=2pt =1.5pt

#  Perfect state transfer, Equitable partition
and
Continuous-time quantum walk based search

Yusuke Ide, Akihiro Narimatsu Department of Mathematics, College of Humanities
and Sciences, Nihon University, Setagaya, Tokyo, 156-8550, Japan, E-mail:
<EMAIL_ADDRESS>for Mathematical and Data Scienses, The
University of Fukuchiyama, Fukuchiyama, Kyoto, 620-0886, Japan, E-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider a continuous-time quantum walk based search
algorithm. We introduce equitable partition of the graph and perfect state
transfer on it. By these two methods, we can calculate the success probability
and the finding time of the search algorithm. In addition, we gave some
examples of graphs that we can calculate the success probability and the
finding time.

Keywords: quantum walks, quantum search algorithms, perfect state transfer,
equitable partition

##  1 Introduction

The study of Quantum walks (QWs) has been paid much attention in the past two
decades because of its applications[12, 13, 14, 17, 18, 21], especially in
quantum information. The quantum search algorithm is one of the applications
of QWs in quantum information [1, 2, 3, 4, 5, 6, 16, 19].

In the previous study [10], we proved that a graph partitioning method called
equitable partition [9] for the given graph reduce the size of the problem of
the search algorithm. Under equitable partition, vertices of a graph are
clustered and identified by distance from the marked vertex and edge
connection with other vertices. We should remark that the initial state of the
search algorithm is uniform state. Although the initial state of the search
algorithm is uniform, the difference of the numbers of vertices in each
partition causes non-uniform effects for the uniform initial state.

In this paper, we introduce perfect state transfer (PST) which is a state
transfer phenomenon related to quantum walks. PST has been actively studied in
mathematics [8, 15, 20] and physics [11]. In mathematics, there are many
studies from the viewpoint of spectral graph theory and several properties are
revealed [7]. For example, when PST from a vertex $u$ to a vertex $v$ at time
$\tau$ occurs, PST from the vertex $v$ to the vertex $u$ at time $\tau$ also
occurs. Considering this under equitable partition, we obtain our main result
about quantum search algorithm. In addition, we propose some meaningful
example of the theorem as follows. Let $G=(V(G),E(G))$ be a simple graph with
its vertex set $V(G)=\\{0,1,\dots,N-1\\}$ and edge set $E(G)\subset V(G)\times
V(G)$. The matrix $\overline{H}_{G}$ is defined in section $2.3$, Eq.(4).

######  Example 1.1.

When $\overline{H}_{G}$ is given by

$\displaystyle\overline{H}_{G}=|0\rangle\langle
0|+\frac{1}{N-2}\begin{bmatrix}0&\sqrt{N-1}\\\
\sqrt{N-1}&N-2\end{bmatrix}=\begin{bmatrix}1&\frac{\sqrt{N-1}}{N-2}\\\
\frac{\sqrt{N-1}}{N-2}&1\end{bmatrix},$

we have the success probability of the search algorithm $\mathbb{P}_{N,T}(0)$
at time
$\displaystyle{T=\frac{(N-2)\pi}{2\sqrt{N-1}}={\mathcal{O}}(\sqrt{N})}$ is

$\displaystyle\mathbb{P}_{N,T}(0)=1-\frac{1}{N}.$

This situation corresponds to the case where $G$ is a complete graph,
$G_{\overline{0}}=(\\{0\\},\emptyset)$ is a graph consists of the marked one
vertex $0$, $G_{\overline{1}}$ is a complete graph with $N-1$ vertices and the
partition $(G_{\overline{0}},G_{\overline{1}})$ is equitable partition of $G$.

######  Example 1.2.

When $\overline{H}_{G}$ is given by

$\displaystyle\overline{H}_{G}$ $\displaystyle=|0\rangle\langle
0|+\frac{1}{2}\begin{bmatrix}0&(2k+1)\sqrt{2}&0\\\
(2k+1)\sqrt{2}&2&(2k+1)\sqrt{2}\\\ 0&(2k+1)\sqrt{2}&2\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1&\frac{(2k+1)\sqrt{2}}{2}&0\\\
\frac{(2k+1)\sqrt{2}}{2}&1&\frac{(2k+1)\sqrt{2}}{2}\\\
0&\frac{(2k+1)\sqrt{2}}{2}&1\end{bmatrix},$

where $k$ is a positive integer, we have the success probability of the search
algorithm $\mathbb{P}_{N,T}(0)$ at time $T=\pi$,

$\displaystyle\mathbb{P}_{N,T}(0)\approx 1-\frac{1}{\sqrt{N}}.$

This situation corresponds to the case where
$G_{\overline{0}}=(\\{0\\},\emptyset)$ is a graph consists of the marked one
vertex $0$, $G_{\overline{1}}$ is a cycle graph with $2(2k+1)^{2}$ vertices,
$G_{\overline{2}}$ is a cycle graph with $\\{2(2k+1)^{2}\\}^{2}$ vertices.

The rest of this paper is organized as follows: Section 2 is devoted to the
definition of our model and the introduction of a related result. Section 3
gives our result and introduction of the examples. Section 4 summerizes our
paper.

##  2 Preliminaries

###  2.1 Continuous-time quantum walk

Let $G=(V(G),E(G))$ be a simple undirected graph without self-loops and
multiple edges, with its vertex set $V(G)=\\{0,1,\dots,N-1\\}$ and edge set
$E(G)\subset V(G)\times V(G)$. As an example, for the complete graph on $N$
vertices $K_{N}$, which is the simple graph with $\begin{pmatrix}N\\\
2\end{pmatrix}$ edges and the edge set is defined by $E(K_{N})=\\{(j,k)\ :\
0\leq j<k\leq N-1\\}$. Continuous-time quantum walk (CTQW) is a unitary time
evolution process on the graph with the Hilbert space,

$\displaystyle{\mathcal{H}}={\rm Span}\\{|j\rangle:j\in V(G)\\}.$

On this Hilbert space, we define the unitary time evolution operator of CTQW.
We choose an $N$-dimensional Hermitian matrix $M_{G}$ with $(M_{G})_{j,k}\neq
0$ when $(j,k)\in E(G)$ and $(M_{G})_{j,k}=0$ when $(j,k)\notin E(G)$. The
component $(M_{G})_{j,k}$ is regarded as “the weight of the edge $(j,k)\in
E(G)$”. One of the typical $M_{G}$ is the adjacency matrix

$\displaystyle(A_{G})_{j,k}=\begin{cases}1\quad(j,k)\in E(G),\\\ 0\quad{\rm
otherwise.}\end{cases}$

of the graph $G$.

For the time $t\geq 0$, we define the time evolution operator

$\displaystyle
U_{M_{G}}(t)=\exp(itM_{G})=\sum_{k=0}^{\infty}\frac{(it)^{k}}{k!}M_{G}^{k},$
(1)

of CTQW on $G$ corresponding to $M_{G}$. We should note that CTQW is a quantum
dynamics determined by Schrödinger equation with its Hamiltonian $M_{G}$. We
set a vector $|\Psi_{N,0}\rangle\in{\mathcal{H}}(\||\Psi_{N,0}\rangle\|=1)$ as
initial state. Then the time evolution is written by

$\displaystyle|\Psi_{N,t}\rangle=U_{M_{G}}(t)|\Psi_{N,0}\rangle,$

for $t\geq 0$. $|\Psi_{N,t}\rangle$ describes the probability amplitude at
time $t$. We put the finding probability of the quantum walker on $s\in V(G)$
at time $t$ as

$\displaystyle\mathbb{P}_{N,t}(s)=|\langle s|\Psi_{N,t}\rangle|^{2}.$ (2)

###  2.2 The quantum walk based search algorithm

In this subsection, we consider CTQW based search on simple graph
$G=(V(G),E(G))$ with $V(G)=\\{0,1,\dots,N-1\\}$. The Hermitian matrix $H_{G}$
for CTQW search on $G$ is defined by

$\displaystyle H_{G}=|w\rangle\langle w|+\gamma A_{G},$

where $w\in V(G)$ is the marked vertex. Now we consider the CTQW on $G$ with
its time evolution operator $U_{H_{G}}$. The main task for CTQW search is
finding a suitable $\gamma\in\mathbb{R}$ and a time
$T\leq{\mathcal{O}}(\sqrt{N})$ such that we can attain

$\displaystyle\mathbb{P}_{N,T}(w)\approx 1\quad(N\to\infty),$

with the uniform initial state

$\displaystyle|\Psi_{0}\rangle=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}|j\rangle.$

###  2.3 Equitable partition

This subsection is a review of [10], section $4$ and deals with a partition of
the graph, called equitable partition for CTQW search. Without loss of
generality, we assume that the marked vertex is $0\in V(G)$. We consider the
partition $(G_{\overline{0}},G_{\overline{1}},\dots,G_{\overline{J-1}})$ of
$G=(V(G),E(G))$ which satisfies the following $3$ conditions:

  1. (1)

$V(G_{\overline{0}})=\\{0\\}$.

  2. (2)

$V(G)=\bigcup_{\overline{j}=0}^{J-1}V(G_{\overline{j}}),$
$V(G_{\overline{j}})\cap
V(G_{\overline{k}})=\emptyset(\overline{j}\neq\overline{k})$.

  3. (3)

For each $\overline{0}\leq\overline{j},\overline{k}\leq\overline{J-1}$, there
exists a non-negative integer $d_{\overline{j},\overline{k}}$ such that
$d_{\overline{j},\overline{k}}=|\\{w\in V(G_{\overline{k}}):(v,w)\in E(G)\\}|$
for each $v\in V(G_{\overline{J}})$.

