Super-resolution in Imaging High Contrast Targets from the Perspective of Scattering Coefficients
aa r X i v : . [ m a t h - ph ] O c t Super-resolution in Imaging High Contrast Targets from thePerspective of Scattering Coefficients ∗ Habib Ammari † Yat Tin Chow ‡ Jun Zou ‡ Abstract
In this paper we consider the inverse scattering problem for high-contrast targets. We mathe-matically analyze the experimentally-observed phenomenon of super-resolution in imaging the targetshape. This is the first time that a mathematical theory of super-resolution has been establishedin the context of imaging high contrast inclusions. We illustrate our main findings with a varietyof numerical examples. Our analysis is based on the novel concept of scattering coefficients. Thesefindings may help in developing resonant structures for resolution enhancement.
Mathematics Subject Classification (MSC 2000): 35B30, 35R30
Keywords : inverse scattering, super-resolution, scattering coefficients
The aim of this work is to mathematically investigate the mechanism underlying the experimentally-observed phenomenon of super-resolution in reconstructing targets of high contrast from far-field mea-surements. Our main focus is to explore the possibility of breaking the diffraction barrier from the far-fieldmeasurements using the novel concept of scattering coefficients [4, 5, 6]. This diffraction barrier, referredto as the Abbe-Rayleigh or the resolution limit, places a fundamental limit on the minimal distance atwhich we can resolve the shape of a target [2]. It applies only to waves that have propagated for a distancesubstantially larger than its wavelength [10, 11].Since the mid-20th century, several approaches have aimed at pushing this diffraction limits. Res-olution enhancement in imaging the target shape from far-field measurements can be achieved usingsub-wavelength-scaled resonant media [7, 20, 21, 22, 23] and single molecule imaging [19]. Another inno-vative method to overcome the diffraction barrier has been proposed after some experimental observationsin [8]. In their work, resolution enhancement in shape reconstruction of the inclusion was experimentallyshown when the contrast value is very high. In the reconstructed images from far-field measurements, theobserved resolution is smaller than half of the operating wavelength, 2 π/k , where k is the wave number.This encouraging observation suggests a possibility of breaking the resolution limit with high permittivityof the target. It is therefore the purpose of this work to prove that the higher the permittivity of thetarget is, the higher the resolving power is in imaging its shape.For the transmission problem of a strictly convex domain, it was proved in [25] that there existsan infinite sequence of complex resonant frequencies located at the upper half plane. These resonancesconverge to the real axis exponentially fast, and the real part of these resonances correspond to the quasi-resonant modes introduced as in [25]. Quasi-resonance occurs when the wavelength inside the inclusion ∗ This work was supported by the ERC Advanced Grant Project MULTIMOD–267184 and the Hong Kong RGC grants(projects 405513 and 404611). † Department of Mathematics and Applications, Ecole Normale Sup´erieure, 45 Rue d’Ulm, 75005 Paris, France.([email protected]). ‡ Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong ([email protected],[email protected]).
1s larger than the wavelength in the background media and is such that it reaches the real part of one ofthese true resonant frequencies. In this paper, we have shown, via the analysis of the shape derivativeof the scattering coefficients, that these resonant state of the inclusion actually has a signature in thefar-field and can be used for super-resolved imaging from far-field data. To be more exact, we have provedthat, in the shape derivative of the scattering coefficients for a circular domain, there are simple poles atthe complex resonant states, and therefore peaks corresponding to the the real parts of these resonances.Henceforth, as the material contrast increases to infinity and is such that it is equal to the real part ofa resonance, the sensitivity in the scattering coefficients becomes large and super-resolution for imagingbecomes possible.Throughout this paper, we consider the following scattering problem in R , (cid:18) ∆ + k (cid:16) q ( x ) (cid:17)(cid:19) u = 0 , (1.1)where u is the total field, q ( x ) > k is the wave number.We consider an inclusion D contained inside a homogeneous background medium, and assume that D is an open bounded connected domain with a C ,α -boundary for some 0 < α <
1. Suppose that thefunction q is of the form q ( x ) = ε ∗ χ D ( x ) , (1.2)where χ D denotes the characteristic function of D and ε ∗ > (cid:12)(cid:12) ∂∂ | x | u s − iku s (cid:12)(cid:12) = O ( | x | − ) as | x | → ∞ , (1.3)where u s := u − u i is the scattered field and u i is the incident field. The solution u to the system (1.1)-(1.3) represents the total field due to the scattering from the inclusion D corresponding to the incidentfield u i .Following the work of [3, 5, 6], the scattering coefficients provide a powerful and efficient tool forshape classification of the target D . Therefore, we aim at exhibiting the mechanism underlying the super-resolution phenomenon experimentally-observed in [8] in terms of the scattering coefficients correspondingto high-contrast inclusions.In [5], it is proved that the scattering coefficient of order ( n, m ) decays very quickly as the orders | n | , | m | increase. Nonetheless, it is shown in [6] that the scattering coefficients can be stably reconstructedfrom the far-field measurements by a least-squares method. The stability of the reconstruction in thepresence of a measurement noise is analyzed and the resolving power of the reconstruction in terms ofthe signal-to-noise-ratio is estimated. It is the purpose of this paper to use the scattering coefficients toestimate the resolution limit for imaging high contrast targets from far-field measurements as function ofthe material contrast, and to prove that the higher the permittivity is inside the target, the better theresolution is for imaging its shape from far-field measurements.In order to achieve this goal, in this work, we first give a decay estimate of the scattering coefficientsin arbitrary shaped domains, and then in the particular case of a circular domain. Our estimate showsdifferent behaviors of the scattering coefficients of different orders as the material contrast increases.Then we provide a sensitivity analysis of the scattering coefficients, which clearly shows that, in thelinearized case, the scattering coefficient of order ( n, m ) of a circular domain contains information aboutthe ( n − m )-th Fourier mode of the shape perturbation. Afterwards, we establish the asymptotic behaviorof eigenvalues of an important family of integral operators closely related to the scattering coefficients.Series representations of the scattering coefficients and their shape derivatives in the case of a circulardomain are given based on this asymptotic behavior. From these series representations, we prove that asthe material contrast increases and moves close to the reciprocal of the eigenvalues, the shape derivativesof the scattering coefficients behave like simple poles. This explains the better conditioning of the inver-sion process of higher Fourier modes of inclusions with large material contrast, and hence an enhanced2esolution of reconstructing the perturbation using the scattering coefficients. Numerical examples illus-trate that the relative magnitudes of higher order scattering coefficients grow as the medium coefficientsgrow and move close to the reciprocals of the eigenvalues, therefore providing more information aboutthe shape of the domain with a fixed signal-to-noise ratio. Our approach provides a good and promisingdirection of understanding towards the super-resolution phenomenon for high-contrast targets.This paper is organized as follows. In section 2 we give a brief review of the concept of scatteringcoefficients. We also prove a fundamental expression of the scattering coefficients in terms of a familyof important integral operators. Sensitivity analysis of the scattering coefficients with a fixed contrastis then presented in section 3, which shows that the shape derivative can also be represented by thefamily of integral operators introduced in section 2. Section 4.1 briefly recalls Riesz decomposition ofcompact operators. Asymptotic behavior of eigenvalues and eigenfunctions of the introduced integraloperators will be studied in section 4.2. Section 4.3 provides a series representation of the scatteringcoefficients and their shape derivative. A mathematical explanation of the super-resolution phenomenonis given. Numerical results are reported in section 5 to illustrate the phenomenon of super-resolution asthe material contrast increases. In this section, we estimate the behavior of the scattering coefficients. Without loss of generality,from now on, we normalize the wave number k in (1.1) to be k = 1 by a change of variables.To begin with, we first recall the definition of the scattering coefficients W nm ( D, ε ∗ ) from [3, 5]. Forthis purpose, we introduce the following several notions. The fundamental solution Φ to the Helmholtzoperator ∆ + 1 in two dimensions satisfying(∆ + 1)Φ( x ) = δ ( x ) , (2.1)where δ is the Dirac mass at 0, with the outgoing Sommerfeld radiation condition: (cid:12)(cid:12) ∂∂ | x | Φ − i Φ (cid:12)(cid:12) = O ( | x | − ) as | x | → ∞ , is given by Φ( x ) = − i H (1)0 ( | x | ) , (2.2)where H (1)0 is the Hankel function of the first kind of order zero.Now, given an incident field u i satisfying the homogeneous Helmholtz equation, i.e.,∆ u i + u i = 0 , (2.3)the solution u to (1.1) and (1.3) can be readily represented by the Lippmann-Schwinger equation as u ( x ) = u i ( x ) − ε ∗ Z D Φ( x − y ) u ( y ) dy , x ∈ R , (2.4)and hence, the scattered field reads u s ( x ) = − ε ∗ Z D Φ( x − y ) u ( y ) dy , x ∈ R . (2.5)Let S ∂D be the single-layer potential defined by the kernel Φ( · ), i.e., S ∂D [ φ ]( x ) = Z ∂D Φ( x − y ) φ ( y ) ds ( y ) (2.6)for φ ∈ L ( ∂D ). Let S √ ε ∗ +1 ∂D be the single-layer potential associated with the kernel Φ (cid:0) √ ε ∗ ( · ) (cid:1) .3 efinition 2.1. The scattering coefficient W nm ( D, ε ∗ ) for n, m ∈ Z is defined as follows: W nm ( D, ε ∗ ) = Z ∂ Ω J n ( r x ) e − inθ x φ m ( x ) ds ( x ) , (2.7) where x = r x (cos θ x , sin θ x ) in polar coordinates and the weight function φ m ∈ L ( ∂D ) is such that thepair ( φ m , ψ m ) ∈ L ( ∂D ) × L ( ∂D ) satisfies the following system of integral equations: ( S √ ε ∗ +1 ∂D [ φ m ]( x ) − S ∂D [ ψ m ]( x ) = J m ( r x ) e imθ x , ∂∂ν S √ ε ∗ +1 ∂D [ φ m ] | − ( x ) − ∂∂ν S ∂D [ ψ m ] | + ( x ) = ∂∂ν ( J m ( r x ) e imθ x ) . (2.8)Here + and − in the subscripts respectively indicate the limit from outside D and inside D to ∂D alongthe normal direction, and ∂/∂ν denotes the normal derivative.According to [3, 5], the scattering coefficients W nm ( D, ε ∗ ) are basically the Fourier coefficients of thefar-field pattern (scattering amplitude) which is 2 π -periodic function in two dimensions. The far-fieldpattern A ∞ ( b d, b x ), when the incident field is given by e i b d · x for a unit vector b d , is defined to be( u − u i )( x ) = ie − πi/ e i | x | p π | x | A ∞ ( b d, b x ) + O ( | x | − ) as | x | → ∞ , with b x := x/ | x | . We have, recalling from [3, 5], that W nm ( D, ε ∗ ) = i n − m F θ d ,θ x [ A ∞ ( b d, b x )]( − m, n ) , (2.9)where b x = (cos θ x , sin θ x ) and b d = (cos θ d , sin θ d ) in polar coordinates and F θ d ,θ x [ A ∞ ( b d, b x )]( m, n ) denotesthe ( m, n )-th Fourier coefficient of the far-field pattern A ∞ ( b d, b x ).Our first objective is then to work out an explicit relation between the far-field pattern and thecontrast ε ∗ so as to obtain the behavior of the scattering coefficients when ε ∗ is large.In view of (2.4), we introduce the following operator for the subsequent analysis. Definition 2.2.
