Angular average of time-harmonic transport solutions

Transcription

1 Angular average of time-harmonic transport solutions Guillaume Bal, Alexandre Jollivet, Ian Langmore, François Monard November 4, Abstract We consider the angular averaging of solutions to time-harmonic transport equations. Such quantities model measurements obtained for instance in optical tomography, a medical imaging technique, with frequency-modulated sources. Frequency modulated sources are useful to separate ballistic photons from photons that undergo scattering with the underlying medium. This paper presents a precise asymptotic description of the angularly averaged transport solutions as the modulation frequency ω tends to. Provided that scattering vanishes in the vicinity of measurements, we show that the ballistic contribution is asymptotically larger than the contribution corresponding to single scattering. Similarly, we show that singly scattered photons also have a much larger contribution to the measurements than multiply scattered photons. This decomposition is a necessary step toward the reconstruction of the optical coefficients from available measurements. Introduction Transport linear Boltzmann) equations offer accurate descriptions for the propagation of particles in scattering media such as e.g. photons in human tissues in the application of optical tomography []. In the latter medical imaging modality, optical parameters are reconstructed from available photon density readings at detectors. Such detectors are typically optical fibers, which collect photons coming from different directions. The optical sources are also typically optical fibers and photons are thus emitted over a larger range of directions. When steady state sources are used, the reconstruction of the optical parameters from available measurements is severely ill-posed; see e.g. [7] as well as [4] for a review of inverse transport theory in several regimes of interest in optical tomography. When time dependent sources are being used instead, some optical coefficients can be reconstructed from stably from angularly averaged measurements as described in [6]. However, because light speed is very large, measurements in the time domain are typically not available. An intermediate regime consists of using frequency modulated sources, for instance sources of the form + cosωt + φ))sx, v) where φ is a constant phase shift, x is position and v direction of propagation. Heuristically, large values of ω correspond to good temporal sampling inversely proportional to ω) in the time domain. As ω increases, we thus expect to be able to achieve accurate reconstructions of some) optical parameters as in the time domain. That the reconstruction of the optical parameters is greatly improved when ω increases was demonstrated in the numerical simulations performed in e.g. [5, 6]; see also [] for a similar behavior in the diffusive regime. See also [] for an approach to the time-dependent) inverse problem using highly-oscillatory solutions. The main objective of this paper is to give a precise description of the asymptotic behavior of the angular averaging of transport solutions in the limit of large ω. The reason why reconstructions are more stable for large values of ω is that scattering is damped, in a sense that will be made precise, as ω increases. We decompose the transport solution into three components: the ballistic component, the single scattering component, and the multiple scattering component. These contributions exhibit different behaviors for large values of ω. By using stationary phase techniques, we present a precise asymptotic description of these three components as ω. Although we shall not do so here, such a decomposition can then be used to obtain stable reconstructions of some) optical coefficients; see also [3]. Department of Applied Physics and Applied Mathematics, olumbia University Laboratoire de Physique Théorique et Modélisation, NRS UMR 889/Université de ergy-pontoise

2 That stationary phases appear in the analysis may be understood as follows. As time harmonic photons propagate from the source located at a point x ) to the detector located at a point x c ), they accumulate a phase described by e iωd, where d is the distance traveled by the photon along its possibly complicated) path. For ballistic photons, the distance is fixed and given by the distance between the source and the detector. The amplitude of ballistic photons is not affected by the modulation frequency ω. Scattered photons, however, may scatter at different locations and thus arrive at the detectors location with interfering phases. The precise averaging of phases is then described by stationary phase. It turns out that the points where the phase is stationary are precisely the points on the line segment joining the source to the detector. There is therefore a continuum of stationary points, which is the main mathematical difficulty we overcome in this paper by a careful estimate of the remainders that appear in standard stationary phase expansions. Multiple scattering is then handled similarly and shown to have a contribution that is asymptotically negligible compared to that of the ballistic and single scattering components. The rest of the paper is structured as follows. Our main results are stated in section. The decomposition of the angularly averaged transport solution defined in.4) below) into ballistic, single scattering, and multiple scattering contributions is presented in.6) below. The asymptotic behavior of these terms for large values of ω is described in Theorems.,.5 and.6. Section 3 introduces notation on transport theory and decomposes the transport solution into terms involving increasing orders of scattering. Section 4 gives a proof of lemma 3., which states how the so-called subcriticality conditions in the steady-state regime can be relaxed in the time-harmonic regime. Finally, sections 5 and 6-7 give proofs for the single and multiple scattering estimates, respectively, using a careful stationary phase expansion. Statement of the main results Let X R n, n be an open convex bounded domain with boundary X and diameter >. Denote the incoming and outgoing boundaries Γ ± = { x, v) X S n v S n } x,±, where S n x,± := {v S n : ±ν x v > },.) where ν x is the outer normal to X at x X. We consider the transport equation in the time-harmonic regime for the density ψx, v), with isotropic ingoing boundary conditions v ψx, v) + σx, v) + iω)ψx, v) = kx, v, v)ψx, v ) dv, x, v) X S n, S n.) ψx, v) = gx), x, v) Γ, where ω and the input function g takes the form gx) = δx x ), x, x) X) call it g x ). By δx x ) we mean the delta distribution that satisfies for each smooth function φ defined at the boundary: δx x )φx) dµx) = φx ),.3) X where dµx) is the standard measure on the boundary. The coefficient σx, v) accounts for particles that were absorbed by the medium, and kx, v, v) accounts for particles that scattered at point x from direction v to direction v. It is customary to write σx, v) = σ a x, v) + σ p x, v), where σ a is the intrinsic absorption of the medium, and σ p x, v) = kx, v, v ) dv, S n represents the loss of particles that have scattered at x from direction v to other directions. An existence theory for.) is given in the next section. From the solution ψ of.), we consider the angularly averaged outgoing measurements: T ω x, x c ) = S n xc,+ ψ Γ+ x c, v) v ν xc dv..4)

