlePoseCNN: A convolutional neural network for 6D object
pose estimation in cluttered scenes
journaltitlearXiv preprint arXiv:1711.00199

booktitleEuropean Conference on Computer Vision (ECCV)
titleOpenDR: An approximate differentiable renderer

booktitleProceedings of the IEEE Conference on Computer Vision and
Pattern Recognition(CVPR)
titleNeural 3D Mesh Renderer

booktitleAdvances in Neural Information Processing Systems
titleUnsupervised learning of 3D structure from images

titleSimple statistical gradient-following algorithms for
connectionist reinforcement learning
journaltitleMachine learning

booktitleSIGGRAPH Asia 2018 Technical Papers
titleDifferentiable monte carlo ray tracing through edge sampling

titleSoft Rasterizer: Differentiable Rendering for Unsupervised
Single-View Mesh Reconstruction
journaltitlearXiv preprint arXiv:1901.05567

booktitle2016 IEEE/RSJ International Conference on Intelligent
Robots and Systems (IROS)
titleReal-time height map fusion using differentiable rendering

booktitleProceedings of the IEEE International Conference on
Computer Vision (CVPR)
titleMOFA: Model-based deep convolutional face autoencoder for
unsupervised monocular reconstruction

booktitleProceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR)
title3D-RCNN: Instance-level 3D object reconstruction via

booktitleEuropean Conference on Computer Vision (ECCV)
titleOvercoming occlusion with inverse graphics

booktitleProceedings of the European Conference on Computer Vision
titleDeepIM: Deep iterative matching for 6D pose estimation

titleRecovering intrinsic scene characteristics
journaltitleThe Journal of Computer and System Sciences

booktitleAdvances in neural information processing systems
titleRecovering intrinsic images from a single image

booktitleEuropean conference on computer vision (ECCV)
titleIntrinsic images by entropy minimization

titleSelf-supervised visual descriptor learning for dense
journaltitleIEEE Robotics and Automation Letters (RA-L)

titleLight-weight refinenet for real-time semantic segmentation
journaltitlearXiv preprint arXiv:1810.03272

booktitleProceedings of the IEEE conference on computer vision and
pattern recognition (CVPR)
titleDeep residual learning for image recognition

titleDense Object Nets: Learning Dense Visual Object Descriptors
By and For Robotic Manipulation
journaltitlearXiv preprint arXiv:1806.08756

booktitleProceedings of the International Conference on Learning
Representations (ICLR)
titleExplaining and harnessing adversarial examples

booktitleProceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR)
titleWhat do Deep Networks Like to See?

booktitleProceedings of the European Conference on Computer Vision
titleMaking deep heatmaps robust to partial occlusions for 3D
object pose estimation