# Inversion of limited-aperture Fresnel experimental data using orthogonality
sampling method with single and multiple sources

Won-Kwang Park<EMAIL_ADDRESS>Department of Information Security,
Cryptography, and Mathematics, Kookmin University, Seoul, 02707, Korea

###### Abstract

In this study, we consider the application of orthogonality sampling method
(OSM) with single and multiple sources for a fast identification of small
objects in limited-aperture inverse scattering problem. We first apply the OSM
with single source and show that the indicator function with single source can
be expressed by the Bessel function of order zero of the first kind, infinite
series of Bessel function of nonzero integer order of the first kind, range of
signal receiver, and the location of emitter. Based on this result, we explain
that the objects can be identified through the OSM with single source but the
identification is significantly influenced by the location of source and
applied frequency. For a successful improvement, we then consider the OSM with
multiple sources. Based on the identified structure of the OSM with single
source, we design an indicator function of the OSM with multiple sources and
show that it can be expressed by the square of the Bessel function of order
zero of the first kind an infinite series of the square of Bessel function of
nonzero integer order of the first kind. Based on the theoretical results, we
explain that the objects can be identified uniquely through the designed OSM.
Several numerical experiments with experimental data provided by the Institute
Fresnel demonstrate the pros and cons of the OSM with single source and how
the designed OSM with multiple sources behave.

###### keywords:

Orthogonality sampling method , limited-aperture inverse scattering problem ,
Bessel functions of the first kind , experimental data

##  1 Introduction

Development of an effective and stable technique for retrieving unknown object
from measured scattered field or scattering parameter data is an old but still
interesting research subject to nowadays scientists and engineers because this
subject is highly related to modern human life such as biomedical imaging [1,
2] including breast cancer detection [3, 4], through-wall imaging for defect
recognition [5, 6], damage detection of concrete structure [7, 8], land mine
detection [9, 10], synthetic-aperture radar imaging [11, 12], ground
penetrating radar [13, 14]. We further refer to related studies [15, 16, 17,
18, 19, 20, 21, 22] for various applications. Let us notice that most of
algorithms are based on Newton-type iteration schemes so that one must
generate good initial guess which is close enough to the unknown objects.

Instead of iterative based algorithm, alternative non-iterative techniques
have been investigated to retrieve unknown object. Throughout several
researches about the bifocusing method [23, 24, 25], direct sampling method
[26, 27, 28], factorization method [29, 30, 31], MUltiple SIgnal
Classificiation [32, 33, 34], migration techniques [35, 36, 37], and
topological derivative [38, 39, 40], it has been turned out that although
complete information of objects such as material properties cannot be
retrieved, they are very effective techniques for identifying the existence,
location, and outline shape of objects.

Orthogonality sampling method (OSM) is classified as a non-iterative imaging
technique in both inverse scattering problem and microwave imaging. From the
beginning study of the OSM by Potthast [41], it has been applied various
inverse scattering problems. Owing to the several studies [42, 43, 44, 45, 46,
47], it has been confirmed that the OSM is very fast, stable, and effective
imaging technique in inverse scattering problem. Unfortunately, most of
studies performed the numerical simulation to show the effectiveness of the
OSM with synthetic data. In some researches [43, 47], the OSM was applied in
real-world inverse scattering problem with experimental datasets produced by
the Institute Fresnel, France [48]. Although the OSM has demonstrated its
applicability and robustness for retrieving a set of small objects from
experimental data, an appropriate mathematical theory to explain some
phenomena (for example, ① why the imaging performance is significantly on the
location of source, ② why the application of low and high frequencies is not
appropriate for retrieving multiple objects), and to design alternative
technique for improving the imaging performance has not been established yet.

In this paper, we consider the application of the OSM for identifying a set of
objects from experimental Fresnel data. First, we introduce the traditional
indicator function for the OSM and reveal its mathematical structure by
establishing a relationship with the Bessel function of order zero of the
first kind, infinite series of Bessel function of nonzero integer order of the
first kind, range of signal receiver, and the location of emitter. Based on
the structure, we explain some intrinsic properties of the OSM and provide
theoretical answers to the unexplained phenomena mentioned above. We then
exhibit simulation results with experimental data to demonstrate the
theoretical result and fundamental limitation of object detection.

Next, we consider the OSM with multiple sources to improve the imaging
performance for a proper detection of objects. To this end, we adopt the
traditional indicator function with multiple sources introduced in [41] and
propose another indicator function. In order to show the applicability,
effectiveness, improvement of the proposed indicator function, and unique
determination, we show that it can be expressed by the square of the Bessel
function of order zero of the first kind an infinite series of the square of
Bessel function of nonzero integer order of the first kind. We then exhibit
simulation results to support established structure, discovered certain
properties of the designed indicator function, and compare the
imaging/detection performances.

The rest of this paper is organized as follows. In Section 2, we briefly
survey the direct scattering problem in the presence of a set of small objects
and introduce the traditional indicator function of the OSM. In Section 3,
mathematical structure of the indicator function with single source is
explored by establishing a relationship with an infinite series of the Bessel
functions, range of receivers, and the location of emitter. In Section 4, a
set of simulation results with experimental Fresnel dataset is exhibited to
confirm the theoretical result and examine the influence of the location of
emitter and frequencies at operation. In Section 5, we introduce the
traditional and design a new indicator functions with multiple sources,
establish mathematical structure, discover some intrinsic properties of the
designed indicator function including unique determination., and exhibit
simulation results. Conclusions and perspectives are included in Section 6.

##  2 Direct scattering problem and orthogonality sampling method

Let $\Omega$ be a two-dimensional homogeneous region, $D_{s}\subset\Omega$,
$s=1,2,\ldots,S$, be a two-dimensional small object, and $D$ be the collection
of $D_{s}$. Throughout this paper, we assume that all $D_{s}$ are well-
separated from each other and $\Omega$ is a subset of interior of an anechoic
chamber so that the values of background conductivity, permeability, and
permittivity are set to $\sigma_{\mathrm{b}}\approx 0$,
$\mu_{\mathrm{b}}=4\pi\times${10}^{-7}\text{\,}\mathrm{H}\mathrm{/}\mathrm{m}$$,
and
$\varepsilon_{0}=$8.854\text{\times}{10}^{-12}\text{\,}\mathrm{F}\mathrm{/}\mathrm{m}$$,
respectively, refer to [21]. Correspondingly, every $D_{s}$ and $\Omega$ are
characterized by the value of dielectric permittivity at given angular
frequency $\omega=2\pi f$. Let $\varepsilon_{s}$ and
$\varepsilon_{\mathrm{b}}$ as the value of permittivity of $D_{s}$ and
$\Omega$, respectively, and
$k_{\mathrm{b}}=\omega\sqrt{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}$ be the
background wavenumber. With this, we introduce the following piecewise
constant