We use the notation $G_{\overline{j}}\sim G_{\overline{k}}$ if
$d_{\overline{j},\overline{k}}\neq 0$.

The partition $(G_{\overline{0}},G_{\overline{1}},\dots,G_{\overline{J-1}})$
of $G=(V(G),E(G))$ consists of $\overline{J}$ subgraphs. For each subgraph
$G_{\overline{j}}$ is $d_{\overline{j},\overline{j}}$-regular graph.
Especially, $G_{\overline{0}}$ consists of only the marked vertex $0\in
V(G)(d_{\overline{0},\overline{0}}=0)$, which is called the null graph. In
addition, each of the vertices $v\in V(G_{\overline{j}})$ connected to the
same number $d_{\overline{j},\overline{k}}$ of the vertices in
$V(G_{\overline{k}})$.

From now on, we consider the CTQW search on simple and connected graph $G$
with equitable partition
$(G_{\overline{0}},G_{\overline{1}},\dots,G_{\overline{J-1}})$ with
$n_{\overline{j}}=|V(G_{\overline{j}})|$ of $G$. We define the uniform states
related to the partition as follows.

$\displaystyle|\overline{j}\rangle=\frac{1}{\sqrt{n_{\overline{j}}}}\sum_{j\in
V(G_{\overline{j}})}|j\rangle,$

for $0\leq\overline{j}\leq J-1$. By direct calculation, we have

$\displaystyle A_{G}|\overline{j}\rangle$
$\displaystyle=d_{\overline{j},\overline{j}}|\overline{j}\rangle+\sum_{G_{\overline{j}}\sim
G_{\overline{k}}}d_{\overline{k},\overline{j}}\sqrt{\frac{n_{\overline{k}}}{n_{\overline{j}}}}|\overline{k}\rangle=d_{\overline{j},\overline{j}}|\overline{j}\rangle+\sum_{G_{\overline{j}}\sim
G_{\overline{k}}}\sqrt{d_{\overline{k},\overline{j}}}\sqrt{\frac{n_{\overline{k}}d_{\overline{j},\overline{k}}}{n_{\overline{j}}}}|\overline{k}\rangle$
$\displaystyle=d_{\overline{j},\overline{j}}|\overline{j}\rangle+\sum_{G_{\overline{j}}\sim
G_{\overline{k}}}\sqrt{d_{\overline{j},\overline{k}}d_{\overline{k},\overline{j}}}|\overline{k}\rangle$
$\displaystyle=\sum_{\overline{0}\leq\overline{J-1}}\sqrt{d_{\overline{j},\overline{k}}d_{\overline{k},\overline{j}}}|\overline{k}\rangle,$

where
$n_{\overline{j}}d_{\overline{j},\overline{k}}=n_{\overline{k}}d_{\overline{k},\overline{j}}$
coming from simplicity of the graph. Here we define a $J\times J$ matrix
$\overline{A}_{G}$ as

$\displaystyle(\overline{A}_{G})_{\overline{j},\overline{k}}=\sqrt{d_{\overline{j},\overline{k}}d_{\overline{k},\overline{j}}}.$
(3)

When we define a $J\times J$ matrix $\overline{H}_{G}$ as

$\displaystyle\overline{H}_{G}={\rm
diag}(1,0,\dots,0)+\gamma\overline{A}_{G},$ (4)

the action of the Hermitian matrix $H_{G}$ is closed on the subspace
Span$\\{|\overline{j}\rangle:\overline{0}\leq\overline{j}\leq\overline{J-1}\\}$
as follows.

$\displaystyle
H_{G}|\overline{j}\rangle=\overline{H}_{G}|\overline{j}\rangle,$

for $\overline{0}\leq\overline{j}\leq\overline{J-1}$. We also have
$\displaystyle{|\Psi_{0}\rangle=\frac{1}{\sqrt{N}}\sum_{\overline{j}=\overline{0}}^{\overline{J-1}}\sqrt{n_{\overline{j}}}|\overline{j}\rangle}\in$Span$\\{|\overline{j}\rangle:\overline{0}\leq\overline{j}\leq\overline{J-1}\\}$.
Then we obtain

$\displaystyle
H_{G}|\Psi_{0}\rangle=\overline{H}_{G}\Biggl{(}\frac{1}{\sqrt{N}}\sum_{\overline{j}=\overline{0}}^{\overline{J-1}}\sqrt{n_{\overline{j}}}|\overline{j}\rangle\Biggr{)},$

thus we have

$\displaystyle{\rm exp}(itH_{G})|\Psi_{0}\rangle={\rm
exp}(it\overline{H}_{G})\Biggl{(}\frac{1}{\sqrt{N}}\sum_{\overline{j}=\overline{0}}^{\overline{J-1}}\sqrt{n_{\overline{j}}}|\overline{j}\rangle\Biggr{)}.$
(5)

###  2.4 Perfect state transfer

In this subsection, we consider PST related to $M_{G}$. PST related to $M_{G}$
from a vertex $j$ to a vertex $k$ at time $\tau$ with phase $\lambda$ occurs
if and only if

$\displaystyle\bigl{(}U_{M_{G}}(\tau)\bigr{)}_{k,j}=\lambda,$ (6)

where $j,k\in\\{0,1,\dots,N-1\\}$, $|\lambda|=1$. Combining Eq.(1) with Eq.(6)
gives

$\displaystyle\langle j|\Psi_{N,0}\rangle=\lambda\langle
k|\Psi_{N,\tau}\rangle.$

Noting $|\lambda|=1$, this equation implies that the state on a vertex $j$
moves to a vertex $k$ after time $\tau$.

Since $M_{G}$ be a Hermitian matrix, eigenvalues of $M_{G}$ and the each
component of the corresponding eigenvectors can be real numbers. Let
$\theta_{0}\geq\theta_{1}\geq\dots\geq\theta_{N-1}$ be eigenvalues of $M_{G}$
and $|v_{0}\rangle,|v_{1}\rangle,\dots,|v_{N-1}\rangle$ be corresponding
${\mathbb{R}}^{N}$ valued eigenvectors. Previous study [7, 8] revealed the
following claim.

######  Proposition 2.1.

A necessary and sufficient condition that the time evolution operator must
satisfy for perfect State Transfer between a vertex $j$ and a vertex $k$ at
time $\tau$ with phase $\lambda$ to occur is following three properties.

  1. (1)

For all $\ell\in\\{0,\dots,N-1\\}$, eigenvalues $\theta_{\ell}$ and
corresponding eigenvectors $|v_{\ell}\rangle$ satisfy $\langle
v_{\ell}|j\rangle=\pm\langle v_{\ell}|k\rangle$.

  2. (2)

If $\langle v_{\ell}|j\rangle=\langle v_{\ell}|k\rangle$, there exist
$n\in\mathbb{Z}$ such that $\theta_{0}-\theta_{\ell}=2n\pi$.

  3. (3)

If $\langle v_{\ell}|j\rangle=-\langle v_{\ell}|k\rangle$, there exist
$n\in\mathbb{Z}$ such that $\theta_{0}-\theta_{\ell}=(2n+1)\pi$.

Under these conditions, $\lambda=e^{i\tau\theta_{0}}.$

Using this proposition, the three properties can be rewritten as given in
appendix.

##  3 Result

This section is devoted to our theorem and some examples refered in Section
$1$.

###  3.1 Theorem

######  Theorem 3.1.

If perfect state transfer between vertex $\overline{0}$ and a vertex
$\overline{j}$ at time $\tau$ with phase $\lambda$ occurs for CTQW using
$\overline{H}_{G}$, success probability of the search algorithm is given by

$\displaystyle\mathbb{P}_{N,\tau}(0)=\frac{n_{\overline{j}}}{N}.$

proof. We should note that the initial state $|\Psi_{0}\rangle$ can be written
as

$\displaystyle|\Psi_{0}\rangle=\frac{1}{\sqrt{N}}\sum_{\overline{j}=\overline{0}}^{\overline{J-1}}\sqrt{n_{\overline{j}}}|\overline{j}\rangle.$

Combining Eq.(2) with Eq.(5) gives

$\displaystyle\mathbb{P}_{N,\tau}(0)$ $\displaystyle=\bigl{|}\langle
0|\exp(i\tau H_{G})|\Psi_{0}\rangle\bigr{|}^{2}$
$\displaystyle=\Biggl{|}\langle\overline{0}|\exp(i\tau\overline{H}_{G})\Biggl{(}\sum_{\overline{k}=\overline{0}}^{\overline{J-1}}\frac{\sqrt{n_{\overline{k}}}}{\sqrt{N}}|\overline{k}\rangle\Biggr{)}\Biggr{|}^{2}.$

Since perfect state transfer between $\overline{0}$ and $\overline{j}$ occur,
we have

$\displaystyle\mathbb{P}_{N,\tau}(0)$
$\displaystyle=\Biggl{|}\langle\overline{0}|\lambda\frac{\sqrt{n_{\overline{j}}}}{\sqrt{N}}|\overline{0}\rangle\Biggr{|}^{2}=\Biggl{|}\lambda\sqrt{\frac{n_{\overline{j}}}{N}}\Biggr{|}^{2}=\frac{n_{\overline{j}}}{N}.$

$\square$

###  3.2 Examples

In this subsection, we consider examples $1.1.$ and $1.2.$, which are the
examples of our theorem.