3 Here and below, x X will denote the emitter s position and x c X, the captor s position. The measurement function.4) can be seen as the distributional kernel of the operator M ω f, g) := T ω x, x c )fx )gx c ) dµx ) dµx c ), f, g) L X)),.5) X X defined and studied in [7] in the case ω =. Now, as will be justified in the next section, assuming that ω is large enough, the measurement operator defined in.4) admits the following decomposition T ω x, x c ) = T ω x, x c ) + T ω x, x c ) + T ω +x, x c ),.6) where T ω accounts for ballistic particles emitted at x and captured at x c, T ω /T+ ω accounts for particles that scattered once/multiple-times inside the domain, respectively. This sort of decomposition was introduced in [3] in three dimensions and later in arbitrary dimensions for the stationary problem [9], and the timedependent problem [8]. For the work that follows, and in order to avoid effects of scattering at the boundary, we formulate the following crucial hypothesis, Hypothesis.. Define the spatial support of the scattering function supp X k = then we make the following assumption: { x X, sup kx, v, v ) > v,v S n D := dist X, supp X k) >. Let us now give the main results. The ballistic term is given by T ω x, x c ) = e iω xc x Ex, x c ) x c x n ν x e ν xc e,.7) where we have defined e := x c x and where Ex, x c ) is defined by 3.4) given below. Here and below x := x/ x is the unit vector in the direction of x. One can readily see from expression.7) that the magnitude of the ballistic part is unaffected by the frequency ω. This is because the ballistic particles are the first ones to reach the detector, and there is only one ballistic path from x to x c, hence no interference due to a difference of paths. The single scattering term admits the following expression T ω x, x c ) = e iωϕx,x,xc) Ex, x, x c )kx, x x, x c x)cx, x, x c ) dx, where X cx, x, x c ) := x x x x c ) n+ ν x x x ν xc x c x, ϕx, x, x c ) := x x x x c. where Ex, x, x c ) is defined below by 3.5). We define the following function },.8) fx, x c, x) := Ex, x, x c )kx, x x, x c x)cx, x, x c )..9) Due to the fact that there is a continuum of single scattering paths from x to x c, particles having taken different paths will interfere. As a result, the single scattering becomes an oscillatory integral, the leading behavior of which, as ω, is ruled by the stationary points of the phase function ϕx, x, x c ) := x x x x c. The following theorem gives the leading-order term in the asymptotic expansion of.8) for large ω here and below, we denote by, the floor and ceiling functions, respectively): 3

4 Theorem.. Assume that the integrand defined in.9) satisfies f L X) ; n+3 X)) and assume hypothesis.. Then there exists a constant such that the following decomposition holds for every x, x c ) X, x x c denote d = x c x and e = x c x ) and for every ω > π T ω x, x c ) = e iωd d ω d ) n kx + ue, e, e ) ud u)) n e in ) π 4 Ex, x c ) ν x e ν xc e du + R ω x, x c ),.) where the remainder R ω belongs to L X X) and satisfies the estimate R ω ω n+ f n+3..) Remark.3. A sufficient condition for the integrand f to satisfy the regularity prescribed in theorem. is to assume that the optical coefficients σ, k) are of class n+3. Remark.4. Besides increasing the ballistic-scattering separation in the measurements, the leading-order of.) contains a weighted integral transform of k which will motivate a mildly ill-posed reconstruction formula for k. This is to be compared with [7], where the same inverse problem was investigated in the case ω = and was shown to be severely ill-posed. This weighted integral transform also appears in the time-dependent setting [6]. Finally, the multiple scattering part of the measurements is a bounded function and satisfies the following estimate: Theorem.5. Let σ, k) n+ X S n ) n+ X S n S n ), and assume hypothesis.. Then there exists a frequency ω and a constant such that for every ω ω the multiple scattering T ω + L X X) and satisfies the estimate ω, n =, T+ ω ω lnω), n = 3, ω n+, n 4..) These results are summarized in the following theorem, using the most regularity of the optical coefficients that is required for theorems. and.5 to hold: Theorem.6. Assume that σ, k) n+3 X S n ) n+3 X S n S n ), and assume hypothesis.. Then there exists a frequency ω and a constant such that for every ω ω the measurement function T ω admits the following singular decomposition T ω x, x c ) = T ω x, x c ) + T ω x, x c ) + T ω +x, x c ),.3) for a.e. x, x c ) X X, where T ω is given by.8) and has the asymptotic behavior described in.) and 3 Forward theory T ω x, x c ) := e iω xc x Ex, x c ) x c x n ν x e ν xc e,.4) ω, n =, T+ ω L X X) and T+ ω ω lnω), n = 3,.5) ω n+, n 4. Let us now return to the forward model.) and present the necessary results that will be useful for the subsequent sections. 4

