Reconstructions for some coupled-physics inverse problems

Transcription

1 Reconstructions for some couple-physics inerse problems Guillaume al Gunther Uhlmann March 9, 01 Abstract This letter announces an summarizes results obtaine in [8] an consiers seeral natural extensions. The aforementione paper proposes a proceure to reconstruct coefficients in a secon-orer, scalar, elliptic equation from knowlege of a sufficiently large number of its solutions. We present this eriation an exten it to show which parameters may or may not be reconstructe for seeral hybri (also calle couple physics) imaging moalities incluing photo-acoustic tomography, thermo-acoustic tomography, transient elastography, an magnetic resonance elastography. Stability estimates are also propose. 1 Introuction Consier a general secon-orer, linear elliptic equation of the form a u j + b u j + cu j = 0, x X, u j = f j x X, 1 j J, (1) for X a smooth open omain in R n, with n spatial imension, an (a, b, c) possibly complexalue, symmetric secon-orer tensor, ector fiel, an scalar coefficient, respectiely. We assume that a is elliptic, the real part of a is coercie an boune, an c is such that the aboe equation amits a unique solution. We also assume that (a, b, c, a) are of class C 0,α for some α > 0. We construct J solutions of the aboe equation for ifferent bounary conitions. Seeral recent hybri inerse problems aim to reconstruct the unknown coefficients (a, b, c) from knowlege of internal functionals of the coefficients an of the elliptic solutions (u j ) 1 j J. Concretely, we assume knowlege of the following functionals H j (x) = (x)u j (x), x X, () with (x) a scalar coefficient that is a priori also unknown. What may be reconstructe from (a, b, c) from knowlege of (H j ) 1 j J in the setting 1 is analyze in [8]. We present the reconstruction proceure of the aforementione reference in section. Such a reconstruction is base on the aailability of ratios of solutions H j H k = u j u k. This preliminary step is then use in section 3 to show that (a, b, c, ) can be reconstructe up to explicit obstructions that take the form of gauge transformations. We also proie stability estimates for the reconstructions. The inerse problems with internal functionals of the form () are part of a larger class referre to as hybri inerse problems or couple-physics inerse problems. For recent results an reiews on this topic, we refer the reaer to [1, 4, 10, 1, 11, 13, 14, 16, 15, 17] an their references. Department of Applie Physics an Applie Mathematics, Columbia Uniersity, New York NY, 1007; gb030@columbia.eu Department of Mathematics, Uniersity of Washington, Seattle, WA, an Uniersity of California, Irine, CA, 9697; gunther@math.washington.eu 1

2 Reconstruction proceure For the rest of the paper, we assume the existence of u 1 0 on X. We refer to [8] for conitions on f 1 that ensure such a property, either globally, in faorable cases, or at least locally. This allows us to efine the known quantities j = H j+1 H 1 = u j+1 u 1, 1 j J 1. (3) Using the notation A : = Tr(A) for symmetric matrices A an, we fin that α τ : j + β τ j = 0 x X, j = f j+1 f 1 x X, (4) where for an arbitrary complex-alue non-anishing function τ(x) on X, we hae α τ = τu 1a, β τ = τu 1b + τ au 1. (5) Note that the equation (4) is inariant by multiplication by a non-anishing scalar coefficient so that (α τ, β τ ) may at best be reconstructe up to a multiplicatie scalar coefficient. The result in [8] shows that this is the only obstruction to the reconstruction of (α τ, β τ ). More precisely, let us assume that ( 1,..., n ) form a basis of R n for all x X. We istinguish the case a scalar from the case a a secon-orer tensor. When a is scalar an J = n + 1, then (α τ, β τ ) are reconstructe up to the multiplicatie scalar τ. This is equialent to saying that a 1 b is uniquely reconstructe. Inee, we hae j + β τ α τ j = j + b a j = 0 so that, efining H ij = i j an H ij the coefficients of H 1, we hae a 1 b = H ij (a 1 b j ) i = H ij j i, (6) where we hae use the conention of summation oer repeate inices an the fact that for any ector F, we hae F = H ij F j i. When a is tensor-alue, we nee J = I n := 1 n(n + 3) = n M n, M n = 1 n(n + 1) 1. (7) For 1 j I n 1 an 1 m M n, let us efine the coefficients θ m j such that I n 1 j=1 θ m j j = 0 an the symmetric matrices M m = I n 1 j=1 θ m j j, (8) such that (M m ) 1 m Mn form a free family of symmetric matrices. Sufficient conitions are presente in [8] to guaranty that ( j ) 1 j n an (M m ) 1 m Mn are free families for the choice θj m = H jk m+n k for 1 j n, θj m = 1 for j = n + m an θj m = 0 otherwise. The aboe construction allows us to obtain the following constraints: α τ : M m = 0, 1 m M n. (9) This implies that α τ = M 0, where (M 0 ) is a matrix in the one-imensional orthogonal complement to (M m ) 1 m Mn for the inner prouct for symmetric matrices (A, ) = Tr(A ). Thus α τ is reconstructe up to a multiplicatie scalar coefficient. From (4), we euce that β τ = H ij α τ : j i. (10)

