Determining both sound speed and internal source in thermo- and photo-acoustic tomography

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1 Inverse Problems Inverse Problems (05) (0pp) doi:0.088/06656//0/05005 Determining both sound speed and internal soure in thermo and photoaousti tomography Hongyu Liu,,5 and Gunther Uhlmann,4 Department of Mathematis, Hong Kong aptist University, Kowloon Tong, Hong Kong SA, Peopleʼs epubli of China HKU Institute of esearh and Continuing Eduation, Virtual University Park, Shenzhen, Peopleʼs epubli of China Department of Mathematis, University of Washington, Seattle, WA 9895, USA 4 Institute for Advaned Stu, Hong Kong University of Siene and Tehnology, Hong Kong SA, Peopleʼs epubli of China hongyu.liuip@gmail.om and gunther@math.washington.edu eeived June 05, revised August 05 Aepted for publiation 7 August 05 Published 5 September 05 Abstrat This paper onerns thermoaousti tomography and photoaousti tomography, two ouplephysis imaging modalities that attempt to ombine the high resolution of ultrasound and the high ontrast apabilities of eletromagneti waves. We give suffiient onditions to reover both the sound speed of the medium being probed and the soure. Keywords: thermoaousti and photoaousti tomography, sound speed, internal soure, uniqueness. Introdution In this paper we onsider thermoaousti tomography (TAT) and photoaousti tomography (PAT), two oupledphysis modalities that attempt to ombine the high resolution of ultrasound and the high ontrast apabilities of eletromagneti waves. In the latter a laser pulse is used to probe a medium and in the former a lower frequeny wave is used. As a result the medium is heated and produes an elasti expansion that in turns produes a sound wave that is measured outside the medium. This is used to find optial properties of the medium in the ase of PAT or eletromagneti properties in the ase of TAT. These two imaging 5 Author to whom any orrespondene should be addressed /5/ $ IOP Publishing Ltd Printed in the UK

2 Inverse Problems (05) modalities and other oupledphysis imaging tehniques have the potential of signifiant linial and biologial appliations [, 8,, ]. We next present the mathematial formulation of the PAT and TAT problem desribed above. Let WÌ be a ompat set suh that W is onneted. Let () x Î L ( ) be suh thatsupp( ) Ì W, and () x 0 for x ÎW, where 0 is a positive onstant. Consider the following Cauhy problem of the wave equation for ux (, t), ( x, t) Î +, t ux (, t) D ux (, t) = 0 in, + () x (.) ux (,0) f( x), tux (,0) 0 for x = = Î, where f () x Î L ( ) with supp ()ÌW f. (x) represents the wave speed in the bo Ω and f (x) represents an internal soure produed by the tissue volume expansion. The wave measurements are given by L f, ( x, t) = u( x, t), ( x, t) Î W +. (.) The inverse problem is to reover ( f, ) by knowledge of Lf, ( x, t). In what follows, for simpliity, ( f, ) shall be referred to as a TAT/PAT onfiguration. The uniqueness issue for the TAT/PAT problem has gained extensive attention in the literature. However, most of the available results intend to reover either f or, by assuming that the other one is known. For the reovery of f by assuming that is known, we refer to [4] and the referenes therein for the onstant sound speed ase; and [9] for the variable sound speed ase. In [7], the reovery of a smooth sound speed by assuming that the soure f is known was onsidered. Uniqueness was established in [7] with one measurement under the geometri assumption that the domain is foliated by stritly onvex hypersurfaes with respet to the iemannian metri g = d x, where dx denotes the Eulidean metri. In [0], the instability of the linearized problem in reovering both the internal soure and the sound speed was established. In [6], stability estimate was derived in reovering f if there is a modeling error of the variable sound speed. In [], an interesting onnetion was observed between the TAT/PAT problem and the interior transmission eigenvalue problem. In this work, we show that in ertain pratially interesting senarios, one an reover both the sound speed and the internal soure by a single measurement. Our argument relies on the temporal Fourier transform, onverting the TAT/PAT problem into the frequeny domain. Then by using the low frequeny asymptotis, we derive two internal identities involving the unknowns. The main uniqueness that we ould derive from the two integral identities is that if the sound speed is onstant and the internal soure funtion is independent of one variable, then both of them an be reovered. We also point out some other uniqueness results in our stu. Moreover, our argument requires the least regularity assumptions on the parameters as well as the domain. To our best knowledge, this is the first uniqueness result in the literature on jointly determining the sound speed and the internal soure. Our stu makes the first step toward solving this hallenging problem. The rest of the paper is organized as follows. In setion, we present the main theorems of the urrent stu. Setions is devoted to the proofs.. Statement of the main results We first introdue the admissible lass of TAT/PAT onfigurations ( f, ) for our stu. Let ( f, ), Ω and ux (, t) be given in (.). Our argument relies essentially on the temporal Fourier transform of the funtion ux (, t) defined by

