A Bayesian level set method for an inverse medium scattering problem in acoustics
MManuscript submitted to doi:10.3934/xx.xxxxxxx
AIMS’ Journals
Volume X , Number , XX pp. X–XX
A BAYESIAN LEVEL SET METHOD FOR AN INVERSEMEDIUM SCATTERING PROBLEM IN ACOUSTICS
Jiangfeng Huang
School of Mathematical Sciences, University of Electronic Science and Technology of China,Sichuan 611731, China
Zhiliang Deng
School of Mathematical Sciences, University of Electronic Science and Technology of China,Sichuan 611731, China
Liwei Xu ∗ School of Mathematical Sciences, University of Electronic Science and Technology of China,Sichuan 611731, China
Abstract.
In this work, we are interested in the determination of the shape ofthe scatterer for the two dimensional time harmonic inverse medium scatteringproblems in acoustics. The scatterer is assumed to be a piecewise constantfunction with a known value inside inhomogeneities, and its shape is repre-sented by the level set functions for which we investigate the information usingthe Bayesian method. In the Bayesian framework, the solution of the geo-metric inverse problem is defined as a posterior probability distribution. Thewell-posedness of the posterior distribution would be discussed, and the Markovchain Monte Carlo (MCMC) methods will be applied to generate samples fromthe arising posterior distribution. Numerical experiments will be presented todemonstrate the effectiveness of the proposed method. Introduction.
The inverse scattering problems have been extensively investi-gated because of their great importance and broad applications in radar and sonar,geophysical exploration, medical imaging, and to name a few [12, 11]. One of themain goals of the inverse scattering problems is to determine the unknown scatterer,such as location, geometry, or material property etc. [47, 39, 38]. This kind of in-verse scattering problems can be regarded as inverse medium scattering problems(IMSP). The IMSP are ill-posed and nonlinear admitting great theoretical and com-putational challenges, which have attracted attention of many researchers in pastdecades.There have been many numerical methods being proposed to solve the IMSP[41, 3, 5]. Classical methods for the IMSP can be roughly classified into two cat-egories: direct methods and indirect methods. The direct methods mainly re-cover the support or the shape of the scatterer, such as linear sampling methods[37, 10, 24], multiple signal classification methods [23, 22], and factorization meth-ods [1, 2]. The indirect methods attempt to determine the unknown representation
Mathematics Subject Classification.
Primary: 65N21, 62F15, 78A46.
Key words and phrases.
Inverse medium scattering problems; Helmholtz equations; Bayesianlevel set method; Markov chain Monte Carlo (MCMC) methods; ∗ Corresponding author: [email protected]. a r X i v : . [ m a t h . NA ] J a n J. HUANG AND Z. DENG AND L. XU of the scatterer by applying regularization techniques, including recursive lineariza-tion methods [6, 4], level set methods [16, 19], and Gauss-Newton methods [42, 43].Among these indirect methods, the level set method is a good methodology forthe computation of evolving boundaries and interfaces [15]. The level set methodwas originally designed to track propagating interfaces through topological changes[40], and more recently it has been found applications in inverse problems involvingobstacles [19, 46, 28].In addition to the classical methods, another kind of methods lies in the classof statistical methods, and one of those is known as the Bayesian approach. TheBayesian method has attracted considerable attention for inverse problems due to itsability of uncertainty quantification [9, 45, 31]. Recently, it has been widely appliedto solve the inverse scattering problems [48, 29, 26, 34, 33, 36]. In the Bayesiansetting, the Gaussian measures are favorable options of the prior distributions, whichplay central roles in