3 Note that the aboe is nothing but (6) when a is a scalar coefficient. This shows that (α τ, β τ ) are uniquely reconstructe up to the multiplicatie coefficient τ. Note that aitional information of the form H k = u k for u k solution of (1) with u k = f k on X oes not proie any new information. Inee, H k H 1 is a solution of the elliptic equation (4) with known bounary conition u k u 1 on X. 3 Reconstruction of (a, b, c, ) up to gauge transforms Reconstruction up to gauge transforms. The aboe eriation shows that all that can be extracte from an arbitrary large number of functionals of the form H k = u k is (α τ, β τ, H 1 ) augmente with the equation for u 1. Let us ecompose a = â for â a matrix with eterminant equal to 1. We assume here to simplify that such a ecomposition is ali globally on X (which is obious in the case where a is real-alue an positie-efinite). Since α τ = τu 1 â is known, we euce that â is known. We compute âα 1 τ (β τ α τ ) = b ( ln τ) â. Moreoer, efining = u 1 = H 1, we fin that = â + c. Thus, we obtain after elimination of τ an u 1 that knowlege of (α τ, β τ, H 1 ) an the equation for u 1 is equialent to knowlege of ( â, b + â ln, H1 H 1 = â + c ). (11) No aitional information may be extracte from functionals of the form H k = u k since knowlege of the aboe coefficients uniquely etermines the functionals H k. The imension of the unknown coefficients in (11) is n(n+1) 1 + n + 1 = I n = 1 n(n + 3), which is the number of functionals use to reconstruct them. The imension of (a, b, c, ) is n(n+1) +n+1+1 = I n +. There are therefore two gauge parameters that remain unetermine. Moreoer, (a, b, c, ) are reconstructe up to any transformation that leae the coefficients in (11) inariant. Applications to meical imaging moalities. In the setting of Transient Elastography an Magnetic Resonance Elastography, we may assume that is known (an equal to 1) an that b = 0. We thus obtain a (reunant) transport equation for (or equialently for the gauge τ) an then an explicit expression for c. Therefore, (a, c) is uniquely reconstructe. More generally, when (a 1 b) is known, we obtain an elliptic equation for or equialently for τ. Then (a, a 1 b, c) is uniquely reconstructe. In the setting of quantitatie photo-acoustic tomography (QPAT), we may assume that b = 0 an that = Γc. We again obtain that = Γc is known, an hence q =, is known. The reconstruction of (, c, Γ) is unique up to any transformation that leaes ( Γc, â + c ) inariant. When Γ is known, then (, c) are uniquely reconstructe [5, 6, 9]. A similar result may be obtaine in the imaging moality calle quantitatie thermoacoustic tomography (QTAT), where = Γ(Ic)u 1 ; see [, 3, 7] for a eriation of such a moel for H j = Γ(Ic)u j u 1. Assuming again that b = 0, or more generally that (a 1 b) is known so that τ, or equialently is known, then = u 1 is known. In this setting, we 3

4 thus fin that (, c, Γ) are reconstructe up to any transform that leaes ( ΓIc, â + c ) inariant. Note that when a is real-alue, then Γ is uniquely reconstructe an (, c) are reconstructe up to a transform that leaes â + c inariant [3]. Note that, more generally, one conition on the fiel b is sufficient to uniquely reconstruct the gauge τ or equialently. Inee, we obsere that the secon known quantity in (11) is equialent to knowlege of a 1 b + ln. Thus, knowlege of one component of a 1 b, or of a 1 b, for instance, again proies an equation that allows us to uniquely reconstruct an, hence, a 1 b. In such a setting, q = with = u 1 = H 1 is known an (, c, ) can then be reconstructe up to any transform that leaes (, â + c ) inariant. 4 Sufficient conitions an stability estimates Sufficient conitions. The results of the preceing section exactly characterize which coefficients in (a, b, c, ) can be reconstructe. Such reconstructions hinge on the solutions (u j ) to be sufficiently inepenent. More precisely, we assume that u 1 0 on X, ( j ) 1 j n is a basis of R n at eery point x X, an that the matrices M m are linearly inepenent on X. In some situations, for instance when complex geometric optics (CGO) solutions can be constructe, the aboe conitions are shown to hol for an open set of well-chosen bounary conition (f j ) 1 j In [8]. Howeer, in the general situation where a is possibly complex-alue an anisotropic, such CGO solutions are not aailable. The linear inepenences mentione aboe can be shown to hol locally on subomains on X. More precisely, it is shown in [8] that for a finite coering K k=1 X k of X, then for an open set of bounary conitions (f j ) 1 j J=K In, we can construct a non-anishing solution u k,1 on X k, linearly inepenent graients ( u k,j u k,1 ) j n+1 an linearly inepenent matrices M k,m as constructe in (8). Stability estimates. The proceure leaing to the reconstruction of (11) is explicit an allows one to estimate how errors in the functionals (H j ) propagate into errors in the reconstructe coefficients. Let us assume that is known an smooth an that b = 0 for concreteness. Similar results can be obtaine in more general cases. We obsere that the construction of the matrices M m inole taking two eriaties of the functionals H j. The reconstruction of â therefore inoles ifferentiating (H j ) twice. When b = 0, we obsere that the reconstruction of or equialently τ from the (secon) ector fiel in (11) also inoles ifferentiating (H j ) twice. Once (â, ) are known, then (11) proies a formula for c. Howeer, some simplifications occur. From (5), we obsere that a is reconstructe from ifferentiating (H j ) twice (an not thrice). Then with u 1 known since is known, we reconstruct c irectly from (1) with again a loss of two eriaties. This yiels the result (a, c, a) (ã, c, ã) C 0,α C (H j H j ) 1 j J C,α, for some positie constant C, where H j is constructe as H j in () with the coefficients (a, b, c) in (1) replace by (ã, 0, c). Similar stability estimates may be obtaine in the more general case with an b unknown; see [8] for aitional results. Acknowlegment G was partially fune by grants NSF DMS an DMS GU was partially fune by the NSF an a Rothschil Distinguishe Visiting Fellowship at the Newton Institute. 4

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