Optical Tomography on Simple Riemannian Surfaces

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1 Communications in Partial Differential Equations, 3: 379 4, 25 Copyright Taylor & Francis, Inc. ISSN print/ online DOI:.8/ Optical Tomography on Simple Riemannian Surfaces STEPHEN R. McDOWALL Department of Mathematics, Western Washington University, Bellingham, Washington, USA Optical tomography means the use of near-infrared light to determine the optical absorption and scattering properties of a medium. In the stationary Euclidean case the dynamics are modeled by the radiative transport equation, which assumes that, in the absence of interaction, particles follow straight lines. Here we shall study the problem in the presence of a Riemannian metric where particles follow the geodesic flow of the metric. In particular, we study the problem in dimension two, where the analysis is more delicate than in the higher dimensional cases. Keywords Boltzmann equation; Riemannian metric; Transport equation. 99 Mathematics Subject Classification Primary 35R35; Secondary 35Q99.. Introduction Optical tomography means the use of near-infrared light to determine the optical absorption and scattering properties of a medium. One prescribes a distribution of particles photons in this case entering the body at its boundary and measures the resulting flux of particles leaving the body. One then seeks to determine the absorption and scattering properties interior to the medium from knowledge of the albedo operator, the map from the incoming to the outgoing distributions of particles. Optical tomography is beginning to be applied to the problems of medical imaging; see Arridge 999 for a review article. In the stationary Euclidean case, the dynamics are modeled by the radiative transport equation, which assumes that, in the absence of interaction, particles follow straight lines. We are concerned here with the situation of particles moving in an ambient field represented by a Riemannian metric. The consequence is that in the absence of interaction a particle will now follow the geodesics of the metric. We first describe the problem in some generality. Let M be a bounded open domain in n with smooth boundary, and let g be a Riemannian metric on M. If fx v represents the density of particles at position x with velocity vector v in the Received July 2, 24; Accepted March 24, 25 Address correspondence to Stephen R. McDowall, Department of Mathematics, Western Washington University, Bellingham, Washington, , USA; stephen.mcdowall@wwu.edu 379

2 38 McDowall unit tangent sphere at x, x M, then the stationary linear transport equation is fx v a x vfx v + kx v vfx v dv x = x M The operator is the derivative along the geodesic flow see 3 below which in the case of g being Euclidean is simply fx v = v x fx v. The coefficient a is the absorption coefficient, and k is the scattering kernel; a describes the probability of a particle with position x and velocity v being absorbed, and k describes that of a particle with position x and velocity v being scattered to velocity v. We restrict ourselves to the case where all particles travel at unit speed and hence use the unit sphere bundle M rather than the full tangent bundle TM. The measure dv x in is the Euclidean volume form on the tangent sphere x M determined by the metric g at x see definition 2 below. Define the incoming and outgoing bundles ± = x v Mx M ±v > on the boundary M of M, where is the outward unit normal vector to M and = gx is the inner product with respect to the metric g at x. Iff is a distribution of particles defined on, let f be the solution to, should it exist, with boundary condition f = f. Then the albedo operator is the map which maps f to the outgoing flux of particles: f = f + The inverse problem, the problem of optical tomography, is to determine uniquely a and k from knowledge of. When the ambient metric is Euclidean, the inverse problem described has been well studied. Choulli and Stefanov 999 show that the most singular part of the distributional kernel of the albedo operator determines the X-ray transform of the absorption coefficient and that the next most singular part determines the collision kernel k in dimensions three and greater. Their method fails to determine k from in the two-dimensional case. Stefanov and Uhlmann 23 performed a more precise analysis of the second term in the singular expansion of the distributional kernel to obtain a unique determination of k in dimension two, under the assumption that k is small relative to a, with an explicit constant given. They also give stability estimates. Also in dimension two, Tamasan 22 shows a determination of k that is homogeneous in x, and Tamasan 23 proves a determination of a kernel that is weakly anisotropic. The time-dependent analog of and its inverse problem were considered in Choulli and Stefanov 996. Stability estimates have also been obtained in Romanov 996 and Wang 999. In the Riemannian setting, an inverse source problem for the stationary transport equation is addressed in Sharafutdinov 999 see also Sharafutdinov, 994 where kx v v = kx v v is assumed to depend on the angle between v and v and the object of interest is the reconstruction of an isotropic source term. In Ferwerda 999 the time dependent radiative transfer equation is derived for a medium with spatially varying refractive index and with scattering kernel k independent of position. Such a refractive index is represented by a Riemannian metric. In McDowall 24 the author proved the analogous result to that of Choulli and Stefanov 999, namely that in the presence of a Riemannian metric