the theory of the Bayesian approach [45]. Samples from theGaussian priors are generated by solving a related stochastic differential equation orusing the Karhunen-Lo`eve expansion based on the eigenfunctions and eigenvaluesof covariance operators of the prior distributions [27, 17, 30, 32]. The solution tothe Bayesian inversion, i.e. the original problem, is a posterior distribution. Toexplore the information of the posterior distribution, sampling methods such as theMarkov chain Monte Carlo (MCMC) methods are usually employed [8, 20, 7].In this work, we are mainly interested in solving the IMSP by the Bayesian levelset method, which is a coupling of the Bayesian method and the level set method.Assuming that the scatterer is a piecewise constant function with known values,we characterize the shape of the scatterer by the level set functions. There are fewliteratures on the numerical solution of the inverse scattering problems by using theBayesian level set method. In [27], it establishes the mathematical foundations ofthe Bayesian level set method, and its hierarchical extension has been developedin [17]. In [13, 14], the Bayesian method and the ensemble Kalman filter approachbased on level set parameterization are introduced for acoustic source identificationusing multiple frequency information, respectively. Actually, when the level setmethod is coupled with the Bayesian approach, there are several advantages for theshape reconstruction. First of all, there are no needs on the implementation of theFr´echet derivative of the forward map as well as the corresponding adjoint operator.Secondly, one needs no considering the evolution of the level set functions governedby a Hamilton-Jacobi type equation. Finally, the Bayesian level set method notonly can provide with point estimates of the solution, such as the maximum aposterior (MAP) estimate and the conditional mean (CM) estimate, but also canprovide with a systematic framework for quantifying the uncertainty. In this paper,we consider the Whittle-Mat´ern Gaussian random fields as the prior[44, 35], withwhich the level sets of the Gaussian random fields have Lebesgue measure zero [27].We will also discuss the well-posedness of the posterior distribution based on Bayes’theorem. Applying the MCMC methods, we will show the numerical results via theCM estimates.The rest of the paper is organized as follows. In Section 2, we simply descible theforward model, and employ the Dirichlet-to-Neumann finite element method (DtN-FEM) as the forward solver [25, 21]. We discuss the Bayesian level set approachsolving the IMSP with the proposed prior and the well-posedness theory of theposterior distribution in Section 3. In Section 4, the numerical results are presentedto illustrate the effectiveness of the proposed method.
BAYESIAN LEVEL SET METHOD FOR AN IMSP IN ACOUSTICS 3
Figure 1.
The geometry setting for the scattering problem2.
Direct Scattering Problem.
In this section, we introduce the propagation oftime harmonic acoustic waves in two dimensions. The scatterer formed by an inho-mogeneous medium is embedded in an infinite homogeneous background medium.2.1.
A Model Problem.
The scattering problem under consideration is modeledby ∆ u + k (1 + q ( x )) u = 0 , in R , (1a)lim r →∞ r ( ∂u s ∂r − iku s ) = 0 , r = | x | , (1b)where k > u = u s + u i is the total field, u i is the planeincident field, and u s is the scattered field which satisfies the Sommerfeld radiationcondition (1b) uniformly in all directions. q ( x ) > − B R := { x ∈ R : | x | < R } , which is boundedby an artificial boundary Γ R := { x ∈ R : | x | = R } with R being sufficiently largeto enclose the scatterer inside (see Fig. 1). In particular, considering the planeincident wave u i = e ikx · d with the incident direction d ∈ { x ∈ R : | x | = 1 } , we canwrite the equation (1a) into∆ u s + k (1 + q ( x )) u s = − k q ( x ) u i in R . (2)2.2. Equivalent Formulation.
