This document describes using the linear sampling method to reconstruct the shape of two-dimensional dielectric targets from scattered field measurements. The linear sampling method solves an integral equation to determine if a test point is inside the target support. Regularization is used to address ill-posedness. Numerical results show the reconstructed shape varies slightly with frequency and accuracy improves with more transmitters/receivers. The method provides fast reconstruction of target support but not material properties. Future work includes extending to 3D imaging and using linear sampling method results to initialize other reconstruction algorithms.

1. SHAPE RECONSTRUCTION OF
TWO-DIMENSIONAL DIELECTRIC
TARGETS BY USING LINEAR
SAMPLING METHOD
Presented by
N GANGABHAVANI YASWANTH KALEPU
09EC6313
Under the guidance of
Prof.A.Bhattacharya

2. Introduction
In recent years, the interest of the scientific community in inverse scattering
problems has grown significantly, together with the development of numerous
new techniques in remote sensing and non-invasive investigation. Important
electromagnetic inverse scattering problems arise, in particular, in
1)Military applications, for example when it is desired to identify hostile
objects by means of radar;
2)Medical imaging,
3)Non-destructive testing, such as the case when small cracks are looked for
inside metallic or plastic structures.
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3. The capability of retrieving the geometrical features of a system of unknown
targets from the measures of the scattered fields is important in many
noninvasive diagnostics applications.
However, as the scattered fields depend on both the electrical and geometrical
properties of the scatterers, this task is not anyway simple.
In the literature, this difficulty has been approached by introducing
approximate models to cope with a linear problem, or by solving simpler
auxiliary problems.
Methods belonging to this second class are particularly attractive, since they
do not entail any approximation and usually require a reduced computational
burden.
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4. ILL-POSEDNESS
A problem satisfying the requirements of uniqueness, existence and
continuity is called well-posed.
 The problems which are not well-posed are called ill-posed or also
incorrectly posed or improperly posed.
Therefore an ill-posed problem is a problem whose solution is not unique or
does not exist for arbitrary data or does not depend continuously on the data.
Typical property of inverse problems is ill-posedness.
Small oscillating data produce large oscillating solutions.
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5. In any inverse problem data are always affected by noise which can be
viewed as a small randomly oscillating function. Therefore the solution
method amplifies the noise producing a large and wildly oscillating function
which completely hides the physical solution corresponding to the noise-free
data.
This property holds true also for the discrete version of the ill-posed problem.
Then one says that the corresponding linear algebraic system is ill-conditioned:
even if the solution exists and is unique, it is completely corrupted by a small
error on the data.
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6. Linear Sampling-Method: Basics and
Interpretation
Linear Sampling Method (LSM) is claimed to be able to retrieve the support
of dielectric or metallic scatterers even when the support is not convex and/or
not connected (that is in the case of multiple targets). Notwithstanding these
attractive features, contributions on LSM are mostly, if not completely,
restricted to the mathematical community wherein they have been originally
proposed.
In particular, we show by means of examples that the LSM can be
Assimilated to the problem of focusing into a point the field radiated by a
collection of sources, so that it can be revised and investigated exploiting
widely usual arguments of applied electromagnetics.
Notably, the interpretation as focusing problems gives a physical basis to the
application of LSM in those cases wherein no mathematical proof is yet
available.
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Figure: Configuration

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 is the region under test, is the (possibly not connected) support of the
dielectric scatterers and is the contrast .
Measurement probes and primary sources lie on a curve located in the scatterers far-
zone (i.e., at a distance R >2d2/λ ,d being the size of and λ the wavelength), so that
incident fields can be well approximated by plane waves
 let k be the background wave number and the scattered far-field pattern as
measured on in the direction , when a unit amplitude plane wave impinges from
the direction . For a generic point , the LSM consists in solving the “far-field”
integral equation in the unknown
Wherein the right-hand side is the far-field radiated on Г by an
elementary source located in .
[4]

