23rd Annual Review of Progress in Applied Computational Electromagnetics
March 19-23, 2007 - Verona, Italy ©2007 ACES
The Linear Sampling Method, a New Regularized Solution and Real Data J. Coyle1 , M. Brignone2 and R. Aramini3 1
Department of Mathematics
Monmouth University, West Long Branch, New Jersey, USA
[email protected] 2
Dipartimento di Matematica Universit` a di Genova, via Dodecaneso 35, 16146 Genova, Italy
[email protected] 3
Dipartimento di Matematica Universit` a di Trento, via Sommarive 14, 38050 Povo di Trento, Italy
[email protected]
Abstract: A modified version of the Linear Sampling Method is presented and applied to the Ipswich data. In particular, two incompatibility measures are considered leading to two generalized discrepancy functions and, consequently, a blended regularized solution. This new technique leads to promising results that are illustrated in an application to two real targets. 1.
Introduction
A true measure of the capabilities as well as limitations of a technique designed to remotely estimate properties of an object is testing it with real data. One such method in electromagnetic or acoustic scattering, the Linear Sampling Method (LSM), is based on constructing regularized solutions to a discretized version of the far-field equation in order to determine characteristics of the unknown scatterer (see [3] and the references therein). In this paper we are specifically interested in finding the support of an object and begin by considering the two-dimensional associated direct scattering problem of determining u from the following equations: 4u + k 2 u u u = ui √ ∂us s lim r − iku r→∞ ∂r
= 0 = 0 + us
in R2 \ D, on Γ, in R2 \ D,
= 0
(1) (2) (3) (4)
where k is the wavenumber, ui(x) = eikx·d , d ∈ Ω = {v ∈ R2 : |v| = 1} is the direction vector of the incident plane wave, (4) is known as the Sommerfeld radiation condition and D denotes the bounded scatterer with boundary Γ. The asymptotic behavior of us can be written as eik|x| x, d) + O |x|−3/2 us (x) = p u∞ (ˆ |x|
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as |x| → ∞,
23rd Annual Review of Progress in Applied Computational Electromagnetics
March 19-23, 2007 - Verona, Italy ©2007 ACES
x where u∞ is the so called far-field pattern, and x ˆ = |x| ∈ Ω. The far-field operator F : L2 (Ω) → L2 (Ω) is then defined as Z F g (ˆ x) = u∞ (ˆ x, y)g(y)dsy . (5) Ω
Finally, the fundamental solution, Φ, to (1) is given by i (1) Φ(x, z) = H0 (k|x − z|) 4 (1)
where H0 given by
x, z ∈ R2 ,
is the Hankel function of the first kind and of order zero. The far-field behavior is π
ei 4 −ikˆx·z e Φ∞ (ˆ x, z) = √ , 8πk where x ˆ ∈ Ω and z ∈ R2 is the source point. The far-field equation can then be defined as F gz (·) = Φ∞ (·, z)
(6)
for any gz (·) ∈ L2 (Ω) and z ∈ R2 . This equation is an ill-posed integral equation of the first kind and is numerically unstable when solving for gz (·). However, it can be shown [2] that there exist approximate solutions such that their L2 -norm becomes unbounded as z approaches the boundary, Γ, from the inside and remains large when z is outside Γ. This leads to a process, the LSM, of sampling a region (by varying z) where the scatterer is thought to be located and plotting the norm of the regularized solution. Following [1], we can express the infinitely many algebraic systems of the traditional implementation of the linear sampling method by a single functional equation in the Hilbert space N L [L2(T )]N = L2 (T ), where T is a rectangle containing the scatterer, while N denotes the i=1
number of incidence and observation angles. To this end, we introduce the linear operator N N Fh : [L2 (T )] → [L2 (T )] given by (N −1 )N −1 X [Fh g(·)] (·) = (Fh )ij gj (·) j=0
N ∀g(·) = (g0 (·), . . . , gN −1(·)) ∈ L2 (T ) ,
(7)
i=0
where (Fh )ij = (F + H)ij are the elements of the noisy version of the far-field matrix, in which F is obtained by discretizing the far-field operator F , and H is the noise matrix. If, as usual, we denote the operatorial norm of a continuous operator by k · k and regard the matrix H as a linear continuous operator in CN , then we assume that kHk ≤ h. This formulation leads to the linear inverse problem [Fh g(·)](·) = Φ∞ (·), (8) x0, ·), . . . , Φ∞ (ˆ xN −1, ·)); here x ˆi = (cos (2πi/N ) , sin (2πi/N )) for any i = where Φ∞ (·) = (Φ∞ (ˆ 0, . . . , N − 1. Numerical results concerning the application of this process to real data will be shown in section 3. 1928