In the following, let L ( B R ) be the usual Hilbertspace of all square integrable functions, and let H ( B R ) be the Sobolev spaceequipped with the norm (cid:107) u (cid:107) H ( B R ) = ( (cid:107) u (cid:107) L ( B R ) + (cid:107)∇ u (cid:107) L ( B R ) ) . Define the trace space H s (Γ R ), s ∈ R , as H s (Γ R ) = { u ∈ L (Γ R ) | (cid:107) u (cid:107) H s (Γ R ) < ∞} equipped with the norm (cid:107) u (cid:107) H s (Γ R ) = ( | a | (cid:88) n ∈ Z (1 + n ) s ( | a n | + | b n | )) , where a n and b n are Fourier coefficients of u ∈ H s (Γ R ). J. HUANG AND Z. DENG AND L. XU
In the domain R \ B R , the solution of equation (2) has the form in the polarcoordinates as follows [12]: u s ( r, θ ) = (cid:88) n ∈ Z H (1) n ( kr ) H (1) n ( kR ) ˆ u s n e inθ , (3)where ˆ u s n = (2 π ) − (cid:90) π u s ( R, θ ) e − inθ dθ, and H (1) n is the Hankel function of the first kind with order n .On the artificial boundary Γ R , we can define the Dirichlet-to-Neumann (DtN)operator T : H / (Γ R ) → H − / (Γ R ) as follows: for any u s ∈ H / (Γ R ), ∂u s ∂ν | Γ R = T u s := k (cid:88) n ∈ Z H (1) (cid:48) n ( kR ) H (1) n ( kR ) ˆ u s n e inθ , (4)where ν is the unit outward normal to Γ R ([6]). Alternatively, the DtN operator T can be expressed as ∂u s ∂ν | Γ R = T u s := ∞ (cid:88) n =0 kH (1) (cid:48) n ( kR )2 πH (1) n ( kR ) (cid:90) π u s ( R, ϕ ) cos( n ( θ − ϕ )) dϕ. (5)The original scattering problem (1a)-(1b) defined on R can be equivalently re-duced to the following problem defined on the bounded domain [3], ∆ u s + k (1 + q ( x )) u s = − k q ( x ) u i in B R , (6a) ∂u s ∂ν = T u s on Γ R . (6b)Then, we have the weak formulation of the boundary value problem (6): find u s ∈ H ( B R ) such that A ( u s , v ) = (cid:96) ( v ) ∀ v ∈ H ( B R ) , (7)where the bilinear form A ( · , · ) : H ( B R ) × H ( B R ) → C is defined by A ( u s , v ) = (cid:90) B R ∇ u s · ∇ vdx − k (cid:90) B R (1 + q ( x ) u s vdx − (cid:90) Γ R T u s vdS, and the linear functional (cid:96) ( · ) : H ( B R ) → C is defined by (cid:96) ( v ) = k (cid:90) B R q ( x ) u i vdx. Finally, we point out that given the incident field u i and the scatterer q ( x ), thedirect scattering problem is to determine the scattered field u s .3. Bayesian Level Set Inversion.
In this section, we adopt level set functionsto reformulate the inverse medium scattering problem as a shape reconstructionproblem.
BAYESIAN LEVEL SET METHOD FOR AN IMSP IN ACOUSTICS 5
The Inverse Problem.Definition 3.1.
The scatterer q ( x ) ∈ L ∞ ( B R ) is said to be admissible if thereexists a compact domain D ⊂⊂ B R such that q ( x ) = (cid:26) b, for x ∈ D ,0 , for x ∈ R \ D ,where b > A ( B R ).For the problem (6), we assume that M different wavenumbers k := k m , m =1 , · · · , M , are given. For each of these wavenumbers, there correspond to the in-cident waves u i mj = e ik m x · d j , j = 1 , · · · , J . Thus, for a given wavenumber k m and a given incident wave u i mj , we define the forward operator G mj : X → Y by u s mj = G mj ( q ( x )), where q ( x ) ∈ A ( B R ) := X , u s mj ∈ H ( B R ) := Y . On the otherhand, we denote the observation data with noise by y mj = O mj ◦ G mj ( q ( x )) + η mj , (8)where O mj : Y → C N denotes the collection of N linear functionals on Y , y mj ∈ C N , and η mj ∼ N (0 , Σ ) is the additive Gaussian noise with covariance matrixΣ ∈ R N × N . Gathering all the observations, one can rewrite (8) as y = O ◦ G ( q ( x )) + η, (9)where y := ( y , · · · , y MJ ) T ∈ C MJN := Y , O ◦ G := ( u s11 ( x ) , · · · , u s MJ ( x N )) T ∈ C MJN denotes the noise-free data with observation points { x n } Nn =1 ⊆ Γ R , and η := ( η , · · · , η MJ ) ∼ N (0 , Σ) with covariance matrix Σ = diag (Σ , · · · , Σ ) ∈ R MJN × MJN .3.2.