3 Optical Tomography on Simple Riemannian Surfaces 38 uniquely determines a and k in dimensions three and greater, and only a in dimension two. In this article we address the two-dimensional case in the presence of a Riemannian metric, with a smallness assumption on an otherwise general k. One may understand such smallness in k as the body being nearly transparent. Let M be a two-dimensional Riemannian manifold with smooth, strictly convex boundary. We make the following assumptions on the geometry of M g: M. The sectional curvature of M g is bounded above by. M2. If > we assume that M g has no focal points, that is, for every geodesic a b M and every nonzero Jacobi field Jt along satisfying Ja =, we have Jt as a strictly increasing function on a b. Note that if then M g necessarily has no focal points. M3. In the case that > we assume that the diameter A of M g satisfies A < /2. There is no restriction on the diameter of M g when, other than it is finite. It follows that M g is simple. In particular, for any x M the exponential map exp x exp x M M is a diffeomorphism. Consequently M is diffeomorphic to a disk. We make the following assumptions on a k: A. Even in the Euclidean case and when k =, does not uniquely determine a see Choulli and Stefanov, 999; Stefanov and Uhlmann, 23 and so for the inverse problem we assume that a depends only on x. A2. a L M, k L y v v M y M y M, and k L 2diamM. Following the notation of Stefanov and Uhlmann 23 we introduce the class = a x kx w w a L k L and a k satisfies A, A2 2 Theorem. Let M g be a two-dimensional Riemannian manifold with smooth, strictly convex boundary, satisfying assumptions M M3: M g has curvature bounded above by, has no focal points, and in the case that > has diameter bounded by /2. If a k satisfy A, A2 then the metric g is uniquely determined by the associated albedo operator. 2 Given > there exists > such that any pair a k is uniquely determined, within, by the associated albedo operator. Furthermore, can be chosen to be = Ce 2A, where C depends only on M g and A = diamm g. As in Choulli and Stefanov 996, 999, Stefanov and Uhlmann 23, and McDowall 24, we find a singular expansion of x v x v, the distributional kernel of the albedo operator. For dimensions n 3 it was proven by McDowall 24 see also Choulli and Stefanov, 999 that, writing = + + 2, and are delta-type singularities supported on varieties of positive and differing codimensions in +. For dimension n = 2, however, McDowall 24 did not show as distinguishable from the remainder, 2. In the Euclidean case, the more precise analysis in Stefanov and Uhlmann 23 of the singular decomposition allows one to show that the collision

4 382 McDowall kernel k is uniquely determined by ; we shall perform a similar analysis here for a domain with a Riemannian metric. The determination of the metric g follows from the result of Pestov and Uhlmann to appear see also Pestov and Uhlmann, 24. They show that the scattering relation of M g uniquely determines the metric g in dimension two when M g is simple, up to a diffeomorphism that fixes the boundary of M. The scattering relation is the set of pairs x v x v +, where x v is the exit point and direction of the geodesic which enters M at x with direction v. In this work we simply observe that the leading term in the expansion of the kernel of determines this scattering relation of M g. This article is arranged as follows. In Section 2 we outline the construction of the solution to the forward problem and so define the albedo operator. We express in terms of a Neumann series expansion for I + K, where K is an integral operator, and we calculate explicitly the first two terms. In Section 3 we prove part one of Theorem and then outline the method of proof of part two in order to motivate the estimates in Section 4. In Section 4 we obtain estimates for terms involving K 2 and K 3, which we use in Section 5 to complete the proof of Theorem. 2. The Forward Problem and the Albedo Operator We define the following terms and notation. If x v M we denote by xv t the geodesic that has xv = x and xv = v; we introduce the compressed notation xv t = xv t xv t Define the time to boundary functions ± M + by ± x v = mint xv ±t M and set x v = x v + + x v. Since M g is simple, these functions are well-defined and finite. The operator in is the derivative along the geodesic flow and is defined by fx v = t f xv t xv t 3 t= If x i y i n i= are local coordinates for M with the yi with respect to the natural basis x i, then in these coordinates f = f x i yi + f y i yj y k i jk where jk i are the Christoffel symbols of the Levi Civita connection of g. Definition 2. The Liouville volume form is the canonical volume form defined on M, which is preserved under the geodesic flow of g. It is the product of the Riemannian volume dx on M and the surface element dv x on x M induced from