The operator e K D : L ( D ) → L ( D ) is defined by e K D [ φ ]( x ) = Z D Φ( x − y ) φ ( y ) dy , for x ∈ D and φ ∈ L ( D ) ; (2.10) whereas, the operator ee K D : L ( D ) → L ∞ ( R ) is given by ee K D [ φ ]( x ) = Z D Φ( x − y ) φ ( y ) dy , for x ∈ R and φ ∈ L ( D ) . (2.11)It is easy to see from the definition of e K D and the Rellich lemma that e K D is a compact operator. However,it is worth emphasizing that e K D is not a normal operator in L ( D ). Therefore, it is not unitary equivalentto a multiplicative operator. With Definition 2.2, we can rewrite (2.4) as( I + ε ∗ e K D )[ u ]( x ) = u i ( x ) , ∀ x ∈ D , (2.12)hence in L ( D ), u = ( I + ε ∗ e K D ) − [ u i ] . (2.13)From the well-known fact thatΦ( x − y ) = − i H (1)0 ( | x − y | ) = − ie − πi/ e i | x |− i b x · y p π | x | + O ( | x | − ) as | x | → ∞ , (2.14)4e have u s ( x ) = − ε ∗ Z D Φ( x − y ) u ( y ) dy = iε ∗ e − πi/ e i | x | p π | x | Z D e − i b x · y u ( y ) dy + O ( | x | − ) as | x | → ∞ . (2.15)Therefore, the far-field of the scattered field can be written as A ∞ ( θ d , θ x ) := A ∞ ( b d, b x ) = ε ∗ Z D e − i b x · y u ( y ) dy . (2.16)Recall the following well-known Jacobi-Anger identity [26] for any unit vector b d , e − i b d · x = ∞ X n = −∞ ( − i ) n J n ( r ) e in ( θ d − θ ) (2.17)for x = ( r, θ ) in polar coordinates. Using (2.17) and taking the Fourier transform with respect to θ x , weget F θ x [ A ∞ ]( n ) = ( − i ) n ε ∗ h J n ( r ) e inθ , u i L ( D ) = i − n D J n ( r ) e inθ , ( ε ∗− + e K D ) − [ u i ] E L ( D ) . (2.18)Now using u i ( x ) = e i b d · x , it follows from (2.9) and (2.17)-(2.18) that the following theorem holds: Theorem 2.3.
For a domain D and a contrast ε ∗ , the scattering coefficient W nm ( D, ε ∗ ) for n, m ∈ Z can be written in the following form W nm ( D, ε ∗ ) = i ( n − m ) F θ d ,θ x [ A ( θ d , θ x )]( − m, n ) = D J n ( r ) e inθ , ( ε ∗− + ˜ K D ) − [ J m ( r ) e imθ ] E L ( D ) , (2.19) where ˜ K D is defined by (2.10) . The expression (2.19) of the scattering coefficients W nm will be fundamental to the analysis of thebehavior of W nm with respect to ε ∗ .Using (2.19), we can readily obtain an a priori estimate for the coefficients W nm . Let us first recallthe following facts on Schatten-von Neumann ideals; see, for example, [16]. Given a Hilbert space H , welet B ( H ) to be the set of bounded operators on H . We denote by S ∞ ( H ) the closed two-sided ideal ofcompact operators in B ( H ). For K ∈ S ∞ and k ∈ N , let the k -th singular number s k ( K ) be definedas the k -th eigenvalue of | K | = √ K ∗ K ordered in descending order of magnitude and being repeatedaccording to its multiplicity, written as s k ( K ) := λ k ( | K | ). Now, for 0 < p ≤ ∞ , we shall often write thefollowing Schatten-von Neumann quasi-norms (which are norms if 1 ≤ p ≤ ∞ ) as follows : || K || S p ( H ) := ∞ X k =1 s k ( K ) p ! /p for p < ∞ ; || K || S ∞ ( H ) := || K || H , (2.20)whenever they are finite. Now let the Schatten-von Neumann quasi-normed operator ideal S p ( H ) bedefined by S p ( H ) := (cid:8) K ∈ S ∞ : || K || S p ( H ) < ∞ (cid:9) . (2.21)Note that with this convention, S ( H ) is the well-known trace class, S ( H ) is the usual Hilbert-Schmidtclass, and S ∞ ( H ) is the usual class of compact operators in H . Moreover, if H = L ( D ) and K ∈ S ( H )is the integral operator defined by K [ f ]( x ) = Z D K ( x, y ) f ( y ) dy, for x ∈ D and f ∈ L ( D ) , (2.22)5hen it holds that || K || S ( L ( D )) = Z D Z D | K ( x, y ) | dx dy, (2.23)which is always well-defined for any K ∈ S ( L ( D )). We refer the reader to, for example, [16] for moreproperties concerning the Schatten-von Neumann ideals.For a compact operator K , let σ ( K ) := { λ ∈ C | λ − K is singular } denote its spectrum and ( z − K ) − its resolvent operator whenever z ∈ C \ σ ( K ). Now, we have the following resolvent estimate [9]. Theorem 2.4.
For < p < ∞ and K ∈ S p ( H ) , we have the following estimate for the resolvent operator ( z − K ) − that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( z − K ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ≤ d ( z, σ ( K )) exp a p || K || pS p ( H ) d ( z, σ ( K )) p + b p ! , (2.24) where a p , b p are two constants depending on p and d ( z, σ ( K )) is defined by d ( z, σ ( K )) := inf λ ∈ σ ( K ) | z − λ | . (2.25)Now we can apply Theorem 2.4 to get an estimate for W nm ( D, ε ∗ ). In fact, with the logarithmic typesingularity of the function H (1)0 , we readily obtain that || ˜ K D || S ( L ( D )) = Z D Z D | H (1)0 ( | x − y | ) | dx dy < C (1 + R ) (1 + log R ) < ∞ (2.26)whenever D ⊂ B (0 , R ), and hence ˜ K D ∈ S . Therefore, using the Cauchy-Schwartz inequality andapplying (2.24) for H = L ( D ) to (2.19), together with the following well-known asymptotic expressionof J m for large m [1, pp. 365-366 ], J m ( z ) (cid:30) √ πm (cid:16) ez m (cid:17) m → m → ∞ , (2.27)we readily obtain the following inequality (using that a = 1 / , b = 1 / p = 2 [13]): | W nm ( D, ε ∗ ) | = (cid:12)(cid:12)(cid:12)(cid:12)D J n ( r ) e inθ , ( ε ∗− + ˜ K D ) − [ J m ( r ) e imθ ] E L ( D ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ε ∗− + ˜ K D ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( D ) (cid:12)(cid:12)(cid:12)(cid:12) J n ( r ) e inθ (cid:12)(cid:12)(cid:12)(cid:12) L ( D ) (cid:12)(cid:12)(cid:12)(cid:12) J m ( r ) e imθ (cid:12)(cid:12)(cid:12)(cid:12) L ( D ) ≤ d ( − ε ∗− , σ ( ˜ K D )) exp || ˜ K D || S ( L ( D )) d ( − ε ∗− , σ ( ˜ K D )) + 12 ! (cid:12)(cid:12)(cid:12)(cid:12) J n ( r ) e inθ (cid:12)(cid:12)(cid:12)(cid:12) L ( D ) (cid:12)(cid:12)(cid:12)(cid:12) J m ( r ) e imθ (cid:12)(cid:12)(cid:12)(cid:12) L ( D ) ≤ d ( − ε ∗− , σ ( ˜ K D )) exp (cid:18) C ,R d ( − ε ∗− , σ ( ˜ K D )) + 12 (cid:19) C | m | + | n | ,R | m | | m | | n | | n | , where C i,R ( i = 1 ,
2) are some constants, which depend only on the radius R such that D ⊂ B (0 , R ). Wesummarize the above result in the following theorem. Theorem 2.5.
For a given domain D and a contrast ε ∗ , we have the following estimate for the scatteringcoefficient W nm ( D, ε ∗ ) , for n, m ∈ Z , | W nm ( D, ε ∗ ) | ≤ d ( − ε ∗− , σ ( ˜ K D )) exp (cid:18) C ,R d ( − ε ∗− , σ ( ˜ K D )) + 12 (cid:19) C | m | + | n | ,R | m | | m | | n | | n | . (2.28)From Theorem 2.5, we foresee that the magnitude of W nm may grow as ε ∗ increases, and becomes avery large value as ε ∗− is close to the spectrum of the operator ˜ K D .6 .1 The case of a circular domain Now, we consider the operator ˜ K D for a circular domain, i.e., when D = B (0 , R ). In this case, theoperator ˜ K D becomes more explicit. Actually, from the Graf’s formula [26], we have for | x | 6 = | y | that H (1)0 ( | x − y | ) = ∞ X m = −∞ χ {| x | < | y |} J m ( | x | ) e − imθ x H (1) m ( | y | ) e imθ y + χ {| x | > | y |} H (1) m ( | x | ) e − imθ x J m ( | y | ) e imθ y . Therefore, for all f ∈ L ( D ), the operator e K D can be written as e K D [ f ]( y ) = − i ∞ X m = −∞ (cid:20) h J m ( r ) e imθ , f i D T B (0 , | y | ) H (1) m ( | y | ) e imθ y + h H (1) m ( r ) e imθ , f i D \ B (0 , | y | ) J m ( | y | ) e imθ y (cid:21) . The above expression of ˜ K D will be helpful to investigate the behavior of ˜ K D and W nm . Before wecontinue our discussion on the operator ˜ K D , we shall first define some operators. Definition 2.6.
