The linear sampling method and the MUSIC algorithm

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Probles 17 (2001) PII: S (01) The linear sapling ethod and the MUSIC algorith Margaret Cheney Departent of Applied Electronics, Lund University, S Lund, Sweden and Departent of Matheatical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA 1 E-ail: chene@rpi.edu Received 9 Septeber 2000 Abstract This paper gives a short tutorial on the MUSIC algorith (Devaney, Therrien) and the linear sapling ethod of Kirsch, and explains how the latter is an extension of the forer. In particular, for the case of scattering fro a finite nuber of weakly scattering targets, the two algoriths are identical. Section 1 outlines the MUSIC algorith and its use in signal processing and iaging. Section 2 outlines the linear sapling ethod and discusses its siilarity with the MUSIC algorith. The paper ends with section 3, a discussion and listing of open questions. 1. MUSIC MUSIC is an abbreviation for MUltiple SIgnal Classification [10]. Section 1.1 outlines the general MUSIC algorith; section 1.2 explains how it is applied in signal processing (in order to explain the nae); section 1.3 explains how it applies to iaging [2] The basics of MUSIC MUSIC is essentially a ethod of characterizing the range of a self-adjoint operator. Suppose A is a self-adjoint operator with eigenvalues λ 1 λ 2 and corresponding eigenvectors v 1,v 2,... Suppose the eigenvalues λ M+1,λ M+2,... are all zero, so that the vectors v M+1,v M+2,...span the null space of A. Alternatively, λ M+1,λ M+2,...could erely be very sall, below the noise level of the syste represented by A; in this case we say that the vectors v M+1,v M+2,...span the noise subspace of A. We can for the projection onto the noise subspace; this projection is given explicitly by P noise = T v j v j (1) j>m 1 Address for correspondence /01/ $ IOP Publishing Ltd Printed in the UK 591

2 592 M Cheney T where the superscript T denotes the transpose, the bar denotes the coplex conjugate and v j is the linear functional that aps a vector f to the inner product v j,f. The (essential) range of A, eanwhile, is spanned by the vectors v 1,v 2,...,v M. The key idea of MUSIC is this: because A is self-adjoint, we know that the noise subspace is orthogonal to the (essential) range. Therefore, a vector f is in the range if and only if its projection onto the noise subspace is zero, i.e. if P noise f =0. This, in turn, happens only if 1 =. (2) P noise f Equation (2) is the MUSIC characterization of the range of A. We note that, for an operator that is not self-adjoint, MUSIC can be used with the singularvalue decoposition instead of the eigenvalue decoposition The use of MUSIC in signal processing MUSIC is generally used in signal processing probles. In this case, we ake easureents of soe signal x(t) at discrete ties t n = n. The resulting saples x n = x(t n ) are considered rando variables. We for the correlation atrix A n, = E(x n x ), where E denotes the expected value. We consider the special case when the signal is coposed of two tie-haronic signals of different frequencies, plus noise. Thus x n = a 1 e iω 1n + a 2 e iω 2n + w n. We assue that the rando variables w n are identically distributed. The goal is to estiate the frequencies of the signals. Because the different ters of x n are utually independent, the self-adjoint atrix A can be written [10] A = E( a 1 2 )s 1 s 1T + E( a 2 2 )s 2 s 2T + σ 2 0 I (3) where the nth coponent of the vector s j is given by sn j = e iω j n, I denotes the identity operator and σ0 2 = E( w n 2 ). Thus we see that the the range is spanned by s 1 and s 2 ; the orthogonal copleent is the noise subspace. The MUSIC algorith for estiating the frequencies ω 1 and ω 2 is to plot, as a function of ω, the quotient 1 (4) P noise s ω where s ω is the vector whose nth coponent is e iωn. The resulting plot, which has large peaks at the frequencies ω 1 and ω 2, is called the MUSIC pseudospectru. We note that the MUSIC algorith involves applying a test to a large nuber of trial signals s ω. The appropriateness of the nae MUSIC is now clear: MUSIC is a ethod for estiating the individual frequencies of ultiple tie-haronic signals The use of MUSIC in iaging Devaney [2] has recently applied the MUSIC algorith to the proble of estiating the locations of a nuber of pointlike scatterers. The following is an outline of his approach. We consider the atheatical odel in which wave propagation is governed in free space by the Helholtz equation ( 2 + k 2 )ψ = 0 (5)