######  Example 1.1.

When $\overline{H}_{G}$ is given by

$\displaystyle\overline{H}_{G}=|0\rangle\langle
0|+\frac{1}{N-2}\begin{bmatrix}0&\sqrt{N-1}\\\
\sqrt{N-1}&N-2\end{bmatrix}=\begin{bmatrix}1&\frac{\sqrt{N-1}}{N-2}\\\
\frac{\sqrt{N-1}}{N-2}&1\end{bmatrix},$

we have the success probability of the search algorithm $\mathbb{P}_{N,T}(0)$
at time
$\displaystyle{T=\frac{(N-2)\pi}{2\sqrt{N-1}}={\mathcal{O}}(\sqrt{N})}$ is

$\displaystyle\mathbb{P}_{N,T}(0)=1-\frac{1}{N}.$

This situation corresponds to the case where $G$ is a complete graph,
$G_{\overline{0}}=(\\{0\\},\emptyset)$ is a graph consists of the marked
vertex $0$, $G_{\overline{1}}$ is a complete graph with $N-1$ vertices and the
partition $(G_{\overline{0}},G_{\overline{1}})$ is equitable partition of $G$.

proof. By Eqs.(3) and (4), we have $\gamma=1/(N-2)$ and

$\displaystyle\overline{A}_{G}$ $\displaystyle=\begin{bmatrix}0&\sqrt{N-1}\\\
\sqrt{N-1}&N-2\end{bmatrix}=\begin{bmatrix}d_{\overline{0},\overline{0}}&\sqrt{d_{\overline{0},\overline{1}}d_{\overline{1},\overline{0}}}\\\
\sqrt{d_{\overline{1},\overline{0}}d_{\overline{0},\overline{1}}}&d_{\overline{1},\overline{1}}\end{bmatrix}.$

Since $d_{\overline{0},\overline{0}}=0$, we have $|V(G_{\overline{0}})|=1$.
Thus we get $d_{\overline{0},\overline{1}}=N-1$, $|V(G_{\overline{1}})|=N-1$
and $G_{\overline{1}}$ is a complete graph with $N-1$ vertices. Then we obtain
$G$ is a complete graph with $N$ vertices.

Next we consider the success probability. Let $\theta_{\pm}$ be eigenvalues of
$\overline{H}_{G}$ and $|v_{\pm}\rangle$ be corresponding eigenvectors. By
spectral decomposition, we have

$\displaystyle\overline{H}_{G}$ $\displaystyle=\theta_{+}|v_{+}\rangle\langle
v_{+}|+\theta_{-}|v_{-}\rangle\langle v_{-}|$
$\displaystyle=\theta_{+}\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\
1\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\end{bmatrix}+\theta_{-}\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\
-1\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&-1\end{bmatrix}$
$\displaystyle=\frac{\theta_{+}}{2}\begin{bmatrix}1&1\\\
1&1\end{bmatrix}+\frac{\theta_{-}}{2}\begin{bmatrix}1&-1\\\
-1&1\end{bmatrix},$

where

$\displaystyle\theta_{\pm}=1\pm\frac{\sqrt{N-1}}{N-2}.$

Then we obtain

$\displaystyle e^{it\overline{H}_{G}}$
$\displaystyle=e^{it\theta_{+}}\frac{1}{2}\begin{bmatrix}1&1\\\
1&1\end{bmatrix}+e^{it\theta_{-}}\frac{1}{2}\begin{bmatrix}1&-1\\\
-1&1\end{bmatrix}$
$\displaystyle=e^{it}\begin{bmatrix}\cos\Bigl{(}\frac{\sqrt{N-1}}{N-2}t\Bigr{)}&i\sin\Bigl{(}\frac{\sqrt{N-1}}{N-2}t\Bigr{)}\\\
i\sin\Bigl{(}\frac{\sqrt{N-1}}{N-2}t\Bigr{)}&\cos\Bigl{(}\frac{\sqrt{N-1}}{N-2}t\Bigr{)}\end{bmatrix}.$

Therefore when time
$\displaystyle{T=\frac{(N-2)\pi}{2\sqrt{N-1}}={\mathcal{O}}(\sqrt{N})}$,

$\displaystyle
e^{iT\overline{H}_{G}}=e^{iT}\begin{bmatrix}\cos\Bigl{(}\frac{\sqrt{N-1}}{N-2}T\Bigr{)}&i\sin\Bigl{(}\frac{\sqrt{N-1}}{N-2}T\Bigr{)}\\\
i\sin\Bigl{(}\frac{\sqrt{N-1}}{N-2}T\Bigr{)}&\cos\Bigl{(}\frac{\sqrt{N-1}}{N-2}T\Bigr{)}\end{bmatrix}=e^{iT}\begin{bmatrix}0&i\\\
i&0\end{bmatrix}.$

Thus PST occurs from $\overline{1}$ to $\overline{0}$ at time
$\displaystyle{T=\frac{(N-2)\pi}{2\sqrt{N-1}}}$. Then the finding probability
is given by

$\displaystyle\mathbb{P}_{N,T}(0)=\frac{n_{\overline{1}}}{N}=\frac{N-1}{N}=1-\frac{1}{N}.$

$\square$

######  Example 1.2.

When $\overline{H}_{G}$ is given by

$\displaystyle\overline{H}_{G}$ $\displaystyle=|0\rangle\langle
0|+\frac{1}{2}\begin{bmatrix}0&(2k+1)\sqrt{2}&0\\\
(2k+1)\sqrt{2}&2&(2k+1)\sqrt{2}\\\ 0&(2k+1)\sqrt{2}&2\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1&\frac{(2k+1)\sqrt{2}}{2}&0\\\
\frac{(2k+1)\sqrt{2}}{2}&1&\frac{(2k+1)\sqrt{2}}{2}\\\
0&\frac{(2k+1)\sqrt{2}}{2}&1\end{bmatrix},$

where $k$ is a positive integer, we have the success probability of the search
algorithm $\mathbb{P}_{N,T}(0)$ at time $T=\pi$,

$\displaystyle\mathbb{P}_{N,T}(0)\approx 1-\frac{1}{\sqrt{N}}.$

This situation corresponds to the case where
$G_{\overline{0}}=(\\{0\\},\emptyset)$ is a graph consists of the marked one
vertex $0$, $G_{\overline{1}}$ is a cycle graph with $2(2k+1)^{2}$ vertices,
$G_{\overline{2}}$ is a cycle graph with $\\{2(2k+1)^{2}\\}^{2}$ vertices.

proof. By Eqs.(3) and (4), we have $\gamma=1/2$ and

$\displaystyle\overline{A}_{G}$
$\displaystyle=\begin{bmatrix}0&(2k+1)\sqrt{2}&0\\\
(2k+1)\sqrt{2}&2&(2k+1)\sqrt{2}\\\ 0&(2k+1)\sqrt{2}&2\end{bmatrix}$
$\displaystyle=\begin{bmatrix}d_{\overline{0},\overline{0}}&\sqrt{d_{\overline{0},\overline{1}}d_{\overline{1},\overline{0}}}&\sqrt{d_{\overline{0},\overline{2}}d_{\overline{2},\overline{0}}}\\\
\sqrt{d_{\overline{1},\overline{0}}d_{\overline{0},\overline{1}}}&d_{\overline{1},\overline{1}}&\sqrt{d_{\overline{1},\overline{2}}d_{\overline{2},\overline{1}}}\\\
\sqrt{d_{\overline{2},\overline{0}}d_{\overline{0},\overline{2}}}&\sqrt{d_{\overline{2},\overline{1}}d_{\overline{1},\overline{2}}}&d_{\overline{2},\overline{2}}\end{bmatrix}.$

There are some choices for each $d_{\overline{j},\overline{k}}$. We put

$\displaystyle\begin{bmatrix}d_{\overline{0},\overline{0}}&d_{\overline{0},\overline{1}}&d_{\overline{0},\overline{2}}\\\
d_{\overline{1},\overline{0}}&d_{\overline{1},\overline{1}}&d_{\overline{1},\overline{2}}\\\
d_{\overline{2},\overline{0}}&d_{\overline{2},\overline{1}}&d_{\overline{2},\overline{2}}\end{bmatrix}=\begin{bmatrix}0&2(2k+1)^{2}&0\\\
1&2&2(2k+1)^{2}\\\ 0&1&2\end{bmatrix}.$

$d_{\overline{0},\overline{0}}=0$ implies that $|V(G_{\overline{0}})|=1$, or
in other words, $G_{\overline{0}}$ consists of only one marked vertex $0$. By
$d_{\overline{1},\overline{1}}=2$ and $d_{\overline{2},\overline{2}}=2$,
$G_{\overline{1}}$ and $G_{\overline{2}}$ are $2$-regular graph. We chose the
cycle graph for $G_{\overline{1}}$ and $G_{\overline{2}}$ because the cycle
graph is a well known example of $2$-regular graphs. Since
$d_{\overline{0},\overline{1}}=2(2k+1)^{2}$ and $V(G_{\overline{0}})=\\{0\\}$,
$|V(G_{\overline{1}})|=2(2k+1)^{2}$. Therefore, $G_{\overline{1}}$ is a cycle
graph with $2(2k+1)^{2}$ vertices. $d_{\overline{1},\overline{2}}=2(2k+1)^{2}$
and $d_{\overline{2},\overline{1}}=1$, each vertex in $G_{\overline{1}}$ is
connected to $2(2k+1)^{2}$ different vertices in $G_{\overline{2}}$ and each
vertex in $G_{\overline{2}}$ is connected to $1$ vertex in $G_{\overline{1}}$.
Thus we obtain $|V(G_{\overline{2}})|=\\{2(2k+1)^{2}\\}^{2}$ and
$G_{\overline{2}}$ is a cycle graph with $\\{2(2k+1)^{2}\\}^{2}$ vertices.