5 Integral equation: Let us recall some notation. For x, v) X S n ) Γ + Γ, let τ ± x, v) be the distance from x to X traveling in the direction of ±v, and x ± x, v) = x ± τ ± x, v)v be the boundary point encountered when we travel from x in the direction of ±v. We also define τ = τ + + τ. As it is done in many settings, we integrate the PDE in.) along the direction v. We obtain that ψ is a solution of the following integro-differential equation where we have defined, for φ L X S n ) and φ L X) I K ω )ψ = J ω g, 3.) τ x,v) K ω φx, v) := e iωt Ex tv, x) kx tv, v, v)φx tv, v )dv dt, S n 3.) J ω φx, v) := e iωτ x,v) Ex x, v), x) φx x, v)), and ) 3.3) Ex, x) := exp x x For future reference, we also define iteratively σx + s x x, x x ) ds. 3.4) Ex,..., x i+ ) := Ex,..., x i )Ex i, x i+ ). 3.5) The operators K ω and J ω are well-defined and continuous operators in LL X S n )) and LL X), L X S n )), respectively [, 4]. Now we must also make sense of J ω and K ω when the inputs are the singular distributions g x = δx x ) defined in.3). One can show that, in the sense of distributions, we have J ω g x x, v) = Ex, x)e iω x x ν x v δ Sv x x ) x x n, x, v) X S n, where δ S stands for delta distribution on S n. Extending naturally the definition 3.) to distributions in the angular variable and applying it to the previous equality, we obtain that K ω J ω g x is well-defined in L X S n ) and its expression is given by, for a.e. x, v) X S n, K ω J ω g x x, v) = τ x,v) e iωt+ x tv x ) kx tv, x x, v)ex, x tv, x) ν x x tv x dt. 3.6) x tv x n Moreover, taking the L norm of 3.6), we obtain that X S n K ω J ω g x x, v) dx dv k X S n k S n, dt dx dv x tv x n 3.7) and the bound is uniform in x X, where we recall that denotes the diameter of X. We will use this result later on to prove theorem.5. Subcriticality conditions: Like in the stationary setting for ω = see e.g. [9, 7, 5, 7]), equation 3.) is solvable for ψ if the operator I K ω ) is invertible in L := LL X S n )). Such a condition is met when there exists an integer r such that the operator I K r ω) is invertible in L. Then the solution of 3.) admits either one of the following equivalent expressions: r ψ = I Kω) r for any integer N. m= K m ω J ω g = r m= K m ω I K r ω) J ω g = N m= K m ω J ω g + I K ω ) K N+ ω J ω g, 3.8) 5

6 In the stationary setting i.e. ω = ), concrete examples of invertibility of I K r, r =, are given by either one of the following so-called subcriticality conditions [4,, 4] σ σ p, 3.9) τσ p <. 3.) In the present time-harmonic regime, invertibility is still achieved with r = under the above conditions. However, it can also be obtained with r = for large enough ω, independently of any positivity assumption, though we require that the optical coefficients have bounded -norms and k be supported away from the spatial boundary X lemma 3. and corollary 3.3). Lemma 3.. Let σ, k) X S n ) X S n S n ) and assume hypothesis.. Then there exists a constant such that for every ω the following estimates hold σ, k ) ω lnω), n =, Kω L σ, k ) ω lnω)), n = 3, 3.) σ, k ) ω lnω), n 4. Remark 3.. The same result holds even if k is supported up to the boundary X. Lemma 3. will allow us to assess the decay of multiple scattering terms in further sections. From this lemma we obtain the following corollary. orollary 3.3. Let σ, k) X S n ) X S n S n ). Then there exists ω which depends on σ and k, X and n such that I K ω is invertible and I K ω ) L uniformly for ω ω. can be chosen to be + S n k ). The uniform estimate of the norm of I K ω ) at large ω given in corollary 3.3 will also be useful to estimate the decay of multiple scattering terms see section 6). Proof of orollary 3.3. From lemma 3., we have that lim ω Kω L =, so there exists ω such that Kω L for ω ω. For such an ω, I Kω is invertible with inverse I Kω) = m= Km ω and in turn, I K ω is invertible with inverse I K ω ) = I + K ω )I Kω) = m= Km ω. In this case, and noting that K ω L S n k, we get This completes the proof. I K ω ) L + K ω L K ω L + S n k ). Derivation of decomposition.6): Whenever subcriticality conditions are satisfied and the solution u of 3.) admits the expressions 3.8), one obtains the following decomposition of the measurement function T ω x, x c ) = for any integer N where Tmx ω, x c ) = S n xc,+ N Tmx ω, x c ) + R N+ x, x c ), 3.) m= [K m ω J ω g x ]x c, v) v ν xc dv, for m, 3.3) and Rmx ω, x c ) = S n xc,+ [I K) K m ω J ω g x ]x c, v) v ν xc dv, for m. From this latter expression 3.3) of the T ω m s, one can deduce the expressions.7),.8) and 6.) of the ballistic, single and multiple scattering components of the measurement function T ω. This can be done by 6

7 following the calculations in [7, Section 3.] with minor modifications. We do not repeat these calculations here. Then the decomposition.6) follows from 3.) and the equality T ω +x, x c ) = N Tmx ω, x c ) + R N+ x, x c ), for any integer N. 3.4) m= 4 Proof of Lemma 3. From 3.) it follows that Kωφx, v) = β ω x, v, x, v )φx, v )dx dv, X S n 4.) for x, v) X S n and for φ L X S n ), where the distributional kernel β ω of K ω is given by β ω x, v, x, v ) = τ x,v) e iωt+ x x tv ) ft, x, v, x, v ) x tv x dt, where 4.) n ft, x, v, x, v ) := Ex, x tv, x)kx tv, v, v)kx, v, v ), v := x tv x, for a.e. x, v, x, v ) X S n X S n. Note that f is uniformly bounded by k. The goal here is to bound the quantity sup β ω x, v, x, v ) dx dv. x,v X S n Let ω. We extend σ resp. k) by to R S n resp. R S n S n ). Then since k is compactly supported inside X S n S n, f is thus extended smoothly to R R n S n R n S n. We now fix x, v, v) X S n ) and study the quantity Ix, v, v, ω) = β ω x, v, x, v ) dx. 4.3) X It is now customary to study the phase function ϕt) = t x tv x the dependency of ϕ on x, x, v) is made implicit here), especially, for what x s the function is stationary. We have ϕ t = x tv x v x tv x ) x tv x x x x x ) v) = x tv x x tv x + v x tv x )). 4.4) Hence t ϕ = whenever x = x + uv, u [, ]. In order to adapt x to the geometry of the problem, we make the change of variable x = x + uv + ρθ, u [, ], ρ [, ], θ S n {v} ). 4.5) In these new variables, we have the following facts dx = ρ n dρ dθ du, x tv x = u t) + ρ ) x x x x ) v) = ρ, t ϕ = t u) + ρ + u t) t u) + ρ ρ. The integral 4.3) becomes Ix, v, v, ω) = S n β ω xu, ρ, θ), v, x, v ) ρ n dρ dθ du. 7