3 Inverse Problems (05) ux ˆ(, k) = ux (, t) eikt d t, ( x, k). (.) p Î + 0 A TAT/PAT onfiguration ( f, ) is said to be admissible if (.) defines an H lo ( ) distribution for k Î ( 0, 0) with a ertain 0 Î +. It is remarked that a suffiient ondition to guarantee the admissibility of a TAT/PAT onfiguration ( f, ) is that is a nontrapping sound speed; see definition.6 and theorem.7 in [] for a onvenient referene on the definition and admissibility of a nontrapping sound speed. In what follows, we let eik x F () x = for x ¹ 0. 4p x Φis the fundamental solution to D k. For the subsequent use, we also set F 0( x) = for x ¹ 0. (.) 4p x Let ( 0) denote a entral ball of radius Î + ontaining Ω. Moreover, for a funtion y Î L ( ) supported in, we denote D ( y)( x) F ( x y y d, y x ) y Î. (.) We shall prove that 0 Theorem.. Let ( f, ), j =,, be two admissibletat/pat onfigurations suh that Then one has j j L f, ( x, t) =Lf, ( x, t), ( x, t) Î W +. (.4) ( f f )() x v() x dx = 0, (.5) where v(x) is an arbitrary harmoni funtion, and whih holds for any x Î. ( y) D f ( y) F0 ( x y) + f y x y 8 p = y f y 0 x y ( ) D F ( ) + f y x y d y,.6 8p It is noted that in (.5) and (.6) the integrations an be taken over Ω or instead of sine both ( f, ) and ( f, ) are supported in Ω. In the sequel, we let x = ( x j) j= Î. Using theorem., one an show that Theorem.. Let ( f, ), j =,, be two admissibletat/pat onfigurations. Set j j j () x xf() x xf() x for x Î W. (.7)

4 Inverse Problems (05) Then one has that j () x = 0 for a. e. x Î W, (.8) if either of the following two onditions is satisfied (i) j () x =Q() x for x ÎW, where Θ is a harmoni funtion in ; (ii) j () x from (.7) is independent of one variable, say x j, j. emark.. y theorem., one readily sees that if an admissible TAT/PAT onfiguration ( f, ) is suh that either f is the restrition of a harmoni funtion to Ω, or f is a funtion independent of one variable x j, j, then f is uniquely determined by Lf, ( x, t) for ( x, t) Î W +. Using this result, one an further infer that if the sound speed is known, then the internal soure f is uniquely determined; whereas if f is known in advane, then the sound speed supp()is f uniquely determined. Next, we present a pratial senario in uniquely determining both the sound speed and the internal soure. Theorem.. Let ( f, ) be an admissible TAT/PAT onfiguration supported in Ω suh that is onstant, and f is independent of one variable x j, j. Furthermore, we assume that f () x 0 for a. e. x ÎW and f () x dx > 0. (.9) Then both f and are uniquely determined by Lf, ( x, t) for ( x, t) Î W +. emark.. assumption.9 on the internal soure f means that there exists an open portion of Ω on whih f is stritly positive. Sine f is generated by the absorbed energy, it is a pratially reasonable ondition. y generalizing theorem., we an further prove a bit more general unique determination result as follows. W Theorem.4. Let ( f, ) be an admissible TAT/PAT onfiguration suh that () x = () x + g, x Î W, (.0) b S where b is a positive bakground sound speed whih is supposed to be known in advane, and γ is a positive onstant denoting an anomalous inlusion supported in SÌW. Furthermore, we assume that both b and f are independent of the variable x j, j and D ( f)() x v() x d x ¹ 0 (. ) S for a harmoni funtion v. Then both f and γ are uniquely determined by ( x, t) Î W +. Lf, ( x, t) for From our subsequent argument, one an see that if b and Σ in theorem.4 are taken to be 0 and Ω, and moreover v º, then theorem. is a partiular ase of theorem.4. A simple example for theorem.4 is that both f and b are nonzero onstants, then both f and γ an be uniquely determined. 4

