CUBO A Mathematical Journal
Vol.11, No¯ 05, (71–97). December 2009

Optical Tomography for Media with

Variable Index of Refraction

Stephen McDowall

Department of Mathematics,

Western Washington University,

Bellingham, Washington, USA.

email : stephen.mcdowall@wwu.edu

ABSTRACT

Optical tomography is the use of near-infrared light to determine the optical absorption and

scattering properties of a medium M ⊂ R
n
. If the refractive index is constant throughout

the medium, the steady-state case is modeled by the stationary linear transport equation

in terms of the Euclidean metric and photons which do not get absorbed or scatter travel

along straight lines. In this expository article we consider the case of variable refractive index

where the dynamics are modeled by writing the transport equation in terms of a Riemannian

metric; in the absence of interaction, photons follow the geodesics of this metric. The data

one has is the measurement of the out-going flux of photons leaving the body at the boundary.

This may be knowledge of both the locations and directions of the exiting photons (fully

angularly resolved measurements) or some kind of average over direction (angularly averaged

measurements). We discuss the results known for both types of measurements in all spatial

dimensions.

RESUMEN

Tomografia óptica es el uso de luz infrarroja próxima para determinar la absorción óptica y

las propriedades de dispersión de un medio M ⊂ R
n
. Si el indice de refración es constante

a través del medio, el caso estado “steady” es modelado por la ecuación de transporte linear



72 Stephen McDowall CUBO
11, 5 (2009)

estacionaria en terminos de la métrica Euclideana y fatores los cuales não são obsorvidos o

dispersados viajando a lo largo de lineas retas. En este artículo expositorio consideramos el

caso de indice de refración variable donde las dinámicas son modeladas escribiendo la ecuación

de transporte e terminos de la metrica Riemanniana; en la ausencia de interacción, fotones

siguen las geodesicas de esta métrica. Los datos que uno tiene es la medida del flujo out-going

de fotones dejando el cuerpo en la frontera. Esto puede ser conocido de la localización y las

direciones de los fotones existentes (mediciones completamente angulares) o alguna clase de

promedio sobre dirección ( mediciones angularmente promediadas). Nosotros discutimos los

resultados conocidos para ambos tipos de mediciones en todas las dimensiones espaciales.

Key words and phrases: Boltzmann equation, integral geometry, optical tomography, Rieman-

nian metric, transport equation.

Math. Subj. Class.: 35R35, 35Q99.

1 Introduction

The stationary linear transport equation models, among other things, the propagation of the energy

density of waves in heterogeneous media [B3, Cha, RPK, vanR-Nh], neutrons in nuclear reactors

[Mo], and near-infrared photons in tissue. The term “linear” refers to the fact that the equation

models only scattering of particles from the material and assumes that the density of particles is

low enough that particle-to-particle interaction may be neglected. Recently, photon propagation

has been applied in optical tomography for use in medical imaging [A, Ch-Al, RBH].

In the absence of scattering, propagation is usually assumed to be along straight lines, as would

be the case for light propagation in a medium with constant index of refraction. This amounts to

writing the transport equation in terms of the Euclidean metric. Here we consider the situation

where, in the absence of scattering, propagation is along geodesics of a Riemannian metric. We

consider this a model for optical tomography in a medium with continuously varying refractive

index.

When prescribed in-going, and measured out-going, fluxes are allowed to depend on both po-

sition and direction (in other words, measurements are fully “angularly resolved”), the problem of

optical tomography in the Euclidean setting has been well studied. In many instances restrictions

are placed on the scattering kernel, for example that it be small, or that it be independent of direc-

tion. For recent results see [B1, SU, T1, T2]. For dimensions three and greater, these restrictions

are relaxed in [CS2] in the stationary case, and [CS1] in the time-dependent case. Analogous results

are proven in the Riemannian setting in [M1, M2]; see also [KLU]. Stability of the reconstruction

of material parameters has been studied in [BJ, SU, W] for the Euclidean case.

Such measurements are, however, unrealistic since angular resolution of out-going flux is dif-

ficult to achieve. In [L], Langmore introduces an angularly averaged measurement operator and

proves that angularly averaged measurements taken at every point on the boundary of the medium

enables reconstruction of the extinction coefficient σ; once σ is known, measurements made at a



CUBO
11, 5 (2009)

Optical Tomography for Media ... 73

single boundary point allow unique determination of the scattering kernel under some additional

assumptions. In particular, it is assumed that scattering is supported away from the boundary

of the body, the scattering phase function is known, and the scattering kernel is small. It is also

assumed that σ and the phase function are “close to” real-analytic. These results are extended to

the Riemannian case in [LM].

In this article we shall present results for the inverse problem of optical tomography in the

presence of a Riemannian metric. These results have appeared in [LM, M1, M2].

Let M ⊂ Rn be an open bounded set with smooth boundary ∂M and let g be a smooth

Riemannian metric on M̄. We shall make geometric assumptions on the Riemannian manifold

(M,g) in due course. If u(x,v) represents the density of particles at position x with velocity v in

the unit tangent sphere at x, ΩxM, then the stationary linear transport equation is

Tu(x,v) = −Du(x,v) − σ(x,v)u(x,v) +

∫

ΩxM

k(x,v′,v)u(x,v′) dv′ = 0. (1.1)

The functions σ and k are the material specific parameters we seek to determine from boundary

measurements. The first of these, σ is the extinction coefficient and σ(x,v)u(x,v) accounts for loss

of energy in direction v due to both absorption at x as well as scattering into other directions.

The second, k(x,v′,v) is the scattering kernel and is proportional to the probability of a particle

with position x and velocity v′ ∈ ΩxM being scattered to velocity v ∈ ΩxM. The operator D is

the derivative along the geodesic flow (see (2.2) below) which in the case of g being Euclidean is

simply Du(x,v) = v · ∇xu(x,v). The measure dv
′ is the volume form on ΩxM induced from the

Euclidean volume form on TxM determined by g at x; here TxM is the full tangent space to M at

x. While u is strictly speaking a density, for large numbers of particles it is reasonable to represent

u as, say, an L1 function.

In [B2] a transport equation is derived as a limiting case of Maxwell’s equations with non-

constant, but isotropic, permeability resulting in a varying refractive index n(x). If the metric

of (1.1) is conformal to the Euclidean metric, gij(x) = c
−2
0 n(x)

2δij with c0 the speed of light in

a vacuum, then (1.1) can be shown to correctly describe energy density propagation in isotropic

material. While it has not been shown that a general anisotropic index of refraction leads to a

limiting transport equation of the form (1.1), we consider it as a model for this case. We point out

that the same model appears in [Sh2].

Define the “incoming” and “outgoing” bundles on ∂M, the boundary of M,

Γ± = {(x,v) : x ∈ ∂M, and ± 〈v,νx〉 > 0}

where νx is the unit outer normal vector to the boundary ∂M at x and 〈·, ·〉 is the inner product,

each with respect to g at x.

If u− is an incoming flux of particles on Γ− let u be the solution, should it exist, to Tu = 0

with the boundary condition u|Γ− = u−. Denote this solution operator by T
−1, i.e. u = T −1u−.

The measurement operator corresponding to fully angularly dependent measurements is the albedo



74 Stephen McDowall CUBO
11, 5 (2009)

operator

A : u− 7→ u|Γ+.

It is from knowledge of A that we seek to determine uniquely the material parameters σ and k. In

section 5 we will present the corresponding operator for angularly averaged measurements.

Determination of σ and k relies upon invertibility of certain geodesic ray transforms (integrals

along geodesics). To ensure injectivity of these transforms we assume that the metric is “simple”:

Assumption 1.1. (M,g) is simple: M is strictly convex, and for any x ∈ M̄ the exponential map

expx : exp
−1
x (M̄) → M̄ is a diffeomorphism (and consequently M is diffeomorphic to a ball).