8 In this set of variables, the phase is non-stationary for ρ >, so we split the above integral into two parts I := I + I by chopping the integral in ρ: dρ = ε dρ + dρ, respectively. The small parameter ε ε / will be chosen at the end in terms of ω, in order to get the optimal estimate. Let us estimate I first. Note that using 4.) we obtain β ω xu, ρ, θ), v, x, v ) k τ x +uv+ρθ) dt t u) + ρ ) n k dt t + ρ ) n, thus I x, v, v, ω) k 4 k S n du dθ S n ε ε dρ ρ n dt t + ρ ) n ρ n t + ρ ) n dt dρ. Estimating the integral on the above right-hand side, we write t, ρ) = r cos α, r sin α), with α [, π ] and r [, rα)], where rα) = cos α if α α := tan ε and rα) = ε sin α if α α. The above double integral becomes, after simplification: ε ρ n t + ρ ) n α sin n α dt dρ = cos α π { dα + ε sin n 3 ε ln ε, n =, α dα, α ε, n 3, whence the bound on I x, v, v, ω). Now, let us estimate I. For ρ ε, t ϕ is never zero, so we can write e iωϕ = by parts the right-hand-side of 4.). Doing so, we write iω tϕ t eiωϕ and integrate where we have defined β ω = iω β,ω + β,ω + β 3,ω ), β,ω x, v, x, v ) = e iωτ x,v)+ x τ x,v)v x ) fτ x, v), x, v, x, v ) x τ x, v)v x n x τ x, v)v x + τ x, v) x x ) v) β,ω x, v, x, v e iω x x f, x, v, x, v ) ) = x x n x x x x ) v), 4.7) τ x,v) β 3,ω x, v, x, v ) = e iωt+ x tv x ) d ft, x, v, x, v ) ) dt x tv x n x tv x + t x x. 4.8) ) v) The term β,ω is identically zero provided that k is supported away from the boundary hypothesis.). We now estimate β,ω. From 4.7) it follows that 4.6) β,ω x, v, x, v k ) x x n x x x x ) v) k x x + x x ) v x x n x x x x ) v) ) k x x n 3 x x x x ) v) ) 4.9) k u + ρ ) n 3 ρ, 8

9 where we expressed the change of variable 4.5) in the last line. We get β,ω xu, ρ, θ), v, x, v ) ρ n dρ dθ du S n ε 4 k S n 4 k S n ρ n 4 dρ du 4.) ε u + ρ ) n 3 ε, n =, lnε), n = 3, 4.) π, n 4. Finally we estimate β 3,ω. Plugging the change of variable 4.5) into 4.8) and using the fact that τ x + uv + ρθ, v) = τ x + ρθ, v) + u, we get Hence, we have that Finally, β 3,ω xu, ρ, θ), v, x, v ) = τ x +ρθ,v)+u ρ β 3,ω xu, ρ, θ), v, x, v ) f ρ e iωϕ d t u ) t u) + ρ ft, xu, ρ, θ), v, x, v ) dt. dt t u) + ρ ) n τ x +ρθ,v)+u dt t u) + ρ ) n f ρ β 3,ω xu, ρ, θ), v, x, v ) ρ n dρ dθ du S n ε 4 f S n ρ n 4 dρ dt f ε t + ρ ) n S n To sum up, we have the following estimates: { ε ln ε, n =, I ε, n 3,, I dt t + ρ ) n ε, n =, π ln ε), n = 3, π ωε), n =, ω ln ε), n = 3, ω ln ε, n 4. ln ε, n 4.. hoosing ε = ω the proof. when n = and ε = ω when n 3 finally gives the estimate 3.). This completes 5 Proof of single scattering asymptotics theorem.) In order to prove theorem., we will use the following Lemma 5.. Let X R d be an open, bounded set containing, m d + and f m+d X) such that f). onsider the following oscillatory integral I = x m fx)e iωϕ x ) dx, 5.) R d where ϕr) = r gr ) with g a smooth nonnegative function such that g), and r = is the only point satisfying ϕ r) =. Then the following estimate holds I ω m+d f. m+d 9

10 Proof of lemma 5.. Define f r) := frˆθ) dsθ). Note that f S d is an even function of r and has compact support in [, ), where denotes the diameter of X. It is also m+d times continuously differentiable and for every α m+d there exist a constant α such that Applying a polar change of variable to 5.), we get I = f α [, ) α f α X). r m+d f r)e iωϕr) dr. Now notice that the ratio hr) = r m+d ϕ r) is a smooth, bounded function and that by means of integrations by parts with zero boundary terms, one can write I as I = ) m+d iω d dr ) m+d ϕ r m+d f r) ) e iωϕr) dr. We now split cases: if m + d is even, then m+d = m+d and we have that d dr hr) ) m+d r r m+d f r) ) = rf r), where f [, )) after counting derivatives. Thus we can integrate by parts one more time and get that I = in which case we get the bound ) m+d { } d f )h) iω iω dr f h) e iωϕ dr, I ω m+d f m+d. Now if m + d is odd, then m+d = m+d. We call f 3 r) := the expression I = d dr ϕ ) m+d f 3 r)e iωϕr) dr. iω ) m+d r m+d f r) ), and I has Using Morse lemma, there is an interval [, η] where the change of variable vr) = ϕr) is a diffeomorphism. Let χ [, +)) be identically one on a neighborhood of and χs) = for s η. Then I can be split into two integrals: I = ) m+d iω vη) χrv))f 3 rv)) dr dv e iωv dv + χr))f 3 r)e iωϕr) dr ), 5.) where we applied the change of variable v = ϕr) in the first integral. In the first integral, we integrate by parts and obtain that vη) χrv))f 3 rv)) dr dv e iωv dv = d χrv))f ω 3 rv)) dr ) H ωv)dv, dv dv where the function Ht) := t dv is related to the Fresnel integral and is known to be uniformly e iv bounded on [, ). For the second integral in 5.) the integrand is supported away from zero, so we can write e iωϕr) = d iωϕ r) dr e iωϕr) and integrate by parts. We obtain that χr))f 3 r)e iωϕr) dr = iω d dr χr))f3 r) ϕ r) ) e iωϕr) dr.