Level Set Parameterization.
The scatterer q ( x ) is characterized by q ( x ) = L (cid:88) i =1 b i I B i ( x ) , (10)where { B i } Li =1 are L subdomains such that B i ∩ B j = ∅ , ∀ i (cid:54) = j and ∪ Li =1 B i = B R , I denotes the indicator function of a set, and the { b i } Li =1 are known constants, b i ∈ { b, } . In this setting, the unknown scatterer would be determined by thedomains B i , i = 1 , · · · , L . It is natural to make use of the level set representationof the domains through a continuous real-valued function φ : B R → R . To thispurpose, we define B i ⊆ B R by the level set function φ , B i = { x ∈ B R | c i − ≤ φ ( x ) < c i } , i = 1 , · · · L, (11)where c i are constants with −∞ = c < c < · · · < c L = ∞ , i ∈ N . We define thelevel sets as B i = ( i (cid:91) j =1 B j ) ∩ B i +1 = { x ∈ B R | φ ( x ) = c i } , i = 1 , · · · L − . (12)It is evident that the same domain B i can be represented by different level setfunctions φ and φ , and however different domains can not be determined by thesame level set representation. Therefore, we can use the level set representation to J. HUANG AND Z. DENG AND L. XU uniquely specify the domain B i by an associated level set function, i = 1 , · · · , L .Let X = C ( B R , R ), F : X → X is the level set map described by( F φ )( x ) → q ( x ) = L (cid:88) i =1 b i I B i ( x ) . (13)Then we modify our operator O ◦ G into G = O ◦ G ◦ F : X → Y . As a result, theinverse problem can be reformulated as: for given y , find φ such that y = G ( φ ) + η. (14)3.3. Bayesian Inference.
In the Bayesian framework, all quantities in (14) areviewed as random variables. Since it is assumed that η ∈ R MJN is additive Gauss-ian, it is typically straightforward to write the likelihood function, i.e. the proba-bility density of y given φ , π ( y | φ ) ∝ exp( − |G ( φ ) − y | ) , (15)where |·| Σ := | Σ − ·| denotes the weighted norm in terms of the Euclidean norm |·| .In the following, we denote |G ( φ ) − y | by the potential Φ( φ ; y ). We assume thatthe prior measure of the unknown φ is µ , the posterior measure µ y is representedas the Radon-Nikodym derivative with respect to the prior measure µ : dµ y dµ ( φ ) = 1 Z exp( − Φ( φ ; y )) . (16)where Z = (cid:82) X π ( y | φ ) µ ( dφ ) is a normalization constant. The equation (16) can beviewed as the Bayes’ rule in the infinite-dimensional setting.3.3.1. Whittle-Mat´ern Gaussian Random Field Prior.
We now introduce Gaussianprior of Whittle-Mat´ern type with covariance [44] c ( x, y ) = σ − α Γ( α −
1) ( | x − y | l ) α − K α − ( | x − y | l ) , x, y ∈ R , (17)where K α − is the modified Bessel function of the second kind of order α − σ > l > · ) is the Gammafunction. We generate the samples from the Whittle-Mat´ern prior by solving astochastic partial differential equation( I − l ∆) α φ = l √ ςξ, (18)where ( I − l ∆) α is a pseudo-difference operator defined by its Fourier transform, ξ is a Gaussian white noise, and ς = σ π Γ( α )Γ( α − is a constant. Set τ = l − >
0, wehave the stochastic partial differential equation C − α,τ φ = ξ, (19)where C α,τ = ςτ α − ( τ I − ∆) − α denotes the covariance operator of prior distri-bution µ , α controls the regularity of the samples, and τ represents the inverselength scale of the samples. In what follows, assume that A := ∆ is Laplacian withNeumann boundary conditions on B R , and its domain is given by B R ( A ) := { φ : B R → R | φ ∈ H ( B R ; R ) , ∂φ∂ν = 0 on ∂B R } . (20) BAYESIAN LEVEL SET METHOD FOR AN IMSP IN ACOUSTICS 7
Figure 2.