5 Optical Tomography on Simple Riemannian Surfaces 383 the Euclidean volume form in T x M defined by the metric g at x see, for example, Helgason, 978; Katok and Hasselblatt, 995. Following the notation of McDowall 24 we set T f = f a f T fx v = kx v vfx v dv so that the transport Equation reads T f + T f =. We begin by proving solvability of the forward problem and in the process obtain a series expansion for the albedo operator. In this section we do not need assumption A. Since we seek the distribution kernel of, it will suffice to study solvability of the boundary value problem for the transport equation x M T f + T f = in M 4 f = f with smooth boundary functions f C. We rewrite this as an integral equation. In the absence of scattering, the homogeneous boundary value problem has solution Lf where T f = in M 5 f = f Lf x v = Ex v x vf xv x v 6 and here we introduce the notation { t Ex v s t = exp s } a xv rdr The exponent in Ex v s t equals the total absorption along the geodesic xv r between r = s and r = t; note in particular that when t<s, the exponent is negative. Note further that E is always a positive number. Next, the inhomogeneous Dirichlet problem T f = g in M f = 7 has solution T g, where T gx v = xv To see this, one need only notice that Ex v t x vg xv t x vdt 8 Ex v x vfx v = Ex v x vfx v + a x vfx v = Ex v x vt f

6 384 McDowall Thus, if we define the operator K by Kfx v = xv we have that K = T T and 4 is equivalent to Now Ex v t x vt f xv t x vdt I + Kf = Lf Kf L M 2 diammk L Mf L M and so if 2diamMk L <, then I + K is invertible on L M. Iff C then Lf clearly has a well-defined trace Lf xv + ; repeated application of K preserves this and so by considering the series representation of I + K we see that f = I + K Lf has a well-defined trace and that maps C into L +. Write the solution f as f = Lf KLf + I + K K 2 Lf 9 We wish to express f as fx v = x v x v f x v dx v That is to say, we will express f in terms of an expansion of the fundamental solution to the transport equation with a delta boundary condition with T + T = in M = x v x v = n x v x v x v Here, x v x v is the Dirac delta distribution with respect to the measure dx v. The measure dx v is obtained via pulling back the Liouville volume form of Definition 2 to ± by the geodesic flow. See McDowall 24 for a more explicit description. Formally, we have = L = KL and 2 = I + K K 2 L 2 and the kernel of is given by = xv +, and j = j xv +, j = 2 3. It was demonstrated in Proposition 3. of McDowall 24 that Lf x v = x v x v f x v dx v 3

7 Optical Tomography on Simple Riemannian Surfaces 385 with x v x v = + x v Ex v x v xv x v tdt where xv y w is the delta distribution with respect to the Liouville measure dw y dy on M. From this we have = xv +. In Proposition 4 below, we shall perform a change of variables of integration defined in terms of the geodesic flow of g. Lemma 3 facilitates the computation of the Jacobian of this change of variables. Let x v M be fixed, and let s be such that y = x v s M. Define v so that v v is a positive orthonormal basis for T x M. Define the geodesic flow s M M, s ˆx ˆv = ˆxˆv s defined for ˆx ˆx near x v. In the lemma below, D t denotes covariant differentiation along the curve x v and RU VW = U V W V U W UV W is the curvature tensor with respect to the Levi Civita connection of g. Lemma 3. Let J H be a vector field along x v satisfying the Jacobi equation D 2 t J H + RJ H x v x v = J H = v D t J H = and let J V be the vector field along x v satisfying the same Jacobi equation but with initial conditions J V = D t J V = v Then d s x v H = J H s J H s d s x v V = J V s J V s Proof. For general T x v M it is proven in Paternain 999 that d s x v can be described in terms of Jacobi fields whose initial conditions are given in terms of the horizontal and vertical projections of. The claim of the lemma is this result applied to the vectors H and V defined as follows. The vector H T x v y M belongs to the so-called horizontal subbundle: Let M be a curve adapted to v x M, i.e., = v. Let Zr be the parallel transport of v along, and define M by r = r Zr. Then H =. Similarly, we define V T x v x M to be the vector in the vertical subbundle given as the tangent vector to the curve TM, r = x v + rv. From this point on in the exposition we will frequently not present the exact point in M 2 where E is evaluated, choosing instead to simply write E. We will not need the omitted information. Proposition 4. The second term in 9 is given by KLf x v = x v x v f x v dx v 4