11 This finally shows that The proof is complete. I ω m+d f 3 ω m+d f m+d+. Note also the following two lemmas, useful to get uniform bounds in theorem. Lemma 5.. Under hypothesis., there exists δ > such that for all x, x c ) X) x x c δ dist [x, x c ], supp X k) > δ. 5.3) Lemma 5.3. Assume hypothesis. and let δ > be given by lemma 5.. Then there exists δ > such that for all x, x, x c ) supp X k X) with x x c δ, we have denote e := x c x and P x := x + [x x ) e ]e the orthogonal projection of x onto the segment [x, x c ]) dist x, [x, x c ]) δ min P x x, P x x c ) > δ. 5.4) Proof of theorem.. Pick x, x c ) X) with x x c, and define e := x c x as well as d := x x c. The stationary points of ϕ with respect to x solve the equation x x c + x x =, which characterizes exactly the points located on the segment [x, x c ]. Hence it is convenient to introduce the notation xx, x c, u, v) = x + ue + v, where u R and v e. Then {v =, u [, d ]} is a parametrization of the stationary segment. In this new set of variables the phase function is given by ϕu, v) = u + v ) d u) + v ), 5.5) and the function fx, x c, x) defined in.9)) with x = xx, x c, u, v) is now supported in { u [, ], v e, v, s.t. xx, x c, u, v) supp X k }. In the sequel, we make the dependency on x and x c implicit for readability. Let δ, δ be given by lemmas 5., 5.3 respectively, denote δ := minδ, δ ) >, and let χ R n ), χ, χ on { v δ } and χ on { v δ}. We then write T ω x, x c ) = I x, x c ) + I x, x c ), where I x, x c ) := fx, x c, u, v)χv)e iωϕu,v) dv du I x, x c ) := v v fx, x c, u, v) χv))e iωϕu,v) dv du. The integral I contains stationary points whereas I does not. Let us take care of I first. We start by saying that if x x c δ, then lemma 5. ensures that χv)fx, x c, u, v) = everywhere, hence I u) = in this case. Now assume x x c δ. Then lemma 5.3 ensures that for any u, v) in the support of χ )fx, x c,, ), we have that v δ δ < minu, d u), therefore both squareroots in 5.5) are analytic in v, and ϕ can be identically written as ϕu, v) = d + qu) v + v 4 gu, v ), where g is a smooth function, d = ϕu, ) and the quadratic term in v comes from the Hessian matrix with respect to v at v = H ϕ u, ) = d ud u) I n := qu)i n,

12 where I n is the n ) n ) identity matrix. Since u and d u are bounded away from zero uniformly over the support of χ )fx, x c,, ), the Hessian matrix is never degenerate, i.e. the function qu) is bounded away from zero. Following Hörmander s approach to the stationary phase method [], for s [, ], we define the phase ϕ s u, v) := d + qu) v + s v 4 gu, v ), so that ϕ u, v) = d + qu) v and ϕ s = ϕ for s =. We now write a Taylor formula with integral remainder with respect to the parameter s: χv)fu, v)e iωϕu,v) dv = χv)fu, v)e iωϕu,v) dv v + iω v v := I u) + I u). v 4 gu, v )χv)fu, v)e iωϕsu,v) dv ds The term I u) is dealt with by applying lemma 5. with m, d) = 4, n ), so m+d can bound by the following, everything being continuous with respect to the parameter s: = n+3. Therefore we I u) fx ω n+, x c,, ) n ) For the first integral I u), we write for u [δ, d δ ] and v R n χv)fu, v) = fu, )χv) + v f u, v)χv), f u, v) := v fu, tv) dt, and I becomes I u) = fu, ) e iωϕu,v) dv fu, ) R n χv))e iωϕu,v) dv R n + χv)v f u, v)e iωϕu,v) dv v := I u) + I u) + I 3 u). 5.7) The first term in 5.7) will give the leading-order term of the asymptotic expansion: I u) = ) n π ω e iωd e i π 4 σhϕu,)) fu, ), det H ϕ u, ) with σh ϕ u, )) = n ) the signature of H ϕ u, ), det H ϕ u, ) = d n [ud u)] n, and we used the following formula see [, Sections 3.4, 7.6]) R d e i Bx,x dx = e i π 4 σb) π) d det B where B is a non-degenerate quadratic form. The second term in 5.7) has rapid decay in ω because χ is supported away from v =. We see that by noting the identity iωqu) v v v e iωϕ ) = e iωϕ, so by means of n+ integrations by parts, I u) becomes ) n+ I u) = fu, ) iωqu) R n e iωϕ, )) n+ v v v [ χv)] dv,

13 which we can bound by I u) ω f n+ χ n+. The third term is dealt with by integration by parts with zero boundary term: I 3 u) = e iωd iωqu) v v χf ))e iω qu) v dv, and we are now in the setting of lemma 5. with m, d) =, n ), so we can bound Noting that we have I 3 u) ω ω n v χf ) n fx ω n+, x c,, ) n+3. fx, x c, u, ) = Ex, x c )kx + ue, e, e ) νx ) e νx) e [ud u)] n+, we integrate relation 5.7) together with the estimate 5.6) in u and obtain the estimate I x, x c ) = e iωd Ex, x c ) νx ) e νx) e with R x, x c ) d ω n+ kx + ue, e, e ) [d ud u)] n fx, x c, ) n+3. ) n π e in ) π 4 ω du + R x, x c ), For the integral I, let us introduce the differential operator L := vϕ v ϕ v such that iω L e iωϕ) = e iωϕ the exponent denotes the formal adjoint). The function χv))fx, x c, u, v) is supported where v δ and vϕ 4 δ < there, so we can integrate by parts n+ times and get that I x, x c ) = ) n+ e iωϕu,v) L n+ [ χv))fx, x c, u, v)] dv du, iω v where Lg := v g vϕ vϕ ), from which we deduce the estimate This completes the proof. I x, x c ) ω fx, x n+ c, ) n+. 6 The multiple scattering estimate For m, the multiple scattering of order m is given by Tmx ω, x c ) = X m e iω P m i= xi xi+ + xm xc ) Ex, x,..., x m, x c )Π m i= kx i, v i, v i ) Π m i= x i x i+ n x m x c n v ν x v m ν xc dx... dx m, 6.) where v i = x i+ x i, i =... m and v m = x c x m. The following theorem bounds each Tm ω separately. 3