Next we consider the success probability. Let
$\theta_{0},\theta_{1},\theta_{2}$ be eigenvalues of $\overline{H}_{G}$ and
$|v_{0}\rangle,|v_{1}\rangle,|v_{2}\rangle$ be corresponding eigenvectors. By
spectral decomposition, we obtain

$\displaystyle\overline{H}_{G}$ $\displaystyle=\theta_{0}|v_{0}\rangle\langle
v_{0}|+\theta_{1}|v_{1}\rangle\langle v_{1}|+\theta_{2}|v_{2}\rangle\langle
v_{2}|$ $\displaystyle=\theta_{0}\frac{1}{2}\begin{bmatrix}1\\\ \sqrt{2}\\\
1\end{bmatrix}\frac{1}{2}\begin{bmatrix}1&\sqrt{2}&1\end{bmatrix}+\theta_{1}\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\
0\\\ -1\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&-1\end{bmatrix}$
$\displaystyle+\theta_{2}\frac{1}{2}\begin{bmatrix}1\\\ -\sqrt{2}\\\
1\end{bmatrix}\frac{1}{2}\begin{bmatrix}1&-\sqrt{2}&1\end{bmatrix}$
$\displaystyle=\frac{\theta_{0}}{4}\begin{bmatrix}1&\sqrt{2}&1\\\
\sqrt{2}&2&\sqrt{2}\\\
1&\sqrt{2}&1\end{bmatrix}+\frac{\theta_{1}}{2}\begin{bmatrix}1&0&-1\\\
0&0&0\\\
-1&0&1\end{bmatrix}+\frac{\theta_{2}}{4}\begin{bmatrix}1&-\sqrt{2}&1\\\
-\sqrt{2}&2&-\sqrt{2}\\\ 1&-\sqrt{2}&1\end{bmatrix},$

where $\theta_{0}=2(k+1),\theta_{1}=1,\theta_{2}=-2k,$ with $k$ is a positive
integer. Then we obtain

$\displaystyle e^{iT\overline{H}_{G}}$
$\displaystyle=e^{iT\theta_{0}}\frac{1}{4}\begin{bmatrix}1&\sqrt{2}&1\\\
\sqrt{2}&2&\sqrt{2}\\\
1&\sqrt{2}&1\end{bmatrix}+e^{iT\theta_{1}}\frac{1}{2}\begin{bmatrix}1&0&-1\\\
0&0&0\\\ -1&0&1\end{bmatrix}$
$\displaystyle+e^{iT\theta_{2}}\frac{1}{4}\begin{bmatrix}1&-\sqrt{2}&1\\\
-\sqrt{2}&2&-\sqrt{2}\\\ 1&-\sqrt{2}&1\end{bmatrix}.$

When $T=\pi$, we obtain

$\displaystyle e^{i\pi\overline{H}_{G}}$ $\displaystyle=e^{i\pi
2(k+1)}\frac{1}{4}\begin{bmatrix}1&\sqrt{2}&1\\\ \sqrt{2}&2&\sqrt{2}\\\
1&\sqrt{2}&1\end{bmatrix}+e^{i\pi\cdot 1}\frac{1}{2}\begin{bmatrix}1&0&-1\\\
0&0&0\\\ -1&0&1\end{bmatrix}$
$\displaystyle+e^{i\pi(-2k)}\frac{1}{4}\begin{bmatrix}1&-\sqrt{2}&1\\\
-\sqrt{2}&2&-\sqrt{2}\\\ 1&-\sqrt{2}&1\end{bmatrix}=\begin{bmatrix}0&0&1\\\
0&1&0\\\ 1&0&0\end{bmatrix}.$

Thus PST occurs from $\overline{2}$ to $\overline{0}$ at time $T=\pi$. Then
the finding probability is given by

$\displaystyle{\mathbb{P}}_{N,\pi}(0)=\frac{n_{\overline{2}}}{N}=\frac{\\{2(2k+1)^{2}\\}^{2}}{\\{2(2k+1)^{2}\\}^{2}+2(2k+1)^{2}+1}\approx
1-\frac{1}{\sqrt{N}}.$

$\square$

When $k=1$, Example $1.2.$ includes the case in [8].

##  4 Summary

In this paper, we analyzed the continuous-time quantum walk based search
algorithm. We obtained the success probability of the search algorithm by
using PST on equitable partition of the graph. In addition, we proposed some
graphs as examples of our theorem. One of the interesting future problem is to
reduce the conditions of equitable partition.

##  Appendix A Appendix

Let each component of the eigenvector of Hermitian matrix $M_{G}$ be

$\displaystyle|v_{\ell}\rangle=\begin{bmatrix}|v_{\ell}(0)|e^{\tilde{\theta}_{\ell}(0)}\\\
|v_{\ell}(1)|e^{\tilde{\theta}_{\ell}(1)}\\\ \vdots\\\
|v_{\ell}(N-1)|e^{\tilde{\theta}_{\ell}(N-1)}\end{bmatrix}.$

######  Corollary A.1.

A necessary and sufficient condition that the time evolution operator must
satisfy for PST from a vertex $j$ to a vertex $k$ at time $\tau$ to occur is
following.

$\displaystyle(\theta_{0}-\theta_{\ell})\tau\equiv\tilde{\theta}_{\ell}(k)-\tilde{\theta}_{\ell}(j)-\\{\tilde{\theta}_{0}(k)-\tilde{\theta}_{0}(j)\\}.\quad(\bmod\
2\pi)$

proof. By Eq.(6), we have

$\displaystyle U_{M_{G}}(\tau)|j\rangle=\lambda|k\rangle.$

By direct calculation, we get

$\displaystyle\Biggl{(}\sum_{\ell=0}^{N-1}e^{i\tau\theta_{\ell}}|v_{\ell}\rangle\langle
v_{\ell}|\Biggr{)}|j\rangle=\lambda\Biggl{(}\sum_{\ell=0}^{N-1}|v_{\ell}\rangle\langle
v_{\ell}|\Biggr{)}|k\rangle,$

comparing each component of the left hand side with that of the right hand
side, we obtain

$\displaystyle e^{i\tau\theta_{\ell}}\langle v_{\ell}|j\rangle=\lambda\langle
v_{\ell}|k\rangle\quad({\rm for\ all\ }\ell=0,\dots,N-1),$

computing this, we have

$\displaystyle
e^{i\tau\theta_{\ell}}|v_{\ell}(j)|e^{-i\tilde{\theta}_{\ell}(j)}=\lambda|v_{\ell}(k)|e^{-i\tilde{\theta}_{\ell}(k)}\quad({\rm
for\ all\ }\ell=0,\dots,N-1).$

Noting that $|\lambda|=1$, we get

$\displaystyle\begin{cases}e^{i\tau\theta_{\ell}}e^{-i\tilde{\theta}_{\ell}(j)}=\lambda
e^{-i\tilde{\theta}_{\ell}(k)}\\\ |v_{\ell}(j)|=|v_{\ell}(k)|.\end{cases}$ (7)

When $\ell=0$, Eq.(7) gives

$\displaystyle\lambda=e^{i\tau\theta_{0}}e^{i(\tilde{\theta}_{0}(k)-\tilde{\theta}_{0}(j))}$
(8)

Substituting Eq.(8) into Eq.(7), we obtain

$\displaystyle
e^{i(\tilde{\theta}_{\ell}(k)-\tilde{\theta}_{\ell}(j))}=e^{i\tau(\theta_{0}-\theta_{\ell})}e^{i(\tilde{\theta}_{0}(k)-\tilde{\theta}_{0}(j))}$

calculating this, we have

$\displaystyle
e^{i\\{(\tilde{\theta}_{\ell}(k)-\tilde{\theta}_{\ell}(j))-(\tilde{\theta}_{0}(k)-\tilde{\theta}_{0}(j))\\}}=e^{i\tau(\theta_{0}-\theta_{\ell})}.$

Thus we get

$\displaystyle(\theta_{0}-\theta_{\ell})\tau\equiv\tilde{\theta}_{\ell}(k)-\tilde{\theta}_{\ell}(j)-\\{\tilde{\theta}_{0}(k)-\tilde{\theta}_{0}(j)\\}.\quad(\bmod\
2\pi)$