Given an integer m ∈ Z , the operators e K ( i ) m : L ((0 , R ) , r dr ) → L ((0 , R ) , r dr ) for i = 1 , are defined as e K ( i ) m [ φ ]( h ) = − i (cid:18) Z h rJ m ( r ) φ ( r ) dr (cid:19) H ( i ) m ( h ) − i (cid:18) Z Rh rH ( i ) m ( r ) φ ( r ) dr (cid:19) J m ( h ) (2.29) for h ∈ (0 , R ) and φ ∈ L ((0 , R ) , r dr ) , and their extensions ee K ( i ) m : L ((0 , R ) , r dr ) → L ∞ ((0 , + ∞ )) for i = 1 , as ee K ( i ) m [ φ ]( h ) = − i (cid:18) Z h rJ m ( r ) φ ( r ) dr (cid:19) H ( i ) m ( h ) − i (cid:18) Z Rh rH ( i ) m ( r ) φ ( r ) dr (cid:19) J m ( h ) (2.30) for h ∈ (0 , + ∞ ) and φ ∈ L ((0 , R ) , r dr ) . With this notion, we can readily see that if f ∈ L ( D ) has the form f = φ ( r ) e imθ , then we have in polarcoordinates by the orthogonality of { e imθ } m ∈ Z on L ( S ) that e K D [ f ]( h, θ ) = − i (cid:18) Z h rJ m ( r ) φ ( r ) dr (cid:19) H (1) m ( h ) e imθ − i (cid:18) Z Rh rH (1) m ( r ) φ ( r ) dr (cid:19) J m ( h ) e imθ = e K (1) m [ φ ]( h ) e imθ , (2.31)and e K ∗ D [ f ]( h, θ ) = e K (2) m [ φ ]( h ) e imθ . Furthermore, we can directly see that σ ( ˜ K (2) m ) = σ ( ˜ K (1) m ). Moreover,using the following relations for all m ∈ Z , J − m ( z ) = ( − m J m ( z ) and H (1) − m ( z ) = ( − m H (1) m ( z ) , (2.32)we immediately infer the properties for the integral operators: e K ( i ) − m = e K ( i ) m and ee K ( i ) − m = ee K ( i ) m . (2.33)Substituting (2.31) into Theorem 2.3, we obtain the following simplified expressions of the scatteringcoefficients when D = B (0 , R ). 7 heorem 2.7. For a domain D = B (0 , R ) for some R > and a contrast value ε ∗ , the scatteringcoefficient W nm ( D, ε ∗ ) , n, m ∈ Z , can be written in the following form W nm ( D, ε ∗ ) = δ nm (cid:28) J n , (cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:29) L ((0 ,R ) ,r dr ) , (2.34) where δ nm is the Kronecker symbol. As a consequence of Theorem 2.7, we easily see that W nm = 0 for n = m . Moreover, we readilyhave the following a priori estimate for the coefficients W nm by the same arguments as those in Theorem2.5. In order to obtain the desired estimate, we consider the asymptotic expression of Y m as m → ∞ [1,pp. 365-366 ]: Y m ( z ) (cid:30)r πm (cid:16) ez m (cid:17) − m → . (2.35)Together with (2.27) and the logarithmic type singularity of Y , we have from the definitions of e K ( i ) m for i = 1 , || e K ( i ) m ) || S ( L ((0 ,R ) ,r dr )) ≤ C m (1 + R ) (1 + log R ) < ∞ . (2.36)Consequently, following the same arguments as the ones for (2.28), we arrive at the estimate: | W nm ( D, ε ∗ ) | = δ nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) J n , (cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:29) L ((0 ,R ) ,r dr ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ δ nm d (cid:16) − ε ∗− , σ ( ˜ K (1) m ) (cid:17) exp C m C ,R d (cid:16) − ε ∗− , σ ( ˜ K (1) m ) (cid:17) + 12 C | m | + | n | ,R | m | | m | | n | | n | , where C m is a constant depending only on m and C i,R , i = 1 , , are constants only depending on theradius R such that D ⊂ B (0 , R ). Theorem 2.8.
For a circular domain D = B (0 , R ) and a contrast ε ∗ , we have the following estimatefor the scattering coefficient W nm ( D, ε ∗ ) , for n, m ∈ Z , | W nm ( D, ε ∗ ) | ≤ δ nm d (cid:16) − ε ∗− , σ ( ˜ K (1) m ) (cid:17) exp C m C ,R d (cid:16) − ε ∗− , σ ( ˜ K (1) m ) (cid:17) + 12 C | m | + | n | ,R | m | | m | | n | | n | . (2.37)In the next section we perform a sensitivity analysis of the scattering coefficients in order to obtain aquantitative description of what piece of information is provided by the scattering coefficients of differentorders. In this section, for a given contrast ε ∗ , we calculate the shape derivative D W nm ( D, ε ∗ )[ h ] of thescattering coefficient W nm ( D, ε ∗ ) along the variational direction h ∈ C ( ∂D ) when ∂D is of class C .From the shape derivative, we will clearly understand what piece of information is provided by thescattering coefficients of different orders, and how the knowledge of scattering coefficient of order ( n, m )is related to the resolution of the reconstructed shapes.Before going into the sensitivity analysis, we will consider the inclusion of the operators and spectrabetween operators for the subsequent analysis. To do so, we define the following inclusion maps.8 efinition 3.1. For a given domain D , suppose that the bounded linear operator ˜ K D ∈ B (cid:0) L ( D ) (cid:1) isdefined as in (2.10) . Consider any domain b D such that D ⊂ b D , we shall often write ι ( ˜ K D ) ∈ B (cid:16) L ( b D ) (cid:17) as the following operator: ι ( ˜ K D ) [ f ] ( x ) = ˜˜ K D [ χ D f ] ( x ) for any f ∈ L ( b D ) , (3.1) where χ D is the characteristic function of D . Likewise, for a given radius R > , assume the boundedlinear operators ˜ K ( i ) m ∈ B (cid:0) L ((0 , R ) , r dr ) (cid:1) ( m ∈ Z , i = 1 , , which are defined in (2.29) . Then wewrite ι ( ˜ K ( i ) m ) [ f ] ( x ) = ˜˜ K ( i ) m (cid:2) χ (0 ,R ) f (cid:3) ( x ) for any f ∈ L ((0 , b R ) , r dr ) . (3.2)Then the operators ι ( ˜ K D ) and ι ( ˜ K ( i ) m ) , i = 1 , , are compact on L ((0 , b R ) , r dr ). Moreover, we havethe following relations between the spectra of ˜ K D and ι ( ˜ K D ), as well as between ˜ K ( i ) m and ι ( ˜ K ( i ) m ) for m ∈ Z , i = 1 , Lemma 3.2.
Let ˜ K D and ι ( ˜ K D ) be defined as in (2.10) and (3.1) , respectively. Then, the followingsimple relationship between the spectra of ˜ K D and ι ( ˜ K D ) holds: σ ( ι ( ˜ K D )) = σ ( ˜ K D ) [ { } . (3.3) Likewise, for m ∈ Z , i = 1 , , we have σ ( ι ( ˜ K ( i ) m )) = σ ( ˜ K ( i ) m ) [ { } . (3.4) Proof.
For a given λ , suppose that the pair ( λ, e λ ) is an eigenpair of ˜ K D over L ( D ). If λ = 0, we denoteby e e λ ∈ L ( b D ) the following function e e λ := 1 λ gg K D [ e λ ] . (3.5)If λ = 0, we write e e λ ∈ L ( b D ) as the extension by zero of the function e λ outside the domain D , i.e., e e λ ( x ) := ( e λ ( x ) if x ∈ D , . (3.6)Then we readily check from the definition of ι ( ˜ K D ) that ι ( ˜ K D )[ e e λ ] = λ e e λ and hence the pair ( λ, e e λ ) isan eigenpair of ι ( ˜ K D ) over L ( b D ). As any function f ∈ L ( b D \ D ) is a zero eigenfunction of ι ( ˜ K D ), hencewe know σ ( ˜ K D ) S { } ⊂ σ ( ι ( ˜ K D )).Conversely, if a pair ( λ, e e λ ) is an eigenpair of ι ( ˜ K D ) over L ( b D ), then, by writing e λ := e e λ | D , it iseasy to see form the definition of ˜ K D that ( λ, e λ ) is an eigenpair of ˜ K D . Hence, σ ( ι ( ˜ K D )) ⊂ σ ( ˜ K D ). Theproof of σ ( ι ( ˜ K ( i ) m )) = σ ( ˜ K ( i ) m ) S { } is the same.Lemma 3.2 and the Fredholm alternative yield that ε ∗− + ι ( ˜ K D ) is invertible over L ( b D ) if and onlyif ε ∗− + ˜ K D is invertible over L ( D ). Moreover, from the definition, we can show as in section 2 that ι ( ˜ K D ) ∈ S ( L ( b D )) and then apply (2.24) to obtain the following resolvent estimate for ε ∗− + ι ( ˜ K D )that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ε ∗− + ι ( ˜ K D ) (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( b D ) ≤ d (cid:16) − ε ∗− , σ ( ι ( ˜ K D )) (cid:17) exp C ,R d (cid:16) − ε ∗− , σ ( ι ( ˜ K D )) (cid:17) + 12 = 1 d (cid:16) − ε ∗− , σ ( ˜ K D ) (cid:17) exp C ,R d (cid:16) − ε ∗− , σ ( ˜ K D ) (cid:17) + 12 . (3.7)9ere the last equality comes from Lemma 3.2 and the fact that σ ( e K D ) must have zero as its accumulationpoint, since L ( D ) is infinite dimensional. The above argument also applies to the operators ι ( ˜ K ( i ) m ) for m ∈ Z , i = 1 ,