Samples from the prior with α = 2 , ,
4, for τ = 10. Figure 3.
Samples from the prior with inverse length scale τ =10 , ,
5, for α = 3.In Fig. 2 and Fig. 3, we display random samples obtained from (19) regarding todifferent values of the inverse length scale τ and the regularity α . These samplesare constructed in the domain B R with R = 1.3.3.2. Well-Posedness of the Posterior Distribution.
We now discuss the well-posednessof the posterior distribution for the IMSP. It is clear that the level set map is dis-continuous, and due to this fact, we get that the map G is discontinuous. Althoughthe well-posedness of Bayesian inversion relies on the continuity of the map G , itdemonstrates ([27]) that the discontinuity set is a probability zero event under theGaussian prior. As a result, F will be almost surely continuous under the prior,and it will be given in the following Lemma 3.2. Thus, we are able to get the mea-surability required in the Bayes’ theorem [45]. Furthermore, we need to verify someregularity properties of the potential Φ( φ ; y ) which satisfy the assumptions of theBayes’ theorem. Prior to the presentation, we define a complete probability space( X , Ξ , µ ) for the unknown φ , where X denotes a separable Banach space, and Ξ isthe σ -algebra. Lemma 3.2.
Define the map F : X → X given by (13) . Let φ ∈ X be suchthat m ( B i ) = 0 , for i = 1 , · · · , L − . Assume that { φ (cid:15) } (cid:15)> ⊆ C ( B R ) denotesan approximate sequence of level set functions such that (cid:107) φ (cid:15) − φ (cid:107) ∞ → . Then F ( φ (cid:15) ) → F ( φ ) in measure. Remark 1.
Here m ( B i ) = 0 denotes the Lebesgue measure of the set B i . Theproof of this lemma is almost identical to that of Proposition 3.5 in [18], and weomit the details here. J. HUANG AND Z. DENG AND L. XU
Proposition 1.
The potential Φ( φ ; y ) and probability measure µ on the measurespace ( X , Ξ) satisfy the following properties:(1) for every r > , there is a K = K ( r ) such that, for all φ ∈ X and y ∈ Y with | y | Σ < r , ≤ Φ( φ ; y ) ≤ K ; (21) (2) for any fixed y ∈ Y , Φ( · ; y ) : X → R , is continuous µ -almost surely on theprobability space ( X , Ξ , µ ) ;(3) for every r > , there exists a K = K ( r ) such that, for all φ ∈ X , and y , y ∈ Y with max {| y | Σ , | y | Σ } < r , | Φ( φ ; y ) − Φ( φ ; y ) | ≤ K | y − y | Σ . (22) Proof. (1) From the problem (6), it can be observed that the map G is nonlinearwith respect to q . We know that the following estimate holds [3] (cid:107) u s (cid:107) H ( B R ) = (cid:107) G ( q ) (cid:107) H ( B R ) ≤ C (cid:107) q (cid:107) L ∞ ( B R ) (cid:107) u i (cid:107) L ( B R ) . (23) O : Y → Y is the bounded linear map, and (cid:107) F ( φ ) (cid:107) ∞ is bounded uniformly over φ ∈ X . Hence |G ( φ ) | Σ = | O ◦ G ◦ F ( φ ) | Σ ≤ C. (24)Note that Φ( φ ; y ) = 12 | y − G ( φ ) | ≤ | y | + |G ( φ ) | . (25)Then, for any y ∈ Y with | y | Σ < r , we obtain the boundΦ( φ ; y ) ≤ C (1 + r ) =: K . (26)(2) It is known that the map G : X → Y is continuous [3], i.e. (cid:107) G ( q ) − G ( q ) (cid:107) H ( B R ) ≤ C (cid:107) q − q (cid:107) L ∞ ( B R ) (cid:107) u i (cid:107) L ( B R ) , and O : Y → Y is the linear continuous map. Therefore, the discontinuity set of G is determined by the discontinuity set of the level set map F . However, since weassume that φ ∼ µ is a Gaussian measure, it follows from the Proposition 2.8 in[27] that m ( B i ) = 0, µ -almost surly, i = 1 , · · · , L −
1. In brief, the level sets of theGaussian random field have Lebesgue measure zero. By Lemma 3.2, we can obtainthat (cid:107) φ (cid:15) − φ (cid:107) ∞ → F ( φ (cid:15) ) → F ( φ ) in measure. Therefore, Φ( · ; y ) iscontinuous µ -almost surely .(3) Let φ ∈ X and y , y ∈ Y with max {| y | Σ , | y | Σ } < r . It follows that | Φ( φ ; y ) − Φ( φ ; y ) | = | | y − G ( φ ) | − | y − G ( φ ) | | = 12 |(cid:104) y − y , y + y − G ( φ ) (cid:105) Σ |≤ ( | y | Σ + | y | Σ + 2 |G ( φ ) | Σ ) | y − y | Σ ≤ ( r + 2 |G ( φ ) | Σ ) | y − y | Σ =: K | y − y | Σ . Definition 3.3.
Let ν be a common reference measure. The Hellinger distancebetween µ and µ (cid:48) with common reference measure ν is d Hell ( µ, µ (cid:48) ) = (cid:115) (cid:90) ( (cid:114) dµdν − (cid:114) dµ (cid:48) dν ) dν . (27) BAYESIAN LEVEL SET METHOD FOR AN IMSP IN ACOUSTICS 9
Theorem 3.4.
Assume that φ ∼ µ := N (0 , C α,τ ) . Then the following results hold:(1) The posterior measure µ y exists and is absolutely continuous with respect to µ ,i.e. µ y (cid:28) µ , with Radon-Nikodym derivative given by (16).(2) µ y is locally Lipschitz in the data y , with respect to the Hellinger distance:if µ y and µ y (cid:48) are two measures with data y and y (cid:48) , then for all y and y (cid:48) with max {| y | Σ , | y (cid:48) | Σ } < r , there exists a constant C = C ( r ) such that d Hell ( µ y , µ y (cid:48) ) ≤ C | y − y (cid:48) | Σ . (28) Proof.
From the Proposition 1 (2), we get that Φ( · ; y ) is continuous µ -almost surely.Using the µ -almost surely continuity, it establishes the measurability in Lemma 6.1([27]). Then, the first result follows from the Theorem 6.29 in [45]. For the Lipschitzcontinuity of the µ y , it could be proved by the Theorem 4.5 in [9]. Therefore, weomit the details here.4. Numerical Experiments.
In this section, some numerical results are presentedto demonstrate performance of the proposed method. In particular, we comparethe results of the Bayesian level set method with those of the regular Bayesianapproach.4.1.
Sampling Algorithm.
The Markov chain Monte Carlo (MCMC) methodsare usually applied to draw the samples from the posterior distribution µ y definedabove. In this work, we employ the preconditioned Crank-Nicolson (pCN) algorithm[8], which is described in Algorithm 1. We adopt the proposal variance parameter β ∈ (0 ,
1] to control the stepsize in numerical implementations. We take the choiceof β = 0 .