14 Theorem 6.. Let σ, k) n+ X S n ) n+ X S n S n ), and assume hypothesis.. Then for any integer m there exists a constant m such that for any ω, Tmx ω, x c ) is uniformly bounded in X X by Tmx ω, x c ) m ω, n =, m ω lnω), n = 3, m ω n+, n 4. We relegate the proof of this theorem to the next section. We must now prove that one can control the remainder R N+ of the decomposition 3.) for N large enough. In order to do that, let us define the operator K ω for φ L X S n ) as K ω φx) = S n x,+ 6.) K ω φ) Γ+ x, v) ν x v dv, x X. 6.3) Then the following lemma shows that K ω is well-defined in L, := LL X S n ), L X)). Lemma 6.. Under hypothesis. and assuming that k is bounded, the operator K ω L, and satisfies the estimate independent of ω) K ω L, D n k. 6.4) Proof. Let φ L X S n ) and x X. Starting from 6.3), we have τ x,v) K ω φx) = e iωt Ex tv, x) kx tv, v, v)φx tv, v ) ν x v dv dt dv S n = S n x,+ X Sn e iω x x x x n kx, v, x x )Ex, x)φx, v ) ν x x x dx dv, where we changed variables x = x tv, v S n x,+, t [, τ x, v)]. The result is immediate since Ex, x) ν x x x and x x D for all x X and x supp X k. Proof of Theorem.5. Let ω ω, ω, where ω is given in orollary 3.3. decomposition: T ω +x, x c ) = α+ m= T ω mx, x c ) + R α+ x, x c ), R α+ x, x c ) = K ω K α ω I K ω ) K ω J ω g x x c ), where We write the following and where the integer α will be chosen at the end. For the first finite number of terms in 6.5), we bound each of them separately using Theorem 6. and obtain α+ ω, n =, T ω mx, x c ) ω lnω), n = 3, 6.6) m= ω n+, n 4. For the remainder term R α+ x, x c ), sup R α+ x, x c ) K ω L, Kω α L I K ω) L K ω J ω g x L X S n ). x c X D n k K ω J ω g x L X S n ) ω α ln ω) α, n =, ω α ln ω) α, n = 3, ω α ln ω) α, n 4, where we used lemma 6. and corollary 3.3. hoosing α = 3 for n =, 3 and α = n+3 if n 4 ensures that the latter decays are faster than those in 6.6). Taking the supremum over x X satisfies the same estimate because K ω J ω g x L X S n ) is uniformly bounded in x as stated in 3.7), and the proof is complete. 6.5) 4

15 7 Proof of theorem 6. hanges of variable: Let us put ourselves in the appropriate geometry: denote e := x c x always well-defined and real analytic for x supp X k, since x c x D), and write the following changes of variables: x σ, Ω ) = x + σ Ω e ) + d Ω e ) + ) e + σ σ + d Ω Ω e )e ), 7.) with σ, ), Ω S n. Formula 7.) is an elliptic change of variable, where for given σ, x describes the ellipsoid of equation x x + x x c = σ + d. Moreover we have the following equalities x x = σ + d + Ω e )) ), d := x x c = σ + d Ω e )) ), 7.) dx = n σ n σ + d ) n 3 σ + d ) d Ω e ) ) dσ dω, 7.3) where dω denotes the standard measure on the sphere. Now for j =,..., m, we write x j = x j + r j Ωj, e + n Ω j,l V l e ) ), r j [, ), Ω j := Ω j,,..., Ω j,n ) S n, 7.4) l= where V l S n x c,+), l =..., n such that v, V v),..., V n v)) is an orthonormal basis of R n for any v S n x,+. When n = one can choose V v) = v, v ) for v = v, v ) S.) Note that the vectors V l e ) vary when e varies. Then at fixed x σ, Ω ) supp X k we have dx j = r n j dr j dω j. Here and below, we will use the following notations r := r,..., r m ), Ω := Ω,..., Ω m ), and for a given integer α m, r α := r,..., r α ) and Ω α := Ω,..., Ω α ). In this new set of variables, we have that m x m x c = x n x c + r j Ωj, e + Ω j,l V l e ) ) j= l= 7.5) m n m ) = d r j Ω j, + r j Ω j,l. Then we define the phase function ϕ on [, +) m S n ) m by j= l= j= ϕσ, r, Ω) = σ + d + Ω e )) ) m + j= + σ + d Ω e )) ) m r j Ω j, r j j= + n m ) r j Ω j,l l= j=. 7.6) The phase function ϕ is a continuous function on [, +) m S n ) m. In addition from 7.), 7.5) and 7.6) it follows that ϕ D) where D := {σ, r, Ω) [, +) m S n ) m ) x σ, Ω ) x c and x m σ, r, Ω) x c }, 7.7) and the first derivatives of ϕ are uniformly bounded on D. The multiple scattering integral becomes Tmx ω, x c ) = e iωϕσ,r,ω) σ n gσ, r, Ω) dσ dr dω, 7.8),) m S n ) m 5