23rd Annual Review of Progress in Applied Computational Electromagnetics
March 19-23, 2007 - Verona, Italy ©2007 ACES
−1 If rh is the rank of the noisy far-field matrix Fh and {σph , uhp , vph }N p=0 is its singular system, the Tikhonov regularized solution of problem (8) is given by [1]: rh −1
gα (·) =
X p=0
σph (Φ∞ (·), vph)CN uhp , (σph )2 + α
(9)
where α > 0 is the regularization parameter and (Φ∞ (·), vph )CN denotes the element in L2 (T ) defined f.a.a. z ∈ T as (Φ∞ (z), vph )CN . The optimal value α? of α can be determined, according to the generalized discrepancy principle, as the zero of one of the two possible generalized discrepancy functions [7]: ρ(α) = k[Fh gα (·)](·) − Φ∞ (·)k22,N − h2 kgα (·)k22,N − [µh (Φ∞ (·), Fh )]2,
(10)
or 2
2
ρ(α) ˆ = k[Fh gα (·)](·) − Φ∞ (·)k2,N − [hkgα (·)k2,N + µ ˆh (Φ∞ (·), Fh )] ,
(11)
where k · k2,N is the norm induced by the scalar product (·, ·)2,N in [L2(T )]N and µh (Φ∞ (·), Fh ) = µ ˆh (Φ∞ (·), Fh) =
k[Fh g(·)](·) − Φ∞ (·)k2,N
(12)
(k[Fhg(·)](·) − Φ∞ (·)k2,N + hkg(·)k2,N )
(13)
inf
g(·)∈[L2 (T )]N
inf
g(·)∈[L2 (T )]N
are the incompatibility measures. Then gα? (·) is the optimal regularized solution of problem (8) and a visualization of the scatterer can be obtained, for example, by plotting for any z ∈ T (14) Ξ(z) = − log kgα? (z)k2CN , where k · kCN is the classical norm on CN induced by the scalar product (·, ·)CN . Numerical tests suggest to use as the optimal value for the regularization parameter the zero of (10) for high level of noise, while for small noise it seems to be useful the zero of (11). Hence a blending regularized solution g? is then constructed as g?(·) = c gα?o (·) + (1 − c) gα?n (·)
(15)
in [0, 1]. where α?o and α?n are respectively the zero of (10) and (11), while c is a suitable constant ? 2 To visualize the scatterer, following equation (14), we can plot − log kg (z)kCN . 2.
The Data
The numerical results in this paper are run using the Ipswich data provided by Electromagnetics Technology Division, AFRL/SNH, Hanscom AFB. The Ipswich data are single frequency scattered field data measured using a bistatic system. Multiple views, corresponding to incidence angles, are obtained by rotating the target on a separate azimuthal positioner in fixed increments. It is not possible to measure scattering at or even near direct backscatter since the receiver and transmitter cannot be physically coincident. As a result, given an incident angle, the Ipswich 1929
23rd Annual Review of Progress in Applied Computational Electromagnetics
March 19-23, 2007 - Verona, Italy ©2007 ACES
data files do not have a full view of the target. A more detailed discussions of the data and the measurement process can be found in [6]. The Ipswich data as it pertains to the above theory is incomplete in the sense that for each incident angle there is a different set of observation angles. Consider, for example, the data for target Ips009 – an aluminum triangle. Thirty-six data sets are provided corresponding to incident angles 0 − 350 in increments of 10. Each of these sets has only eighteen scattered field measurements given in increments of 10 degrees. In particular, for the incident angle 210 degrees, far-field measurements are provided at the angles 30 − 200 in steps of 10. The data only partially fills in the far-field matrix used in the right hand side of (7) as shown in Figure 1(a). However, using the reciprocity relation of the noise-free far-field pattern [5] u∞ (θi, φo ) = u∞ (φo + π, θi + π) where θi is the incidence angle and φo is the observation angle, we obtain, for an even and equal number of incidence and observation angles, the following block structure inside the noise-free far-field matrix: F11 F12 F= , F21 F22 where F12 and F21 are symmetric and F11 = FT22. Hence, in order to complete the far-field matrix, also in the noisy case, we can use this property. This is illustrated in Figure 1(b). The remaining unknown entries, which correspond to the backscattering measurements, can be filled in by averaging the entries in the same row in the column before and after the unknown entry. Now, for any square matrix M11 M12 M= , M21 M22 with an even number of rows (or columns) define T T M M 22 12 f= M . MT21 MT11 It is straightforward to see that
f kMk = kMk.