3 The linear sapling ethod and the MUSIC algorith 593 where k corresponds to the frequency of the propagating wave. We iagine that we have N antennas or transducers, positioned at the points R 1,R 2,...,R N, that transit spherically spreading waves. If the jth antenna is excited by an input voltage e j, the field produced at the point x by the jth antenna is ψj in(x) = G(x, R j )e j, where G(x, R j ) denotes the outgoing Green s function. We assue that the scatterers, positioned at the points X 1,X 2,...,X M, are sall, weak and well separated, so that they scatter according to the Born approxiation. Thus if the field ψ in is incident on the th scatterer, it produces at x the scattered field G(x, X )τ ψ in (X ), where τ is the strength of the th scatterer. The scattered field fro the whole cloud of scatterers is G(x, X )τ ψ in (X ). Thus the total scattered field due to the field eanating fro the jth antenna is ψj sc(x) = G(x, X )τ G(X,R j )e j. If this field is easured at the lth antenna, the result is ψj sc(r l) = G(R l,x )τ G(X,R j )e j. This expression gives rise to the ulti-static response atrix K, whose (l, j)th eleent is K l,j = G(R l,x )τ G(X,R j ). (6) The ulti-static response atrix aps the vector of input aplitudes (e 1,e 2,...,e N ) T to the vector of easured aplitudes on the N antennas. This atrix can be written K = τ g g T (7) where we have used the notation g = (G(R 1,X ), G(R 2,X ),...,G(R N,X )) T. (8) For siplicity we consider only the case N>M, i.e. ore antennas than scatterers. Because the Green s function is syetric, K is syetric, but it is not self-adjoint. We for a self-adjoint atrix A = K K = KK, where the star denotes the adjoint and the bar denotes the coplex conjugate (which is the sae as the adjoint, since K is syetric). We note that K is the frequency-doain version of a tie-reversed ulti-static response atrix; thus KK corresponds to perforing a scattering experient, tie-reversing the received signals and using the as input for a second scattering experient [1, 5, 7, 8]. The atrix A can be written A = τ g g T τ l g l gl T (9) l fro which we see iediately that the eigenvectors of A are the g. This eans that the range of A is spanned by the M vectors g. Devaney s insight is that the MUSIC algorith can now be used as follows to deterine the location of the scatterers. Given any point p, for the vector g p = (G(R 1, p), G(R 2,p),...,G(R N,p)) T. The point p coincides with the location of a scatterer if and only if P noise g p = 0. (10) Thus we can for an iage of the scatterers by plotting, at each point p, the quantity 1/ P noise g p. The resulting plot will have large peaks at the locations of the scatterers. We note that the condition (10) depends only on the operator P noise and not on the particular basis {g }. 2. The linear sapling ethod The linear sapling ethod is a linear ethod for finding the boundary of one or ore ipenetrable objects fro scattering data.