2, where the resolvent estimate reads (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ε ∗− + ι ( ˜ K ( i ) m )) (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ((0 , b R ) ,r dr ) ≤ d (cid:16) − ε ∗− , σ ( ˜ K (1) m ) (cid:17) exp C m C ,R d (cid:16) − ε ∗− , σ ( ˜ K (1) m ) (cid:17) + 12 . (3.8)Furthermore, we can easily recover the relationship between ι ( e K B (0 ,R ) ) and ι ( ˜ K ( i ) m ) for any D such that B (0 , R ) ⊂ D from their definitions. In fact, for any f ∈ L ( D ) in the form f = φ ( r ) e imθ , where ( r, θ ) ∈ D ,we have in polar coordinates that ι ( e K B (0 ,R ) )[ f ]( h, θ ) = ι ( e K (1) m )[ φ ]( h ) e imθ , ι ( e K ∗ B (0 ,R ) )[ f ]( h, θ ) = ι ( e K (2) m )[ φ ]( h ) e imθ , (3.9)where the operators ι ( e K ( i ) m ) for m ∈ Z , i = 1 , , are the extensions to L ((0 , b R θ ) , r dr ) with the radii b R θ being defined as b R θ := sup { r : ( r, θ ) ∈ D } for different θ ∈ [0 , π ]. Although the extensions ι ( e K ( i ) m ) arenow different for different angles θ , no difficulty will arise in understanding the properties of ι ( e K B (0 ,R ) )via estimating ι ( e K (1) m ), since the conclusions of Lemma 3.2 and (3.8) do not depend on the choice of b R and thus can be applied to different choices of radii.From now on, we will no longer distinguish between the operators ˜ K D and ι ( ˜ K D ) whenever there isno ambiguity, and by an abuse of notation, we denote both operators by ˜ K D , likewise for the operators˜ K ( i ) m and ι ( ˜ K ( i ) m ) for m ∈ Z , i = 1 , D along a perturbation h ∈ C ( ∂D ). Now let ε ∗ be given. For anybounded C -domain D in R , let D δ be a δ -perturbation of D along the variational direction h ∈ C ( ∂D ),i.e., ∂D δ := (cid:26) ˜ x = x + δh ( x ) ν ( x ) : x ∈ ∂D (cid:27) , (3.10)where ν ( x ) is the outward unit normal at ∂D . For such perturbations of the domain D , we investigatethe difference between W nm ( D δ , ε ∗ ) and W nm ( D, ε ∗ ). We first estimate the difference ˜ K D δ − ˜ K D , whereboth operators ˜ K D δ and ˜ K D are regarded as the extended operators on L (cid:0) D δ S D (cid:1) . Indeed, from thefact that the singularity type of the function H (1)0 is logarithmic, there exists a constant C R dependingonly on the radius R such that the estimate || ˜ K D δ − ˜ K D || L ( B (0 ,R )) ≤ C R δ (3.11)holds for δ small enough with R being such that D ⋐ B (0 , R ). Therefore, we can repeatedly apply thefollowing resolvent equalities (cid:16) ε ∗− + ˜ K D δ (cid:17) − − (cid:16) ε ∗− + ˜ K D (cid:17) − = (cid:16) ε ∗− + ˜ K D δ (cid:17) − ( ˜ K D − ˜ K D δ ) (cid:16) ε ∗− + ˜ K D (cid:17) − (3.12)= (cid:16) ε ∗− + ˜ K D (cid:17) − ( ˜ K D − ˜ K D δ ) (cid:16) ε ∗− + ˜ K D δ (cid:17) − (3.13)10o obtain the following expression of the difference of scattering coefficients for any n, m ∈ Z , W nm ( D δ , ε ∗ ) − W nm ( D, ε ∗ )= (cid:28)(cid:16) ε ∗− + ˜ K ∗ D δ (cid:17) − [ J n ( r ) e inθ ] , J m ( r ) e imθ (cid:29) L ( D δ ) − (cid:28)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] , J m ( r ) e imθ (cid:29) L ( D ) = (cid:28) J n ( r ) e inθ , (cid:20)(cid:16) ε ∗− + ˜ K D δ (cid:17) − − (cid:16) ε ∗− + ˜ K D (cid:17) − (cid:21) [ J m ( r ) e imθ ] (cid:29) L ( D ) + (cid:28)(cid:16) ε ∗− + ˜ K ∗ D δ (cid:17) − [ J n ( r ) e inθ ] , sgn( h ) J m ( r ) e imθ (cid:29) L ( D S D δ \ D T D δ ) = − (cid:28)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] , ( ˜ K D δ − ˜ K D ) (cid:16) ε ∗− + ˜ K D δ (cid:17) − [ J m ( r ) e imθ ] (cid:29) L ( D ) + (cid:28)(cid:16) ε ∗− + ˜ K ∗ D δ (cid:17) − [ J n ( r ) e inθ ] , sgn( h ) J m ( r ) e imθ (cid:29) L ( D S D δ \ D T D δ ) = − (cid:28)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] , ( ˜ K D δ − ˜ K D ) (cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:29) L ( D ) + (cid:28)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] , sgn( h ) J m ( r ) e imθ (cid:29) L ( D S D δ \ D T D δ ) + O ( δ ) , (3.14)where the last equality comes from (3.11). Now for any L function f , considering the fact that the shapederivative of the integral I [ D ] = Z D f ( x ) dx (3.15)is given by the following boundary integral D I [ D ]( h ) = Z ∂D f ( x ) h ( x ) ds ( x ) , (3.16)we have for x ∈ D S D δ and φ ∈ L ( D S D δ ) that( ˜ K D δ − ˜ K D )[ φ ]( x ) = − i Z ( D S D δ ) \ ( D T D δ ) sgn( h ) H (1)0 ( | x − y | ) φ ( y ) dy = − δ i Z ∂D H (1)0 ( | x − y | ) h ( y ) φ ( y ) ds ( y ) + O ( δ ) . (3.17)Therefore, by substituting the above expression into (3.14), a direct expansion of the integral togetherwith the Fubini’s theorem yields the following expression for the first term in (3.14): − (cid:28)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] , ( ˜ K D δ − ˜ K D ) (cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:29) L ( D ) = δ i Z D Z ∂D H (1)0 ( | x − y | ) h ( y ) (cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) ( y ) dy (cid:20)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] (cid:21) ( x ) dx + O ( δ )= − δ Z ∂D h ( y ) (cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) ( y ) (cid:20) ˜ K ∗ D (cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] (cid:21) ( y ) dy + O ( δ )= − δ *(cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) (cid:20) ˜ K ∗ D (cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] (cid:21) , h + L ( ∂D ) + O ( δ ) . (3.18)11ikewise, for the second term in (3.14), we derive that (cid:28)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] , sgn( h ) J m ( r ) e imθ (cid:29) L ( D S D δ \ D T D δ ) = δ Z ∂D h ( y ) (cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) ( y )[ J n ( r ) e inθ ] ( y ) dy + O ( δ )= δ *(cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) (cid:2) J n ( r ) e inθ (cid:3) , h + L ( ∂D ) + O ( δ ) . (3.19)Therefore, combining the above two estimates shows that W nm ( D δ , ε ∗ ) − W nm ( D, ε ∗ )= δε ∗− *(cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) (cid:20)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] (cid:21) , h + L ( ∂D ) + O ( δ ) . (3.20)Hence, if we define the following L ( ∂D )-duality gradient function ∇ W nm ( D, ε ∗ ) of the form of ∇ W nm ( D, ε ∗ ) := ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) (cid:20)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] (cid:21) , (3.21)then the shape derivative of the scattering coefficient W nm ( D, ε ∗ ) along the variational direction h isgiven by D W nm ( ε ∗ , D )[ h ] = h∇ W nm ( ε ∗ , D ) , h i L ( ∂D ) . (3.22)In particular, for the case where D is a circular domain D = B (0 , R ), we have from the decomposition ofthe operator ˜ K D the following simple expression of ∇ W nm ( D, ε ∗ ): ∇ W nm ( B (0 , R ) , ε ∗ ) = ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K B (0 ,R ) (cid:17) − [ J m ( r ) e imθ ] (cid:21) (cid:20)(cid:16) ε ∗− + ˜ K ∗ B (0 ,R ) (cid:17) − [ J n ( r ) e inθ ] (cid:21) , = ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K (2) m (cid:17) − [ J m ] (cid:21) ( R ) (cid:20)(cid:16) ε ∗− + ˜ K (2) n (cid:17) − [ J n ] (cid:21) ( R ) e i ( n − m ) θ . Consequently, D W nm ( B (0 , R ) , ε ∗ )[ h ] = ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:21) ( R ) (cid:20)(cid:16) ε ∗− + ˜ K (1) n (cid:17) − [ J n ] (cid:21) ( R ) D e i ( n − m ) θ , h E L ( ∂D ) = ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:21) ( R ) (cid:20)(cid:16) ε ∗− + ˜ K (1) n (cid:17) − [ J n ] (cid:21) ( R ) F θ [ h ] ( n − m ) , (3.23)where F θ [ h ] ( n − m ) is the ( n − m )-th Fourier coefficient of the function h on L ( S ). This gives thefollowing key result on the shape derivative of W nm ( D, ε ∗ ) . Theorem 3.3.
Suppose that ε ∗ > is given. For any C -domain D and n, m ∈ Z , the shape derivativeof the scattering coefficient W nm ( D, ε ∗ ) along the variational direction h ∈ L ( ∂D ) is given by D W nm ( D, ε ∗ )[ h ] = h∇ W nm ( D, ε ∗ ) , h i L ( ∂D ) , (3.24) where ∇ W nm is defined by ∇ W nm ( D, ε ∗ ) = ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K D (cid:17) − [ J m ( r ) e imθ ] (cid:21) (cid:20)(cid:16) ε ∗− + ˜ K ∗ D (cid:17) − [ J n ( r ) e inθ ] (cid:21) . (3.25)12 n particular, if the domain D is a circular domain D = B (0 , R ) , then for any D δ as a δ -perturbation of D along the variational direction h ∈ C ( ∂D ) , we have W nm ( D δ , ε ∗ ) − W nm ( D, ε ∗ ) = δ C ( ε ∗ , n, m ) F θ [ h ] ( n − m ) + O ( δ ) , (3.26) with C ( ε ∗ , n, m ) := ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:21) ( R ) (cid:20)(cid:16) ε ∗− + ˜ K (1) n (cid:17) − [ J n ] (cid:21) ( R ) . (3.27)From the above theorem, we obtain in the linearized case that the scattering coefficient W nm gives usprecise information about the ( m − n )-th Fourier mode of the perturbation h .Therefore, the magnitude of the coefficients W nm and C ( ε ∗ , n, m ) shall be responsible for the resolutionin imaging D δ . Note that the function C ( ε ∗ , n, m ) depends now on the spectra of both ˜ K (1) m and ˜ K (1) n .The change and growth of the coefficients W nm and C ( ε ∗ , n, m ) with respect to ε ∗ will be the main focusof the next section. In the previous section, we have obtained a relationship between the coefficients W nm of a perturbedcircular domain D δ and the Fourier coefficients of the perturbation h . In this section, we investigate thedecay of the eigenvalues of ˜ K (1) m and analyze the behavior with respect to ε ∗ of W nm and C ( ε ∗ , n, m ) fordifferent values of n and m . For this purpose, we introduce the following Riesz decomposition. To continue our analysis on the operators e K D and ˜ K (1) m , we first recall the following classical spectraltheorem for compact operators in a Hilbert space [14]. Theorem 4.1.