8 386 McDowall with x v x v = x v x v E E x v x v k x v sx v xv tx v x v sinx v x v where x v x v = if the geodesics x v s and xv t x v intersect for s = sx v > and t = tx v >, and x v x v = otherwise, and where x v x v is the angle between the tangent vectors at the point of intersection. The function is uniformly bounded <m m 2. Thus the second term in the expansion for the distribution kernel of is = xv +. Recall that involves a delta-type singularity; as is evident from the proof which follows, the singularity in in integrable, and thus is distinguishable from. Proof. By definition, KLf x v = xv E ky w xv t x v y M E f yw y wdw dt where y = yt = xv t x v. We wish to rewrite this integral in terms of an integral with respect to the variables x M and v x M. Let us define a family of indicator functions M, parameterized by x v, by if x v s = xv t x v = yt x v x v = for some unique s>t> otherwise. On the support of we have well-defined functions sx v, tx v defined to be the s and t, respectively, which appear in the definition of. Then, also on the support of, the change of variables t w = x v = tx v x v sx v is well-defined and smooth. We now seek to compute the Jacobian of this change of variables. We first specialize to the case where M is perpendicular to x v at x and xv is perpendicular to x v at yt. For a fixed x v supp, x M, let y w = x v sx v and let w y M be such that w w is a positive orthonormal basis for T y M. Let J H = J x v H and J V = J x v V be the Jacobi fields along x v given in Lemma 3 above, where we are taking s = sx v. Extend the metric g to an neighborhood of M near x. Now let x x 2 r be local coordinates near x v = /2 so that x v =, so that x v = x 2 =, and so that x /2 = x v x v x x v x. Here v x x v x is simply the parallel transport of v along x v from x to x v x. Thus the x x 2 coordinates are geodesic normal coordinates along x v. Similarly,

9 Optical Tomography on Simple Riemannian Surfaces 387 let y y 2 r be coordinates local to y w, so that y w = and yw = y 2 =. In these coordinates, J H s = y x y + y 2 x y 2 = y x y since J H is perpendicular along x v and so y 2 = aty. Next, x J H s = r x r + x = x since J H is a Jacobi field from a variation of geodesics, all of which are parameterized at unit speed. Similarly, we have J V s = y y and J V s = Let us define S 3 S, x y y 2 r equal to x s in the y y 2 r local coordinates where we are abusing notation in the use of x s, but where we trust that the meaning is clear to the reader. Then the differential of this map at /2 is given by y x d /2 = y x Notice that detd /2 cannot be zero, for if it were we would have that for some, J H s J H s = J V s J V s, and then J H J V, a contradiction. For the general case let x v = cos xv t x v x v sx v be the angle with respect to the metric g between the geodesics xv and x v intersecting at y = ytx v. IfJ denotes the Jacobian of the change of variables then we have in local coordinates n x v g y J x v sinx v x = n x v g y sinx v x where n x is the outward pointing unit normal vector to M at x M. Thus det J = n x v g y sinx v y = n x v g sinx v x v x say. From the remark above, we have x v. The above calculation was performed with x v fixed. In fact every term in J is dependent on, or parameterized by, x v, and we introduce the notation x v x v and x v x v, where the order in which the variables have been x