(16)
In order to estimate h, one might consider a value that depends on e h k. v = kFh − F It is simple to observe that, by virtue of the reciprocity relation and (16), we have e = 2kHk ≤ 2h. e ≤ kHk + kHk v = kH − Hk As a result h ≥ v/2.
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(17)
23rd Annual Review of Progress in Applied Computational Electromagnetics
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0
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30
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March 19-23, 2007 - Verona, Italy ©2007 ACES
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5
10
15
20
25
30
35
0
5
(a) Given Data
10
15
20
25
30
35
(b) Completed Data
Figure 1: Plot of the entries in the scattering matrix Fh : (a) shows the given data and (b) shows the given data with the unknown entries from (a) that can be filled in using the reciprocity relation. 3.
Numerical Results
The LSM is performed using data for two objects: Ips009 (aluminum triangle) and Ips011 (aluminum circular cavity), both contained in a disc of radius 6 cm. In each case, a region T = [−7, 7] × [−7, 7] is used and we plot − log kg?(z)k2CN for all z ∈ T . For Ips009, v = 0.0964 and for Ips011 v = 0.5046. In both figures 2 and 3 we show the results of employing only (10) to Ips009 and Ips011, respectively, and, following the lower bound for h given by (17), we set h equal to v/2 (in (a)) and v (in (b)). Recall this is equivalent to setting c = 1 in (15). In each case it is clear that the choice of h has an influence on the reconstruction. The next step is to employ (15) with c 6= 1. These results are given in Figure 4 where the reconstructions are somewhat better. More specifically, we choose h = v and the value of c is set to 0.5 in both cases. 4.
Conclusion
The application of the version of the LSM presented in this paper in the reconstruction of the two Ipswich targets leads to promising results. A robust and efficient method for estimating h (depending on the level of noise in the far-field matrix) and c, in (15), remains an open topic.
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23rd Annual Review of Progress in Applied Computational Electromagnetics
(a) h = 0.0482
March 19-23, 2007 - Verona, Italy ©2007 ACES
(b) h = 0.0964
Figure 2: The LSM applied to target Ips009 where h is chosen to be 0.0482 in (a) and 0.0964 in ? 2 (b). In each case we plot − log kg (z)kCN for any z ∈ T with c = 1.
(a) h = 0.2523
(b) h = 0.5046
Figure 3: The LSM applied to target Ips011 where h is chosen to be 0.2523 in (a) and 0.5046 in (b). In each case we plot − log kg?(z)k2CN for any z ∈ T with c = 1.
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23rd Annual Review of Progress in Applied Computational Electromagnetics
(a) Ips009
March 19-23, 2007 - Verona, Italy ©2007 ACES
(b) Ips011
Figure 4: The LSM applied to targets Ips009 (a) and Ips011(b) with h = 0.0964 for Ips009 and h = 0.5046 for Ips011. In each case we plot − log kg? (z)k2CN for any z ∈ T with c = 0.5. References [1] R. Aramini, M. Brignone and M. Piana, “The linear sampling method without sampling”, Inverse Problems, Vol. 22, pp. 2237-2254, 2006. [2] F. Cakoni and D. Colton, “On the Mathematical Basis of the Linear Sampling Method”, Georg. Math. J., Vol. 10, pp. 911 - 925, 2003. [3] D. Colton, J. Coyle and P. Monk, “Recent Developments in Inverse Acoustic Scattering Theory”, SIAM Review, Vol. 42, pp. 369 - 414, 2000. [4] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Inverse Scattering Theory, Springer, Berlin, 1998. [5] D. Colton and P. Monk, “The scattering of electromagnetic waves by a perfectly conducting infinite sylinder”, Mathematical Methods in the Applied Sciences, Vol. 12, pp. 503 - 518, 1990. [6] R. McGahan and R. Kleinman, “Special session on Image reconstruction using real data”, IEEE Antennas and Propagation Magazine, Vol. 38, pp. 39 - 40, 1996. [7] A. Tikhonov, A. Goncharsky, V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995.
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