4 594 M Cheney 2.1. The basics of the linear sapling ethod Kirsch [3] considers the scattering proble in which incident plane waves scatter off one or ore ipenetrable objects. He considers the far-field operator F, which is an integral operator whose kernel is the far-field scattering aplitude. The operator F satisfies a reciprocity condition but is not self-adjoint. Kirsch fors the self-adjoint operator A = F F, and considers the eigenvalues λ 1 λ 2 and corresponding eigenfunctions v 1,v 2,... The linear sapling ethod is based on the theore [3] that the range of A 1/4 coincides with the range of the operator H, which is defined as follows. Suppose ψ is equal to h on the boundary of the object, satisfies (5) in the region exterior to the object, and satisfies an outgoing radiation condition. Then H aps the Dirichlet data h to the far-field pattern of ψ. In the linear sapling ethod, one deterines the boundary of the object by testing points p as follows. We denote by g p the far-field aplitude corresponding to the Green s function G(x, p). If p is inside one of the objects, then in the region exterior to the object, G(x, p) satisfies (5), so g p is in the range of H. But if p is exterior to the object, then because G(x, p) has a singularity at p, it cannot satisfy (5) there, so g p cannot be in the range of H. The range of H, which Kirsch showed is identical to the range of A 1/4, can be deterined fro the eigenvalues and eigenfunctions of A. In particular, the range of A 1/4 is given by { RanA 1/4 = f : v j,f 2 } <. (11) λ j j 1/2 The algorith of the linear sapling ethod is to plot, at each point p, the quantity 1/( j λ j 1/2 v j,g p 2 ). The plot will be identically zero whenever p is outside all the scattering objects, and nonzero whenever p is inside one of the scatterers Linear sapling for M point scatterers To see the connection between MUSIC and linear sapling, let us consider the linear sapling algorith for the sae case as considered in section 1.3, naely when the scattering object is coposed of M weakly scattering pointlike scatterers. Then, fro the arguents of section 1.3, we find that the operator A has a finite-diensional range, so that the eigenvalues λ M+1,λ M+2,... are all zero. In that case, the condition (11) for g p to be in the range of H becoes v j,g p =0, j = M +1,M +2,... (12) which can also be written P noise g p = 0. This is precisely the MUSIC condition (10). Plotting 1/ P noise g p will give an iage with very large values at the locations of the scatterers. 3. Discussion and open questions It appears that the linear sapling ethod is an extension of the MUSIC iaging algorith of [2] to the case of extended objects and infinite-diensional scattering operators. Many questions arise in connection with these algoriths. First, the MUSIC algorith uses only the null space of the operator K. But we know that the eigenvectors of K contain inforation about the scatterers. In particular, the eigenvector corresponding to the largest eigenvalue corresponds to a wave focusing on the strongest scatterer, and the eigenvalue contains inforation about the strength of the scatterer [1, 8]. How can MUSIC be odified to ake use of this inforation? The linear sapling ethod, on the other hand, uses all the eigenvalues and eigenfunctions but produces only the location of the boundary of the scattering object. Yet we know that the

5 The linear sapling ethod and the MUSIC algorith 595 eigenvalues and eigenfunctions contain all the inforation about the scatterer [4, 9]. How can the eigenvalues and eigenfunctions be used to recover ore inforation? Acknowledgents I a grateful to Tony Devaney for sending e his preprint [2], to Michael Silevitch for encouraging our interaction, to Frank Natterer for pointing out (12) to e and to Andreas Kirsch for several helpful coents and corrections. This paper was partially supported by the Office of Naval Research, Rensselaer Polytechnic Institute, Lund Institute of Technology through the Lise Meitner Visiting Professorship, and CenSSIS, the Center for Subsurface Sensing and Iaging Systes, under the Engineering Research Centers Progra of the National Science Foundation (award nuber EEC ). References [1] Cheney M, Isaacson D and Lassas M Optial acoustic easureents SIAM J. Appl. Math. at press [2] Devaney A J 2000 Super-resolution processing of ulti-static data using tie reversal and MUSIC Northeastern University Preprint [3] Kirsch A 1998 Characterization of the shape of a scattering obstacle using the spectral data of the far field operator Inverse Probles [4] Lassas M, Cheney M and Uhlann G 1998 Uniqueness for a wave propagation inverse proble in a half space Inverse Probles [5] Mast T D, Nachan A I and Waag R C 1997 Focusing and iaging using eigenfunctions of the scattering operator J. Acoust. Soc. A [6] Newton R G 1989 Inverse Schrödinger Scattering in Three Diensions (New York: Springer) [7] Prada C and Fink M 1994 Eigenodes of the tie reversal operator: a solution to selective focusing in ultipletarget edia Wave Motion [8] Prada C, Thoas J-L and Fink M 1995 The iterative tie reversal process: analysis of the convergence J. Acoust. Soc. A [9] Ra A 1988 Recovery of the potential fro fixed-energy scattering data Inverse Probles [10] Therrien C W 1992 Discrete Rando Signals and Statistical Signal Processing (Englewood Cliffs, NJ: Prentice- Hall)