5 Inverse Problems (05) Proofs It is straightforward to show that ux ˆ (, k) satisfies ux, k k x ux, k ik D ˆ + ()ˆ = x f () x, ( x, k ) Î 0,. ( 0) p () (.) Moreover, in order to ensure the wellposedness of the transformed equation (.), one imposes the lassial Sommerfeld radiation ondition lim x + ux ˆ(, k) x x i kuˆ( x, k) = 0. (.) Under the temporal Fourier transform, the TAT/PAT measurement data beome L ( x, k) = uˆ( x, k) W ( 0,. (.) 0) The following lemma shall be needed. Lemma.. Let ux ˆ (, k) Î Hlo ( ) be the solution to (.) (.). Then ux ˆ (, k) is uniquely given by the following integral equation ux ˆ(, k) =k ( y) uy, k x y ˆ F( ) ik ( y f y x y d, y x ). (.4) p F Î Moreover, as k + 0, we have ik ux ˆ(, k) = ( y f y x y d y k ) 0, x. p F + Î (.5) Proof. y the ellipti regularity (see, e.g, [5]), we known that u Î H lo ( ). The integral equation (.4) is of Lippmann Shwinger type, and it is uniquely solvable for u Î C( ) (see []). y using the fat that F () x = F () x + () k as k + 0, (.6) 0 uniformly for x Î, one an verify (.5) by straightforward asymptoti estimates. The proof is omplete., We are in a position to present the proof of theorem.. Proof of theorem.. Let uˆ ( x, k) and uˆ ( x, k) denote the aousti wave fields, respetively, orresponding to ( f, ) and ( f, ). y (.4) and (.), one learly has that uˆ ( x, k) = uˆ ( x, k) for ( x, k) Î W ( 0, 0). Sine both uˆ and uˆ satisfy the same equation ( D+ k ) u = 0 in W, one readily has that 5

6 Inverse Problems (05) and therefore uˆ ( x, k) = uˆ ( x, k) for ( x, k) Î W 0,, (.7) ( 0) uˆ ( x, k) 0, u x, k 0,..8 ( ˆ 0) = ( 0) Hene, by (.4) and (.8), we see that for x Î, k ( y) uˆ ( y, k) F( x y) i y f y x y F( ) p k = ( y) uˆ ( y, k) F( x y) i ( y) f ( y) ( x y) d y. (.9) p F Next, by lemma. one has ik uˆ( j x, k) = y f y x y d y k j () j 0, j,, (.0) p F + = for k Î + suffiiently small. Plugging (.0) into (.9) and using the series expansion for e ik xy of F( x y ), one an further show by straightforward alulations (though a bit tedious) that as k + 0 i k y f y x y y y f y y F0 ( ) d d 8 p p i k y f y x y y ( ) D ( ) F0 ( ) d p i k y f y x y k 6 + p i k = y f y 0 x y y f y F ( ) 8 p p i k ( ( y) ) ( f )( y) 0( x y) D F p i k y f y x y d y k,. 6 ( ) + p whih holds for any x Î. From (.), one immediately has (.6) and for any x Î, 0 0 ( y) f ( y) F ( x y) = ( y) f ( y) F ( x y) d y. (.) We proeed to the proof of the orthogonality relation (.5), and first note the following addition formula for F0( x y) with x > y (see []), a y a = åå Y x Y y,. 4 x y x ( ˆ) ( ˆ ) p a + a a b a b + a= 0b=a where xˆ = x x and yˆ = y y, and Y ( xˆ) denotes the spherial harmonis of order a Î È {} 0, for b =a, ¼, a. y inserting (.) into (.), together with straightforward alulations, one then has a b 6