9. By introducing the operator and the function ,
can be synthetically rewritten as
The generalized solution of above equation is such that its
( i.e , the “energy”) becomes unbounded when rp does not belong to
the support of the scatterer .
plays the role of supporting function
[4]
[4]

10. 10
A RELIABLE FOCUSING STRATEGY
A reliable focusing strategy should be such to:
prevent primary sources from having an infinite energy;
avoid non-uniqueness due to the NR primary sources.
These goals can be achieved in a simple fashion by using the Tichonov
regularization,
Wherein is a weighting parameter.
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11. Wherein
denote the singular values
g is the Nv-dimensional vector of the unknowns
f(r p) is the N m-dimensional vector
F is a N mxNv matrix
are the N m-dimensional left singular vectors and alpha
is the Tichonov parameter .we fix it to 0.01 .

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Forward Scattering Problem
The electromagnetic field generated in an unbounded region (free-
space radiation) by the impressed source J0 satisfying wave equations ,
along with the Silver-Muller radiation conditions, can be expressed in
integral form as
Where is the free-space Green dyadic tensor given by (Tai
1971) [3]

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where is the wavenumber of the propagation medium. It is worth
nothing that Green’s dyadic tensor is related to the radiation produced by
an elementary source and provides a solution to the following tensor
equation:
Volume Equivalence Principle
Volume equivalence principle. (a) Real configuration; (b) Equivalent
problem with the equivalent current density.

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By introducing the equivalent sources
Substituting Meq(r) =0

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Where
Two-Dimensional scattering
Where

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Moreover, since the following relation holds (Balanis 1989)[14]

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Discretization of the continuous model
[9]

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Where

20. RESULTS
We have considered the “Marseille” experimental data-set[6]. This data-set is
related to an aspect-limited configuration in which the primary source is
moved along a circumference with a 10o angular step (Nv=36 ), and for each
illumination, the measurement probe is moved with an angular step of 5o along
a 240o arc which excludes the 120o angular sector centered around the
incidence direction.
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A bistatic measurement system is used to measure the EM field

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Description of the targets
Cross sectional dimensions of the metal targets. (a) Metallic target with a rectangular
cross section; (b) metallic target with ‘U-shaped’ cross section.
Circular cross section of radius a=15 mm. The estimation of the real part of the relative permittivity leads to
(a) Single circular dielectric cylinder; (b) two-identical circular cylinders.

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Five configurations of targets
Output of 1st Configuration
(a)3GHz (b)5GHz (c)8GHz

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Output of 2nd Configuration
(a)2GHZ (b)5GHZ (c)8GHZ
Output of 3rd Configuration
(a)4GHZ (b)8GHZ (c)12GHZ
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(a)6GHZ (b)10GHZ (c)14GHZ
Output of 4th Configuration
Output of 5th Configuration
(a)4GHZ (b)8GHZ (c)16GHZ

25. Imaging results using synthetically
generated data
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(a)1GHz (b) 2GHz
(c) 3GHz (d) 4GHz
1) Dielectric cylinders with radius 0.04m and with 36 TX and 36RX

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(e) 5GHz (f) 6GHz
(g) 7GHz (h) 8GHz
(i) 9GHz (j) 10GHz
(k) 11GHz (l) 12GHz

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(m) 13GHz (n) 14GHz
(o) 15GHz (p) 16GHz
(a)1GHz (b) 2GHz
2)Two dielectric cylinders located at (-0.05m, 0) and (0.05m, 0) with radius
0.02m each

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( c ) 3GHz (d) 4GHz
(e) 5GHz (f) 6GHz
(g) 7GHz (h) 8GHz
(i) 9GHz (j) 10GHz

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(k) 11GHz (l) 12GHz
(m) 13GHz (n) 14GHz
(o) 15GHz (p) 16GHz