10 388 McDowall written is chosen to be consistent with the notation elsewhere in this article. Note that the second of these terms is the angle with respect to g between the geodesics x v s and xv t x v at the point of intersection, should it exist for s> and t>. It is not difficult to see that from the compactness of M that there are constants <m <m 2 such that m x v x v m 2 uniformly. Note that this does not rely on the fact that the metric is simple; it relies only on the fact that geodesics are uniquely determined by an element of M. If we denote x = v x Mn x v < and let dx be the induced Riemannian measure on M, we now have KLf x v = x v x v E E M x k x v sx v xv tx v x vf x v x v x v sinx v x v n x v g dv dx = x v x v f x v dx v where x v x v is as in the statement of the proposition and where we have used that n x v g dv dx = dx v. 3. Outline of Proof of Theorem Suppose that we have two albedo operators and coming from measurements made on manifolds M g and M g with material parameters a k and a k, respectively. Suppose that =. Let all the operators and functions T T JEK j defined thus far in terms of a k have their counterparts for a k denoted by T T JẼ K j. With = and assuming A, from McDowall 24 we have that = ; in fact x v x v = Ex v x v xv xvx v and similarly for, and so we see that determines the scattering relation = x v x v + x v x v + x v for M g. Thus = implies that =, where is the relation for M g. In Pestov and Uhlmann to appear see also Pestov and Uhlmann, 24 it is proven that this implies that g = g, of course up to a choice of coordinates, that is, a diffeomorphism M M with M = Id M. This completes the proof of part one of Theorem. To motivate the analysis that follows, and with Proposition 4 at our disposal, we outline the approach we shall take to prove part two of Theorem. With g = g it follows from McDowall 24 that a = a and so T JE= T JẼ.

11 Optical Tomography on Simple Riemannian Surfaces 389 Combining this with Proposition 4 we then have 2 2 x v x v = x v x v = x v x v E E k k x v sx v xv tx v x v sinx v x v 5 for x v x v +. We refer the reader to the statement of Proposition 4 for the definitions of s t, and. Recall that is a function uniformly bounded on +. In what follows we will obtain another estimate for 2 2 of the form sin 2 2 L + Ck k L 6 with C> a constant depending only on the geometry of the manifold M g. Combining 5 and 6, we will obtain k k L Ck k L which implies that k = k if is sufficiently small. The principal observation to make here is that in relating 2 2 x v x v to k k, we need only consider x v x v + such that the geodesics xv and x v intersect. We see this due to the presence of the indicator function, and due to the fact that to identify ky w w at y w w M y M y M we should follow the geodesics yw and yw backward and forward, respectively, to the boundary. Now, following the decomposition in Stefanov and Uhlmann 23, by 2 and the statement immediately following 2 we write 2 2 = 2 2 xv + = I + K K 2 I + K K 2 xv + = I + K K 2 K 2 + I + K K KI + K K 2 xv + 7 Writing K 2 K 2 = KK K+ K K K we see that we must estimate KK K, which is the content of Proposition 7 and Lemma 9 below for x v x v such that the corresponding geodesics intersect as mentioned above. We estimate the final term in Proposition 8 and Lemma. 4. Estimates for K 2 and K 3 With a view toward Proposition 7 below, we introduce the following notation. Let x v be fixed; this is the location of the delta source in. For a given t + x v, let s = t s be the unit speed geodesic with = x v t joining x v t to x. Let d x v t x be the geodesic distance between these two points, so d x v t x = x. Define v = vt x = and ˆv =ˆvt x = d x v t x 8