We must assume that σ depends only on position. If k ≡ 0 and g is Euclidean then determi-

nation of σ from A is simply the usual X-ray transform. If σ depends on direction v, then it is not

uniquely determined from its X-ray transform (see the introduction of [CS2]). Characterization

of non-uniqueness when σ depends on both x and v has recently been investigated in [ST]. We

will also assume that 0 ≤ σ ∈ L∞(M) and 0 ≤ k ∈ L∞(Ω2M) are bounded functions. Here we

introduce the somewhat unconventional notation

Ω
2M := {(x,v,w) : x ∈ M, v,w ∈ ΩxM}.

The forward problem is not necessarily well-posed without some subcriticality assumptions (see

[RS]). Such conditions will be satisfied by smallness assumptions on k to be stated precisely in the

following section.

The content of this article has appeared in [LM, M1, M2]. The purpose here is to collect

together these results into a single summarizing exposition. In section 2 we describe the solution

to the forward problem, thus enabling the definition of the measurement operator A, and expressing

it as a series. This series can be understood intuitively as contributions from ballistic particles

which do not scatter, singularly scattered particles, and multiply scattered particles. In section 3 we

prove unique identifiability of σ for all dimensions and of quite general k for dimensions three and

higher. In section 4 a finer analysis in two dimensions provides determination of scattering kernels

k which are a-priori known to be small relative to σ. The results of sections 3 and 4 assume

knowledge of angularly resolved measurements. In section 5 we restrict to angularly averaged

measurements and prove identifiability of σ and a restricted class of k in arbitrary dimensions.

2 The Forward Problem and the Albedo Operator

The approach utilized in the results of this article is to obtain a series expansion for the distribution

kernel of the boundary measurement operator. In this section we prove solveability of the forward

problem, and in the process obtain such a series expansion for A.



CUBO
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Optical Tomography for Media ... 75

If (x,v) ∈ ΩM we shall denote by γ(x,v)(t) the geodesic satisfying γ(x,v)(0) = x and γ̇(x,v)(0) =

v; we shall also use the compressed notation

~γ(x,v)(t) =
(
γ(x,v)(t), γ̇(x,v)(t)

)
.

Define the “distance-to-boundary” functions τ± : ΩM → R
+ by

τ±(x,v) = min{t > 0 : γ(x,v)(±t) ∈ ∂M}

and set τ = τ− + τ+.

The volume forms on ΩxM and on Γ± are the following: on M we have the naturally defined

volume form of the metric. At any x ∈ M, the volume form dv on ΩxM is the form induced from

the Euclidean volume form on TxM defined by the metric g at x. The resulting form on ΩM is the

Liouville form and is preserved under the geodesic flow of g. We denote by dµ the induced volume

form on Γ± which has the property that dt dµ(x,v) is the pull-back of the Liouville form by the

geodesic flow. Equivalently, we have the induced volume form of ∂M included in M̄; if (x,v) are

local coordinates for ∂M and dx is this volume form on ∂M, then it holds that

dµ(x,v) = |〈v,νx〉| dv dx.

The operator D in (1.1) is the derivative along the geodesic flow and is defined by

Du(x,v) =
∂

∂t

∣∣∣
t=0

u(γ(x,v)(t), γ̇(x,v)(t)). (2.2)

If (xi,yi)ni=1 are local coordinates for ΩM with the (y
i
) with respect to the natural basis

(
∂
∂xi

)

then in these coordinates

Df =
∂f

∂xi
yi +

∂f

∂yi
(−yjykΓijk)

where Γijk are the Christoffel symbols of the Levi-Civita connection of g. If x,y ∈ M then simplicity

of (M,g) ensures there exists a unique geodesic from x to y; let d(x,y) be the geodesic distance

between x and y, and let v(x,y) be the tangent vector at x of this geodesic. Define

E(x,y) := exp
{
−

∫ d(x,y)

0

σ
(
γ(x,v(x,y))(t)

)
dt

}
.

Note, using the fact γ(y,v(y,x))(d(y,x) − s) = γ(x,v(x,y))(s) we have E(x,y) = E(y,x).

Define

T0u = −Du − σu, T1u(x,v) =

∫

ΩxM

k(x,v′,v)u(x,v) dv′,

so that T = T0 + T1. We begin by re-writing this as an integral equation. In the absence of

scattering, the homogeneous boundary value problem T0u = 0, u|Γ− = u− has solution Ju− where

Ju−(x,v) = E
(
x,γ(x,v)(−τ−(x,v))

)
u−

(
γ(x,v)(−τ−(x,v))

)
.



76 Stephen McDowall CUBO
11, 5 (2009)

The inhomogeneous Dirichlet problem T0u = f, u|Γ− = 0 has solution

u(x,v) = T −10 f(x,v) =

∫ τ−(x,v)

0

E
(
x,γ(x,v)(t − τ−(x,v))

)
u
(
~γ(x,v)(t − τ−(x,v))

)
dt.

Defining

Ku(x,v) =

∫ τ−(x,v)

0

E
(
x,γ(x,v)(t − τ−(x,v))

)
T1u

(
~γ(x,v)(t − τ−(x,v))

)
dt,

we obtain the solution to Tu = 0, u|Γ− = u− satisfies

(I − K)u = Ju−. (2.3)

We shall make repeated use of the following immediate lemma.

Lemma 2.1 (Change of Variables). If u ∈ L1(ΩM) then

∫

M

∫

ΩxM

u(x,v) dvx dx =

∫

Γ±

∫ τ∓(x′,v′)

0

u(γ(x′,v′)(t), γ̇(x′,v′)(t)) dt dµ(x
′,v′).

Assume the following smallness condition on k:

with σp(x,v
′
) :=

∫

ΩxM

k(x,v′,v) dv,
∥∥τσp

∥∥
L∞(ΩM,dv dx)

< 1. (2.4)

This subcriticality condition will ensure well-posedness of the boundary value problem (see [RS]).

Proposition 2.1. The operator K is bounded on L1(ΩM,τ−1 dv dx) with operator norm bounded

by ‖τσp‖ < 1 and so (I −K) is invertible on this space. Equation (2.3) and hence (1.1) is uniquely

solvable for u− ∈ L
1
(Γ−,dµ) and the solution u has a well-defined trace u|Γ− . The albedo operator

A : L1(Γ−,dµ) → L
1
(Γ+,dµ) is a bounded map.

Since we have ‖τσp‖ < 1 we may express the solution to (1.1) as a Neumann series,

u =

∞∑

j=0

KjJu− = Ju− + KJu− + (I − K)
−1K2Ju−. (2.5)

We seek the solution, in the sense of distributions, to (1.1) with singular boundary condition

u|Γ− = δ{x0,v0}(x
′,v′); the right hand side is the distribution on Γ− defined by

(δ{x0,v0},ϕ) =

∫

Γ−

δ{x0,v0}(x
′,v′)ϕ(x′,v′) dµ(x′,v′) = ϕ(x0,v0)

for ϕ ∈ C∞0 (Γ−). It is convenient to use parallel translation in what follows: for x,y ∈ M define

P : ΩxM → ΩyM, P(v; x,y) to be the parallel translation of v from x to y (along the unique

geodesic joining x to y).



CUBO
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Optical Tomography for Media ... 77

Proposition 2.2. The three terms in (2.5) can be written

KjJu−(x,v) =

∫

Γ−

uj(x,v,x
′,v′)u−(x

′,v′) dµ(x′,v′)

with

u0(x,v,x
′,v′) =

∫ τ+(x′,v′)

0

E
(
x,γ(x,v)(−τ−(x,v))

)
δ(x,v)(~γ(x′,v′)(t)) dt

u1(x,v,x
′,v′) =

∫ τ+(x′,v′)

0

∫ τ−(x,v)

0

E(x,y(s))E(x′,z(t))k
(
z(t), ż(t),P(ẏ(s); y(s),z(t))

)
δ{y(s)}(z(t)) ds dt

y(s) = γ(x,v)(s − τ−(x,v)), z(t) = γ(x′,v′)(t)

u2 ∈ L
∞

(Γ−,W), W = {f : Df ∈ L
1
(ΩM), τ−1f ∈ L1(ΩM)}.