007 for a compromise between the acceptance rate and the exploration ofthe state space. The proposed pCN-MCMC algorithm is performed with samples N s = 10 . We take the last 2 × samples to compute the conditional mean (CM)estimates. Algorithm 1
The pCN-MCMC algorithm. Collect the scattered field measured data over all frequencies k m , m = 1 , · · · , M and the incident direction d j , j = 1 , · · · J . Set s = 0. Choose an initial state φ (0) ∈ X . for s = 0 to N s do Propose ψ ( s ) = (cid:112) − β φ ( s ) + βξ ( s ) , ξ ( s ) ∼ N (0 , C α,τ ); Draw θ ∼ U [0 , Let a ( φ ( s ) , ψ ( s ) ) := min { , exp(Φ( φ ( s ) ) − Φ( ψ ( s ) )) } ; if θ ≤ a then φ ( s +1) = ψ ( s ) ; else φ ( s +1) = φ ( s ) ; end if end for Data and Parameters.
We consider the case of R = 1 and discretize thedomain with 16512 elements uniformly with meshsize h = 2 . × − . Meanwhile,we adopt a uniform mesh with meshsize (cid:98) h = 2 h for the application of the pCN-MCMC algorithm. The synthetic data is generated by solving the forward modelwith the noise being added, and the data are measured on the boundary Γ R . We assume that the noise is Gaussian, η ∼ N (0 , γ I ), where γ = 0 . d j is taken as J = 5, and the incident directions d j are equallydistributed around Γ R with d = (1 , k = 0 . π to k = 2 . π with M = 6.4.3. Numerical Results.
We test the regular Bayesian approach and the Bayesianlevel set method on the following examples. In both methods, the prior is taken tobe a zero mean Gaussian with Mat´ern covariance, i.e. N (0 , C α,τ ). Example 1.
The true scatterer has the form of q † ( x ) = (cid:26) , x ∈ D, , x ∈ B R \ D, (29)where D = { ( x, y ) ∈ R : x + ( y − √ x ) ≤ } is a love-shaped scatterer, as shownin Fig. 4. In the level set context, we parameterize D in terms of the level setfunction given by D = { x ∈ D | φ ( x ) ≥ } , and the corresponding level set map is F ( φ ) = I D . Fixing α = 3, we apply the regular Bayesian approach and the Bayesianlevel set method to recover the shape of the scatterer with different inverse lengthscales, respectively. The reconstructed results are presented in Fig. 5. One can seefrom Fig. 5 that both methods are effective to recover the shape of the scatterer.However, compared to the regular Bayesian approach, the Bayesian level set methodshows the advantage of identifying the sharp boundary of the scatterer. One of thereasons is that we are able to make good use of the information on q ( x ) under theframework of Bayesian level set method. Figure 4.
The true scatterer q † Example 2.
The expression of the scatterer is the same as that of the Example1, where D is a cross-shaped scatterer plotted in Fig. 6. We define the level setmap (
F φ )( x ) = I D with D = { x ∈ D | φ ( x ) ≥ } , and the zero level set presents theunknown boundary ∂D . Fixing the α = 2, we display the reconstructed results inFig. 7. It can be observed that the Bayesian approach coupled with the level setmethod is a better alternative to recover the boundary of the scatterer. Example 3.
The scatterers are two disjoint domains satisfying q † ( x ) = (cid:26) , x ∈ D or D , x ∈ B R \ ( D ∪ D ) , (30)where D = { ( x, y ) ∈ R : ( x + 0 . + ( y − . ≤ . } and D = { ( x, y ) ∈ R : ( x − . + ( y + 0 . ≤ . } are shown in Fig. 8. For the Bayeian level setmethod, D is characterized by the level set function, and the corresponding levelset map is F ( φ ) = 3 I D + 3 I D with D i = { x ∈ D i | φ ( x ) ≥ } , i = 1 ,
2. We take
BAYESIAN LEVEL SET METHOD FOR AN IMSP IN ACOUSTICS 11
Figure 5.
The reconstructions of a love-shaped scatterer for theregular Bayesian method (top block) and the Bayesian level setmethod (bottom block) with τ = 10 , , α = 3. Figure 6.
The true scatterer q † α = 3, and show the posterior samples with inverse length scale τ = 10 , , Acknowledgments.
The work of DZL is partially supported by the grants (NSFC-11601067, NSFC-117710680), and the work of XLW is partially supported by thegrant (NSFC-117710680).
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