12 39 McDowall Next, let X = X tx be the Jacobi field along d x v t x s s d x v t x satisfying X = and Ẋ =ˆv, where ˆv ˆv form a positive orthonormal basis for T x M. Then we have Proposition 5. For x v M and with the notation defined above, T K x v = T KL x v + x = v E E kx ˆv vk x v t v Xˆv dx x v t dt Proof. For an arbitrary f x v we have xv T KLf x v = E kx v vkyt v ṽ ẏt v x M ytv M Jf yt v ṽdṽdtdv where yt v = xv t x v. We now make the change of variables t v yt v. The change of volume element is computed to be dy =Y xy dx ydtdv, where Y = Y xy is the Jacobi field along x v r, r dx y with Y = and Ẏ = v, where v v is a positive orthonormal basis for T x M. Thus T KLf x v = E kx wx y vky ṽ wx y M y M Jf y ṽ dṽ dy Ydx y where wx y and wx y are the unit tangent vectors at y and x, respectively, of the geodesic joining y to x. Note that the singularity at x = y in the integral above is of the type /r in two dimensions and so is integrable. As in Lemma 2.2 of McDowall 24, and changing variables y ṽ x v t, we have T KLf x v = + x v E kx wx x v t vk x v t wx x v t Ydx x v t Jf x v tdt dx v Finally, using dt dx v =n x v dx dv and setting f =, we obtain + x v T KL x v = E E kx wx x v t v M x M k x v t wx x v t x x v v Ydx x v t dt dv dx + x = v E E kx ˆv vk x v t v Xˆv dx x v t dt where ˆv, v, and Xˆv are as defined above the statement of the proposition.

13 Optical Tomography on Simple Riemannian Surfaces 39 In the estimates proven below we will repeatedly make use of the fact that E. Lemma 6. Assuming M M3 and A, A2, for almost every x v x v M and for almost every x v x v +, cos A k L k L C + + log A T KL x v x v 2 k L k L + log A = k L k L C + log A < > 9 where is the minimal distance of x from the geodesic x v t, t + x v, and where C + = log3/ cos A, C = log3/ cosh A. Recall that A is the diameter of M. Proof. The Rauch comparison lemma gives sin dx x v t Xˆv dx x v t dx x v t sinh dx x v t > = < 2 Let us abbreviate x v by for the moment. Let t + x v be such that t minimizes the geodesic distance from t to x, and let = dx t. Consider the geodesic triangle with vertices x, t, and t, and let be the interior angle between the geodesic and the geodesic joining t and x. We compare this to the triangle on the manifold of constant curvature, which shares the same angle and adjacent side lengths and t t. Then dx t is at least as great as the length of the third side of the comparison triangle, d say. Consider the case when >. The law of cosines on the sphere of radius / reads cos d = cos t t cos + cos sin t t sin cos t t cos where the inequality follows from /2; this is easily seen by considering the possible location of x relative to the endpoints of. From this we obtain sin dx t = cos 2 dx t cos 2 t t cos 2

14 392 McDowall = sin 2 + sin 2 t t cos t t 2 cos 2 A since <A. Combining this with 2 we thus obtain + x v Xˆv dx x v t dt A t t 2 cos 2 A dt A 2 + t 2 cos 2 A dt = cos A log A cos A A 2 cos 2 A cos C A + + log A t where C + = log3/ cos A. Note that the assumption on the diameter A< /2 is used here. The last estimate is obtained as follows: let = cos A. Since and A/, one easily obtains A and so A 3 A2 2 + A A A 2 Combining this estimate with Proposition 5 we obtain 9 for >. When < we use the law of cosines on the hyperbolic plane of constant curvature as above to obtain sinh dx t 2 +t t 2 cosh 2 and then + x v Xˆv dx x v t dt cosh C + log A with C = log3/ cosh A. This yields 9 for <. When =, we have dx t = 2 +t t 2 and + x v Xˆv dx x v t dt 2 + log A Notice that no restriction on A is necessary in the cases.

15 Optical Tomography on Simple Riemannian Surfaces 393 To obtain an estimate of KKL we now apply T, since K = T T. Let us introduce some notation used in the proposition below. As in Proposition 4, x v x v is the indicator function, which is one when there exist <s< + x v and <t< x v such that x v s = xv t x v and equals zero otherwise. In the event that =, we let x v x v be the angle between the tangent vectors of these geodesics at the point of intersection. Proposition 7. For almost every x v x v M such that xv s and x v t intersect for some xv<s<, <t< + x v, and for almost every x v x v + such that xv s and x v t intersect for some x v < s<, <t< + x v, A k L k L cos A KKL x v x v 2 A 2 + log sin A k L k L + C + + log + C + log C + sin C sin > = < 2 where is evaluated at x v x v, C ± C = 2e A / A. are as in Lemma 6, C + =, and Proof. By 8, T T KL = and from Proposition 5, xv T KL xv t x v = + x v E T KL xv t x vdt E E k k Xˆ d xv t x v x o v s ds where ˆ = ˆs xv t x v as in 8, and Xˆ is the Jacobi field along the geodesic joining xv t x v to x v s as described in the paragraph below 8. We shall not need the precise arguments of k and k. From the estimate 9 we thus have k L k xv L cos C A + + log A dt t > T xv T KL x v k L k L + log A dt t = 22 xv k L k L C + log A dt t < where t is the minimum distance between xv t x v and the geodesic x v s, s + x v.