Proof. We present the straight-forward proof for u0 and refer the reader to [M1] for the more

involved proofs for u1 and u2. If ϕ ∈ C
∞
0 (ΩM) then

(Ju−,ϕ) =

∫

Γ−

u−(x
′,v′)

∫ τ+(x′,v′)

0

E(x′,γ(x′,v′)(t))ϕ(~γ(x′,v′)(t)) dt dµ(x
′,v′)

=

∫

M

∫

ΩxM

∫

Γ−

u−(x
′,v′)

∫ τ+(x′,v′)

0

δ(x,v)(~γ(x′,v′)(t)) dt dµ(x
′,v′)ϕ(x,v) dv dx.

From Proposition 2.2 we obtain the distribution kernel α(x,v,x′,v′) of the albedo operator

A.

Theorem 2.1. [M1] The distribution kernel α(x,v,x′,v′) of A is α = α0 + α1 + α2 with

α0 = E
(
x,γ(x,v)(−τ−(x,v))bigr)δ{~γ(x,v) (−τ−(x,v))}(x

′,v′),

α1 = u1
∣∣
(x,v)∈Γ+

,

α2 ∈ L
∞

(Γ−; L
1
(Γ+,dµ)).

Proof. If ϕ− ∈ C
∞
0 (Γ−) then changing variables to (y,w) = ~γ(x′,v′)(t),

∫

Γ−

u0(x,v,x
′,v′)ϕ−(x

′,v′) dµ(x′,v′)

=

∫

M

∫

ΩyM

E
(
x,γ(x,v)(−τ−(x,v))

)
δ(x,v)(y,w)ϕ−

(
~γ(y,w)(−τ−(y,w))

)
dw dy

= E
(
x,γ(x,v)(−τ−(x,v))

)
ϕ−

(
~γ(x,v)(−τ−(x,v))

)

=

∫

Γ−

α0(x,v,x
′,v′)ϕ−(x

′,v′) dµ(x′,v′).

The claim for α2 follows from [M1] Theorem 2.3 which shows that the trace on Γ± is continuous

from W into L1(Γ±,dµ).



78 Stephen McDowall CUBO
11, 5 (2009)

3 Full Measurements in Arbitrary Dimensions

We show here that both the extinction coefficient and the scattering kernel are uniquely determined

in dimensions three and greater. This is achieved by isolating terms in the kernel of the albedo

operator A that differ in strength of singularity. Briefly, α0 determines σ and α1 determines k. In

dimension two, however, α2 is in fact a locally L
1 function and is not immediately distinguishable

from α2. Thus, the method demonstrated in this section fails to determine k in dimension two.

We address the solution to this problem in the next section.

3.1 Determination of σ

We construct an appropriate approximate identity. Let ψ ∈ C∞0 ([0,∞)) be such that ψ(0) = 1

and
∫ ∞
0
ψ(t) dt = 1; define ψε(x) = ψ(x/ε).

Proposition 3.1. The following limit holds in L1(Γ+,dµ(x,v)):

lim
ε→0

∫

Γ−

α(x,v,x′,v′)ψε
(
d
(
x′,γ(x,v)(−τ−(x,v))

))
dµ(x′,v′) = E

(
x,γ(x,v)(−τ−(x,v))

)
.

Proof. When α is replaced by α0 the result is immediate. When α is replaced by α1,

0 ≤

∫

Γ+

∫

Γ−

∫ τ+(x′,v′)

0

∫ τ−(x,v)

0

k
(
z(t), ż(t),P(ẏ(s); y(s),z(t))

)
δ{y(s)}(z(t))ψε

(
d
(
x′,γ(x,v)(−τ−(x,v))

))

× ds dt dµ(x′,v′) dµ(x,v)

=

∫

M

∫

Ω2yM

k(y,w′,w)ψε
(
d
(
γ(y,w′)(−τ−(y,w

′
)),γ(y,w)(τ+(y,w))

))
dw′ dw dy

where (y,w′) = ~γ(x′,v′)(t), (y,w) = ~γ(x,v)(s − τ−(x,v)). Now there exists constant C such that

supp ψε
(
d
(
γ(y,w′)(−τ−(y,w

′
)),γ(y,w)(τ+(y,w))

))
⊂ Wε = {(y,w

′,w) ∈ Ω2M : ‖w′ − w‖g < Cε}

and so

0 ≤

∫

Γ+

∫

Γ−

α1(x,v,x
′,v′)ψε

(
d
(
x′,γ(x,v)(−τ−(x,v))

))
dµ(x′,v′) dµ(x,v) =

∫

Wε

k(y,w′,w) dw′ dw dy → 0

as ε → 0 since k ∈ L1(Ω2M) and the measure of Wε → 0 as ε → 0. Finally, for α2

0 ≤

∫

Γ+

∣∣∣
∫

Γ−

α1(x,v,x
′,v′)ψε

(
d
(
x′,γ(x,v)(−τ−(x,v))

))
dµ(x′,v′)

∣∣∣ dµ(x,v)

≤

∫

Vε

∣∣α2(x,v,x′,v′)
∣∣ dµ(x′,v′)

∣∣∣ dµ(x,v) → 0

as ε → 0. Here

supp ψε(x,v,x
′,v′) ⊂ Vε = {(x,v,x

′,v′) ∈ Γ+ × Γ− : ‖w
′ − w‖g < Cε}

and the limit above holds since by Theorem 2.1 α2 ∈ L
1
(Γ+ × Γ−) and the measure of Vε → 0 as

ε → 0.



CUBO
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Optical Tomography for Media ... 79

Proposition 3.1 enables us to obtain from A the integral of σ along the geodesic between any

two points in the boundary of M. That is, we may determine the geodesic X-ray transform of σ.

For simple Riemannian manifolds this transform is known to be invertible (see [Sh1]). We have

thus proven the following theorem.

Theorem 3.1. [M1] Let M ⊂ Rn, n ≥ 2, be a bounded domain with smooth boundary and let

g be a known simple Riemannian metric on M. Let 0 ≤ σ(x) ∈ L∞(M) depend only on x, and

0 ≤ k ∈ L∞(Ω2M) satisfy (2.4). Then we may determine σ from A.

3.2 Determination of k

Toward determining the scattering kernel k, fix (y,w,w′) ∈ Ω2M, w 6= w′. Let expw′ : Tw′ ΩyM →

ΩyM be the exponential map of the unit tangent sphere based at w
′ ∈ ΩyM. Denote by v̂(v) =

expw′ (v), and let J(y,w′)(v̂) be the determinant of the Jacobian of this change of variables. Let

ϕ1 ∈ C
∞
0 (R

n
) be such that 0 ≤ ϕ1 ≤ 1, ϕ1(0) = 0, ϕ1(v̂) = 0 for |v̂| > ε0 for sufficiently small ε0,

and
∫

Rn−1
ϕ1(v̂) dv̂ = 1. Now define ψε : ΩyM → R by

ψε(v) =
1

εn−1
ϕ1

(
exp

−1
w′ (v)

ε

)
.

Note that if f : ΩyM → R is continuous at w
′ then

∫

ΩyM

f(v)ψε(v) dv =

∫

Rn−1

f
(
exp

−1
w′ (v̂)

) 1
εn−1

ϕ1

(v̂
ε

)
J(y,w′)(v̂) dv̂ → f

(
exp

−1
w′ (0)

)
J(y,w′)(0) = f(w

′
)

as ε → 0.