$\varepsilon(\mathbf{r})=\left\\{\begin{array}[]{ccl}\varepsilon_{s}&\text{for}&\mathbf{r}\in
D_{s}\\\
\varepsilon_{\mathrm{b}}&\text{for}&\mathbf{r}\in\Omega\backslash\overline{D}.\end{array}\right.$

Let us denote $\mathbf{a}_{m}$ and $\mathbf{b}_{n}$ as the location of $m$th
emitter $\mathcal{A}_{m}$ and $n$th receiver $\mathcal{B}_{n}$, respectively.
Following to [48], $\mathbf{a}_{m}$ and $\mathbf{b}_{n}$ can be written as

$\mathbf{a}_{m}=|\mathbf{a}_{m}|(\cos\vartheta_{m},\sin\vartheta_{m})=|\mathbf{a}_{m}|\boldsymbol{\vartheta}_{m}\quad\text{with}\quad|\mathbf{a}_{m}|\equiv|\mathbf{a}|=$0.72\text{\,}\mathrm{m}$,\quad\vartheta_{m}=\frac{2(n-1)\pi}{N}$

and

$\mathbf{b}_{n}=|\mathbf{b}_{n}|(\cos\theta_{n},\sin\theta_{n})=|\mathbf{b}_{n}|\boldsymbol{\theta}_{n}\quad\text{with}\quad|\mathbf{b}_{n}|\equiv|\mathbf{b}|=$0.76\text{\,}\mathrm{m}$,\quad\theta_{n}=\vartheta_{m}+\frac{\pi}{3}+\frac{4(n-1)\pi}{3(N-1)},$

respectively. Here, $\boldsymbol{\vartheta}_{m}\in\mathbb{S}^{1}$ and
$\boldsymbol{\theta}_{n}\in\mathbb{S}_{m}^{1}$, where $\mathbb{S}^{1}$ denotes
the unit circle centered at the origin, and

$\mathbb{S}_{m}^{1}=\left\\{(\cos\theta,\sin\theta):\vartheta_{m}+\frac{\pi}{3}\leq\theta\leq\vartheta_{m}+\frac{5\pi}{3}\right\\}\subset\mathbb{S}^{1}.$

For an illustration, we refer to Figure 1. Then, the incident field at the
fixed point source $\mathbf{a}_{m}$ can be written as follows: for
$\mathbf{r}\in\Omega$,

$u_{\operatorname*{inc}}(\mathbf{r},\mathbf{a}_{m})=-\frac{i}{4}H_{0}^{(1)}(k_{\mathrm{b}}|\mathbf{r}-\mathbf{a}_{m}|):=G(\mathbf{r},\mathbf{a}_{m}),$

where $H_{0}^{(1)}$ denotes the Hankel function of order zero of the first
kind. Correspondingly, the time-harmonic total field
$u(\mathbf{b}_{n},\mathbf{r})$ measured at $n$th receiver $\mathbf{b}_{n}$
satisfies

$\triangle
u(\mathbf{b}_{n},\mathbf{r})+\omega^{2}\mu_{\mathrm{b}}\varepsilon(\mathbf{r})u(\mathbf{b}_{n},\mathbf{r})=0\quad\text{for}\quad\mathbf{r}\in\Omega$

with transmission condition
$u(\mathbf{b}_{n},\mathbf{r})\big{|}_{-}-u(\mathbf{b}_{n},\mathbf{r})\big{|}_{+}=0$
on $\partial D_{s}$, $s=1,2,\ldots,S$. Here, the time harmonic $e^{-i\omega
t}$ is assumed. Let $u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})$ as
the scattered-field corresponding to the incident field. Then based on [19],
$u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})$ can be expressed by the
single-layer potential with unknown density function $\varphi$:

$u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})=u(\mathbf{b}_{n},\mathbf{a}_{m})-u_{\operatorname*{inc}}(\mathbf{r},\mathbf{a}_{m})=\int_{D}G(\mathbf{b}_{n},\mathbf{r})\varphi(\mathbf{r},\mathbf{a}_{m})\mathrm{d}\mathbf{r}.$

Note that the closed form of the density function
$\varphi(\mathbf{r},\mathbf{a}_{m})$ is unknown, it is not appropriate to use
the $u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})$ directly to design
the indicator function of the OSM. Due to this reason, we use the following
asymptotic expansion formula, which is the key formula to design and analyze
the structure of the indicator function.

######  Lemma 2.1 (Asymptotic expansion formula [49, 50]).

For sufficiently large $\omega$,
$u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})$ can be represented as

$\displaystyle\begin{aligned}
u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})&=k_{\mathrm{b}}^{2}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{b}_{n},\mathbf{r})u_{\operatorname*{inc}}(\mathbf{r},\mathbf{a}_{m})\mathrm{d}\mathbf{r}\\\
&=k_{\mathrm{b}}^{2}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{b}_{n},\mathbf{r})G(\mathbf{r},\mathbf{a}_{m})\mathrm{d}\mathbf{r}.\end{aligned}$
(1)

Emitterreceiver

$\displaystyle\frac{\pi}{3}$$\displaystyle\frac{4\pi}{3}$$~{}~{}\vartheta_{m}$Emitterreceiver

Figure 1: Illustration of measurement configuration corresponding to the
location of emitter.