16 394 McDowall Figure. The geodesic triangle used in the proof of Proposition 7. Let s + x v and t x v be such that x v s = xv t x v. Let st be such that x v st is the closest point to xv t achieving the distance t. We consider the geodesic triangle formed by the points x v s = xv t x v, x v st and xv t x v; let t be the angle at x v st. See Figure. First suppose that >. Then t is not less than the length of the corresponding side of the comparison triangle on the 2-sphere of radius /, with the same angle and the same adjacent side-lengths. If is the comparison angle to t then from the spherical law of sines we thus obtain sin x v x v sin t sin sin t t sin t t For t t A</2, and similarly for t, sin t t and sin t t 2 t t so that A t A 2t t sin x v x v and 22 becomes T T KL x v cos A k L k L A C + + log A 2t t sin x v x v dt 23 Now A C + + log t = + A 2t t sin C + + log dt A t A dt 2t sin = + C + A + A t A log 2A t sin + t log A + C + + log sin A 2t sin

17 Optical Tomography on Simple Riemannian Surfaces 395 since the expression is majorized at t = A/2. Combining this with 23 gives 2 in the case >. For <, we apply the hyperbolic law of sines and the estimates sinhz z and, for z A, sinhz e A /Az to obtain A t e A t t sin x v x v As in the case for >, this yields 2 for <. Finally, if =, t t t sin, and upon integrating with respect to t as above we obtain 2 for =. We shall also need the following estimate. Proposition 8. We have K 3 L M and K 3 L + with norms bounded by Ck 3 L, where C depends only on M g. Proof. We present the proofs in the cases ; the case = is handled in the same manner of comparison with flat Euclidean space as demonstrated in previous proofs. Since K 3 = T T T T KL, from Lemma 6 we have K 3 x v x v Ĉ xv ytw ± k C ± + log A ds dw dt yt M s 24 where Ĉ + = k 2 L / cos A, Ĉ =k 2 L, yt = xvt x v, and s is the distance from ytw s yt w to the geodesic r = x v r, r + x v. Fix t. For the time being denote yt simply by y. Consider the following geodesics: = x v ; is the geodesic from x v through y described by yv s, y v s + y v, thus defining v y M; 2 is the geodesic from x v + x v through y described by yv 2 s, y v 2 s + y v 2, thus defining v 2. Choose coordinates 2 for y M such that v = and < v 2 <. We compute the integral 2 R C ± + log A dr d 25 dexp y r where R = y and dw is the geodesic distance from w to. Of course, by we mean the vector in y M corresponding to the coordinate. We split the integral into three pieces two of which overlap: v 2 v 2 v Let v 2 and suppose that >. The geodesic y r intersects at, say, r = R. For <r R consider the geodesic triangle a b c = y r y R = x v s c, where c is the point on closest to a. Denote by the angle at the vertex b. See Figure 2. Now let ã b c be the comparison triangle on the 2-sphere of radius /, which has angle at b and has adjacent sides lengths d b c = db c and d b ã = db a. Then da c d ã c.

18 396 McDowall Figure 2. The geodesic triangle a b c in the proof of Proposition 8. If is the angle at c, then from the law of sines on the sphere we have sin d ã c sin = sin R r sin sin R r so that da c sin d ã c 2 R r sin 26 For R r y the analogous argument gives the same estimate 26. Thus for 25 we obtain R R A + C + + log dr 2 R A C + + log dexp y r dr = 2A + C + + log A 2r sin C sin and this holds for v 2. When <, we apply the hyperbolic law of sines as in Proposition 7 to obtain the estimate 27 with C + and C + replaced by C and C. Next, let v 2 v 2 +. Note that 2 t, y v 2 t + y v 2 divides M into two regions with in one region and y s, y s in the other. For any r y let a = y r, b be the closest point on to a, c be the point of intersection of the geodesic from a to b with 2, and d be the closest point on 2 to a. See Figure 3. Then we have da b = da c + dc b da c da d Consider the geodesic triangle y a d and suppose that > ; comparing this with the corresponding triangle ỹ ã d on the sphere of radius / with the same angle at y and adjacent side lengths, we obtain da d sin d ã d = sin v 2 sin r 2 r sin v 2