$\square$

## References

  * [1] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, One-dimensional quantum walks, Proceedings of the $33$rd Annual ACM Symposium on Theory of Computing, pp. 37-49, 2001.
  * [2] A. Ambainis, J. Kempe, and A. Rivosh, Coins make quantum walks faster, Proceedings of the $16$th ACM-SIAM Symposium on Discrete Algorithms, pp. 1099-1108, 2005.
  * [3] S. Chakraborty, L. Novo, A. Ambainis, and Y. Omar, Spatial search by quantum walk is optimal for almost all graphs, Phys. Rev. Lett., vol. 116, 100501, 2016.
  * [4] A.M. Childs, Universal computation by quantumwalk, Phys. Rev. Lett., vol. 102, 180501, 2009.
  * [5] A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D.A. Spielman, Exponential algorithmic speedup by quantum walk, Proceedings of the $35$rd Annual ACM Symposium on Theory of Computing, pp. 59-68, 2003.
  * [6] A.M. Childs and J. Goldstone, Spatial search by quantum walk, Phys. Rev., vol. A70, 022314, 2004.
  * [7] G. Coutinho, Quantum State Transfer in Graphs. PhD Thesis, University of Waterloo, Waterloo, ON, Canada, 2014.
  * [8] C. Godsil, State transfer on graphs, Discrete Math., 312, vol. 1, pp. 129-147, 2012.
  * [9] C. Godsil and G.F. Royle, Algebraic Graph Theory. Springer-Verlag, New York, 2001.
  * [10] Y. Ide, Partition of graphs and quantum walk based search algorithms, Nonlinear Theory and Its Applications, IEICE, vol. 10, Issue 1, pp. 16-27, 2019.
  * [11] A. Kay, Perfect, efficient, state transfer and its application as a constructive tool, International Journal of Quantum Information 8.04, pp.641-676, 2010.
  * [12] J. Kempe, Quantum random walks - an introductory overview, Contemporary Physics, vol. 44, pp. 307-327, 2003.
  * [13] V. Kendon, Decoherence in quantum walks - a review. Math. Struct, Comp. Sci., vol. 17, pp. 1169-1220, 2007.
  * [14] N. Konno, Quantum walks, In: Quantum Potential Theory, U. Franz and M. Schürmann, Eds., Lecture Notes in Mathematics: vol. 1954, pp. 309-452, Springer-Verlag, Heidelberg, 2008.
  * [15] S. Kubota and E. Segawa, Perfect state transfer in Grover walks between states associated to vertices of a graph, Linear Algebla Appl., vol. 646, pp. 238-251, 2022.
  * [16] N.B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Universal quantum computation using the discrete-time quantum walk, Phys. Rev., vol. A 81, 042330, 2010.
  * [17] K. Manouchehri and J.B. Wang, Physical Implementation of Quantum Walks, Springer-Verlag, Heidelberg, 2013.
  * [18] R. Portugal, Quantum walks and search algorithms, Springer-Verlag, New York, 2013.
  * [19] N. Shenvi, J. Kempe, and K.B. Whaley, Quantum random-walk search algorithm, Phys. Rev., vol. A 67, 052307, 2003.
  * [20] H. Tanaka, M. Sabri, and R. Portugal, Spatial search on Johnson graphs by continuous-time quantum walk. Quantum Inf. Process., vol. 21, 74, 2022.
  * [21] S.E. Venegas-Andraca, Quantum walks: A comprehensive review, Quant. Inform. Process., vol. 11, pp. 1015-1106, 2012.


# Attention and Pooling based Sigmoid Colon Segmentation in 3D CT images

Md Akizur Rahman1, Sonit Singh1, Kuruparan Shanmugalingam1, Sankaran Iyer1,
Alan Blair1, Praveen Ravindran2, Arcot Sowmya1
Email<EMAIL_ADDRESS>1 School of CSE, University of New South
Wales, Sydney, Australia  2 Department of Colorectal Surgery, Sydney Adventist
Hospital, Sydney, Australia

###### Abstract

Segmentation of the sigmoid colon is a crucial aspect of treating
diverticulitis. It enables accurate identification and localisation of
inflammation, which in turn helps healthcare professionals make informed
decisions about the most appropriate treatment options. This research presents
a novel deep learning architecture for segmenting the sigmoid colon from
Computed Tomography (CT) images using a modified 3D U-Net architecture.
Several variations of the 3D U-Net model with modified hyper-parameters were
examined in this study. Pyramid pooling (PyP) and channel-spatial Squeeze and
Excitation (csSE) were also used to improve the model performance. The
networks were trained using manually annotated sigmoid colon. A five-fold
cross-validation procedure was used on a test dataset to evaluate the
network’s performance. As indicated by the maximum Dice similarity coefficient
(DSC) of $56.92\pm 1.42$%, the application of PyP and csSE techniques improves
segmentation precision. We explored ensemble methods including averaging,
weighted averaging, majority voting, and max ensemble. The results show that
average and majority voting approaches with a threshold value of 0.5 and
consistent weight distribution among the top three models produced comparable
and optimal results with DSC of $88.11\pm 3.52$%. The results indicate that
the application of a modified 3D U-Net architecture is effective for
segmenting the sigmoid colon in Computed Tomography (CT) images. In addition,
the study highlights the potential benefits of integrating ensemble methods to
improve segmentation precision.

###### Index Terms:

Image segmentation, deep learning, sigmoid colon segmentation, Computed
Tomography, diverticulitis.

##  I Introduction

The _sigmoid colon_ is a segment of the large intestine that is situated in
the lower left region of the abdomen. Abnormalities in the sigmoid colon, such
as inflammation or tumours, can result in serious health complications,
including diverticulitis and colon cancer. Timely identification and
management of these ailments can potentially save lives. Diverticular diseases
affect the colon and encompass a range of clinical conditions, from the mere
presence of diverticula to complicated diverticulitis [1]. Diverticula refer
to small pouches that protrude from the inner lining of the digestive system.
Diverticulitis occurs when the pouches become infected. Developing
diverticulitis is associated with certain risk factors such as a diet that is
high in fat and low in fibre, as well as low levels of physical activity. It
is important to note that although these factors are linked with
diverticulitis, not everyone who has these risk factors will necessarily
develop the condition. The presence of air bubbles and gas outside the sigmoid
colon, along with diverticulitis, during bleeding is often an indication of a
more complex and severe medical condition. Based on these factors, it appears
that the diverticulitis has progressed to advanced stages and may result in
complications. Figure 1 shows colon anatomy, having diverticulitis and its
features outside on the sigmoid colon.

Figure 1: Colon Anatomy and diverticulitis. Figure taken from [2]

Medical imaging techniques, such as Computed Tomography (CT) scans, are
frequently utilised to detect abnormalities in the colon [3, 4]. Manually
segmenting the sigmoid colon in medical images is a tedious and time-consuming
process that demands specialised expertise.

Figure 2: Block diagram showing sigmoid colon segmentation.

The precise segmentation of the sigmoid colon can aid in the detection of
diverticula by determining their existence, position, and scope. The
utilisation of this technique aids in the evaluation of inflammation,
distinguishing it from other medical conditions, devising a treatment plan,
and tracking the advancement of the disease.

Automated segmentation of the sigmoid colon can be utilised as an alternative
to the arduous and time-consuming manual task. Although there is lacking
research on sigmoid colon segmentation [4], recent studies have exhibited
encouraging outcomes through the utilisation of Deep Learning methodologies,
specifically Convolutional Neural Networks (CNNs) for medical image
segmentation [5, 6]. The field of computer-aided diagnosis is gaining
popularity due to advancements in medical imaging and deep learning. The focus
is on creating systems that can precisely segment various organs from medical
images. According to recent studies [7, 8], automated segmentation systems
possess the capability to improve diagnostic precision, decrease the workload
on medical professionals, and ultimately enhance patient results. The
development of deep learning-based sigmoid colon segmentation has encountered
challenges despite some preliminary efforts [4]. Managing variations in shape,
size, multi-curve structure, and location of the sigmoid colon between
patients, as well as resolving noise and artefacts in medical images, are
among these challenges. To address such barriers, developing sophisticated and
robust deep-learning models capable of segmenting the sigmoid colon in medical
images is necessary. This will result in improved patient diagnosis and
treatment outcomes. This study introduces an innovative deep learning
architecture to segment the sigmoid colon. The architecture was built using 3D
U-Net with Pyramid pooling (PyP) and channel-spatial Squeeze and Excitation
(csSE), called PyP3DUcsSENet. An overview of the approach is in Figure 2. This
research aims to evaluate and compare the segmentation performance of numerous
3D U-Net variants on 3D Computed Tomography (CT) images of the sigmoid colon.
The proposed network was evaluated using a publicly available 3D CT images of
colon cancer dataset. A comparative analysis was conducted to assess the
performance of the proposed network in comparison to several variants of 3D
U-Net. The results suggest that the PyP3DUcsSENet network is an efficient
approach for segmenting the sigmoid colon. The PyP3DUcsSENet demonstrated
superior precision and efficiency in comparison to alternative variants. Also,
the ensemble technique results indicate that it is an effective approach for
improving sigmoid colon segmentation performance.

The rest of the paper is organised in the following Section II provides a
comprehensive overview of the literature on medical image segmentation and
colon segmentation. Then describes the employed materials and methods,
including a thorough overview of the dataset, the proposed PyP3DUcsSENet
architecture, and ensemble methods in Section III. Next, the experimental
setup is detailed in Section IV. In Section V presents the study’s findings
and evaluates the performance of various variants of the 3D U-Net model
utilising PyP3DUcsSENet and ensemble methodologies. Discusses the findings and
their implications in Section VI, followed by recommendations for future
research. Finally, in Section VII includes a summary of the paper’s essential
contributions and potential applications of this strategy.

##  II Related Work

Medical image analysis requires accurate segmentation of various organs for
diagnosis, treatment planning, and disease monitoring. Various techniques,
such as patch-based approaches [9], multi-atlas techniques [10], and rule-
based techniques [11, 12], have been proposed for organ segmentation. Although
the methods have demonstrated satisfactory performance, segmenting extensive
anatomical structures remains challenging. Deep learning techniques are
gaining popularity for medical image segmentation as a result of their
superior outputs and efficacy [13, 14, 15]. Using deep learning techniques,
various organs including the liver [16, 6, 17, 18, 19, 20], lung [21, 22],
pancreas [23], prostate [24, 25], and multiple organs [5, 26, 27, 28, 29, 30,
31, 32, 33, 34] have been successfully segmented.

