SHAPE RECONSTRUCTION OF TWO-DIMENSIONAL DIELECTRIC

TARGETS BY USING LINEAR SAMPLING METHOD

Presented byN GANGABHAVANI YASWANTH KALEPU

09EC6313

Under the guidance ofProf.A.Bhattacharya

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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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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ILL-POSEDNESSA 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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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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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]

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]

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A RELIABLE FOCUSING STRATEGYA 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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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 ProblemThe 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

RESULTSWe 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.

07/05/2011 20A 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

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

14 0.036 -10 0.038 -5

15 0.041 2.5 0.042 5

16 0.042 5 0.042 5

Table1 Reconstructed images radius with different frequencies for dielectric cylinder located at center with radius 0.04m.

Advantages of the LSMA Notable Computational SpeedThe 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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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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ConclusionFormulation 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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Future workMost 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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References

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1) Mario Bertero and Patrizia Boccacci, “Introduction to Inverse Problems in Imaging,” IOP Publishing Ltd, 1998.2) Matteo Pastorino, “Microwave Imaging,”John Wiley & sons, inc., publication, 2010.3) Chen To Tai, “Dyadic Green Functions in Electromagnetic Theory,” IEEE, Inc., New York, 1993.4) I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, “On the effect of support estimation and of a new model in 2-D inverse scattering problems,” IEEE Trans. on Antennas and Propagation, vol. 55, pp. 1895–1899, Jun 2007.5) D. Colton, H. Haddar, and M. Piana, “The linear sampling method in inverse electromagnetic scattering theory,” Inverse Problems, vol. 19, pp. 105–137, 2003.

6) K. Belkebir and M. Saillard, “Special section: Testing inversion algorithms against experimental data,” Inverse Problems, vol. 17, pp.1565–2028, 2001.7) Colton D and Kress R, Inverse Acoustic and Electromagnetic Scattering Theory .Berlin: Springer 1992.8) 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 section shape,” IEEE Trans. Antennas Propagation, vol. 13, pp. 334-341,1965.10) Peterson, Andrew F, “Computational methods for electromagnetic,” IEEE Press, New York, 1998.11)Gupta, S L , “Fundamental real analysis,” Vikas Publishing House, 4th Edition, 2002.12)Colton, David, “Integral Equations Methods In Scattering theory,” John Wiley, New York, 1983.

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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.163-173, 2004.16) I. Catapano, L. Crocco, M. D’Urso, and T. Isernia, “On the effect of support estimation and of a new model in 2-D inverse scattering problems,” IEEE Trans. Antennas Propagation., vol. 55, no. 6, pp. 1895–1899, Jun. 2007.17) G.H.Golub, “Matrix computations: texts and readings in mathematics,” Hindustan Book Agency, 3rd Edition, 2007.

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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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