16 where the integrand g is given by gσ, r, Ω) = Ex, x,..., x m, x c ) m x c x m n ν x x x ν xc x c x m kx j, x j x j, x j+ x j ) n σ + d ) n 3 σ + d ) d Ω e ) ) j= 7.9) with x = x σ, Ω ) and x j = x j σ, r j, Ω j ) for j =,..., m defined by 7.) and 7.4) and x m+ = x c. Moreover g n+ [, +) m S n ) m ) and g is compactly supported in the set { σ, r, Ω), ), ) m S n ) m}. More precisely, { supp g = σ, r, Ω) [, ) [, ) m S n ) m ) x σ, Ω ) supp X k, } x j σ, r j, Ω j ) supp X k, j =,..., m. The n+ norm of g is bounded by a constant depending on the n+ norm of σ and k. We now give properties of the phase function in terms of σ and r m. First note that we have ϕ σ = σφ, where Φ := + x c x x c x m, for σ, r, Ω) D. 7.) The next lemma shows that the function Φ is uniformly bounded away from zero on the set x, x m, x c ) supp X k) X. Lemma 7.. Assuming hypothesis., there exists a constant D > such that for all x, x m, x c ) supp X k) X. + x c x x c x m D 7.) Proof. By contradiction, assume that 7.) does not hold. Then using compactness of supp X k) and X, and the continuity of inner products, one can construct limit points x, x m, x c) supp X k) X satisfying + x c x x c x m =. This means that x c [x, x m], which by convexity of X implies x / X or x m / X, and this is a contradiction. Hence the result. where Let us introduce the differential operator L σ defined by L σ f = i σ f σφ ). 7.) In addition straightforward computations show that the phase function ϕ satisfies the following equality ϕσ, r, Ω) = ϕ, r, Ω) + σ Ψσ, r, Ω), σ, r, Ω) D, 7.3) Ψσ, r, Ω) = x mσ, r, Ω) x c Φσ, r, Ω) + x m r, Ω) x c x m r, Ω) x c ) e x m σ, r, Ω) x c + x m r, Ω) x c 7.4) where x m r, Ω) = x + d + Ω e ) + m j= r ) jω j, e + j=...m r j Ω j,l V l e ). We have x m r, Ω) = l=...n x m, r, Ω) when, r, Ω) D. Note that α Ψ σ α estimates. D), α+ Ψ σ α r m L loc D) for any α N. In addition the function Ψ satisfies the following 6

17 Lemma 7.. Assuming hypothesis., we have < D D 4 where D is the constant on the right-hand side of 7.). Ψσ, r, Ω), for σ, r, Ω) supp g, 7.5) Proof of lemma 7.. The estimate Ψσ, r, Ω), for σ, r, Ω) supp g, follows from 7.4) and the fact that Φ uniformly in σ, r, Ω). From 7.4) and from Lemma 7. it also follows that Ψσ, r, Ω) for σ, r, Ω) supp g. Then note that x m r, Ω) x c = m d Ω e ))) r j Ω j, σ D x m σ, r, Ω) x c x m σ, r, Ω) x c + x m r, Ω) x c, 7.6) j= j= + n m ) r j Ω j,l + m n m σ + d Ω e ))) r j Ω j, + ) r j Ω j,l σ + x m σ, r, Ω) x c 3, 7.7) for σ, r, Ω) supp g we use the estimate σ ). The estimate 7.4) follows from 7.6),7.7) and from the estimate x m x c D for x m supp X k. l= j= l= j= For further use, we also rewrite Tmx ω, x c ) = Ir m, Ω m ) dr m dω m, where Ir m, Ω m ) =, ) m S n ) m S n e iωϕσ,r,ω) σ n gσ, r, Ω) dω m dr m dσ. 7.8) ontrol of the integral in r m, Ω m ) onsider the following integral I m σ, r m, Ω m ) = e iωϕ hσ, r, Ω) dω m dr m, 7.9) S n for given σ, r m, Ω m ), which implies that x,..., x m are fixed. The function h is an arbitrary bounded integrand whose dependency on r m is compactly supported in [, ), and h satisfies the following bounds, uniform in all variables σ, r, Ω): ) max h, h r m <. For σ, r, Ω) D, the phase function satisfies ϕ = + Ω m x m x c, x m = x m σ, r, Ω), Ω m = Ω m, e + r m n Ω m,j V j e ) ), 7.) so ϕ = whenever Ω m = e m := x c x m. This illustrates the fact that x m x m + x m x c does r m not depend on r m = x m x m when x m [x m, x c ]. This fact motivates the change of variables Ω m = cos θ e m + sin θ w m, θ, π), w m S n e m ), 7.) j= 7

18 with change of volume dω m = sin n θ dθ dw m. In these new variables, we have that rm ϕ = d md m r m + d m cos θ) d m sin θ = h mr m, θ) sin, 7.) θ with h m a smooth function on the support of h. The integral 7.9) becomes π I m σ, r m, Ω m ) = sin n θ e iωϕ hσ, r, Ω m, Ω m θ, w m )) dr m dθ dw m 7.3) S n where S fw)dw := f ) + f)). We now split cases according to the dimension n. ase n = : we split the integral in θ as follows: π = ε + π ε + π ε π ε, < ε. In the second term, we write e iωϕ = iω rm ϕ e iωϕ and integrate by parts in r m. We get r m ε π I m = e iωϕ h dr m dθ dw m + e iωϕ h dr m dθ dw m S S π ε + { π ε [hm iω S ε sin he iωϕ] } r θ h m h) e iωϕ dr m dθ dw m. m= r m Thus ) I m max h, h r m ε + ) εω max ) h, h r m ω by choosing ε = ω. ase n = 3: splitting the integral in a similar way to n =, we get ε π I m = sin θ e iωϕ h dρ dθ dw m + sin θ e iωϕ h dρ dθ dw m S S π ε + { π ε [hm he iωϕ] } iω sin θ r m= h m h) e iωϕ dr m dθ dw m. r m S ε Thus ) I m max h, h r m ε + ) ln ε max ω ) h, h ln ω r m ω by choosing ε = ω. ase n 4: we integrate by part without splitting the integral, and obtain I m = { π [hm sin n 4 θ he iωϕ] } iω r m= h m h) e iωϕ dr m S n r m ) I m max h, h r m ω. dθ dw m, so Summary: To summarize, we have that ) I m max h, h ω n =, r m ω ln ω n = 3, ω n ) 8