Let K be a compact operator on a Hilbert space H and σ ( K ) its spectrum. Then thefollowing results hold:1. λ ∈ σ ( K ) if and only if λ is an eigenvalue (Fredholm alternative).2. For all λ ∈ σ ( K ) , there exists a smallest m λ such that Ker ( λ − K ) m λ = Ker ( λ − K ) m λ +1 . Denotingthe space Ker ( λ − K ) m λ by E λ , we have dim ( E λ ) < ∞ .3. σ ( K ) is countable and is the only accumulation point of σ ( K ) for dim ( H ) = ∞ .4. The map z ( z − K ) − admits poles at z ∈ σ ( K ) . Applying the above theorem to e K D , which is compact but not normal, we can decompose L ( D ) = M λ ∈ σ ( e K D ) E λ with E λ = L ≤ i ≤ N λ E iλ for some N λ such that the operator e K D can be written as e K D = X λ ∈ σ ( e K D ) X ≤ i ≤ N λ e K i,λ , (4.1)13here the operators e K i,λ : E iλ → E iλ admit the action of the following Jordan block under a choice ofbasis e iλ in E iλ : J iλ := λ . . . . . . λ . . . . . . . . . λ . . . . . . . . . λ , (4.2)as matrices of size smaller than or equal to m λ . For the sake of simplicity, for a given n ∈ N and agiven Riesz basis w , i.e., a complete frame in L ( D ), supposing that v is a finite subset of w , we shalloften write, for any φ ∈ L ( D ), ( φ ) v ,L ( D ) ∈ C n as the coefficients of φ in front of the vectors in v whenexpressed in the Riesz basis w , i.e., if φ = X w i ∈ w b i w i , (4.3)for coefficients b i ∈ C and v = ( w k , w k , . . . , w k n ), then ( φ ) v ,L ( D ) = ( b k , b k , . . . , b k n ). Also, for any a = ( a , . . . , a n ) ∈ C n , and any given finite frame v = ( v , v , . . . , v n ) in L ( D ), we write v T a := n X i =1 a i v i , (4.4)and, for any φ ∈ L ( D ), the L inner product of v and φ as h v , φ i L ( D ) := (cid:0) h v , φ i L ( D ) , h v , φ i L ( D ) , . . . , h v n , φ i L ( D ) (cid:1) ∈ C n . (4.5)With these notations, we can write (4.1) in terms of the frame S λ ∈ σ ( ˜ K D ) S ≤ i ≤ N λ e λ as follows: e K D = X λ ∈ σ ( e K D ) X ≤ i ≤ N λ (cid:0) e iλ (cid:1) T J iλ ( · ) e iλ ,L ( D ) , (4.6)where the superscript T denotes the transpose as described in (4.4). Therefore, substituting the aboveexpression of e K D into (2.19), we have W nm ( D, ε ∗ ) = D J n ( r ) e inθ , ( ε ∗− + ˜ K D ) − [ J m ( r ) e imθ ] E L ( D ) = X λ ∈ σ ( e K D ) X ≤ i ≤ N λ h(cid:10) J n ( r ) e inθ , e iλ (cid:11) L ( D ) i T [ J iε ∗− + λ ] − (cid:2) J m ( r ) e imθ (cid:3) e iλ ,L ( D ) . (4.7)The above expression gives a general decomposition of the scattering coefficient W nm ( D, ε ∗ ).Next, we consider the special domain D = B (0 , R ). Then from Theorem 2.7 we shall focus on theoperators e K (1) m for m ∈ Z . Similarly to the previous argument, we can see that the operators e K (1) m arecompact on L ((0 , R ) , r dr ). Then by Theorem 4.1 there exists a complete basis S λ S ≤ i ≤ N mλ e im,λ over L ((0 , R ) , r dr ) with each e im,λ spanning the subspace E im,λ such that e K m admits the action of a Jordanblock, denoted by J im,λ , with respect to the basis when acting on the invariant subspace E im,λ . Moreover,adopting the same notations as previously introduced, we can write (cid:16) ε ∗− + ˜ K (1) m (cid:17) − = X λ ∈ σ ( e K (1) m ) X ≤ i ≤ N mλ (cid:0) e im,λ (cid:1) T [ J im,ε ∗− + λ ] − ( · ) e im,λ ,L ((0 ,R ) ,r dr ) , (4.8)14nd a similar expansion holds for ˜ K (2) m . Now, using the orthogonality of { e imθ } m ∈ Z on L ( S ), for a givencontrast ε ∗ such that − ε ∗− is not an eigenvalue of ˜ K (1) m , we have that W nm ( D, ε ∗ )= δ nm (cid:28) J n , (cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:29) L ((0 ,R ) ,r dr ) = δ nm X λ ∈ σ ( e K (1) m ) X ≤ i ≤ N mλ [ h J n ( r ) , e im,λ i L ((0 ,R ) ,rdr ) ] T [ J im,ε ∗− + λ ] − ( J m ( r ) ) e im,λ ,L ((0 ,R ) ,r dr ) . (4.9)Finally, the following remarks are in order. For D = B (0 , R ), the action of ˜ K D on each of the subspace E im,λ e imθ of L ( D ) is invariant and admits the same Jordan block representation as e K (1) m acting on E im,λ of L ((0 , R ) , rdr ). Hence, the decomposition L ( D ) = M m ∈ Z M λ ∈ σ (cid:16) e K (1) m (cid:17) M ≤ i ≤ N mλ E im,λ e imθ coincides with the original Jordan block decomposition of ˜ K D , L ( D ) = M λ ∈ σ ( e K ) M ≤ i ≤ N λ E iλ . Therefore, we readily get S m ∈ Z σ ( e K (1) m ) = σ ( ˜ K D ), and the sum (4.9) constitutes a part of the sum (4.7)with all the other terms in (4.7) being zero. In the next section, we will focus on the decay of theeigenvalues of ˜ K m and the asymptotic expansion for the eigenvalues and eigenfunctions of the operators.This will allow us to better understand the behavior of W nm and C ( ε ∗ , n, m ). e K ( i ) m Intuitively we can expect that the eigenvalues of e K (1) m are distributed closer to 0 as | m | increases forthe following reason. Considering (2.33) together with the asymptotic expressions (2.27) and (2.35) of J m and Y m as m → ∞ , we have the following bound for the operator norm of ˜ K m for m ∈ Z : || e K (1) m || L ((0 ,R ) rdr ) ≤ C ′ R m (4.10)for some constant C ′ R depending on R . Then we obtain the estimate for the spectral radius of ˜ K m fromthe Gelfand theorem: sup λ ∈ σ ( e K (1) m ) | λ | = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) e K (1) m (cid:17) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ≤ C ′ R m . (4.11)This implies that the spectrum σ ( e K (1) m ) actually lies inside σ ( ˜ K ) T B (0 , C ′ R m ).However, the above argument is a bit heuristic, and we intend to obtain a formal asymptotic expansionof the eigenvalues for the operators ˜ K m . For this purpose, we first restrict ourselves to the discussion ofthe operators for m ∈ N , and consider the equation ˜ K (1) m f = λf with λ = 0. Since we have (cid:18) r ∂ r r∂ r + 1 − m r (cid:19) (cid:16) ˜ K (1) m f (cid:17) e imθ = (∆ + 1) ˜ K (1) (cid:0) f e imθ (cid:1) = f e imθ , (4.12)we obtain for m = 0 the following equivalence˜ K (1) m f = λf ⇔ (cid:16) r ∂ r r∂ r + 1 − λ − m r (cid:17) f = 0 ,f (0) = 0 ,f ( R ) = − i R R rJ m ( r ) f ( r ) drH ( i ) m ( R ) . (4.13)15numerating the eigenvalues λ of ˜ K (1) m as λ m,l in descending order of their magnitudes, and writing e im,l as the unique eigenfunction in the Jordan basis e im,λ m,l for each i , we are bound to have the followingform for the eigenpair of the operator for all i ,( λ m,l , e im,l ) = λ m,l , J m s − λ m,l r !! . (4.14)The above statement implies that the geometric multiplicities of all the eigenvalues of ˜ K (1) m should be N λ = 1 (while the algebraic multiplicities are still unknown for the time being). For the sake of simplicity,we denote the frame e m,λ m,l by e m,l , and also the eigenfunction e m,l by e m,l . Substituting (4.14) into(4.13), together with the following well-known property of Lommel’s integrals [1] that for all n ∈ N andfor all a, b > a = b : Z R [ J n ( ar )] rdr = R J n ( aR ) − J n − ( aR ) J n +1 ( aR )] , (4.15) Z R J n ( ar ) J n ( br ) rdr = Ra − b [ bJ n ( aR ) J n − ( bR ) − aJ n − ( aR ) J n ( bR )] , (4.16)we get the following equation for λ m,l : J m s − λ m,l R ! = − i Z R rJ m ( r ) J m s − λ m,l r ! drH ( i ) m ( R )= i Rλ m,l H ( i ) m ( R ) (cid:20)s − λ m,l J m ( R ) J m − s − λ m,l R ! − J m − ( R ) J m s − λ m,l R ! (cid:21) . (4.17)Now since λ m,l → l → ∞ , from the following well-known asymptotic of J n [1] for all n : J n ( z ) = r πz cos (cid:18) z − n + 14 π (cid:19) + O ( z − / ) , (4.18)we obtain the following estimate for m, n, l ∈ N : J n s − λ m,l R ! = vuut πR q − λ m,l cos s − λ m,l R − n + 14 π ! + O ( | λ m,l | / ) . (4.19)Hence, substituting this expression into (4.17), we shall directly infer that the eigenvalues λ m,l satisfythe following bound: J m s − λ m,l R ! = O ( | λ m,l | / ) , (4.20)which has a decay order higher than the one in (4.19). With this observation, we shall expect that theterms q − λ m,l R should be close to the l -th zeros of the Bessel functions of J m as l grows, which isindeed the case following the argument below. 16or the sake of exposition, we shall often denote by a m,l the zeros of the m -th Bessel function ofthe first kind, i.e., J m ( a m,l ) = 0, arranged in ascending order. Then it follows from (4.18), the inversefunction theorem and the Taylor expansion that (cid:12)(cid:12)(cid:12)(cid:12) a m,l − m + 4 l − π (cid:12)(cid:12)(cid:12)(cid:12) < C ( m + 2 l ) − / → l → ∞ . (4.21)Then, again from (4.18), we have J ′ m ( a m,l ) − ( − l s πa m,l = O ( a m,l − / ) , (4.22)which, combined with (4.20), leads to R s − λ m,l − a m,l = O ( a m,l − / ) . (4.23)This gives us the following estimate for λ m,l : R s − λ m,l (cid:30) (cid:18) ( m + 2 l ) π − π (cid:19) → l → ∞ . (4.24)Therefore, we obtain the following decay rate of the eigenvalues, λ m,l (cid:30) (cid:18) R π m + 2 l ) (cid:19) → − l → ∞ . (4.25)Moreover, using (4.23) and the fact that J m is holomorphic, we have the following uniform estimate forthe eigenfunctions: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J m s − λ m,l r ! − J m (cid:16) a m,l R r (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ((0 ,R )) ≤ C || J ′ m || L ∞ ((0 ,R )) a m,l − / < C ( m + 2 l ) − / . (4.26)Note that the set { J m (cid:0) a m,l R r (cid:1) } ∞ l =1 forms a complete orthogonal basis in L ((0 , R ) , r dr ). Hence, the aboveestimate actually implies that the eigenfunctions of ˜ K (1) m approach in the sup-norm to an orthogonal basisin L ((0 , R ) , r dr ) for all m ∈ N . From (2.33), together with the fact that a − m,l = a m,l from (2.32), theabove analysis also holds for ˜ K (1) − m .The following theorem summarizes the main eigenvalue and eigenfunction estimates for the operator˜ K (1) m . Theorem 4.2.