Define y(s) = γ(y,w)(s − τ−(y,w)), −τ+(y,w) ≤ s ≤ τ−(y,w). Define β(s) ∈ ∂M to be the

unique point in the boundary for which it holds that γ(β(s),P(w′;y,β(s)))(·) contains the point y(s),

and define v(s) ∈ Ωy(s)M to be its tangent vector there. Since w and w
′ are independent, β′(0) 6= 0.

Now let ϕ ∈ C∞0 (R) be such that 0 ≤ ϕ ≤ 1, ϕ(0) = 1,
∫

R
ϕ(x) dx = ‖β′(0)‖−2g ; let

ϕη(x) =
1

η
ϕ
(x
η

)
.

Define h1 : ∂M → R by h1(x
′
) = 〈exp−1

β(0)
(x′),β′(0)〉. Note that, restricted to the curve β(s), for

sufficiently small s, h1(β(s)) = 0 if and only if s = 0. Furthermore,

d

ds

∣∣∣
s=0

h1(β(s)) = ‖β
′
(0)‖2.

The construction of β(s) for fixed (y,w,w′) is now repeated for arbitrary (z,ξ,ξ′) ∈ Ω2M and

we denote this by β(z,ξ,ξ′)(s). Define h2 : ∂M × Ω
2M by h2(x

′,z,ξ,ξ′) = dist
(
x′,β(z,ξ,ξ′)(·)

)
. Let

µ ∈ C∞0 (R) with µ(0) = 1 and define µδ(t) = µ(t/δ). Let

W = {(z,ξ,ξ′) ∈ Ω2M : ξ 6= ξ′}.



80 Stephen McDowall CUBO
11, 5 (2009)

Proposition 3.2. If n ≥ 3 and (y,w,w′) ∈ Ω2M with w 6= w′ then

lim
η→0

lim
δ→0

lim
ε→0

∫

Γ−

ψε(P(v
′
; x′,y))ϕη(h1(x

′
))µδ(h2(x

′,y,w,w′))α(~γ(y,w)(τ+(y,w)),x
′,v′) dµ(x′,v′)

= E
(
y,γ(y,w)(τ+(y,w))

)
E

(
y,γ(y,w′)(−τ−(y,w

′
))

)
k(y,w′,w)

the limit holding in L1(W).

Proof. Replacing α by α0 and integrating with respect to dµ(x
′,v′), the integrand gets evaluated

at (x′,v′) = ~γ(y,w)(−τ−(y,w)) and so we obtain a multiple of

ψε
(
P

(
~γ(y,w)(−τ−(y,w))

)
; γ(y,w)(−τ−(y,w)),y

)
= ψε(w) = 0

for all sufficiently small ε since w 6= w′. Replacing α by α1,

I1 : =

∫

Γ−

ψε(P(v
′
; x′,y))ϕη(h1(x

′
))µδ(h2(x

′,y,w,w′))α1(~γ(y,w)(τ+(y,w)),x
′,v′) dµ(x′,v′)

=

∫ τ(y,w)

0

∫

Ωy(s)M

ψε
(
P(γ̇(y(s),v̂)(−τ−(y(s), v̂)); γ(y(s),v̂)(−τ−(y(s), v̂)),y)

)

× ϕη
(
h1

(
γ(y(s),v̂)(−τ−(y(s), v̂))

))
µδ

(
h2(γ(y(s),v̂)(−τ−(y(s), v̂)),y,w,w

′
)
)

× E
(
y(s),γ(y,w)(τ+(y,w))

)
E

(
y(s),γ(y(s),v̂)(−τ−(y(s), v̂))

)
k(y(s), v̂, ẏ(s)) dv̂ ds

Now define ṽ(v̂,s) = P
(
γ̇(y(s),v̂)(−τ−(y(s), v̂)); γ(y(s),v̂)(−τ−(y(s), v̂)),y

)
∈ ΩyM and denote by

dv̂
dṽ

the change of volume element of this change of variables. We obtain

I1 =

∫ τ(y,w)

0

∫

ΩyM

ψε(ṽ)ϕη
(
h1

(
γ(y(s),v̂(ṽ,s))(−τ−(y(s), v̂(ṽ,s)))

))

× µδ
(
h2(γ(y(s),v̂(ṽ,s))(−τ−(y(s), v̂(ṽ,s))),y,w,w

′
)
)
E(·)E(·)k(y(s), v̂(ṽ,s), ẏ(s))

dv̂

dṽ
dṽ ds

→ I2 :=

∫ τ(y,w)

0

ϕη
(
h1(β(y,w,w′)(s))

)
µδ

(
h2(β(y,w,w′)(s),y,w,w

′
)
)
E(·)E(·)

× k(y(s),v(s), ẏ(s))
dv̂

dṽ

∣∣∣
v̂=v(s)

ds

as ε → 0. Note that µδ
(
h2(β(y,w,w′)(s),y,w,w

′
)
)

= 1. Now define s̃(s) = h1(β(y,w,w′)(s)). Then

s̃ = 0 if and only if s = 0 (for sufficiently small s), and ds̃
ds

∣∣
s=0

= ‖β′
(y,w,w′)

(0)‖2. So, for sufficiently

small s̃0,

I2 =

∫ s̃0

−s̃0

ϕη(s̃)E(·)E(·)k
(
y(s(s̃)),v(s(s̃)), ẏ(s(s̃))

)dv̂
dṽ

∣∣∣
v̂=v(s)

ds

ds̃
ds̃

→ E
(
y,γ(y,w)(τ+(y,w))

)
E

(
y,γ(y,w′)(−τ−(y,w

′
))

)
k(y,w′,w)

as η → 0 since
∫

R
ϕ2(x) dx = ‖β′(y,w,w′)(0)‖

−2
g .



CUBO
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Optical Tomography for Media ... 81

Finally, we must show that the integral vanishes when we replace α by α2. Let χ(z,ξ,ξ
′
) ∈

C∞0 (W). Then

lim
ε→0

∫

M

∫

Ω2
z
M

∣∣∣
∫

Γ−

ψε(P(v
′
; x′,z))ϕη(h1(x

′
))µδ(h2(x

′,z,ξ,ξ′))α2(~γ(z,ξ)(τ+(z,ξ)),x
′,v′)χ(z,ξ,ξ′)

× dµ(x′,v′)
∣∣∣ dξ′ dξ dz

≤
1

η

∫

Γ+

∫ τ−(x,v)

0

∫

Ωγ
(x,v)

(s−τ
−

(x,v))M

∫

∂M

µδ(h2(x
′,~γ(x,v)(s − τ−(x,v)),ξ

′
))

× |χ(~γ(x,v)(s − τ−(x,v)),ξ
′
)||〈P(ξ′;~γ(x,v)(s − τ−(x,v)),x

′
),νx′〉| dx

′ dξ′ ds dµ(x,v)

→ 0

as δ → 0 since the support of µδ is a (3n − 1)-dimensional variety in the (4n − 3)-dimensional

domain of integration, and since α2 ∈ L
∞

(Γ−; L
1
(Γ+,dµ)).

Combining Proposition 3.2 with Theorem 3.1 we first determine the function E, and then

obtain k. We thus have the following.

Theorem 3.2. [M1] Let M ⊂ Rn, n ≥ 3, be a bounded domain with smooth boundary and let

g be a known simple Riemannian metric on M. Let 0 ≤ σ(x) ∈ L∞(M) depend only on x, and

0 ≤ k ∈ L∞(Ω2M) satisfy (2.4). Then we may determine k from A.

4 Full Measurements in Dimension Two

In this section we present the results of [M2] where it is proven that knowledge of the albedo

operator on a simple Riemannian surface uniquely determines the unknown metric g, and the

coefficients σ and k. We shall refer the reader to [M2] for most of the proofs. We impose two

additional conditions on the manifold (M,g).