A comprehensive analysis of colon segmentation techniques was performed,
encompassing approaches that do not rely on diverticulitis diseases. A method
for colon segmentation was proposed in [35]. The objective of the method is to
identify polyps or growth and remove opacified fluid for virtual colonoscopy.
The segmentation of air pockets was achieved through the utilisation of 3D
Seeded Region Growing methodology. The seed was positioned in the axial slice
that encompasses the rectum. The region growing process was executed using
6-connected neighbourhood connectivity. The segmentation of opacified fluid
portions was achieved through the utilisation of fuzzy connectedness. The 3D
rendering of the segmented colon was achieved by concatenating the segmented
colon and opacified fluid. The study incorporated a total of 15 datasets,
consisting of 7 datasets obtained from The Cancer Imaging Atlas (TCIA) [36]
and 8 real-time datasets sourced from Imaging Centre, Coimbatore. The method
proposed in this study attained a 98.73% accuracy rate, indicating its
efficacy in virtual colonoscopy for colon segmentation. In their study, Guachi
et al. [3] presented an automated technique for colon tissue segmentation. The
technique is based on binary classification and utilises the LeNet-5 network,
which was initially developed for handwritten digit recognition. In this
study, the authors utilised the aforementioned technique to detect colon
tissue at every individual pixel location. A CNN was trained using patches
extracted from a dataset of CT images consisting of 100 slices. The
convolutional LeNet-5 architecture proposal comprises convolution layers, ReLU
activation functions, pooling layers, and fully connected layers. The method
proposed in this study attained a colon tissue segmentation accuracy of
$97.83\%$. This result indicates that the proposed approach is effective for
colon tissue segmentation. In their study, Gonzalez et al. [4] presented an
iterative 2.5D deep learning methodology for the segmentation of the sigmoid
colon. The approach employed a U-Net-like network structure. The approach
employed a 3-D convolutional layer and down-sampling via max-pooling layers to
process inputs slice by slice. Skip connections were utilised to facilitate
the exchange of information between the descending and ascending branches of
the network. The proposed methodology demonstrated encouraging results in both
scenarios, i.e., with and without prior knowledge of other organs, for the
automated segmentation of the sigmoid colon in CT images intended for
radiotherapy treatment planning. The report is deficient in providing adequate
information regarding their proposed architecture, thereby impeding the
assessment of their approach’s efficacy and replication of the outcomes.

The reviewed studies indicate that deep learning, specifically 3D U-Net
variants, have the capability to perform precise and effective segmentation of
medical images. Although 3D U-Net has demonstrated exceptional performance in
medical image segmentation, there remain certain issues that necessitate
further consideration. The complicated shapes of certain organs and their
closeness to other organs continue to be challenging issues to solve. In
addition, the presence of notable differences in the size and structure of
organs may affect the precision of segmentation. Furthermore, there exists a
requirement for enhancing the segmentation performance of tiny organs. It is
recommended that future research endeavours focus on addressing the challenges
associated with 3D U-Net in medical image segmentation and developing novel
techniques to improve its overall performance.

##  III Materials and Methods

We describe the dataset and different variants of the 3D U-Net model employed
in this study. Specifically, we have made modifications to the network hyper-
parameters and incorporated additional modules, including Pyramid pooling
(PyP) [25] and channel-spatial Squeeze and Excitation (csSE) [37] to enhance
the performance of 3D U-Net on sigmoid colon segmentation. Additionally, we
provide comprehensive information on the ensemble techniques employed in this
study.

###  III-A Dataset

For this study, used a publicly available dataset of CT scans of colon cancer,
collected from the medical segmentation decathlon challenge [38]. At Memorial
Sloan Kettering Cancer Centre in New York, $95$ CT scans were collected from
patients having primary colon cancer resection. The scans were taken using an
X-ray tube operating at $100-140$ kVp for $500-1782$ ms of exposure time at a
current of $100-752$ mA. The reconstructed scans have diameters of between
$274$ and $5000$ millimetres and slice thicknesses of between $1$ and $7.5$
millimetres. Although most of the cases in the dataset are those of colon
cancer, we think it is a good place to begin our research with sigmoid colon
segmentation in diverticulitis since the two diseases have many
characteristics. Once CT scans were taken, the sigmoid colon was marked by
hand on a 3D viewer. A colorectal expert from Liverpool Hospital checked the
annotations for accuracy, and an example of this data is shown in Figure 3.

Figure 3: (a) Original abdominal CT slice. (b) Sigmoid colon annotation. (c)
Ground-truth (annotated) mask.

###  III-B 3D U-Net with Pyramid Pooling and Channel-Spatial Squeeze and
Excitation (PyP3DUcsSENet)

In the field of image segmentation, 3D U-Net, a CNN architecture, has seen
extensive application [3, 4]. It is an expansion of the 2D U-Net framework,
optimised for dealing with 3D volumes. There are two nodes in the network: an
encoder and a decoder. Multiple convolutional layers are used in the encoder,
followed by a max pooling operation to decrease the input’s spatial dimensions
while simultaneously increasing the number of feature channels. To catch the
complex feature details, this process is performed many times. In contrast,
the decoder is built using up-convolutional layers that are concatenated with
the relevant feature map from the encoder. The number of feature channels is
decreased while the spatial dimensions of the input are increased by the up-
convolutional layers. This allows the network to maintain the high-level
characteristics learnt by the encoder while reconstructing the input image at
its original resolution. As an efficient deep learning architecture for 3D
medical image segmentation, the 3D U-Net model has shown state-of-the-art
performance on several datasets.

Figure 4: Architecture of the 3D U-Net with Pyramid Pooling and Channel-
Spatial Squeeze and Excitation (PyP3DUcsSENet). The number of channels is
denoted in the box.

In this study, started with the fundamental 3D U-Net and investigated various
modifications to its hyperparameters, including the number of filters in each
layer, kernel size, dropout rate, and batch normalisation. In addition, we
implemented two novel techniques known as Pyramid pooling (PyP) and channel-
spatial Squeeze and Excitation (csSE) to boost performance. PyP is a technique
that captures features with varying resolutions by aggregating at multiple
dimensions. This procedure allows the network to extract finer details from
the input images. csSE, on the other hand, is a technique that concentrates on
improving the model’s spatial and channel-wise attention. It enables the model
to selectively emphasise the most pertinent features while suppressing the
less significant ones, resulting in improved performance. This study achieved
enhanced segmentation results for sigmoid colon data by implementing various
improvements and strategies in the 3D U-Net model. The integration of csSE and
PyP techniques significantly contributed to the improved performance. Figure 4
shows the modified 3D U-Net architecture incorporating these enhancements.

###  III-C Ensemble methods

This investigation employed ensemble techniques, wherein the three highest-
performing 3D U-Net iterations were utilised to improve the precision of 3D CT
image segmentation of the sigmoid colon. The objective was to integrate the
prognostications of multiple models with the aim of enhancing the outcome
beyond that of any individual model. This study conducted tests using the
Average [39], Weighted Average [40], Majority Voting [41], and Max Ensemble
[41] techniques, for a total of four ensemble techniques. The final
segmentation was obtained by averaging the SoftMax values produced by each
model on a given image, which is what the Average Ensemble does. To create the
Weighted Average Ensemble, we first gave a value to each model’s SoftMax
output and then averaged the results. We utilised model weights of $0.33\%$,
$0.33\%$, and $0.34\%$ for the top three models, respectively, based on their
performance. We utilised the predicted binary masks from each model in the
Majority Voting Ensemble to calculate the final segmentation by calculating
the voxel with the greatest number of votes. To complete the segmentation, we
used the Max Ensemble, which averaged the SoftMax output values for each voxel
across all models.

##  IV Experimental Setup

In this work, we evaluated various 3D U-Net architectures and we also
introduce a unique architecture called PyP3DUcsSENet . In the latter, the
analysis path includes consecutive $3\times 3\times 3$ convolutions followed
by a ReLU activation function and a $2\times 2\times 2$ max pooling layer with
strides in each dimension for each layer. The range of filters that are used
is from $32$ to $512$. In the same way, in the synthesis path, each layer
consists of a $2\times 2\times 2$ up-convolution, which is then followed by
$3\times 3\times 3$ convolutions and a ReLU activation function. For the
purpose of transferring high-resolution characteristics from the analysis path
to the synthesis path, we utilised concatenation connections. In addition, in
order to avoid the problem of overfitting, a dropout layer that had a
probability of $0.4$ was included at the end of each ReLU that was included in
each layer. In the last layer, we used a 1×1×1 convolution along with the
sigmoid activation function to decrease the number of output channels to one.
This number corresponds to the total number of labels. We utilised the Adam
optimizer with a learning rate of $1e-4$ for all of the 3D U-Net variants and
$1e-6$ for PyP3DUcsSENet. The momentum was set at $0.90$, and we ran the
algorithm for $100$ iterations. During training, we used the Dice loss as our
loss function, and DSC matrix was utilised to assess the final predicted
segmentation outcomes.

The implementation of each of the 3D U-Net architectures was done in Python
using the Keras framework, which is a high-level API for TensorFlow 2.1.0. The
most powerful supercomputer in Australia, NCI Gadi, was used to carry out this
studies. This machine allowed to access High Performance 24 CPUs and 2 GPUs
(each Tesla V100 having 32 GB VRAM), together with 240 GB of Memory.

##  V Experimental Results

A total of $80$ sigmoid colon annotated data samples were utilised, which were
obtained from colon cancer datasets. Five-fold cross-validation was utilised
for the segmentation procedure, using $64$ training samples and $16$ test
samples in each fold. The experimental outcomes for several versions of the 3D
U-Net are presented in Table I.