For all m ∈ Z \{ } , the eigenpairs of the operator ˜ K (1) m are of the form ( λ m,l , e m,l ) = λ m,l , J m s − λ m,l r !! for l ∈ N , (4.27) where the eigenvalues λ m,l satisfy the following asymptotic behavior λ m,l (cid:30) (cid:18) R π | m | + 2 l ) (cid:19) → − as l → ∞ . (4.28) Moreover, the eigenfunctions also have the following uniform estimate: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J m s − λ m,l r ! − J m (cid:16) a m,l R r (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ((0 ,R )) = O (( | m | + 2 l ) − / ) . (4.29)17his theorem is very important for the analysis of the behaviors of W nm and C ( ε ∗ , n, m ). Figure 1shows the distribution of eigenvalues of ˜ K (1) m for R = 10 with different values of m . It not only illustratesthat the spectral radius decreases as the value of m increases (which agrees with the estimate (4.11));but also that, for a fixed number l ∈ N , the magnitude of the l -th eigenvalue of ˜ K (1) m decreases in generalmonotonically with respect to increment of m (which agrees with (4.28)). Eigenfunctions of ˜ K (1) m forsome values of m are also plotted in Figure 2 for a better illustration of the behaviour of eigenfunctions. (a) (b)Figure 1: (a) Spectral radius of ˜ K (1) m for m = 0 , , . . . ,
11. (b) Norms of eigenvalues λ m,l , l = 1 , , . . . , , for operators ˜ K (1) m , m = 0 , , . . . , , as in the legend. (1a) (1b)18 (2a) (2b) (3a) (3b)Figure 2: Real and imaginary parts of the first 4 eigenfunctions of ˜ K (1) m , m = 1 , ,
3. (1a) Real parts ofeigenfunctions of ˜ K (1)1 ; (1b) imaginary parts of eigenfunctions of ˜ K (1)1 ; (2a) real parts of eigenfunctionsof ˜ K (1)2 , and so forth. W nm and C ( ε ∗ , n, m ) and thesuper-resolution phenomenon In this subsection, we deduce very useful information on the behaviors of W nm and C ( ε ∗ , n, m ) fromthe asymptotic behaviors of eigenpairs of ˜ K (1) m derived in the previous subsection. W nm We first focus on the scattering coefficients W nm ( D, ε ∗ ) when D = B (0 , R ). Form (2.34), it is knownthat W nm = 0 when n = m , therefore the only interesting case is when n = m . Again, we shall firstconsider m ∈ N . From the analysis in the previous subsection that the geometric multiplicities of all theeigenvalues of ˜ K (1) m are N mλ = 1 , we already obtain from (4.9) that W mm ( D, ε ∗ ) = ∞ X l =0 [ h J m ( r ) , e m,l i L ((0 ,R ) ,rdr ) ] T [ J m,ε ∗− + λ m,l ] − ( J m ( r ) ) e m,l ,L ((0 ,R ) ,r dr ) . g λ m,l := 11 − a m,l R and g e m,l := J m (cid:16) a m,l R r (cid:17) . (4.30)From (4.15) and (4.29), together with the completeness and orthogonality of g e m,l in L ((0 , R ) , r dr ) andthe Parseval’s identity, we readily obtain that, fixing any m ∈ N and for any given ǫ , there exists N ( m )such that for all i > N ( m ), we have (cid:12)(cid:12)(cid:12)(cid:12) h e m,i , g e m,j i L ((0 ,R ) ,rdr ) − δ ij R J m +1 ( a m,j ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ ij , (4.31)where P j ǫ ij < ǫ . Therefore, for a large N ( m ), the span of { e m,l } ∞ l = N ( m ) has a finite dimensionalorthogonal complement. This follows that there exists a large N ( m ) > N ( m ) such that the algebraicmultiplicity of λ m,l is 1. Therefore, we directly obtain W mm ( D, ε ∗ ) = S ,m ( ε ∗ ) + S ,m ( ε ∗ ) , (4.32)where the sums S i,m ( ε ∗ ), i = 1 , , are defined by S ,m ( ε ∗ ) := N ( m ) X l =0 [ h J m ( r ) , e m,l i L ((0 ,R ) ,rdr ) ] T [ J m,ε ∗− + λ m,l ] − ( J m ( r ) ) e m,l ,L ((0 ,R ) ,r dr ) (4.33) S ,m ( ε ∗ ) := ∞ X l = N ( m )+1 α m,l ε ∗− + λ m,l , (4.34)with the coefficients α m,l being defined, for all m, l , as α m,l := h J m ( r ) , e m,l i L ((0 ,R ) ,rdr ) ( J m ( r ) ) e m,l ,L ((0 ,R ) ,r dr ) . (4.35)Note that for any ε ∗ ≥ − (cid:16) λ − m,N ( m ) (cid:17) , we have | S ,m ( ε ∗ ) | < C m for some constant C m . Therefore, ifwe want to investigate the behavior of (4.32) for large ε ∗ , we shall focus on the term S ,m ( ε ∗ ). For thispurpose, we analyse the limiting behavior of α m,l as l increases. Now, from (4.16) and (4.29), we havethe following estimate for the inner product: h J m ( r ) , e m,l i L ((0 ,R ) ,rdr ) − g λ ml a m,l J m ( R ) J m − ( a m,l ) = O ( a − / m,l ) . (4.36)From (4.18) we get J m ± ( a m,l ) − ( − l s πa m,l = O ( a m,l − / ) , (4.37)and hence it follows that h J m ( r ) , e m,l i L ((0 ,R ) ,rdr ) (cid:30) ( − l g λ m,l a / m,l r π J m ( R ) → l → ∞ . (4.38)From (4.31), we obtain that the coefficient of J m ( r ) of e m,l with respect to the Jordan basis approachesto the orthogonal project of J m ( r ) on the subspace e m,l , whence the following holds( J m ( r ) ) e m,l ,L ((0 ,R ) ,r dr ) (cid:30) h J m ( r ) , e m,l i L ((0 ,R ) ,rdr ) R [ J m − ( a m,l )] → l → ∞ . (4.39)20ombining the above several limiting behaviors (4.38) and (4.39) yields α m,l (cid:30) g λ m,l a m,l J m ( R ) R → l → ∞ , (4.40)which can further be reduced to the following asymptotic behavior by combining (4.23),(4.25) and (4.30), α m,l (cid:30) λ m,l J m ( R ) → − l → ∞ . (4.41)From (2.32) and (2.33), the conclusions also hold for the case with − m ∈ N .The above analysis can be summarized in the following theorem. Theorem 4.3.