Assumption 4.1. If the maximal sectional curvature κ0 of (M,g) is positive then we assume that

there are no focal points. That is, for every geodesic γ : [a,b] → M and every non-zero Jacobi field

J(t) along γ with J(a) = 0 it holds that ‖J(t)‖ is strictly increasing on [a,b].

Assumption 4.2. In the case that κ0 > 0 we assume that the diameter of (M,g) satisfies

diam(M,g) < π/(2
√
κ0).

Theorem 4.1. [M2] Let (M,g) be a two-dimensional Riemannian manifold satisfying assumptions

1.1 and 4.1. Then the albedo operator A determines the metric g up to a diffeomorphic change of

coordinates which is the identity on ∂M.

Proof. From Theorem 3.1 we see that A determines the so-called scattering relation of (M,g), that

is, the set {(x,v,γ(x,v)(−τ−(x,v)), γ̇(x,v)(−τ−(x,v))}. It is a result of [PU1] (see also [PU2]) that

for simple manifolds with no focal points, g is uniquely determined by this scattering relation.



82 Stephen McDowall CUBO
11, 5 (2009)

We now present the precise statement for the determination of σ and k. Let

UΣ,ε = {(σ(x),k(x,w
′,w)) : ‖σ‖L∞ ≤ Σ, ‖k‖L∞ ≤ ε}.

Theorem 4.2. [M2] Let (M,g) be a two-dimensional Riemannian manifold satisfying assumptions

1.1 and 4.2. Given Σ > 0 there exists ε > 0 such that any pair (σ,k) ∈ UΣ,ε is uniquely determined,

within (σ,k) ∈ UΣ,ε, by the associated albedo operator A. One may take ε = Ce
−2 diam(M,g)Σ where

C depends only on (M,g).

From Theorem 3.1 we may assume that σ (and hence E) is known. The main starting point

for determination of k is a more precise expression for α1. Throughout we are restricting to the

case n = 2. We will frequently omit the exact points of evaluation of E writing instead “E(·).” We

will not need the omitted information.

Proposition 4.1. The second term α1 in the series expansion for the kernel of A has the expression

α1(x,v,x
′,v′) = χ(x,v,x′,v′)E(·)E(·)J (x,v,x′,v′)

×
k
(
~γ(x′,v′)(s(x

′,v′)), γ̇(x,v)(t(x
′,v′) − τ−(x,v))

)

| sin ψ|

where χ = 1 if the geodesics γ(x′,v′)(s(x
′,v′)) and γ(x,v)(t(x

′,v′) − τ−(x,v)) intersect for s =

s(x′,v′) > 0 and t = t(x′,v′) > 0, and χ = 0 otherwise, and where ψ is the angle between the

tangent vectors at this point of intersection. The function J is uniformly bounded 0 < m1 ≤ J ≤

m2.

Proof. (Sketch) By definition, if ϕ− ∈ C
∞
0 (Γ−) then

KJϕ−(x,v) =

∫ τ−(x,v)

0

E(·)

∫

Ωy(t)M

k(y(t),w, ẏ(t))E(·)ϕ−
(
~γ(y(t),w)(−τ−(y(t),w))

)
dw dt

where y(t) = γ(x,v)(t − τ−(x,v)). We re-write this integral in terms of an integral over Γ−. Define

the family of indicator functions χ : ΩM → {0, 1}, parameterized by (x,v), by

χ(x,v,x′,v′) =

{
1 if γ(x′,v′)(s) = γ(x,v)(t − τ−(x,v)) = y(t) for some s > 0, t > 0,

0 otherwise.

On the support of χ we have well-defined functions s(x′,v′), t(x′,v′); on this support the change

of variables

(t,w) = Φ(x′,v′) =
(
t(x′,v′), γ̇(x′,v′)(s(x

′,v′))
)

is well-defined and smooth. In [M2] it is shown that if JΦ is the Jacobian of this change of variables

then

| det Jφ| =
|〈v′,νx′〉|

| sin ψ|
J (x,v,x′,v′)

where J 6= 0 and J is bounded as in the statement of the proposition. The expression for α1

follows immediately.



CUBO
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Optical Tomography for Media ... 83

Suppose now that we have two identical manifolds (M,g) with material parameters (σ,k) and

σ,k̃). Let A and à be their respective albedo operators and let αj, α̃j be the terms in their series

expansions. Then it follows that

(α2 − α1)(x,v,x
′,v′) = χ(x,v,x′,v′)E(·)E(·)J (·)

(k̃ − k)
(
~γ(x′,v′)(s(x

′,v′)), γ̇(x,v)(t(x
′,v′) − τ−(x,v))

)

| sin ψ|
,

and this in turn implies

χ|(k̃ − k)(y,w,ŵ)| ≤ Ce2 diam(M,g)Σχ| sin ψ||(α2 − α̃2)(x,v,x
′,v′)| a.e., (4.6)

where y is the point of intersection of the geodesics γ(x′,v′) and γ(x,v) and w, w
′ are their tangent

vectors there. In what follows we outline the proof of another estimate of α2 − α̃2 of the form

‖ sin ψ(α2 − α̃2)‖L∞ ≤ Cε‖k − k̃‖L∞. (4.7)

Once this is established, we combine it with (4.6) to obtain

‖k − k̃‖L∞ ≤ Cε‖k − k̃‖L∞

so that for sufficiently small ε we have k = k̃. This completes the proof of Theorem 4.2.

Toward proving (4.7) let ϕ− be the Dirac delta distribution on Γ− with respect to the measure

dµ(x′,v′):

ϕ−(x
′,v′) =

1

|〈v′,νx′〉|
δ(x0,v0)(x

′,v′).

Then

α2 − α̃2 = (I − K)
−1K2Jϕ− − (I − K̃)

−1K̃2Jϕ−

= (I − K)−1
[
K(K − K̃) + (K − K̃)K̃

]
Jϕ− + (I − K̃)

−1
(K − K̃)(I − K)−1K̃2Jϕ−.

(4.8)

From this we see that we must obtain estimates for K2 and K3. These are obtained in [M2] and

we present them here without proof.

Proposition 4.2. For almost every (x,v,x′0,v
′
0) ∈ Γ+ × Γ− such that there exist s > 0, t > 0 with

γ(x,v)(s − τ−(x,v)) = γ(x′
0
,v′

0
)(t)

|K̃KJϕ−(x,v,x
′
0,v

′
0)| ≤





C‖k̃‖L∞ ‖k‖L∞
cos(

√
κ0 diam(M,g))

(
1 + log

1
| sin ψ|

)
κ0 > 0,

C‖k̃‖L∞‖k‖L∞
(
1 + log

1
| sin ψ|

)
κ0 ≤ 0,

where κ0 is the maximal sectional curvature of (M,g) and where ψ is the angle between the tangent

vectors of the intersecting geodesics at the point of intersection.



84 Stephen McDowall CUBO
11, 5 (2009)

Although the proof of this proposition is not presented here, we point out that a recurring

theme in the proof is comparison of geodesic triangles in (M,g) to triangles with (for example) the

same side-angle-side property on either the sphere or the hyperbolic plane of constant curvature

κ0. Such constant curvature manifolds represent the “worst case” scenario with regard to how

intersecting geodesics might get close to intersecting a second time. The final estimate needed is

the following.

Proposition 4.3. It holds that K3Jϕ− ∈ L
∞

(Γ+ × Γ−) with norm bounded by C‖k‖
3
L∞.

We now complete the proof of (4.7) via (4.8).