##  3 Indicator function with single source

In this section, we consider the design an indicator function with single
source. Let us denote $\mathbf{E}(\mathbf{a}_{m})$ as the following
arrangement of measurement data

$\mathbf{E}(\mathbf{a}_{m})=\Big{(}u_{\operatorname*{scat}}(\mathbf{b}_{1},\mathbf{r}),u_{\operatorname*{scat}}(\mathbf{b}_{2},\mathbf{r}),\ldots,u_{\operatorname*{scat}}(\mathbf{b}_{N},\mathbf{r})\Big{)}.$
(2)

Now, applying the mean-value theorem to (1) yields

$u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r}_{s})=\sum_{s=1}^{S}k_{\mathrm{b}}^{2}\operatorname*{area}(D_{s})\left(\frac{\varepsilon_{s}-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{b}_{n},\mathbf{r}_{s})G(\mathbf{r}_{s},\mathbf{a}_{m}),$

thus, we can design the indicator function of the OSM based on the
orthogonality relation between the
$u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})$ and
$G(\mathbf{b}_{n},\cdot)$. To this end, let us introduce a test vector: for
each $\mathbf{r}^{\prime}\in\Omega$,

$\mathbf{G}(\mathbf{r}^{\prime})=\Big{(}G(\mathbf{b}_{1},\mathbf{r}^{\prime}),G(\mathbf{b}_{2},\mathbf{r}^{\prime}),\ldots,G(\mathbf{b}_{N},\mathbf{r}^{\prime})\Big{)}$

and corresponding indicator function

$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})=|\mathbf{E}(\mathbf{a}_{m})\cdot\overline{\mathbf{G}(\mathbf{r}^{\prime})}|=\left|\sum_{n=1}^{N}u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})\overline{G(\mathbf{b}_{n},\mathbf{r}^{\prime})}\right|.$

Then, map of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ will
contain peaks of large magnitude at $\mathbf{r}^{\prime}\in D_{s}$ thereby, it
will be possible to recognize the existence or outline shape of $D_{s}$,
$s=1,2,\ldots,S$. In order to discover the feasibility and some properties of
the $\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$,
we derive the following result.

######  Theorem 3.1.

Let $\boldsymbol{\vartheta}_{m}=(\cos\vartheta_{m},\sin\vartheta_{m})$,
$\boldsymbol{\theta}_{n}=(\cos\theta_{n},\sin\theta_{n})$,
$\boldsymbol{\theta}=(\cos\theta,\sin\theta)$, and
$\mathbf{r}^{\prime}-\mathbf{r}=|\mathbf{r}^{\prime}-\mathbf{r}|(\cos\phi,\sin\phi)$.
Then, for sufficiently large $N$ and $\omega$,
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ can
be represented as follows:

$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})=\left|\frac{k_{\mathrm{b}}}{6|\mathbf{b}|}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})\left[J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{3}{\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})\right]\mathrm{d}\mathbf{r}\right|,$
(3)

where $J_{p}$ denotes the Bessel function of order $q$ and

$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})=\sum_{p=1}^{\infty}\frac{(-i)^{p}}{p}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)\cos\big{(}p(\vartheta_{m}-\phi)\big{)}\sin\left(\frac{2p}{3}\pi\right).$
(4)

###### Proof.

Since $4k_{\mathrm{b}}|\mathbf{r}-\mathbf{b}_{n}|\gg 1$ for $n=1,2,\cdots,N$,
the following asymptotic form holds (see [19] for instance)

$H_{0}^{(1)}(k_{\mathrm{b}}|\mathbf{b}_{n}-\mathbf{r}|)=\frac{(1-i)e^{ik_{\mathrm{b}}|\mathbf{b}_{n}|}}{\sqrt{k_{\mathrm{b}}|\mathbf{b}_{n}|\pi}}e^{-ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot\mathbf{r}}.$
(5)

Thus, we can examine that

$u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})\approx-\frac{k_{\mathrm{b}}^{2}(1+i)e^{ik_{\mathrm{b}}|\mathbf{b}_{n}|}}{4\sqrt{k_{\mathrm{b}}|\mathbf{b}_{n}|\pi}}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})e^{-ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot\mathbf{r}}\mathrm{d}\mathbf{r}$

and correspondingly, we have

$\displaystyle\sum_{n=1}^{N}u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})\overline{G(\mathbf{b}_{n},\mathbf{r}^{\prime})}$
$\displaystyle\approx-\sum_{n=1}^{N}\left(\frac{k_{\mathrm{b}}^{2}(1+i)e^{ik_{\mathrm{b}}|\mathbf{b}_{n}|}}{4\sqrt{k_{\mathrm{b}}|\mathbf{b}_{n}|\pi}}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})e^{-ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot\mathbf{r}}\mathrm{d}\mathbf{r}\right)\frac{(-1+i)e^{-ik_{\mathrm{b}}|\mathbf{b}_{n}|}}{4\sqrt{k_{\mathrm{b}}|\mathbf{b}_{n}|\pi}}e^{ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot\mathbf{r}^{\prime}}$
$\displaystyle=\frac{k_{\mathrm{b}}}{8|\mathbf{b}|\pi}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})\left(\sum_{n=1}^{N}e^{ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot(\mathbf{r}^{\prime}-\mathbf{r})}\right)\mathrm{d}\mathbf{r}.$

Since $N$ is sufficiently large,
$\boldsymbol{\theta}\cdot(\mathbf{r}^{\prime}-\mathbf{r})=|\mathbf{r}^{\prime}-\mathbf{r}|\cos(\theta-\phi)$,
and the following relation holds uniformly (see [51] for instance),

$\int_{\alpha}^{\beta}e^{ix\cos(\theta-\phi)}\mathrm{d}\theta=(\beta-\alpha)J_{0}(x)+4\sum_{p=1}^{\infty}\frac{i^{p}}{p}J_{p}(x)\cos\frac{p(\beta+\alpha-2\phi)}{2}\sin\frac{p(\beta-\alpha)}{2},$
(6)

we can evaluate

$\displaystyle\sum_{n=1}^{N}e^{ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot(\mathbf{r}^{\prime}-\mathbf{r})}$
$\displaystyle\approx\int_{\mathbb{S}_{m}^{1}}e^{ik_{\mathrm{b}}\boldsymbol{\theta}\cdot(\mathbf{r}^{\prime}-\mathbf{r})}\mathrm{d}\boldsymbol{\theta}$
$\displaystyle=\int_{\theta_{1}=\vartheta_{m}+\pi/6}^{\theta_{N}=\vartheta_{m}+5\pi/6}e^{ik_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|\cos(\theta-\phi)}\mathrm{d}\theta$
$\displaystyle=\frac{4\pi}{3}J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+4\sum_{p=1}^{\infty}\frac{(-i)^{p}}{p}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)\cos\big{(}p(\vartheta_{m}-\phi)\big{)}\sin\left(\frac{2p}{3}\pi\right).$

Hence,

$\sum_{n=1}^{N}u_{\operatorname*{scat}}(\mathbf{b}_{n},\mathbf{r})\overline{G(\mathbf{b}_{n},\mathbf{r}^{\prime})}=\frac{k_{\mathrm{b}}}{6|\mathbf{b}|}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})\left[J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{3}{\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})\right]\mathrm{d}\mathbf{r}$

and correspondingly, (3) can be derived. ∎

Based on the Theorem 3.1, we can examine some properties of the indicator
function.