19 ompletion of the proof: For fixed r m, Ω m ), let us study the integral Ir m, Ω m ) from expression 7.8) and study cases according to the dimension n. ase n even: Using the differential operator L σ defined in 7.), we integrate by parts w.r.t. σ with zero boundary terms n times. One can show that we can write and the integral I becomes Ir m, Ω m ) = ω n Then we perform the change of variables L n σ σ n g ) = g σ, r, Ω), S n e iωϕσ,r,ω) g σ, r, Ω) dσ dω m dr m. 7.5) η = σ Ψσ, r, Ω) = ϕσ, r, Ω) ϕ, r, Ω)), 7.6) for the integral over the σ variable where Ψ is defined by 7.3) and satisfies 7.5). We have Φσ, r, Ω) Ψσ, r, Ω) dσ = dη, where the function Φ is defined by 7.) and satisfies 7.) on supp g. And we obtain Ir m, Ω m ) = e iωϕ,r,ω) e iω η ω n g η, r, Ω) dη dω m dr m, 7.7) S n ) where g [, ) m S n ) m ), max α g α=, η, α+ g α η α r m g n + for some constant. The function g is also compactly supported in [, ) [, ) m S n ) m. Then integrating by parts in the η variable on the right hand side of 7.7) we obtain π Ir m, Ω m ) = e i π 4 e iωϕ,r,ω) g ω n, r, Ω) dη dω m dr m S n + H ωη)e iωϕση,r,ω),r,ω) g ω n S n η η, r, Ω) dη dω m dr m, 7.8) e i t where Hs) = + dt, s [, +), is uniformly bounded on [, +), and where ση, r, Ω) is the s unique solution of 7.6) at fixed r, Ω) we also used the equality dη = π e iω η e i π 4 ω ). Hence the integral in σ in 7.5) is the sum of two terms. To each of these terms, we apply the estimate 7.4) to the integral in the r m, Ω m ) variables, and obtain I max α n α g σ α, +α ) { g ω n =, σ α r m ω n+ n 4, n even. 7.9) ase n odd: Using again the operator L σ defined in 7.), we integrate by parts w.r.t. σ with zero boundary terms n 3 times. One can show that we can write and the integral I becomes Ir m, Ω m ) = ω n 3 L n 3 σ σ n g ) = σg 3 σ, r, Ω), S n e iωϕσ,r,ω) σg 3 σ, r, Ω) dω m dr m dσ. 9

20 Integrating by parts once again, we arrive at Ir m, Ω m ) = { [ e iωϕ g ] 3 ω n S iφ n σ= + e iωϕ σ g3 ) iφ dσ } dω m dr m. We then use the estimate 7.4) to treat the integral in the r m, Ω m ) variables, and obtain ) { α g I max σ, +α g ω ln ω n = 3, α σ α r m ω n+ n 4, n odd. α n 3, α n 7.3) Finally, integrating 7.3) or 7.9) over r m, Ω m ) [, ) m S n ) m does not change the estimates. The last precision to make is that the operators σ and satisfiy the following equalities: r m = x m xm = Ω m xm, and r m r m m σ = j= x j σ x j = m v j σ) xj, j= where v σ) = σω e )e + σ + d σ + d Ω Ω e )e ), v j σ) = v σ) + j m= Ω m, σ x c x + n l= ) Ω m,l σ V l x c x ), j, are infinitely smooth on the support of g. This in turn allows us to bound estimates 7.3) and 7.9) by the suitable α norms in the initial variables x,..., x m. This concludes the proof of Theorem 6.. Acknowledgment This work was supported in part by NSF grant DMS and DOE grant DE-FG5-8NA8779. References [] S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 5 999), pp. R4 R93. [] S. R. Arridge and W. R. B. Lionheart, Nonuniqueness in diffusion-based optical tomography, Optics Letters, 3 998), pp [3] G. Bal, Inverse transport from angularly averaged measurements and time harmonic isotropic sources, in Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy, Y. ensor, M. Jiang, and A. Louis, eds., RM Series, Scuola Normale Superiore Pisa, Italy, 8, pp [4], Inverse transport theory and applications, Inverse Problems, 5 9), p. 53. [5] G. Bal and A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging, 4) 8), pp [6], Time-dependent angularly averaged inverse transport, Inverse Problems, 5 9), p. 75. [7] G. Bal, I. Langmore, and F. Monard, Inverse transport with isotropic sources and angularly averaged measurements, Inverse Probl. Imaging, ) 8), pp [8] M. houlli and P. Stefanov, Inverse scattering and inverse boundary value problems for the linear Boltzmann equation, omm. P.D.E. 5&6)996),

21 [9], Reconstruction of the coefficients of the stationary transport equation from boundary measurements, Inverse Problems, 996), pp. L9 L3. [] R. ipolatti, Identification of the collision kernel in the linear Boltzmann equation by a finite number of measurements on the boundary, omputational and Applied Mathematics, 5 6) pp [] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol.6, Springer Verlag, Berlin, 993. [] L. V. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer Verlag, 983. [3] E. W. Larsen, Solution of thee-dimensional inverse transport problems., Transp. Theory Stat. Phys ), pp [4] M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory, World Scientific, Singapore, 997. [5] K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, Algorithm for solving the equation of radiative transfer in the frequency domain, Optics Letter, 96) 4), pp [6] K. Ren, G. Bal, and A. H. Hielscher, Frequency domain optical tomography based on the equation of radiative transfer, SIAM J. Sci. omput., 8 6), pp [7] P. Stefanov and A. Tamasan, Uniqueness and non-uniqueness in inverse radiative transfer, Proc. Amer. Math. Soc., 37 9), pp