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Frequency
(GHz)
Measured
radius with
36 TX and RX
(m)
% error
with 36 TX
and RX
Measured
radius with
72 TX and RX
(m)
% error
with 72 TX
and RX
1 0.04 0 0.032 -20
2 0.044 10 0.044 10
3 0.029 -27.5 0.031 -22.5
4 0.037 -7.5 0.039 -2.5
5 0.035 -12.5 0.037 -7.5
6 0.04 0 0.042 5
7 0.04 0 0.042 5
8 0.042 5 0.044 10
9 0.041 2.5 0.043 7.5
10 0.04 0 0.042 5
11 0.04 0 0.043 7.5
12 0.041 2.5 0.042 5
13 0.042 5 0.043 7.5
Table1 Reconstructed images radius with different frequencies for dielectric
cylinder located at center with radius 0.04m.

31. Advantages of the LSM
A Notable Computational Speed
The implementation is computationally simple, since it only requires the
solution of a finite number of ill-conditioned linear systems. This also implies
that the numerical instability due to the presence of noise on the far-field data
can be easily handled by applying the classical algorithms of regularization
theory for ill-conditioned linear systems.
Very little a priori information on the scatterers is required.
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32. Disadvantage
The method, of course, also has some disadvantages. The main one is that,
in the case of inhomogeneous scattering, it only provides a reconstruction of
the support of the scatterer and it is not possible to infer information about the
point values of the index of refraction.
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33. Conclusion
Formulation for scattering of a 2D dielectric targets with a plane wave
incident has been discussed.
From the scattering field, shape reconstruction of dielectric material by using
Linear Sampling Method has been presented.
From the observation it has been concluded that shape reconstruction using
Linear Sampling Method varies in small amount with respect to frequency.
 It is observed that the accuracy of the results increase with increase in
number of transmitters and receivers. And care should be taken in estimating
the number of transmitters and receivers.
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34. Future work
Most of the work in microwave imaging is limited to 2D-imaging.So; it can
be extended to 3D-imaging.
In many microwave imaging methods initial guess is very crucial. So, we can
use the LSM output as an initial guess to get better results. I.e. to implement
hybrid techniques.
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35. References
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1) Mario Bertero and Patrizia Boccacci, “Introduction to Inverse Problems in
Imaging,” IOP Publishing Ltd, 1998.
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5) D. Colton, H. Haddar, and M. Piana, “The linear sampling method in
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105–137, 2003.

36. 6) K. Belkebir and M. Saillard, “Special section: Testing inversion algorithms
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8) Harrington, R.F “Time-Harmonic Electromagnetic Fields,” New York:
McGraw- Hill, pp. 254-338, 1961.
9) J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross
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341,1965.
10) Peterson, Andrew F, “Computational methods for electromagnetic,” IEEE
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11)Gupta, S L , “Fundamental real analysis,” Vikas Publishing House, 4th
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12)Colton, David, “Integral Equations Methods In Scattering theory,” John
Wiley, New York, 1983.
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37. 13)Stoll, R. R.,” Linear Algebra and Matrix Theory,” Mcgraw-Hill, New York,
1952.
14)Balanis, C. A., Advanced Engineering Electromagnetics, wiley, New York,
1989.
15)T. Arens, “Why linear sampling works” Inverse Problems, vol.20, pp.163-
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16) I. Catapano, L. Crocco, M. D’Urso, and T. Isernia, “On the effect of
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17) G.H.Golub, “Matrix computations: texts and readings in mathematics,”
Hindustan Book Agency, 3rd Edition, 2007.
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38. Questions ?
And
Suggestions for
improvement..,

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Thank you

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Singular value decomposition

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Singular value decomposition for solving linear problems

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Regularized solution of a linear system using singular
value decomposition

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Elemental two-dimensional Source

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Infinitely long filament of constant ac-current along z axis, as shown in Figure.
The field will be TM to z, expressible in terms of an A having only a z
component Ψ. From Symmetry, Ψ should be independent of and z. To
represent outward-traveling waves, we choose[4]

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