Lemma 4.1. For almost every (x,v,x′0,v
′
0) ∈ Γ+ × Γ− such that γ(x,v) and γ(x′0,v

′

0
) we have

|(I − K)−1(K2 − K̃2)Jϕ−(x,v,x
′
0,v

′
0)| ≤ C‖k − k̃‖L∞ (‖k‖L∞ + ‖k̃‖L∞ )

(
1 + log

1

| sin ψ|

)

and

|(I − K̃)−1(K − K̃)(I − K)−1K̃2Jϕ−(x,v,x
′
0,v

′
0)| ≤ C‖k − k̃‖L∞‖k̃‖

2
L∞ (1 + ‖k‖L∞ ).

Proof. For the first estimate, we re-write

(I − K)−1
[
K(K−K̃) + (K − K̃)K̃

]
Jϕ− = (I + (I − K)

−1K)
[
K(K − K̃) + (K − K̃)K̃

]
Jϕ−

=
[
K(K − K̃) + (K − K̃)K̃

]
Jϕ− + (I − K)

−1
[
K2(K − K̃) + K(K − K̃)K̃

]
Jϕ−.

The contribution from the first term is estimated by Proposition 4.2 (with K or K̃ replaced by

K −K̃). For the final term, K2(K −K̃) + K(K −K̃)K̃Jϕ− is an L
∞ function by Proposition 4.3,

and (I − K)−1 preserves this space.

For the second estimate, express (I − K)−1K̃2 = K̃2 + (I − K)−1KK̃2 and apply Proposition

4.3.

Combining the estimates of Lemma 4.1 with (4.8) we obtain

Cχ| sin ψ||(α2 − α̃2)(x,v,x
′,v′)| ≤ C′ε‖k − k̃‖L∞,

and this together with (4.6) yields

‖k − k̃‖L∞(Ω2M) ≤ Cε‖k − k̃‖L∞(Ω2M). (4.9)

Thus, for sufficiently small ε, it must hold that k = k̃. Although it has not been carefully tracked

in this article, in [M2] it is shown that ε can be taken to be ε = Ce−2 diam(M,g)Σ with C depending

only on (M,g).



CUBO
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Optical Tomography for Media ... 85

5 Angularly Averaged Measurements

We now consider the problem of parameter determination with less information. The results

presented here appear in [LM]. Instead of knowing the angularly resolved measurement u(x,v) on

Γ+ we will assume only knowledge of an average over outgoing directions of u at x ∈ ∂M. This is

motivated in part by the fact that in practice it is very difficult and perhaps impossible to measure

the angular resolution of exiting photons. To be more precise, for x ∈ ∂M, define

Ω
±
x M = {v ∈ ΩxM : ±〈v,νx〉 > 0}

(and so Γ± are the disjoint unions of the Ω
±
x M over x ∈ ∂M).

Definition 5.1. The data of which we will be assuming knowledge consists of angularly averaged

measurements on ∂M, weighted with respect to a prescribed function m(x,v). Specifically, given

u− on Γ− and u = T
−1u− we define M : L

1
(Γ−,dµ) → L

1
(∂M) by

Mu−(x) :=

∫

Ω
+
x M

u(x,v)m(x,v) dv.

We require that m ∈ L∞(Γ+), that m does not vanish on Γ+, and that |m(x,v)/〈v,νx〉| be bounded

on Γ+.

The function m corresponds to the limitations of the measurement apparatus. It may represent

a limited aperture or, for example, when m(x,v) = 〈v,νx〉, the measurement is power flux exiting

the boundary.

We assume here that the metric g is known. With this limited data, we are still able to

determine the extinction coefficient σ, but are only able to determine scattering kernels of a more

restricted class. We assume that k is of the form k(x)Θ(x,w,w′) where Θ is assumed to be known.

This corresponds to knowing the angular behavior of the scattering events, but not the density

of scatterers (which is quantified by k(x)). There are also assumptions of analyticity of various

coefficients. To be precise we present the main theorems.

If H ⊂ Γ−, then by Γ(H) = {γ(x′,v′)(t) : (x
′,v′) ∈ H, 0 ≤ t ≤ τ+(x

′,v′)} we mean the set of

geodesics with initial data in H. With constants Cκm and CκM to be defined later, we have the

following.

Theorem 5.1. [LM] Suppose that

‖k‖L∞ < min
{
[(CκmCκM )

n−1
diam M|Sn−1|]−1, [diam M|Sn−1|]−1

}
,

and that Hσ ⊂ Γ− is open and such that the geodesic X-ray transform restricted to Γ(Hσ) is

injective. Suppose that σ = σ(x) depends on position only. Then σ is uniquely determined by

{(u−,Mu−) : u− ∈ L
1
(Hσ)}.

Fixing D > 0 and making the definition KDε := {k ∈ L
∞

(M) : dist(supp(k),∂M) >

D,‖k‖L∞ ≤ ε} we also have the following uniqueness result for k.



86 Stephen McDowall CUBO
11, 5 (2009)

Theorem 5.2. Suppose the scattering kernel has the form k(x)Θ(x,v′,v), with (g,m,σ, Θ) known

and real analytic, and with both m and Θ non-vanishing. Let Hk ⊂ Γ− be open, and assume that

for every (x,v) ∈ TM \ {0}, there exists γ ∈ Γ(Hk) through x and normal to v at x. Then there

exits ε sufficiently small such that for a.e. x ∈ ∂M, knowledge of {(u−,Mu−(x)) : u− ∈ L
1
(Hk)}

uniquely determines k within the class KDε .

Furthermore, ε may be chosen such that this result holds in some C2 neighborhood of (m,σ),

and some C3n neighborhood of (Θ,g).

We begin by deriving an expression for the averaged albedo operator M as an infinite sum of

the operators Mi (see Definition 5.2), and where we give expressions for the Schwartz kernel of

Mi, i ≥ 1. In order to simplify the presentation of what follows, we introduce some notation. If

x1, . . . ,xj are points in M, then

E(x1,x2, . . . ,xj) := E(x1,x2)E(x2,x3) · · ·E(xj−1,xj) =

j−1∏

i=1

E(xi,xi+1). (5.10)

If y ∈ M, consider z = zy(t,v) = γ(y,v)(t) defined from the “polar” coordinates (t,v) ∈ R × ΩyM.

Let Jy(z) denote the Jacobian determinant | det ∂z/∂(t,v)|
−1 of this change of variables. For a

given z, let (t,v) be its polar coordinate expression, and let {v1, . . . ,vn} be an orthonormal basis

for TyM with v
1

= v. Let Yy,z,i be the Jacobi field along γ(y,v)(·) with Yy,z,i(0) = 0, Ẏy,z,i = v
i,

2 ≤ i ≤ n. Then Jy(z) is given by the expression

Jy(z) =

n∏

i=2

|Yy,z,i(d(y,z))|
−1. (5.11)

Analogous to (5.10) we introduce the notation

J (y1, . . . ,yj) :=

j∏

i=2

Jyi (yi−1). (5.12)

By considering comparison theorems for Jacobi fields, one easily obtains the following.

Lemma 5.1. With Jy(z) defined as above, there exists a constant CκM , such that for all y,z ∈ M

Jy(z) ≤
CκM

d(y,z)n−1
. (5.13)

Further, there exists a constant Cκm such that if Y is any Jacobi field along a geodesic γ(t) with

Y (0) = 0 and Ẏ (0) ∈ γ̇⊥(0) ⊂ Ωγ(0)M, then

|Y (t)|

t
≤ Cκm (5.14)

for all 0 ≤ t ≤ τ+(γ(0), γ̇(0)).



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Optical Tomography for Media ... 87

The notation CκM and Cκm is chosen because these constants are determined in terms of

minimal and maximal sectional curvatures of (M,g).

Definition 5.2. For each i ≥ 0 we define the operator Mi by

Miu−(x) :=

∫

Ω
+
x M

KiJu−(x,v)m(x,v) dv.