TABLE I: Performance of six different architectures for sigmoid colon segmentation on 3D CT images along with the average DSC obtained from 5-fold cross validation. Model Name | Average DSC (%) $\pm$ SD (%)
---|---
3D U-Net | $45.60\pm 3.13$
3D U-Net++ | $48.57\pm 4.11$
3D wU-Net | $48.82\pm 4.43$
3D ResU-Net | $55.33\pm 1.79$
3D DenseU-Net | $55.61\pm 1.70$
PyP3DUcsSE-Net | $\mathbf{56.92\pm 1.42}$

The outcomes of six different models that were trained for sigmoid colon
segmentation in 3D CT images are shown in Table I, along with the average DSC
values obtained from 5-fold cross-validation. With DSC scores of $55.33\pm
1.79(\%)$ and $55.61\pm 1.70(\%)$, respectively, the 3D Res-U-Net and 3D Dense
U-Net models outperformed other fundamental 3D U-Net versions in terms of
performance. A DSC of $56.92\pm 1.42(\%)$ was attained using the proposed
PyP3DUcsSENet model, which significantly enhanced performance. Interestingly,
all architectures did not perform any better after adding the PyP and csSE
modules, except for the fundamental 3D U-Net model, which showed a modest
increase in DSC.

The results of the ensemble methods are recorded in Table II. Based on the DSC
scores, we can see that the Average Ensemble and Majority Voting Ensemble
techniques performed the best, with an average DSC score of $88.11\pm
3.52(\%)$.

TABLE II: Performance of three models in ensemble for boosting sigmoid colon segmentation DSC on 3D CT images obtained from 5-fold cross validation. Ensemble Technique | Average DSC (%) $\pm$ SD (%)
---|---
Average Ensemble | $\mathbf{88.11\pm 3.52}$
Weighted Average Ensemble | $88.00\pm 3.61$
Majority Voting Ensemble | $\mathbf{88.11\pm 3.52}$
Max Ensemble | $78.58\pm 4.22$

The Weighted Average Ensemble technique demonstrated favourable performance,
exhibiting an average DSC score of $88.00\pm 3.61(\%)$. In contrast, the Max
Ensemble technique exhibited notably inferior performance, as evidenced by an
average DSC score of $78.58\pm 4.22(\%)$. This outcome suggests that the Max
Ensemble technique may not be an appropriate approach for this particular
task. This implies that the utilisation of the top three performing models
results in an enhanced accuracy of sigmoid colon segmentation across all three
techniques. In summary, the findings indicate that the utilisation of ensemble
technique can serve as a valuable mechanism for enhancing the efficacy of
segmentation models in the context of medical imaging applications.

##  VI Discussion

The results from PyP3DUcsSENet show varying levels of accuracy in sigmoid
colon segmentation, as demonstrated in Figure 5 for some of the cases.

Figure 5: Individual cases visual comparison for the sigmoid colon
segmentation.

In Figure 5, case a), the predicted segmentation achieved a DSC score of
$78.69\%$. This indicates a high level of accuracy because there was
significant overlap observed between the predicted segmentation and the ground
truth. In contrast, the predicted segmentation for case b) obtained a DSC
score of $65.99\%$, indicating a moderate level of inconsistency with the
ground truth. The DSC scores obtained for the predicted segmentations in cases
c) and d) were notably lower, measuring $44.54\%$ and $37.68\%$, respectively.
The scores show a notable decline in accuracy when compared to cases a) and
b). This is because the predicted segmentations in cases c) and d) demonstrate
significant differences in shape and size, resulting in noticeable
discrepancies when compared to the actual ground truth. Figure 6 shows the
average ensemble performance of both the best case and worst case.

Figure 6: Ensemble performance of best and worst cases for sigmoid colon
segmentation.

In Figure 6, demonstrates a noteworthy enhancement in the performance of
sigmoid colon segmentation due to the implementation of the ensemble model. In
the best-case scenario (a), the ensemble model achieves an impressive DSC
score of $94.72\%$, which indicates highly accurate segmentation. The ensemble
model achieves a decent DSC score of $54.11\%$, even in the worst-case
scenario. This observation implies that the network faces difficulties in
learning the sigmoid colon efficiently because of the differences in its size
and shape. The sigmoid colon is a relatively small and highly variable
structure within the human body. The shape of the object exhibits a high
degree of complexity and may exhibit considerable inter-patient variability.
Insufficient manual annotation can lead to imprecisions and inconsistencies in
the ground truth data, thereby exacerbating the difficulty of achieving
accurate segmentation. The precision of segmentation of the sigmoid colon may
exhibit significant variability, despite the utilisation of sophisticated
deep-learning models. The results of this investigation demonstrate the
existence of this diversity.

To improve the applicability of our segmentation model, forthcoming research
will focus on expanding the dataset to include a broader spectrum of patient
cohorts. Additionally, future studies will be directed towards improving the
precision of manual annotation through the involvement of numerous specialists
in the process of annotation. The implementation of these enhancements has the
potential to yield segmentation outcomes that are more precise and reliable,
rendering them particularly advantageous in medical settings.

##  VII Conclusion

The study presents the successful achievement of sigmoid colon segmentation in
CT images through the utilisation of a modified 3D U-Net architecture. The
modifications incorporated in the architecture encompassed the integration of
pyramid pooling, channel-spatial squeezing and excitation algorithms, and
pyramid pooling. The study demonstrates the potential benefits of employing
ensemble methods to enhance precision, wherein the average ensemble and
majority voting ensemble techniques yielded the most favourable results. The
proposed method demonstrates the capacity to facilitate precise segmentation
of the sigmoid colon.

## Acknowledgement

This research was facilitated by National Computational Infrastructure (NCI),
which is supported by the Australian Government. We are also grateful for the
financial support provided by The Australian Government Research Training
Program (RTP) Scholarship at the University of New South Wales (UNSW) Sydney.

## Declaration of competing interest

The authors declare no conflicts of interest.