######  Remark 3.1 (Availability and limitation of object detection).

Since $J_{0}(0)=1$ and $J_{p}(0)=0$ for $p=1,2,\ldots$, the resulting plot of
indicator function
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ is
expected to exhibit peaks of magnitudes
$\frac{k_{\mathrm{b}}}{6|\mathbf{b}|}\left(\frac{\varepsilon_{s}-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)|G(\mathbf{r},\mathbf{a}_{m})|\operatorname*{area}(D_{s})$
at the $\mathbf{r}^{\prime}=\mathbf{r}\in D_{s}$ sought.

Notice that the since
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})\propto|G(\mathbf{r},\mathbf{a}_{m})|$,
imaging performance of the
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ will
be significantly dependent on the position of the emitter $\mathcal{A}_{m}$.
Specially, if one applies extremely high frequency then the value of
$|G(\mathbf{r},\mathbf{a}_{m})|$ becomes negligible because

$|G(\mathbf{r},\mathbf{a}_{m})|=\left|\frac{(1-i)e^{ik_{\mathrm{b}}|\mathbf{b}_{n}|}}{\sqrt{k_{\mathrm{b}}|\mathbf{b}_{n}|\pi}}e^{-ik_{\mathrm{b}}\boldsymbol{\theta}_{n}\cdot\mathbf{r}}\right|\longrightarrow
0\quad\text{as}\quad\omega\longrightarrow\infty+.$

Correspondingly, the value of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$
becomes negligible so that it will be unable to distinguish between unknown
objects and several artifacts in the map of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$.
Hence, we conclude that application of extremely high frequency does not
guarantee the detection of unknown object through the OSM with single source.

######  Remark 3.2 (Detection of multiple objects).

Suppose that there exists two objects $D_{1}$ and $D_{2}$ located at
$\mathbf{r}_{1}$ and $\mathbf{r}_{2}$, respectively. Then, the following
relation must satisfy to distinguish objects through the map of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$

$|\mathbf{r}_{1}-\mathbf{r}_{2}|>\frac{\lambda}{2},$ (7)

where $\lambda$ denotes positive wavelength.

In Section 4, the scattered field data were generated in the presence of two
circular shaped dielectric objects centered at
$\mathbf{r}_{1}=$0.045\text{\,}\mathrm{m}$$ and
$\mathbf{r}_{2}=-$0.045\text{\,}\mathrm{m}$$, i.e.,
$|\mathbf{r}_{1}-\mathbf{r}_{2}|=$0.09\text{\,}\mathrm{m}$$ (see [48] for
instance). Notice that if $f=$1\text{\,}\mathrm{GHz}$$ then
$\lambda/2=$0.1499\text{\,}\mathrm{m}$$ because $k=20.9585$ so that two
objects cannot be distinguished through the map of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$
because $\lambda$ does not satisfies the relation (7). If
$f\geq$2\text{\,}\mathrm{GHz}$$ then $D_{1}$ and $D_{2}$ can be distinguished
because $\lambda/2\leq$0.0749\text{\,}\mathrm{m}$$. Hence, we conclude that
application of low frequency does not guarantee the detection of unknown
objects through the OSM with single source.

######  Remark 3.3 (Effect of the factor
$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$).

Based on the (4), the factor
$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$ not only does not
contribute to the identification of objects but also disturb the
identification by generating several artifacts. In order to examine the
influence of the $\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$,
we consider the following quantity

$\mathcal{D}_{1}(x)=\frac{3}{\pi}\left|\sum_{p=1}^{10^{6}}\frac{(-i)^{p}}{p}J_{p}(k_{\mathrm{b}}|x|)\sin\left(\frac{2p}{3}\pi\right)\right|.$

This is similar to the value of
$3\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})/\pi$ for
$\vartheta_{m}=0$ (i.e., $\mathbf{a}_{m}=(1,0))$ and $\mathbf{r}=(0,0)$. By
comparing the oscillating properties with $J_{0}(k_{\mathrm{b}}|x|)$, we can
say that due to the oscillating properties of the
$J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)$ and
$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$, maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ will
contain several artifacts and it will disturb the recognization of the
existence of objects, refer to Figure 2.

$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$$|J_{0}(k_{\mathrm{b}}x)|$$\mathcal{D}_{1}(x)$
Figure 2:  Plots of $|J_{0}(k_{\mathrm{b}}x)|$ and $\mathcal{D}_{1}(x)$ for
$-1\leq x\leq 1$ and $f=$2\text{\,}\mathrm{GHz}$$.

##  4 Simulation results with experimental data

Here, we exhibit simulation results using experimental data [48]. The emitters
and receivers are placed on the circles centered at the origin with radii
$|\mathbf{a}_{m}|=$0.72\text{\,}\mathrm{m}$$ and
$|\mathbf{b}_{n}|=$0.76\text{\,}\mathrm{m}$$, respectively, the the imaging
region $\Omega$ was chosen as a square
$(-$0.1\text{\,}\mathrm{m}$,$0.1\text{\,}\mathrm{m}$)\times(-$0.1\text{\,}\mathrm{m}$,$0.1\text{\,}\mathrm{m}$)$
to satisfy the relation $4k_{\mathrm{b}}|\mathbf{r}-\mathbf{a}_{m}|$,
$4k_{\mathrm{b}}|\mathbf{r}-\mathbf{b}_{n}|\gg 1$ for $m=1,2,\ldots,M$ and
$n=1,2,\ldots,N$. The range of receivers is restricted from
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ to
$300\text{\,}\mathrm{\SIUnitSymbolDegree}$, with step size of
$5\text{\,}\mathrm{\SIUnitSymbolDegree}$ based on each location of emitters.
We refer to Figure 1 again for an illustration of measurement configuration.

The objects are composed of two filled dielectric cylinders $D_{1}$ and
$D_{2}$ with circular cross section of radius $0.015\text{\,}\mathrm{m}$ and
permittivity $\varepsilon_{s}=(3\pm 0.3)\varepsilon_{\mathrm{b}}$ centered at
$\mathbf{r}_{1}=(0.045,0.010)$\text{\,}\mathrm{m}$$ and
$\mathbf{r}_{2}=(-0.045,0)$\text{\,}\mathrm{m}$$, respectively111In [48],
$\mathbf{r}_{1}=(0.045,0)$\text{\,}\mathrm{m}$$ was given but throughout
several results [52, 53, 54, 28, 24, 55], accurate location of $D_{1}$ seems
$\mathbf{r}_{1}=(0.045,0.010)$\text{\,}\mathrm{m}$$.. With this setting, we
generated the imaging results
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ with
$m=1$, $10$, and $25$, refer to Figure 3 for illustration.

emitterreceiver (a) Simulation setup with $\mathcal{A}_{1}$

emitterreceiver (b) Simulation setup with $\mathcal{A}_{10}$

emitterreceiver (c) Simulation setup with $\mathcal{A}_{25}$

Figure 3: Illustration of antenna arrangement with transmitters
$\mathcal{A}_{1}$, $\mathcal{A}_{10}$, and $\mathcal{A}_{25}$.