7 Inverse Problems (05) åå a= 0b=a = a Y x y f y y Y y ( ˆ ) ( ˆ) a + a a b a a b + a Y x y f y y Y y d y,.4 ( ˆ ) ( ˆ ) a + a a b a a b + åå a= 0b=a whih holds for any x Î. Sine { Ya b ( )} a= 0,, ¼, b=a, ¼, a is a omplete orthonormal basis of L ( ), where denotes the unit sphere, one immediately has from (.4) that ( y) f ( y) y a Y yˆ = n m ( y) f ( y) y a Y yˆ d y, (.5) for a = 0,,, ¼, and b =a, ¼, a. We note that y ay (ˆ) y for a = 0,,, ¼, and b =a, ¼, a yield all the homogeneous harmoni polynomials. Hene, (.5) readily implies (.5). The proof is omplete., n m a b Proof of theorem.. For ase (i), one an simply take v(x) to be the harmoni funtion Q() x. Then by (.5) in theorem., one immediately has that j º 0. For ase (ii), without loss of generality, we assume that j () x is independent of x for x = ( x, x, x) Î. Let vx () = e iz x xî, (.6) where z = x + i x, x = x, x, 0 Î, x = 0, 0, x Î,.7 ^ ^ ^ satisfying x + x = ( x ^ ). It is straightforwardly verified that v(x) in (.6) (.7) defines a harmoni funtion. y taking v(x) in (.5) to be the one defined in (.6), we have 0 = j( x, x ) eiz x d x = j( x, x ) eiz x dx ix x ^ = e j ( x dx e x ) x d x, (.8) { x; ( x, x) Î } where x = ( x, x) Î and x = ( x, x) Î. In (.8), we have made use of the fat that j is supported in. Clearly, from (.8), one readily has ix x e j ( x ) dx = 0, (.9) whih holds for any x Î. The lhs of (.9) is the Fourier transform of the funtion j, and hene one easily infers from (.9) that j = 0. The proof is omplete., Proof of theorem.. Without loss of generality, we assume that f is independent of the variable x and write it as f ( x ) for x = ( x, x) Î. y ontradition, we assume that there exists another TAT/PAT onfiguration ( f ( x ), ) supported in Ω with being a 7

8 Inverse Problems (05) onstant suh that L ( x, t) =L ( x, t) for ( x, t) Î W +. (.0) f, f, Sine f f is independent of the variable x, we have by theorem. that Therefore, and f = f. (.) f y x y = f () y x y (.) D f =D f. (.) Next, by (.6) in theorem., together with the use of (.) and (.), there learly holds ( ) D ( f)( y) F0 ( x y) = 0, (.4) where x Î. Then, by the assumption (.9) and remark., it is diretly shown that D ( f)() x > 0 for x Î W. (.5 ) Using (.5), one an easily verify that D ( f)() x F 0( x y) > 0, x Î, (.6) whih together with (.) immediately implies that =. (.7) Finally, by (.7) and (.), one has f = f. The proof is omplete., emark.. We note that the assumption (.9) is only needed to guarantee that (see (.4) and (.6)) D ( f)() x F 0( x y) ¹ 0, x Î. (.8) y using a ompletely similar argument to the proof of theorem., one an show that if f is nonpositive on Ω but stritly negative on an open subset. Then (.8) also holds true. ut as remarked in remark., (.9) is a pratial ondition. The ondition (.) is more general, and this an be observed by simply taking v º. Proof of theorem.4. The proof follows from a similar argument to that of theorem.. Without loss of generality, we an assume that f and b are independent of the variable x, and write them as b ( x ) and f ( x ) for x = ( x, x) Î. y ontradition, we assume that there exists another TAT/PAT onfiguration ( f ( x ), ( x )) supported in Ω with ( x ) given as ( x x ) = ( ) + g, (.9) b S 8

9 Inverse Problems (05) where g is a positive onstant, suh that L ( x, t) =L ( x, t) for ( x, t) Î W +. (.0) f, f, Noting that f f is independent of the variable x, we have by theorem. that f = f. (.) Similar to (.4), we have from (.6) in theorem. that 0 ( y ) D f ( y) F ( x y) = ( y ) D f ( y) F ( x y) (.) for any x Î. Using a ompletely similar argument in proving (.5) in the proof of theorem., one an show from (.) that ( y ) D f ( y) v( y) = ( y ) D f ( y) v( y) (.) for any harmoni funtion v. Next, by ombining (.0), (.9), (.) and (.), we have ( g g ) D ( f)() x v() x d x = 0 (.4 ) S for any harmoni funtion vx. ()Noting (.), we immediately have from (.4) that g = g, whih in turn, together with (.), implies that f = f. The proof is omplete., 0 Aknowledgments The work of H Liu was supported by the FG grants of Hong Kong aptist University, Hong Kong esearh Grant Counil GF grant HKU 045, and the NSF grant of China No. 75. The work of G Uhlmann was supported by NSF. eferenes [] Colton D and Kress 998 Inverse Aousti and Eletromagneti Sattering Theory nd edn (erlin: Springer) [] Finh D and Hikmann K S 0 Transmission eigenvalues and thermoaousti tomography Inverse Problems [] Kruger A, eineke D and Kruger G A 999 Thermoaousti omputed tomographytehnial onstrution Med. Phys [4] Kuhment P and Kunyansky L 008 Mathematis of thermoaousti tomography Eur. J. Appl. Math [5] MLean W 000 Strongly Ellipti Systems and oundary Integral Equations (Cambridge: Cambridge University Press) 9

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