## References

  * [1] S. Pecere, G. Gibiino, D. L. Milia, G. Bertè, G. B. Scotti, C. Petruzziello, L. Minordi, R. Manfredi, F. Franceschi, V. Ojetti, and A. Gasbarrini, “Acute uncomplicated diverticulitis: key points for early management. a single-centre retrospective study,” _European review for Medical and Pharmocological Sciences_ , vol. 24, pp. 2710–2718, 2020.
  * [2] M. Clinic, “Overview of diverticulitis,” https://www.mayoclinic.org/diseases-conditions/diverticulitis/symptoms-causes/syc-20371758, accessed: 2023-08-28.
  * [3] L. Guachi, R. Guachi, F. Bini, and F. Marinozzi, “Automatic colorectal segmentation with convolutional neural network,” _Computer-Aided Design and Applications_ , vol. 16, pp. 836–845, 2019.
  * [4] Y. Gonzalez, C. Shen, H. Jung, D. Nguyen, S. B. Jiang, K. Albuquerque, and X. Jia, “Semi-automatic sigmoid colon segmentation in ct for radiation therapy treatment planning via an iterative 2.5-d deep learning approach,” _Medical Image Analysis_ , vol. 68, p. 101896, 2021.
  * [5] P. Hu, F. Wu, J. Peng, Y. Bao, F. Chen, and D. Kong, “Automatic abdominal multi-organ segmentation using deep convolutional neural network and time-implicit level sets,” _International Journal of Computer Assisted Radiology and Surgery_ , vol. 12, no. 3, pp. 399–411, Mar 2017.
  * [6] Q. Dou, L. Yu, H. Chen, Y. Jin, X. Yang, J. Qin, and P.-A. Heng, “3d deeply supervised network for automated segmentation of volumetric medical images,” _Medical Image Analysis_ , vol. 41, pp. 40–54, 2017, special Issue on the 2016 Conference on Medical Image Computing and Computer Assisted Intervention (Analog to MICCAI 2015).
  * [7] X. Zhou, _Automatic Segmentation of Multiple Organs on 3D CT Images by Using Deep Learning Approaches_. Cham: Springer International Publishing, 2020, pp. 135–147.
  * [8] R. O. Dogan, H. Dogan, C. Bayrak, and T. Kayikcioglu, “A two-phase approach using mask r-cnn and 3d u-net for high-accuracy automatic segmentation of pancreas in ct imaging,” _Computer Methods and Programs in Biomedicine_ , vol. 207, p. 106141, 2021.
  * [9] Z. Wang, K. K. Bhatia, B. Glocker, A. Marvao, T. Dawes, K. Misawa, K. Mori, and D. Rueckert, “Geodesic patch-based segmentation,” in _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014_ , P. Golland, N. Hata, C. Barillot, J. Hornegger, and R. Howe, Eds. Cham: Springer International Publishing, 2014, pp. 666–673.
  * [10] R. Wolz, C. Chu, K. Misawa, M. Fujiwara, K. Mori, and D. Rueckert, “Automated abdominal multi-organ segmentation with subject-specific atlas generation,” _IEEE Transactions on Medical Imaging_ , vol. 32, no. 9, pp. 1723–1730, 2013\.
  * [11] C. Chu, M. Oda, T. Kitasaka, K. Misawa, M. Fujiwara, Y. Hayashi, Y. Nimura, D. Rueckert, and K. Mori, “Multi-organ segmentation based on spatially-divided probabilistic atlas from 3d abdominal ct images,” in _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2013_ , K. Mori, I. Sakuma, Y. Sato, C. Barillot, and N. Navab, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 165–172.
  * [12] H. Park, P. Bland, and C. Meyer, “Construction of an abdominal probabilistic atlas and its application in segmentation,” _IEEE Transactions on Medical Imaging_ , vol. 22, no. 4, pp. 483–492, 2003.
  * [13] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in _2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2015, pp. 3431–3440.
  * [14] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015_ , N. Navab, J. Hornegger, W. M. Wells, and A. F. Frangi, Eds. Cham: Springer International Publishing, 2015, pp. 234–241.
  * [15] T. Estienne, M. Vakalopoulou, S. Christodoulidis, E. Battistela, M. Lerousseau, A. Carre, G. Klausner, R. Sun, C. Robert, S. Mougiakakou, N. Paragios, and E. Deutsch, “U-resnet: Ultimate coupling of registration and segmentation with deep nets,” in _Medical Image Computing and Computer Assisted Intervention – MICCAI 2019_ , D. Shen, T. Liu, T. M. Peters, L. H. Staib, C. Essert, S. Zhou, P.-T. Yap, and A. Khan, Eds. Cham: Springer International Publishing, 2019, pp. 310–319.
  * [16] T. Heimann, B. van Ginneken, M. A. Styner, Y. Arzhaeva, V. Aurich, C. Bauer, A. Beck, C. Becker, R. Beichel, G. Bekes, F. Bello, G. Binnig, H. Bischof, A. Bornik, P. M. M. Cashman, Y. Chi, A. Cordova, B. M. Dawant, M. Fidrich, J. D. Furst, D. Furukawa, L. Grenacher, J. Hornegger, D. KainmÜller, R. I. Kitney, H. Kobatake, H. Lamecker, T. Lange, J. Lee, B. Lennon, R. Li, S. Li, H.-P. Meinzer, G. Nemeth, D. S. Raicu, A.-M. Rau, E. M. van Rikxoort, M. Rousson, L. Rusko, K. A. Saddi, G. Schmidt, D. Seghers, A. Shimizu, P. Slagmolen, E. Sorantin, G. Soza, R. Susomboon, J. M. Waite, A. Wimmer, and I. Wolf, “Comparison and evaluation of methods for liver segmentation from ct datasets,” _IEEE Transactions on Medical Imaging_ , vol. 28, no. 8, pp. 1251–1265, 2009.
  * [17] X. Li, H. Chen, X. Qi, Q. Dou, C.-W. Fu, and P.-A. Heng, “H-denseunet: Hybrid densely connected unet for liver and tumor segmentation from ct volumes,” _IEEE Transactions on Medical Imaging_ , vol. 37, no. 12, pp. 2663–2674, 2018\.
  * [18] M. N. Ul Haq, A. Irtaza, N. Nida, M. A. Shah, and L. Zubair, “Liver tumor segmentation using resnet based mask-r-cnn,” in _2021 International Bhurban Conference on Applied Sciences and Technologies (IBCAST)_ , 2021, pp. 276–281.
  * [19] S.-T. Tran, C.-H. Cheng, and D.-G. Liu, “A multiple layer u-net, un-net, for liver and liver tumor segmentation in ct,” _IEEE Access_ , vol. 9, pp. 3752–3764, 2021.
  * [20] S. Singh, S. Hoque, A. Zekry, and A. Sowmya, “Radiological diagnosis of chronic liver disease and hepatocellular carcinoma: A review,” _Journal of Medical Systems_ , vol. 47, no. 1, p. 73, Jul 2023.
  * [21] B. Ait Skourt, A. El Hassani, and A. Majda, “Lung ct image segmentation using deep neural networks,” _Procedia Computer Science_ , vol. 127, pp. 109–113, 2018, pROCEEDINGS OF THE FIRST INTERNATIONAL CONFERENCE ON INTELLIGENT COMPUTING IN DATA SCIENCES, ICDS2017.
  * [22] K. Asipong, S. Gabbualoy, and P. Phasukkit, “Coronavirus infected lung ct scan image segmentation using deep learning,” in _2021 18th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON)_ , 2021, pp. 773–776.
  * [23] M. Huang, C. Huang, J. Yuan, and D. Kong, “A semiautomated deep learning approach for pancreas segmentation,” _Journal of Healthcare Engineering_ , vol. 2021, p. 3284493, Jul 2021.
  * [24] T. Hassanzadeh, L. G. C. Hamey, and K. Ho-Shon, “Convolutional neural networks for prostate magnetic resonance image segmentation,” _IEEE Access_ , vol. 7, pp. 36 748–36 760, 2019.
  * [25] K. Shanmugalingam, A. Sowmya, D. Moses, and E. Meijering, “Attention guided deep supervision model for prostate segmentation in multisite heterogeneous mri data,” in _Proceedings of The 5th International Conference on Medical Imaging with Deep Learning_ , ser. Proceedings of Machine Learning Research, E. Konukoglu, B. Menze, A. Venkataraman, C. Baumgartner, Q. Dou, and S. Albarqouni, Eds., vol. 172. PMLR, 06–08 Jul 2022, pp. 1085–1095.
  * [26] Y. Wang, Y. Zhou, W. Shen, S. Park, E. K. Fishman, and A. L. Yuille, “Abdominal multi-organ segmentation with organ-attention networks and statistical fusion,” _Medical Image Analysis_ , vol. 55, pp. 88–102, 2019\.
  * [27] S. Sultana, A. Robinson, D. Song, and J. Lee, “Automatic multi-organ segmentation in computed tomography images using hierarchical convolutional neural network,” _Journal of Medical Imaging_ , vol. 7, pp. 88–102, 2020\.
  * [28] J. Wu, G. Li, H. Lu, and T. Kamiya, “A supervoxel classification based method for multi-organ segmentation from abdominal ct images,” _Journal of Image and Graphics_ , vol. 9, pp. 88–102, 2021.
  * [29] Y. Liu, Y. Lei, Y. Fu, T. Wang, X. Tang, X. Jiang, W. J. Curran, T. Liu, P. Patel, and X. Yang, “Ct-based multi-organ segmentation using a 3d self-attention u-net network for pancreatic radiotherapy,” _Medical Physics_ , vol. 47, no. 9, pp. 4316–4324, 2020.
  * [30] Y. Chen, D. Ruan, J. Xiao, L. Wang, B. Sun, R. Saouaf, W. Yang, D. Li, and Z. Fan, “Fully automated multiorgan segmentation in abdominal magnetic resonance imaging with deep neural networks,” _Medical Physics_ , vol. 47, no. 10, pp. 4971–4982, 2020.
  * [31] C. Jia and J. Wei, “Amo-net: abdominal multi-organ segmentation in mri with a extend unet,” in _2021 IEEE 4th Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC)_ , vol. 4, 2021, pp. 1770–1775.
  * [32] H. Huang, L. Lin, R. Tong, H. Hu, Q. Zhang, Y. Iwamoto, X. Han, Y.-W. Chen, and J. Wu, “Unet 3+: A full-scale connected unet for medical image segmentation,” in _ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2020, pp. 1055–1059.
  * [33] Z. Zhou, M. M. Rahman Siddiquee, N. Tajbakhsh, and J. Liang, “Unet++: A nested u-net architecture for medical image segmentation,” in _Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support_ , D. Stoyanov, Z. Taylor, G. Carneiro, T. Syeda-Mahmood, A. Martel, L. Maier-Hein, J. M. R. Tavares, A. Bradley, J. P. Papa, V. Belagiannis, J. C. Nascimento, Z. Lu, S. Conjeti, M. Moradi, H. Greenspan, and A. Madabhushi, Eds. Cham: Springer International Publishing, 2018, pp. 3–11.
  * [34] Z. Chu, S. Singh, and A. Sowmya, “Tsdnet: A tumour segmentation network with 3d direction-wise convolution,” in _2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI)_ , 2023, pp. 1–5.
  * [35] D. K. ayathri, R. R, and R. Kumar, “Segmentation of colon and removal of opacified fluid for virtual colonoscopy,” _Pattern Analysis and Applications_ , vol. 21, no. 9, pp. 205–219, 2018.
  * [36] N. C. Institute, “The cancer imaging archive,” https://www.cancerimagingarchive.net/, accessed: 2023-08-20.
  * [37] A. G. Roy, N. Navab, and C. Wachinger, “Concurrent spatial and channel ‘squeeze & excitation’ in fully convolutional networks,” in _Medical Image Computing and Computer Assisted Intervention – MICCAI 2018_ , A. F. Frangi, J. A. Schnabel, C. Davatzikos, C. Alberola-López, and G. Fichtinger, Eds. Cham: Springer International Publishing, 2018, pp. 421–429.
  * [38] M. S. Decathlon, “Generalisable 3d semantic segmentation,” http://medicaldecathlon.com/, accessed: 2023-08-20.
  * [39] S. Liu, X. Yuan, R. Hu, S. Liang, S. Feng, Y. Ai, and Y. Zhang, “Automatic pancreas segmentation via coarse location and ensemble learning,” _IEEE Access_ , vol. 8, pp. 2906–2914, 2020.
  * [40] K. Shaga Devan, H. A. Kestler, C. Read, and P. Walther, “Weighted average ensemble-based semantic segmentation in biological electron microscopy images,” _Histochemistry and Cell Biology_ , vol. 158, no. 5, pp. 447–462, Nov 2022.
  * [41] A. E. Kavur, L. I. Kuncheva, and M. A. Selver, “Basic ensembles of vanilla-style deep learning models improve liver segmentation from ct images,” 2020.