Figure 4 shows maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{1})$ with
$\mathbf{a}_{1}=$0.72\text{\,}\mathrm{m}$(\cos$0\text{\,}\mathrm{\SIUnitSymbolDegree}$,\sin$0\text{\,}\mathrm{\SIUnitSymbolDegree}$)$
at several frequencies. Based on the simulation results, it is impossible to
recognize objects through the map of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{1})$ when
$f\leq$2\text{\,}\mathrm{GHz}$$ and $f\geq$6\text{\,}\mathrm{GHz}$$.
Fortunately, peaks of large magnitudes are appeared at $\mathbf{r}_{1}$ and
$\mathbf{r}_{2}$ so that we can recognize the existence of two objects but
their outline shape cannot be determined.

(a) $f=$1\text{\,}\mathrm{GHz}$$ (b) $f=$2\text{\,}\mathrm{GHz}$$ (c)
$f=$3\text{\,}\mathrm{GHz}$$ (d) $f=$4\text{\,}\mathrm{GHz}$$ (e)
$f=$6\text{\,}\mathrm{GHz}$$ (f) $f=$8\text{\,}\mathrm{GHz}$$

Figure 4: Maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{1})$.
White colored circles describes the boundary of objects.

Figure 5 shows maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{10})$ with
$\mathbf{a}_{10}=$0.72\text{\,}\mathrm{m}$(\cos$90\text{\,}\mathrm{\SIUnitSymbolDegree}$,\sin$90\text{\,}\mathrm{\SIUnitSymbolDegree}$)$
at several frequencies. In contrast to the results in Figure 4, it is possible
to recognize the existence of two object at $f=$2\text{\,}\mathrm{GHz}$$ but
it is still impossible to recognize them when $f=$1\text{\,}\mathrm{GHz}$$ and
$f\geq$6\text{\,}\mathrm{GHz}$$. Moreover, it seems to be difficult to
recognize the existence of objects when $f=$4\text{\,}\mathrm{GHz}$$ due to
the appearance of two artifacts with large magnitudes.

(a) $f=$1\text{\,}\mathrm{GHz}$$ (b) $f=$2\text{\,}\mathrm{GHz}$$ (c)
$f=$3\text{\,}\mathrm{GHz}$$ (d) $f=$4\text{\,}\mathrm{GHz}$$ (e)
$f=$6\text{\,}\mathrm{GHz}$$ (f) $f=$8\text{\,}\mathrm{GHz}$$

Figure 5: Maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{10})$.
White colored circles describes the boundary of objects.

Figure 6 shows maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{25})$ with
$\mathbf{a}_{25}=$0.72\text{\,}\mathrm{m}$(\cos$240\text{\,}\mathrm{\SIUnitSymbolDegree}$,\sin$240\text{\,}\mathrm{\SIUnitSymbolDegree}$)$
at several frequencies. In contrast to the results in Figures 4 and 5, the
existence of two objects can be recognized when $f=$2\text{\,}\mathrm{GHz}$$
and only a peak of large magnitude appeared when $\mathbf{r}^{\prime}\in
D_{1}$ when $f=3,$4\text{\,}\mathrm{GHz}$$. Same as previously, it is still
impossible to recognize objects when $f=$1\text{\,}\mathrm{GHz}$$ and
$f\geq$6\text{\,}\mathrm{GHz}$$.

(a) $f=$1\text{\,}\mathrm{GHz}$$ (b) $f=$2\text{\,}\mathrm{GHz}$$ (c)
$f=$3\text{\,}\mathrm{GHz}$$ (d) $f=$4\text{\,}\mathrm{GHz}$$ (e)
$f=$6\text{\,}\mathrm{GHz}$$ (f) $f=$8\text{\,}\mathrm{GHz}$$

Figure 6: Maps of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{25})$.
White colored circles describes the boundary of objects.

Based on the simulation results, and Remarks 3.1 and 3.2, we can conclude that
the imaging performance of the OSM with single source is significantly
dependent on the operated frequency and location of the emitter. Hence, design
of another indicator function of the OSM seems required for a proper
improvement of the imaging performance.

##  5 Indicator function with multiple sources: analysis and simulation
results

Following to the several studies [42, 56, 35, 51, 57, 37, 41], it has been
confirmed that application of multiple sources and/or frequencies successfully
improves the imaging performance. Following to [43, 41], one can examine
several simulation results for the improvement of the multi-frequency OSM.
Hence, we consider the application of multiple sources at a fixed frequency.
Following to [41], the following indicator function (say, OSMM) can be used:
for each $\mathbf{r}^{\prime}\in\Omega$

$\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})=\sum_{m=1}^{M}\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m}).$

Although one can obtain good result via the map of
$\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$, we introduce
another indicator function to obtain a better result. To this end, we denote
$\mathbf{F}(\mathbf{r}^{\prime})$ as the following arrangement

$\displaystyle\mathbf{F}(\mathbf{r}^{\prime})$
$\displaystyle=\Big{(}\Phi(\mathbf{r}^{\prime},\mathbf{a}_{1}),\Phi(\mathbf{r}^{\prime},\mathbf{a}_{2}),\ldots,\Phi(\mathbf{r}^{\prime},\mathbf{a}_{N})\Big{)}$
$\displaystyle=\frac{k_{\mathrm{b}}}{6|\mathbf{b}|}\begin{pmatrix}\vskip 6.0pt
plus 2.0pt minus
2.0pt\displaystyle\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{1})\bigg{[}J_{0}(k|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{3}{\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{1})\bigg{]}\mathrm{d}\mathbf{r}\\\
\displaystyle\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{2})\bigg{[}J_{0}(k|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{3}{\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{2})\bigg{]}\mathrm{d}\mathbf{r}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\vdots\\\
\displaystyle\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})\bigg{[}J_{0}(k|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{3}{\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})\bigg{]}\mathrm{d}\mathbf{r}\end{pmatrix}^{T}.$

where $\Phi(\mathbf{r}^{\prime},\mathbf{a}_{m})$ satisfies
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})=|\Phi(\mathbf{r}^{\prime},\mathbf{a}_{m})|$.
Then, based on the structure of the $\mathbf{F}(\mathbf{r}^{\prime})$, it
seems natural to test the orthogonality relation between the
$\Phi(\mathbf{r}^{\prime},\mathbf{a}_{m})$ and $G(\cdot,\mathbf{a}_{m})$.
Thus, by introducing a test vector,