Theorem 5.3. [LM] Let ‖k‖L∞(Ω2M) <
(
|Sn−1| diam M

)−1
. Then Mi,M : L

1
(Γ−,dµ) →

L1(∂M) continuously, and for almost every x ∈ ∂M, M has the expansion

Mu−(x) =

∞∑

j=0

Mju−(x) =

∞∑

j=0

∫

Γ−

αj(x,x
′,v′)u−(x

′,v′) dµ(x′,v′)

where α0(x, ·, ·) is a distribution supported on a manifold of dimension n − 1, and for j ≥ 1,

αj(x,x
′,v′) are the following Schwartz kernels: when j = 1,

α1(x,x
′,v′) =

∫ τ+(x′,v′)

0

k(~γ(x′,v′)(t), w̄1)E(x
′,γ(x′,v′)(t),x)m(x,ŵ1)Jx(γ(x′,v′)(t)) dt (5.15)

where w̄1, ŵ1 are the initial and final tangent vectors of the geodesic from γ(x′,v′)(t) to x, and E

and Jx are defined in (5.10), and (5.11); for j ≥ 2,

αj(x,x
′,v′) =

∫ τ+(x′,v′)

0

∫

M

· · ·

∫

M

k(~γ(x′,v′)(t), w̄1)

j∏

i=2

k(yi, ŵi−1, w̄i)

E(x′,γ(x′,v′)(t),y2, . . . ,yj,x)J m(x,ŵj) dyj · · · dy2 dt (5.16)

where:

• w̄1, ŵ1 are the initial and final tangent vectors of the geodesic joining γ(x′,v′)(t) to y2,

• w̄i, ŵi are the initial and final tangent vectors of the geodesic joining yi to yi+1 for i =

2, . . . ,j − 1,

• w̄j, ŵj are the initial and final tangent vectors of the geodesic joining yj to x, and

• J = J (γ(x′,v′)(t),y2, . . . ,yj,x).

Proof. We refer the reader to [LM] for the full proof of this theorem. To give a flavor of the proof,



88 Stephen McDowall CUBO
11, 5 (2009)

we present the derivation of M1 here. Let φ− be a function on Γ−. We have

M1φ−(x) =

∫

Ω
+
x M

KJφ−(x,v) dv

=

∫

Ω
+
x M

∫ τ−(x,v)

0

E
(
x,γ(x,v)(t − τ−(x,v))

)
T1Jφ−(~γ(x,v)(t − τ−(x,v)) dt m(x,v) dv

=

∫

Ω
+

x M

∫ τ−(x,v)

0

E
(
x,γ(x,v)(t − τ−(x,v))

)

×

∫

ΩyM

k(y,ŵ,w̄1)E
(
γ(x,v)(t − τ−(x,v)),γ(y,ŵ)(−τ−(y,ŵ))

)

× φ−(~γ(y,ŵ)(−τ−(y,ŵ)) dŵ dt m(x,v) dv

where (y,w̄1) = ~γ(x,v)(t − τ−(x,v))

=

∫

M

∫

ΩyM

E
(
x,y,γ(y,ŵ)(−τ−(y,ŵ))

)
k(y,ŵ,w̄1)φ−

(
~γ(y,ŵ)(−τ−(y,ŵ))

)
dŵ

× m(x,ŵ1)Jx(y) dy

where w̄1, ŵ1 are the initial and final tangent vectors (respectively) of the geodesic from y to x,

=

∫

Γ−

∫ τ+(x′,v′)

0

k(~γ(x′,v′)(t), w̄1)E(x
′,γ(x′,v′)(t),x)m(x,ŵ1)Jx(γ(x′,v′)(t)) dt

× φ−(x
′,v′) dµ(x′,v′).

This proves (5.15).

As seen in the (partial) proof of Theorem 5.3, there appear weakly singular integrals due to

the presence of the functions Jx (see also (5.13)).

Definition 5.3. Let T be the operator with kernel d(x, ·)n−1, that is

T f(x) :=

∫

M

f(y)

d(x,y)n−1
dy. (5.17)

From (5.13) one has T : Lp(M) → Lp(M) continuously, 1 ≤ p ≤ ∞ with ‖T ‖ ≤ (Cκm )
n−1|Sn−1| diam M

(see [T], Prop. 5.1, Appendix A).

We also define the analogous operator T̃ ,

T̃ f(x) :=

∫

M

f(y)Jy(x) dy (5.18)

for which it holds T̃ : Lp(M) → Lp(M), 1 ≤ p ≤ ∞, with ‖T̃ ‖ ≤ |Sn−1| diam M.

We have seen that α0 is more singular than the remaining αj. In the following proposition we

estimate the contribution from the terms for j ≥ 1.



CUBO
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Optical Tomography for Media ... 89

Proposition 5.1. Let p = (n − µ)/(n − 1), 0 < µ < 1, q = (1 − 1/p)−1. With Cκm,CκM defined

as in Lemma 5.1, let ‖k‖L∞(Ω2M) <
[
(CκmCκM )

n−1
diam M|Sn−1|

]−1
. Then for almost every

x ∈ ∂M,

∣∣∣
∞∑

j=1

∫

Γ−

αj(x,x
′,v′)f(x′,v′) dµ(x′,v′)

∣∣∣ ≤ C0‖f‖Lq(Γ−,dµ) (5.19)

where C0 > 0 depends on κm, κM, ‖k‖L∞(Ω2M), ‖m‖L∞(ΩM) and diam M.

Proof. When j = 1,

∣∣∣
∫

Γ−

α1(x,x
′,v′)f(x′,v′) dµ(x′,v′)

∣∣∣

=

∣∣∣
∫

Γ−

∫ τ+(x′,v′)

0

k(~γ(x′,v′)(t), w̄)E(·)m(x,ŵ1)Jx(γ(x′,v′)(t)) dt f(x
′,v′) dµ(x′,v′)

∣∣∣

≤ ‖k‖∞‖m‖∞(CκM )
n−1

∫

M

∫

ΩyM

|f
(
~γ(y,v)(−τ−(y,v))

)
|

d(x,y)n−1
dv dy by (5.13)

≤ (CκM )
n−1‖k‖∞‖m‖∞

(∫

ΩM

1

d(x,y)(n−1)p
dv dy

) 1
p

×
(∫

ΩM

∣∣f
(
~γ(y,v)(−τ−(y,v))

)∣∣q dv dy
) 1

q

by Hölder’s inequality

≤ (CκM )
n−1‖k‖∞‖m‖∞|S

n−1|1/pC
1/p
d (diam M)

1/q‖f‖Lq(Γ−,dµ) see (5.17)

= C‖k‖∞‖m‖∞(diam M)
1/q‖f‖Lq(Γ−,dµ), say.

The computation for the terms j ≥ 2 are slightly different from the above. We refer the reader to

[LM] for the details, where it is also proven that the infinite series converges in L1.

5.1 Determination of σ

The determination of σ is attained via a limiting argument exactly as in Proposition 3.1. The

construction of the approximate identity is slightly different but the spirit is the same. The details

are in [LM]. Let Hσ ⊂ Γ− be such that the geodesic X-ray transform restricted to geodesics in

Γ(Hσ) is invertible.

Theorem 5.4. [LM] Let (σ,k,m) satisfy the hypothesis of Theorem 5.1. For almost every (x∗,v∗) ∈

Hσ

lim
η→0

Mfη(x
∗
) = E

(
γ(x∗,v∗)(−τ−(x

∗,v∗)),x∗
)
m(x∗,v∗)

where fη concentrates to (x
′,v′) = ~γ(x∗,v∗)(−τ−(x

∗,v∗)) as η → 0.

Since m is known we thus know the integrals of σ along every geodesic in Γ(Hσ) and hence σ

is uniquely determined.