Let D = B (0 , R ) be a circular domain. For all m ∈ Z \{ } , there exist constants N ( m ) ∈ N and C m > such that, for any given contrast value ε ∗ > − Re (cid:16) λ − m,N ( m ) (cid:17) , the scatteringcoefficient W mm ( D, ε ∗ ) has the following decomposition W mm ( D, ε ∗ ) = S ,m ( ε ∗ ) + S ,m ( ε ∗ ) , (4.42) where S ,m ( ε ∗ ) has a uniform bound | S ,m ( ε ∗ ) | < C m , (4.43) whereas S ,m ( ε ∗ ) is of the form S ,m ( ε ∗ ) = ∞ X l = N ( m )+1 α m,l ε ∗− + λ m,l , (4.44) where the coefficients α m,l have the following limiting behavior α m,l (cid:30) λ m,l J m ( R ) → − as l → ∞ . (4.45)This decomposition of the coefficient W mm gives us a clear picture of the behavior of W mm as ε ∗ grows. When ε ∗ increases, ε ∗− passes through the values − Re( λ m,l ) ∼ ( | m | + 2 l ) − for large l . If λ m,l ∈ R , ε ∗− directly passes through the pole. Therefore W mm grows from a finite value rapidly to adirectional complex infinity ∞ e iθ for some θ , and then comes back from −∞ e iθ to a finite value after ε ∗− passes through it. Otherwise, if λ m,l / ∈ R , then ε ∗− does not directly hit the pole. However, since λ m,l ∼ − ( | m | + 2 l ) − where ( | m | + 2 l ) − are real, Im( λ m,l ) is very small for large l . Hence, as ε ∗− movesclose to − Re( λ m,l ), it comes close to the pole. Therefore, W mm grows from a comparably small valuevery rapidly to a complex value of very large modulus, and then drops back to a small value after passingthrough − Re( λ m,l ). The behavior of W mm is consequently very oscillatory as ε ∗ grows. Moreover, from(4.45) we have for a fixed pair of m, l that α m,l ε ∗− + λ m,l → − J m ( R ) (4.46)as ε ∗ → ∞ , therefore that there is no hope on any convergence behavior of W mm as ε ∗ grows to infinity.Furthermore, from (4.28) that the asymptotic λ m,l ∼ − ( | m | + 2 l ) − holds and the limit comparisontest, we have for a fixed ε ∗ > − (cid:16) λ − m,N ( m ) (cid:17) that | W nm ( D, ε ∗ ) | ≤ δ nm C m + C ′ m d ( − ε ∗− , σ ( ˜ K (1) m )) J m ( R ) ∞ X l =0 | λ m,l | ! (4.47) ≤ δ nm C m + C ′ m d ( − ε ∗− , σ ( ˜ K (1) m )) R | m | + | n | | m | | m | | n | | n | ! . (4.48)21 orollary 4.4. Let D = B (0 , R ) . For all m ∈ Z \{ } , there exist constants N ( m ) ∈ N and C i,m , i = 1 , such that, for any given contrast value ε ∗ > − Re (cid:16) λ − m,N ( m ) (cid:17) , the scattering coefficient W nm ( D, ε ∗ ) satisfies the following estimate for all n ∈ Z , | W nm ( D, ε ∗ ) | ≤ δ nm C ,m + C ,m d ( − ε ∗− , σ ( ˜ K (1) m )) R | m | + | n | | m | | m | | n | | n | ! . (4.49)This clearly improves the estimate (2.37). C ( ε ∗ , n, m )We now focus on the behaviours of the coefficients C ( ε ∗ , n, m ), which will help us to understand thephenomenon of super-resolution. We first focus on the case when n, m ∈ N . We recall the expression ofthe coefficient C ( ε ∗ , n, m ) in (3.27): C ( ε ∗ , n, m ) := ε ∗− (cid:20)(cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ] (cid:21) ( R ) (cid:20)(cid:16) ε ∗− + ˜ K (1) n (cid:17) − [ J n ] (cid:21) ( R ) . It remains to study the term (cid:16) ε ∗− + ˜ K (1) m (cid:17) − [ J m ]( R ). From the previous subsection, the geometricmultiplicities of all the eigenvalues of ˜ K (1) m are N mλ = 1, and the algebraic multiplicities of eigenvalues λ m,l of ˜ K (1) m are also 1 for l > N ( m ) (see Theorem 4.3). Together with the regularity of J m , we readilyobtain as in the previous subsection that C ( ε ∗ , n, m ) = ε ∗− ( s ,n ( ε ∗ ) + s ,n ( ε ∗ ))( s ,m ( ε ∗ ) + s ,m ( ε ∗ )) , (4.50)where the sums s i,m ( ε ∗ ), ( i = 1 ,
2) are defined by s ,m ( ε ∗ ) := N ( m ) X l =0 ( e m,l ( R )) T [ J m,ε ∗− + λ m,l ] − ( J m ( r ) ) e m,l ,L ((0 ,R ) ,r dr ) , (4.51) s ,m ( ε ∗ ) := ∞ X l = N ( m )+1 β m,l ε ∗− + λ m,l (4.52)with the coefficients β m,l being given for all m, l by β m,l := ( J m ( r ) ) e m,l ,L ((0 ,R ) ,r dr ) J m s − λ m,l R ! . (4.53)Similarly to the previous subsection, for any ε ∗ ≥ − (cid:16) λ − m,N ( m ) (cid:17) , we have | s ,m ( ε ∗ ) | < C m for someconstant C m . Therefore, we can study the behavior of (4.50) for large ε ∗ by investigating the limitingbehavior of β m,l in the series s ,m ( ε ∗ ) .Substituting (4.19), (4.23) and (4.25) into (4.17), we readily derive J m s − λ m,l R ! (cid:30) ( − l i λ m,l a / m,l H (1) m ( R ) J m ( R ) r Rπ → l → ∞ . (4.54)Together with (4.38) and (4.39), we conclude that β m,l (cid:30) i √ Rλ m,l J m ( R ) H (1) m ( R ) → − l → ∞ . (4.55)Combining the above results with (2.32) and (2.33), we obtain the following decomposition of C ( ε ∗ , n, m ).22 heorem 4.5. Let D = B (0 , R ) be a circular domain. For all p ∈ Z \{ } , there exist constants N ( p ) ∈ N and C p > such that, for any n, m ∈ Z \{ } and any contrast value ε ∗ > − n Re (cid:16) λ − n,N ( n ) (cid:17) , Re (cid:16) λ − m,N ( m ) (cid:17)o ,the coefficient C ( ε ∗ , n, m ) (3.27) admits the following decomposition: C ( ε ∗ , n, m ) = ε ∗− ( s ,n ( ε ∗ ) + s ,n ( ε ∗ ))( s ,m ( ε ∗ ) + s ,m ( ε ∗ )) . (4.56) For all p ∈ Z \{ } , s ,p ( ε ∗ ) satisfies the uniform bound | s ,p ( ε ∗ ) | < C p , (4.57) whereas s ,p ( ε ∗ ) is given by s ,p ( ε ∗ ) = ∞ X l = N ( p )+1 β p,l ε ∗− + λ p,l (4.58) where the coefficients β p,l have the following limiting behavior β p,l (cid:30) i √ Rλ p,l J p ( R ) H (1) p ( R ) → − as l → ∞ . (4.59)Similarly to the previous subsection, the aforementioned decomposition of C ( ε ∗ , n, m ) clearly illus-trates the behavior of C ( ε ∗ , n, m ) as ε ∗ grows and ε ∗− passes through the values − Re( λ p,l ) ∼ ( | p | + 2 l ) − with p = n, m . If λ p,l ∈ R , ε ∗− directly hits the pole. Therefore C ( ε ∗ , n, m ) first grows from a finitevalue rapidly to a directional complex infinity ∞ e iθ for some θ , then back from −∞ e iθ to a finite valueafter passing through it. Otherwise if λ p,l / ∈ R and when l is large, ε ∗− does not pass through the pole,but comes very close to it. Hence, C ( ε ∗ , n, m ) grows rapidly from a considerably small value to a complexvalue of very large modulus, then drops to a small value after passing through − Re( λ p,l ). Moreover, fora fixed pair of p, l , we have β p,l ε ∗− + λ p,l → − i √ RJ p ( R ) H (1) p ( R ) (4.60)as ε ∗ → ∞ . Therefore, we can see that C ( ε ∗ , n, m ) has very oscillatory behavior as ε ∗ grows. Although C ( ε ∗ , n, m ) is very oscillatory as ε ∗ grows, the aforementioned behavior and series decom-position of C ( ε ∗ , n, m ) gives a clear explanation of the super-resolution phenomenon for high-contrastinclusions. It is because, what we have actually proved is that, in the shape derivative of the scatteringcoefficients of a circular domain, there are simple poles corresponding to the complex resonant states,and therefore peaks at the real parts of these resonances. Hence, as the material contrast ε ∗ increases toinfinity and is such that it hits the real part of a resonance, the sensitivity in the scattering coefficientsbecomes very large and super-resolution for imaging occurs.To put it more accurately, let us recall (3.26). Suppose D = B (0 , R ), then for any δ -perturbation of D , D δ , along the variational direction h ∈ C ( ∂D ), we have W nm ( D δ , ε ∗ ) − W nm ( D, ε ∗ ) = δ C ( ε ∗ , n, m ) F θ [ h ] ( n − m ) + O ( δ ) . As one might recall from (2.28), W nm ( D δ , ε ∗ ) always decays exponentially as | n | , | m | increase. Hence, itis always of exponential ill-posedness to recover the higher order Fourier modes of the perturbation h .The inversion process to recover the k -th Fourier mode F θ [ h ] ( k ) becomes less ill-posed if C ( ε ∗ , n, m ) islarge for some n, m ∈ Z such that k = n − m . This not only makes the respective scattering coefficientsmore apparent than the others, but also lowers the condition number of the inverse process to reconstruct23he respective Fourier mode. From the analysis in the previous subsection, this can only be made possiblewhen ε ∗− comes close to − Re( λ p,l ) for some p = n, m and for some l ∈ N .Now, suppose ε ∗ is close to the following resonant value (cid:0) Kπ R − π R (cid:1) where K ∈ N is large. Then,from the fact that the eigenvalues λ p,l of the operators e K (1) p follow the asymptotics: − λ − p,l ∼ (cid:18) π ( | p | + 2 l )2 R − π R (cid:19) , (4.61)we see that ε ∗− is close to − Re( λ − p,l ( p ) ) for all p ∈ Z such that | p | + 2 l ( p ) = K for some l ( p ) ∈ N .Therefore, ε ∗− comes close to − Re( λ − K, ) , − Re( λ − K − , ) , − Re( λ − K − , ) , . . . , − Re( λ − K − K ] , [ K ] ) simultane-ously where [ · ] is the floor function. This in turn boosts up the magnitudes of all the terms β p,l ( p ) ε ∗− + λ p,l ( p ) whenever p is of the form p = − K + 2 s , s = 0 , , . . . , K . These terms dominate the series s ,p ( ε ∗ ), hencewe obtain the following approximations of s ,p ( ε ∗ ) for all p = − K + 2 s , s = 0 , , . . . , K : s ,p ( ε ∗ ) ≈ − i √ RJ p ( R ) H (1) p ( R ) ( K − . − − π R − ε ∗− − ( K − . − . Now we see from Theorem 4.5 that the coefficients C ( ε ∗ , n, m ) have the following approximations for n, m ∈ Z when ε ∗ is very close to the resonant values (cid:0) Kπ R − π R (cid:1) for large K : C ( ε ∗ , n, m ) ≈ M n,m,R ( K − . − (cid:0) − π R − ε ∗− − ( K − . − (cid:1) − if both of n, m have the form − K + 2 s , s = 0 , , . . . , K ; ≈ M n,m,R ( K − . − (cid:0) − π R − ε ∗− − ( K − . − (cid:1) − if only one of n, m has the form − K + 2 s , s = 0 , , . . . , K ;is very small otherwise , where M n,m,R are some constants depending only on n, m and R . Here, the term (cid:0) − π R − ε ∗− − ( K − . − (cid:1) − is very large, and makes the Fourier coefficients F θ [ h ] ( n − m ) visible for n, m ∈ {− K + 2 s : s =1 , , . . . , K } for accurate classification of the shapes. The above mechanism is possible only when ε ∗ increases up to one of the resonant values (cid:0) Kπ R − π R (cid:1) when K is large. This explains the increasinglikelihood of obtaining super-resolution as ε ∗ increases.Now, for a given ε ∗ , consider the following bounded linear map over the space l ± ( C ) of two-sidedsequences ( a l ) ∞ l = −∞ such that P ∞ l = −∞ a l < ∞ , A ( ε ∗ ) : l ± ( C ) → l ± ( C ) ⊗ l ± ( C )( a l ) ∞ l = −∞ ( C ( ε ∗ , n, m ) a n − m ) ∞ n,m = −∞ . (4.62)By Theorem 3.3, we know the shape derivative of ( W nm ( D, ε ∗ )) ∞ n,m = −∞ in the variational direction h isgiven by D W ( D, ε ∗ )[ h ] = A ( ε ∗ ) F θ [ h ] . (4.63)It is ready to conclude that the least-squared map[ A ( ε ∗ )] ∗ [ A ( ε ∗ )] : l ± ( C ) → l ± ( C )( a l ) ∞ l = −∞ X n − m = l | C ( ε ∗ , n, m ) | a l ! ∞ l = −∞ (4.64)is a diagonal operator, and the l -th singular value s l ( A ) is of the form s l ( A ) = s X n − m = l | C ( ε ∗ , n, m ) | . (4.65)24herefore, from the above analysis on C ( ε ∗ , n, m ) when ε ∗ is close to the resonant values (cid:0) Kπ R − π R (cid:1) , wecan observe that the singular values s l become large and comparable to each other, making the inversionof many Fourier modes well-conditioned. This implies a much higher resolution of the modes of h , and alsofor reconstructing the geometry of D δ in the linearized case. This provides a good understanding towardsthe recently observed phenomenon of super-resolution in the physics and engineering communities. In this section, we present some numerical experiments on the behaviors of the scattering coefficientsfor some domains as the contrast ε ∗ grows, and numerically illustrate the phenomenon of super-resolution.In the following 2 examples, we consider an infinite domain of homogeneous background medium withits material coefficient being 1. An inclusion D δ is then introduced as a perturbation of a circular domain D = B (0 , R ) for some R > δ > ε ∗ = a m,l /R − m, l such that a m,l ≤ . φ m , ψ m ) of (2.8) for | m | ≤
25 usingrectangular quadrature rule with mesh-size s/ s denotes thelength of the inclusion boundary. The scattering coefficients of D δ of orders ( n, m ) for | n | , | m | ≤
25 arethen calculated as the Fourier transform of the far-field data.In order to test the robustness of the super-resolution phenomenon, we introduce some multiplicativerandom noise in the scattering coefficients in the form: W γnm ( D δ , ε ∗ ) = W nm ( D δ , ε ∗ ) (1 + γ ( η + iη )) , (5.1)where η i , i = 1 , , are uniformly distributed between [ − ,
1] and γ refers to the relative noise level. Inboth examples below, we always set the noise level to be γ = 5%.Since the purpose of our numerical experiments is to illustrate the phenomenon of super-resolutionas ε ∗ increases, we assume that both R and ε ∗ are known and use the following regularized inversionmethod suggested from the linearized problem (3.26) to recover the k -th Fourier mode for | k | ≤
50 fromthe observed noisy scattering coefficients W γnm ( D δ , ε ∗ ) , | n | , | m | ≤ δ F θ [ h ] recovered ( k ) = X n − m = k, | n | , | m |≤ W γnm ( D δ , ε ∗ ) − W nm ( D, ε ∗ ) C ( ε ∗ , n, m ) + α , (5.2)where α is a regularization parameter. The coefficients W nm ( D, ε ∗ ) used in the inversion process arecalculated using the same method as previously mentioned for the forward problem without addingnoise, and the coefficients C ( ε ∗ , n, m ) are calculated by the following approximations C ( ε ∗ , n, m ) ≈ (cid:0) W nm ( D δ ( n − m ) , ε ∗ ) − W nm ( D, ε ∗ ) (cid:1) /δ (5.3)for | n | , | m | ≤
25, where D δ ( k ) are defined as domains with the following boundaries for | k | ≤ ∂D δ ( k ) := { ˜ x = R (1 + δ e ikθ ) : θ ∈ (0 , π ] } , (5.4)with δ chosen to be δ = 0 . Example 1
As a toy example, we first consider a flori-form shape D δ described by the followingparametric form (with δ = 0 . r = 0 . δ cos(3 θ ) + 2 δ cos(6 θ ) + 4 δ cos(9 θ )) , θ ∈ (0 , π ] , (5.5)which is a perturbation of the domain D := B (0 , . D δ and D .25 Figure 3: Inclusion shape in Example 1.The relative magnitudes of the scattering coefficients max | m − n | = k | W nm ( D δ , ǫ ∗ ) | / max m = n | W nm ( D δ , ǫ ∗ ) | are plotted for k = 3 , , ε ∗ grows, the relative magnitude of the scatteringcoefficient corresponding to the ± k -th Fourier mode grows from a smaller magnitude to larger magnitude,and the peaks become apparent when ε ∗ hits the respective zeros of the Bessel functions.
500 1000 1500 2000 2500 3000 350000.20.40.60.811.21.41.61.82 max |m−n| = 3 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| 500 1000 1500 2000 2500 3000 35000.010.020.030.040.050.06 max |m−n| = 6 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| 500 1000 1500 2000 2500 3000 350024681012141618x 10 −6 max |m−n| = 9 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| Figure 4: Relative magnitudes of the scattering coefficients in Example 1.From the relative magnitudes shown in the above figures, we observe that the scattering coefficientsare best conditioned for inversion when ε ∗ = 1971 . , . ε ∗ = 63 . J as a comparison. We notice from the figures that the scattering coefficients corresponding tohigher Fourier modes become more apparent as ε ∗ increases. We then apply the aforementioned inversionprocess, with the regularization parameter chosen as α = 1 × − . The magnitudes of the recoveredFourier modes and the reconstructed domains are shown in Figure 5 (middle) and (right) respectively.We can clearly see that the fine features are more and more apparent as ε ∗ grows along the specificcontrasts that we choose. Notice also that the fine features are of a magnitude smaller than 0 .
4, which ismuch smaller than half of the operating wavelength, π . |W nm (D δ , ε * )| , n ≠ m , ε * = 63.2669 −10 −5 0 5 10−10−8−6−4−20246810 00.010.020.030.040.050.060.07 −10 −5 0 5 1000.20.40.60.811.21.4 |F[h](k)| , −10 ≤ k ≤
10 , ε * = 63.2669 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5 Shape Reconstruction, ε * = 63.2669 W nm (D δ , ε * )| , n ≠ m , ε * = 1971.2481 −10 −5 0 5 10−10−8−6−4−20246810 00.0050.010.0150.020.025 −10 −5 0 5 1000.020.040.060.080.10.12 |F[h](k)| , −10 ≤ k ≤
10 , ε * = 1971.2481 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.3−0.2−0.100.10.20.3 Shape Reconstruction, ε * = 1971.2481|W nm (D δ , ε * )| , n ≠ m , ε * = 3627.456 −10 −5 0 5 10−10−8−6−4−20246810 0.511.522.533.54x 10 −3 −10 −5 0 5 1000.020.040.060.080.10.12 |F[h](k)| , −10 ≤ k ≤
10 , ε * = 3627.456 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.3−0.2−0.100.10.20.3 Shape Reconstruction, ε * = 3627.456 Figure 5: Illustration of super-resolution in Example 1. Left: magnitude of scattering coefficients;middle: magnitude of recovered Fourier coefficients; right: recovered domain.
Example 2
We try the following right-angled isosceles triangle D δ , which is a perturbation of thedomain D := B (0 , . D δ and D . This case is substantially harder, since the perturbation h consists of manyFourier modes and is no longer smooth. −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.25−0.2−0.15−0.1−0.0500.050.10.15 −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 Figure 6: Inclusion shape in Example 2.The relative magnitudes of the scattering coefficients max | m − n | = k | W nm ( D δ , ǫ ∗ ) | / max m = n | W nm ( D δ , ǫ ∗ ) | are plotted for k = 1 , , . . . , , in Figure 7. From this figure, we can see that the relative magnitude of thescattering coefficient corresponding to the ± k -th Fourier mode comes out more often when ε ∗ becomeslarge. |m−n| = 1 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| 1000 2000 3000 4000 5000 6000 7000 80000.30.40.50.60.70.80.91 max |m−n| = 2 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| 1000 2000 3000 4000 5000 6000 7000 80000.10.20.30.40.50.6 max |m−n| = 3 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )|
000 2000 3000 4000 5000 6000 7000 80000.0050.010.0150.020.0250.030.0350.04 max |m−n| = 4 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| 1000 2000 3000 4000 5000 6000 7000 80000.20.40.60.811.21.41.61.82x 10 −3 max |m−n| = 5 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| 1000 2000 3000 4000 5000 6000 7000 80000.511.522.533.54x 10 −4 max |m−n| = 6 |W nm (D δ , ε * )| / max m ≠ n |W nm (D δ , ε * )| Figure 7: Relative magnitudes of the scattering coefficients in Example 2.We observe from the relative magnitudes shown in the above figures that the scattering coefficients arebest-conditioned for inversion when ε ∗ = 5237 . ε ∗ = 143 . J as acomparison. The aforementioned inversion process is then applied with regularization parameter chosenas α = 1 × − . Figure 8 (middle) and (right) respectively show the magnitude of the recovered Fouriermodes and the reconstructed domains. We can see that the shape obtained from ε ∗ = 5237 . ε ∗ = 5237 . |W nm (D δ , ε * )| , n ≠ m , ε * = 143.6006 −10 −5 0 5 10−10−8−6−4−20246810 00.050.10.150.20.250.30.350.4 −10 −5 0 5 1000.20.40.60.811.21.4 |F[h](k)| , −10 ≤ k ≤
10 , ε * = 143.6006 −0.3 −0.2 −0.1 0 0.1 0.2 0.3−0.3−0.25−0.2−0.15−0.1−0.0500.050.1 Shape Reconstruction, ε * = 143.6006|W nm (D δ , ε * )| , n ≠ m , ε * = 5237.1406 −10 −5 0 5 10−10−8−6−4−20246810 246810121416x 10 −3 −10 −5 0 5 1000.050.10.150.20.25 |F[h](k)| , −10 ≤ k ≤
10 , ε * = 5237.1406 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.2−0.15−0.1−0.0500.050.10.15 Shape Reconstruction, ε * = 5237.1406 Figure 8: Illustration of super-resolution in Example 2. Left: magnitude of scattering coefficients;middle: magnitude of recovered Fourier coefficients; right: recovered domain.
In this paper, we have for the first time established a mathematical theory of super-resolution in thecontext of imaging high contrast inclusions. We have found both analytically and numerically that atsome high resonant values of the contrast, super-resolution in reconstructing the shape of the inclusioncan be achieved.Our approach opens many new avenues for mathematical imaging and focusing in resonant media.Many challenging problems are still to be solved. It would be very interesting to generalize our approachin order to justify the fact that super-resolution can be achieved using structured light illuminations2817, 18]. Another challenging problem is to generalize our approach to electromagnetic and elastic waveimaging problems of high contrast inclusions. This would be the subject of a forthcoming publication.
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