$\mathbf{H}(\mathbf{r}^{\prime})=\Big{(}G(\mathbf{r}^{\prime},\mathbf{a}_{1}),G(\mathbf{r}^{\prime},\mathbf{a}_{2}),\ldots,G(\mathbf{r}^{\prime},\mathbf{a}_{M})\Big{)},\quad\mathbf{r}^{\prime}\in\Omega,$

the following indicator function (say, MOSM) with multiple sources can be
introduced

$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})=|\mathbf{F}(\mathbf{r}^{\prime})\cdot\overline{\mathbf{H}(\mathbf{r}^{\prime})}|=\left|\sum_{m=1}^{M}\Phi(\mathbf{r}^{\prime},\mathbf{a}_{m})\overline{G(\mathbf{r}^{\prime},\mathbf{a}_{m})}\right|.$

The map of $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ will
contain peaks of large magnitude at $\mathbf{r}^{\prime}\in D_{s}$ thereby, it
will be possible to recognize the existence or outline shape of $D_{s}$,
$s=1,2,\ldots,S$. In order to discover the feasibility and some properties of
the $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$, we derive the
following result.

######  Theorem 5.1.

Let $\boldsymbol{\vartheta}=(\cos\vartheta,\sin\vartheta)$,
$\boldsymbol{\vartheta}_{m}=(\cos\vartheta_{m},\sin\vartheta_{m})$, and
$\mathbf{r}^{\prime}-\mathbf{r}=|\mathbf{r}^{\prime}-\mathbf{r}|(\cos\phi,\sin\phi)$.
Then, for sufficiently large $M$ and $\omega$,
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ can be represented
as follows:

$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})=\left|\frac{2}{3|\mathbf{a}||\mathbf{b}|}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)\left(J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)^{2}+\frac{3}{\pi}\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})\right)\mathrm{d}\mathbf{r}\right|,$
(8)

where

$\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})=\sum_{p=1}^{\infty}\frac{i^{p}}{p}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)^{2}\sin\left(\frac{2p}{3}\pi\right).$

###### Proof.

Based on (3), we have

$\displaystyle\sum_{m=1}^{M}\Phi(\mathbf{r}^{\prime},\mathbf{a}_{m})\overline{G(\mathbf{r}^{\prime},\mathbf{a}_{m})}$
$\displaystyle=\sum_{m=1}^{M}\left(\frac{k_{\mathrm{b}}}{6|\mathbf{b}|}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)G(\mathbf{r},\mathbf{a}_{m})\left[J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{12}{5\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})\right]\mathrm{d}\mathbf{r}\right)\overline{G(\mathbf{r}^{\prime},\mathbf{a}_{m})}$
$\displaystyle=\frac{k_{\mathrm{b}}}{6|\mathbf{b}|}\int_{D}\left(\frac{\varepsilon(\mathbf{r})-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)\sum_{m=1}^{M}\left(G(\mathbf{r},\mathbf{a}_{m})\overline{G(\mathbf{r}^{\prime},\mathbf{a}_{m})}\left[J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{12}{5\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})\right]\right)\mathrm{d}\mathbf{r}.$

Since $M$ is sufficiently large, applying (5) and (6) yields

$\displaystyle\begin{aligned}
\sum_{m=1}^{M}G(\mathbf{r},\mathbf{a}_{m})\overline{G(\mathbf{r}^{\prime},\mathbf{a}_{m})}&=\sum_{m=1}^{M}\frac{2}{k_{\mathrm{b}}|\mathbf{a}_{m}|\pi}e^{ik_{\mathrm{b}}\boldsymbol{\vartheta}_{m}\cdot(\mathbf{r}^{\prime}-\mathbf{r})}\\\
&=\frac{2}{k_{\mathrm{b}}|\mathbf{a}|\pi}\int_{\mathbb{S}^{1}}e^{ik_{\mathrm{b}}\boldsymbol{\vartheta}\cdot(\mathbf{r}^{\prime}-\mathbf{r})}\mathrm{d}\boldsymbol{\vartheta}\\\
&=\frac{2}{k_{\mathrm{b}}|\mathbf{a}|\pi}\int_{0}^{2\pi}e^{ik_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|\cos(\vartheta-\phi)}\mathrm{d}\vartheta=\frac{4}{k_{\mathrm{b}}|\mathbf{a}|}J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)\end{aligned}$
(9)

and

$\displaystyle\begin{aligned}
&\sum_{m=1}^{M}G(\mathbf{r},\mathbf{a}_{m})\overline{G(\mathbf{a}_{m},\mathbf{r})}\cos\big{(}p(\vartheta_{m}-\phi)\big{)}\\\
&=\frac{2}{k_{\mathrm{b}}|\mathbf{a}|\pi}\int_{0}^{2\pi}\cos\big{(}p(\vartheta-\phi)\big{)}e^{ik_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|\cos(\vartheta-\phi)}\mathrm{d}\vartheta\\\
&=\frac{2}{k_{\mathrm{b}}|\mathbf{a}|\pi}\int_{0}^{2\pi}\cos\big{(}p(\vartheta-\phi)\big{)}\bigg{(}J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+2\sum_{q=1}^{\infty}i^{q}J_{q}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)\cos\big{(}q(\vartheta-\phi)\big{)}\bigg{)}\mathrm{d}\vartheta\\\
&=\frac{4i^{p}}{k_{\mathrm{b}}|\mathbf{a}|}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|).\end{aligned}$
(10)

Based on (9) and (LABEL:Term2), we can examine that

$\sum_{m=1}^{M}G(\mathbf{r},\mathbf{a}_{m})\overline{G(\mathbf{a}_{m},\mathbf{r})}\left[J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)+\frac{3}{\pi}\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})\right]\\\
=\frac{4}{k_{\mathrm{b}}|\mathbf{a}|}\left[J_{0}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)^{2}+\frac{3}{\pi}\sum_{p=1}^{\infty}\frac{i^{p}}{p}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)^{2}\sin\left(\frac{2p}{3}\pi\right)\right]$

and correspondingly, we can derive (8). ∎

Based on the Theorem 5.1, we can examine some properties of the indicator
function $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$.