19 Optical Tomography on Simple Riemannian Surfaces 397 Figure 3. The geometry in the case that v 2 v 2 +. Thus, for v 2 v 2 + and for >, R C + + log When < we have that A dr A + C dexp y r + + log da d A e A r sin v 2 C sin v 2 which yields 28, with C + and C + replaced by C and C. The analogous argument for 2 with 2 in place of gives the same estimate as 28 and its equivalent for <, with v = in place of v 2. We now integrate with respect to. First, for v 2 we have v2 C ± 2A + C ± + log 29 2 sin We wish to change variables to integrate with respect to and so must understand d/d. For fixed, let R and s be such that y R = x v s let J be the Jacobi field along y s, s R = R with J =, and J = either choice of will suffice. Then cos = y R x v s so sin d d d = d y R + d = d y R x v = s + y R x v s ds d

20 398 McDowall d = d y R x v = s = d d y R = cos 2 Now assumption M, that there are no focal points, implies the existence of > such that J uniformly over all Jacobi fields along geodesics of M g satisfying J =, J =. Thus d d = JR The integral 29 becomes v2 2A + C ± + log C ± d 2A 2 sin d d + C ± + log C ± d 2 sin = 2A + C ± + log C ± 3 Integrating 28 over v 2 v 2 + and the analogous expression for 2 we obtain that the integral 25 is bounded above by 2A + + C ± + log C ± Referring back to 24, we integrate with respect to t to obtain that for almost every x v x v M and for almost every x v x v +, K 3 x v x v C ± k 3 L + C ± + log C ± where 2 2 A 2 cos + A C ± = 2A 2 + > < 5. Proof of Theorem We complete the proof of Theorem after first deriving the estimates of Lemmas 9 and. In what follows, C + and C + are the constants in the case > ; C and C are for the case < ; when =, C ± = and C ± = 2. Lemma 9. For almost every x v x v + such that xv t = x v s for some xv<t< and <s< + x v, we have I + K K 2 K 2 x v x v C ± Ck k L k L + k L + C ± + log sin where C depends only on M g, and is the angle of intersection of the geodesics.

21 Optical Tomography on Simple Riemannian Surfaces 399 Proof. Write I + K K 2 K 2 = I I + K K KK K + K K K = KK K + K K K I + K K 2 K K + I + K KK K K The contribution from the first two terms is estimated by 2 from Proposition 7 with either K or K replaced by K K the proof proceeds unaltered. For the final two terms, K 2 K K + KK K K is a function in L by Proposition 8, and then I K preserves this space. Lemma. For almost every x v x v + such that xv t = x v s for some xv<t< and <s< + x v, we have I + K K KI + K K 2 x v x v Ck k L k 2 L +k L where C depends only on M g. Proof. Writing I + K K 2 = K 2 I + K K K 2 and applying Proposition 8 as in the previous lemma yields the claim. Proof of Theorem. By 5, 2 2 x v x v = E E k ky w ŵ sin a.e. so that k ky w ŵ C e 2A sin 2 2 x v x v a.e. 3 where C is a uniform constant depending only on M g, y is the point of intersection y = x v s = xv t x v, w = x v s y M, and ŵ = xv t x v y M. Recall that = x v x v is the angle between w and ŵ at y, and that a L see 2. Referring back to 7 and applying Lemmas 9 and, we also have 2 2 x v x v C ± C 2 k k L + log a.e. sin so that C e 2A sin 2 2 x v x v C 3 e 2A k k L a.e. 32 where again C 3 depends only on M g. Combining 3 and 32 we obtain k k L X C 3 e 2A k k L X where X = y w ŵ M y M y M. The factor appearing on the left-hand side of 3 no longer appears in the final estimate, since is identically one on the

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