######  Remark 5.1 (Availability of object detection).

Similar to the discussion in Remark 3.1, the resulting plot of indicator
function
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$ is
expected to exhibit peaks of magnitudes
$\frac{2}{3|\mathbf{a}||\mathbf{b}|}\left(\frac{\varepsilon_{s}-\varepsilon_{\mathrm{b}}}{\varepsilon_{\mathrm{b}}\mu_{\mathrm{b}}}\right)\operatorname*{area}(D_{s})$
at the $\mathbf{r}^{\prime}=\mathbf{r}\in D_{s}$ sought. Notice that opposite
to the single source case, the imaging performance is independent to the
location of emitters and receivers so that for any frequency of operation, it
will be possible to recognize the existence or outline shape of objects
through the map of $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$.

######  Remark 5.2 (Effect of the factor
$\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$).

Similar to the OSM with single source, the factor
$\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$ of (8) not only does not
contribute to the identification of objects but also disturb the
identification by generating several artifacts. In order to examine the
influence of the $\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$, we consider
the following quantity

$\mathcal{D}_{2}(x)=\frac{3}{\pi}\left|\sum_{p=1}^{10^{6}}\frac{i^{p}}{p}J_{p}(k_{\mathrm{b}}x)^{2}\sin\left(\frac{2p}{3}\pi\right)\right|.$

This is similar to the value of
$3\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})/\pi$ for $\mathbf{r}=(0,0)$.
Opposite to the discussion in Remark 3.3, we can say that although
$J_{0}(k_{\mathrm{b}}|x|)^{2}$ and
$\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$ generate some artifacts, their
magnitudes can be negligible so that their influence of object detection
should be disregarded. We refer to Figure 7 for an illustration of the
oscillating properties of $J_{0}(k_{\mathrm{b}}x)^{2}$ and
$\mathcal{D}_{2}(x)$.

$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$$J_{0}(k_{\mathrm{b}}x)^{2}$$\mathcal{D}_{2}(x)$
Figure 7: Plots of $J_{0}(k_{\mathrm{b}}x)^{2}$ and $\mathcal{D}_{2}(x)$ for
$-1\leq x\leq 1$ and $f=$2\text{\,}\mathrm{GHz}$$.

######  Remark 5.3 (Compare the imaging performance with single source OSM).

Here, we compare the imaging performance of the OSM with single and multiple
sources. Based on the 1-D Plots of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{1})$ and
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ for
$\mathbf{r}^{\prime}=(x,0)$ with $-1\leq x\leq 1$ in Figure 8, we can conclude
that the $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ will yield
better images owing to less oscillation than
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{m})$.

For a more detailed description, since
$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$ and
$\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$ are uniformly convergent, for
each $\epsilon>0$, there exists $N=N(\epsilon)\in\mathbb{N}$ such that

$\left|\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})-\sum_{p=1}^{N}\frac{(-i)^{p}}{p}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)\cos\big{(}p(\vartheta_{m}-\phi)\big{)}\sin\left(\frac{2p}{3}\pi\right)\right|<\epsilon$

and

$\left|\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})-\sum_{p=1}^{N}\frac{i^{p}}{p}J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)^{2}\sin\left(\frac{2p}{3}\pi\right)\right|<\epsilon.$

Suppose that $\mathbf{r}^{\prime}$ is not close to $\mathbf{r}$ such that
$k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|\gg 1/4$. Then since following
asymptotic form holds

$J_{p}(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|)\approx\sqrt{\frac{2}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}}\cos\left(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|-\frac{p\pi}{2}-\frac{\pi}{4}+O\left(\frac{1}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}\right)\right),$

applying Euler–Maclaurin formula yields

$\displaystyle|\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})|$
$\displaystyle\approx\left|\sum_{p=1}^{N}\frac{(-i)^{p}}{p}\sqrt{\frac{2}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}}\cos\left(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|-\frac{p\pi}{2}-\frac{\pi}{4}+O\left(\frac{1}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}\right)\right)\sin\left(\frac{2p}{3}\pi\right)\right|$
$\displaystyle\leq\sqrt{\frac{3}{2k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}}\sum_{p=1}^{N}\frac{1}{p}=\sqrt{\frac{3}{2k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}}\left(\ln
N+\gamma+\frac{1}{2N}-c_{p}\right)$
$\displaystyle\leq\sqrt{\frac{3}{2k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}}\left(\ln
N+\gamma+\frac{1}{2N}\right)$

and

$\displaystyle|\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})|$
$\displaystyle\approx\left|\sum_{p=1}^{N}\frac{i^{p}}{p}\frac{2}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}\cos^{2}\left(k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|-\frac{p\pi}{2}-\frac{\pi}{4}+O\left(\frac{1}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}\right)\right)\sin\left(\frac{2p}{3}\pi\right)\right|$
$\displaystyle\leq\frac{\sqrt{3}}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}\sum_{p=1}^{N}\frac{1}{p}\leq\frac{\sqrt{3}}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}\left(\ln
N+\gamma+\frac{1}{2N}\right),$

where $c_{p}$ satisfies $0\leq c_{p}\leq(8p^{2})^{-1}$ and
$\gamma=0.577215665\ldots$ denotes the Euler–Mascheroni constant. Since

$\frac{1}{k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}<\sqrt{\frac{1}{2k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|}}\quad\text{if}\quad
k_{\mathrm{b}}|\mathbf{r}^{\prime}-\mathbf{r}|\gg 1/4,$

we can explain that the factor $\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$
generates less artifacts than
$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$ because
$\mathcal{M}(\mathbf{r}^{\prime},\mathbf{r})$ has a smaller amplitude than
$\mathcal{E}(\mathbf{r}^{\prime},\mathbf{r},\mathbf{a}_{m})$. We refer to 1-D
plots of $\mathcal{D}_{1}(x)$ and $\mathcal{D}_{2}(x)$ in Figures 2 and 7,
respectively.

$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{1})$$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$
Figure 8: 1-D Plots of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime},\mathbf{a}_{1})$ and
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ for
$\mathbf{r}=(0,0)$ and $\mathbf{r}^{\prime}=(x,0)$, $-1\leq x\leq 1$, at
$f=$2\text{\,}\mathrm{GHz}$$.

Based on the Theorem 5.1 and Remark 5.1, we can also obtain the following
result of unique determination.

######  Theorem 5.2 (Unique determination of small objects).

Assume that the condition of the Theorem 5.1 holds. Then small objects can be
identified uniquely through the map of
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$.

From now on, we consider the simulation results. The simulation configuration
is same as the one in Section 4 except the range of emitters is from
$0\text{\,}\mathrm{\SIUnitSymbolDegree}$ to
$350\text{\,}\mathrm{\SIUnitSymbolDegree}$ with step size of
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$. Figure 9 shows maps of
$\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ and
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$, and Jaccard index
(see [58] for instance) at $f=1,$2\text{\,}\mathrm{GHz}$$. Same as the imaging
result with single source, it is impossible to recognize two objects through
the maps of $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ and
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$1\text{\,}\mathrm{GHz}$$. Fortunately, in contrast to the single source
case, the location and outline shape of two objects were successfully
retrieved. Although the imaging quality of
$\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ and
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ looks similar,
based on the evaluated Jaccard index, we can say that the imaging performance
of $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ is slightly
better than the one of
$\mathfrak{F}_{\operatorname*{OSM}}(\mathbf{r}^{\prime})$. Similar phenomenon
can be examined through the imaging results in Figure 10 at
$f=3,$4\text{\,}\mathrm{GHz}$$.

It is interesting to examine that opposite to the imaging results with single
source, it is possible to recognize the existence of two objects at
$f=6,$8\text{\,}\mathrm{GHz}$$, refer to Figure 11. However, unfortunately,
the imaging quality seems poorer than the ones in Figure 10. Hence, similar to
the Remark 3.1, application of extremely high frequency is not appropriate to
retrieve unknown objects.

(a) $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ at
$f=$1\text{\,}\mathrm{GHz}$$ (b)
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$1\text{\,}\mathrm{GHz}$$ (c) Jaccard index at $f=$1\text{\,}\mathrm{GHz}$$
(d) $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ at
$f=$2\text{\,}\mathrm{GHz}$$ (e)
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$2\text{\,}\mathrm{GHz}$$ (f) Jaccard index at $f=$2\text{\,}\mathrm{GHz}$$

Figure 9: Maps of $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$
and $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$, and Jaccard
index versus threshold.

(a) $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ at
$f=$3\text{\,}\mathrm{GHz}$$ (b)
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$3\text{\,}\mathrm{GHz}$$ (c) Jaccard index at $f=$3\text{\,}\mathrm{GHz}$$
(d) $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ at
$f=$4\text{\,}\mathrm{GHz}$$ (e)
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$4\text{\,}\mathrm{GHz}$$ (f) Jaccard index at $f=$4\text{\,}\mathrm{GHz}$$

Figure 10: Maps of $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$
and $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$, and Jaccard
index versus threshold.

(a) $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ at
$f=$6\text{\,}\mathrm{GHz}$$ (b)
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$6\text{\,}\mathrm{GHz}$$ (c) Jaccard index at $f=$6\text{\,}\mathrm{GHz}$$
(d) $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$ at
$f=$8\text{\,}\mathrm{GHz}$$ (e)
$\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$ at
$f=$8\text{\,}\mathrm{GHz}$$ (f) Jaccard index at $f=$8\text{\,}\mathrm{GHz}$$

Figure 11: Maps of $\mathfrak{F}_{\operatorname*{OSMM}}(\mathbf{r}^{\prime})$
and $\mathfrak{F}_{\operatorname*{MOSM}}(\mathbf{r}^{\prime})$, and Jaccard
index versus threshold.

Finally, we mention that Theorem 5.2 holds based on the imaging results in
Figures 9, 10, and 11.

##  6 Conclusion

In this paper, we have considered the application of the OSM with single
source to identify the existence and outline shape of small dielectric objects
from real-world experimental data. Thanks to the asymptotic expansion formula
for the scattered field data in the presence of small object, we derived an
accurate relationship between the indicator function and infinite series of
the Bessel function of the first kind. On the basis of the derived
relationship, we explored various properties of the indicator function
including the applicability and limitation. We have also verified the
theoretical result with experimental data at various frequencies.

In order to improve the imaging performance of the OSM, we have collected
scattered field data with multiple sources and introduced a new indicator
function. Throughout a careful analysis, we have shown that the imaging
performance of the designed indicator function is independent to the location
of the emitter and better than the traditional ones with single and multiple
sources. Moreover, it can be possible to identify small object uniquely if
applied frequency is not extremely low or high.

In this paper, we considered the application of the OSM with 2D Fresnel
experimental database. An extension to the 3D Fresnel experimental database
[59] will be the forthcoming work.

## Acknowledgments

This research was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (NRF-2020R1A2C1A01005221) and the
research program of the Kookmin University.

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]Minimum Cost Loop Nests for Contraction of a Sparse Tensor with a Tensor Network

Raghavendra Kanakagiri

Department of Computer Science
University of Illinois at Urbana-Champaign

Edgar Solomonik

Department of Computer Science
University of Illinois at Urbana-Champaign

Sparse tensor decomposition and completion are common in numerous applications, ranging from machine learning to computational quantum chemistry.
Typically, the main bottleneck in optimization of these models are contractions of a single large sparse tensor with a network of several dense matrices or tensors (SpTTN).
Prior works on high-performance tensor decomposition and completion have focused on performance and scalability optimizations for specific SpTTN kernels.
We present algorithms and a runtime system for identifying and executing the most efficient loop nest for any SpTTN kernel.
We consider both enumeration of such loop nests for autotuning and efficient algorithms for finding the lowest cost loop-nest for simpler metrics, such as buffer size or cache miss models.
Our runtime system identifies the best choice of loop nest without user guidance, and also provides a distributed-memory parallelization of SpTTN kernels.
We evaluate our framework using both real-world and synthetic tensors. Our results demonstrate that our approach outperforms available generalized state-of-the-art libraries and matches the performance of specialized codes.

<concept_desc>Software and its engineering General programming languages</concept_desc>
<concept_desc>Social and professional topics History of programming languages</concept_desc>

[500]Software and its engineering General programming languages
[300]Social and professional topics History of programming languages

This research has been supported by funding from the United States National Science Foundation (NSF) via grants #1942995 and #1931258, as well as by the Department of Energy (DOE) Advanced Scientific Computing Research program via award DE-SC0023483.
Raghavendra Kanakagiri has been supported by NSF grant #1931258 and the University of Illinois Urbana-Champaign Computer Science Future Faculty Fellows program.
This work used Stampede2 at TACC through allocation
CCR180006 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.

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