90 Stephen McDowall CUBO
11, 5 (2009)

5.2 Determination of k

Throughout this section, the measurement point x will be fixed. As mentioned earlier we assume

that the scattering kernel is of the form k(x)Θ(x,v′,v), where Θ(x,v′,v) is a-priori known. We

prove in this setting that the spatial distribution k(x) is uniquely determined by the averaged

albedo operator M from measurements at the single point x.

Definition 5.4. Given a complete Riemannian manifold (M,g) with geodesics γ(x,v)(t), and func-

tions η : ΩM → R, and β ∈ C∞(Γ−) we may define the weighted geodesic transform by, for

f : M → R,

Iη,βf(x
′,v′) := β(x′,v′)

∫ τ+(x′,v′)

0

f(γ(x′,v′)(t))η(~γ(x′,v′)(t)) dt. (5.20)

We also have the L2 adjoint, for f : Γ− → R,

I∗η,βf(x) =

∫

ΩxM

f
(
~γ(x,v)(−τ−(x,v))

)
β
(
~γ(x,v)(−τ−(x,v))

)
η(x,v) dv. (5.21)

Let χ ∈ C∞0 (M) with χ ≡ 1 on a neighborhood of {y ∈ M : dist(y,∂M) ≥ D}. If x1 ∈ M,

let v̄ = v̄(x1) ∈ Ωx1M and v = v(x1) ∈ Ω
+
x M be the initial and final tangent vectors, respectively,

of the geodesic joining x1 to x. Then for v1 ∈ Ωx1M define the weight function

w(x1,v1) := Θ(x1,v1, v̄)E
(
γ(x1,v1)(−τ−(x1,v1)),x1,x

)
m(x,v)Jx(x1)χ(x1). (5.22)

From (5.15) and (5.20) we see that α1(x,x
′,v′) = Iw,1k(x

′,v′). If g, Θ, σ and m are real-analytic,

then w is a real-analytic, non-vanishing weight function in {y ∈ M : dist(y,∂M) ≥ D}. It is

injectivity of this weighted ray transform which will allow determination of k(x). We make use of

injectivity results of [FSU]. These results allow a possibly proper subsets of geodesics on which

the transform is known. It is the inclusion of the factor β in definition (5.20) that restricts the

transform to a such a subset geodesics.

Definition 5.5. We say that Γ is a regular family of curves (for the metric g) if for any (x,v) ∈

T ∗M\{0} there exists γ ∈ Γ through x, normal to v, and such that γ has no conjugate points

(automatically satisfied for simple metrics).

We say that β ∈ C∞(Γ−) is regular if there exists a set H ⊂ {(x
′,v′) : β(x′,v′) 6= 0} such

that Γ(H) is a regular family.

Suppose that k(x)Θ(x,v,v′) and k̃(x)Θ(x,v,v′) are two scattering kernels with k, k̃ ∈ KDε ;

let M, M̃, and α, α̃ be the averaged albedo operators and Schwarz kernels associated to k and k̃

respectively. We set ∆k = k − k̃ and ∆αj = αj − α̃j, j = 1, 2, . . . .

Lemma 5.2. [FSU] Let (m, Θ,σ,g) be fixed and real analytic, and suppose β ∈ C∞(Γ−) is a

regular function for g. Then there exits C > 0, independent of k, k̃ ∈ KDε such that

‖∆k‖L2(M) ≤ C‖I
∗
w,βIw,β∆k‖H1(M), (5.23)

with the above estimate holding in a C2 neighborhood of (m, Θ,σ), and a C3 neighborhood of g.



CUBO
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Optical Tomography for Media ... 91

Proposition 5.2. Let (m, Θ,σ,g) be fixed and real analytic, and suppose β ∈ C∞(Γ−) is a regular

function for g. Furthermore, suppose Hk ⊂ {(x
′,v′) : β(x′,v′) 6= 0} is such that Γ(Hk) is a regular

set of geodesics. Suppose that M = M̃ on L1(Hk,dµ). Then there exists C > 0 such that for all

k, k̃ ∈ KDε

‖∆k‖L2(M) ≤ C‖I
∗
w,βIw,β∆k‖H1(M) = C‖I

∗
w,ββ∆α1‖H1(M) = C

∥∥∥I∗w,ββ
∞∑

j=2

∆αj

∥∥∥
H1(M)

,

with the above estimate holding in a C2 neighborhood of (m,σ, Θ) and a C3 neighborhood of g.

Proof. The first inequality is (5.23); next, βIw,1 = Iw,β and w has been defined so that Iw,1∆k =

∆α1; finally, since σ = σ̃, α0 = α̃0 and so M = M̃ implies that
∑

∞

j=1 ∆αj = 0 which proves the

final equality.

Analogous to (4.9) we proceed to prove that ‖∆k‖L2(M) ≤ εC‖∆k‖L2(M), and so for sufficiently

small ε > 0, we will have ∆k = 0. The following proposition is an extension of Proposition 4 of

[FSU] to more general weight functions ηj and restrictions βj. The proof is essentially the same

and further details are given in [LM].

Proposition 5.3. There is C > 0 such that for all f ∈ L2(M) with supp f ⊂ {y ∈ M : d(x,∂M) >

D},

‖I∗η1,β1Iη2,β2f‖H1(M) ≤ C‖η1‖C2(ΩM)‖η2‖C2(ΩM)‖β1β2‖C2(Γ−)‖f‖L2(M), (5.24)

with C depending continuously on the C4 norm of g.

We wish to apply (5.24) to each ∆αi, i > 1, and to do so we express each as a sum of weighted

X-ray transforms. This is achieved by expanding one instance of the kernel Θ occurring in ∆αi

in a manner based on spherical harmonic expansions of functions on Sn−1. This expansion is the

content of Lemma 5.3. The proof is postponed to the end of this section.

Lemma 5.3. Let Θ(x,v′,v) ∈ C3n(Ω2M) and g ∈ C3n(M). There exist Θj(x,v
′
), ϕj(x,v) ∈

C3n(ΩM) such that

Θ(x,v′,v) =

∞∑

j=1

Θj(x,v
′
)ϕj(x,v)

with

‖Θj‖C2(ΩM) ≤
C

1 + j2
, and ‖ϕj(x,v)‖L∞(ΩM) ≤ 1,

the above estimate holding in a C3n neighborhood of (Θ,g).



92 Stephen McDowall CUBO
11, 5 (2009)

Proposition 5.4. Fix (m,g, Θ,σ) with m,σ ∈ C2, and g, Θ ∈ C3n. Suppose that ‖k‖∞,‖k̃‖∞ <

[‖Θ‖∞ diam M|S
n−1|]−1, k, k̃ ∈ KDε , and β ∈ C

∞
(Γ−). Then there is C > 0 such that

∥∥∥I∗w,ββ
∞∑

i=2

∆αi

∥∥∥
H1(M)

≤ Cε‖∆k‖L2(M),

with the above estimate holding in a C2 neighborhood of (m,σ), and a C3n neighborhood of (Θ,g).

Proof. For (x,v) ∈ ΩM we define E(x,v) = E(γ(x,v)(−τ−(x,v)),x). From Theorem 5.3,

∆α2(x,x
′,v′)

=

∫ τ+(x′,v′)

0

E(x′,γ(x′,v′)(t))

∫

M

[
∆k(γ(x′,v′)(t))k(y2) − k̃(γ(x′,v′)(t))∆k(y2)

]

× Θ(~γ(x′,v′)(t), w̄1)Θ(y2, ŵ1, w̄2)E(γ(x′,v′)(t),y2)E(y2,x)J (γ(x′,v′)(t),y2,x) dy2 dt

and from Lemma 5.3 we have (formally at least),

∆α2(x,x
′,v′)

=

∞∑

l=1

∫ τ+(x′,v′)

0

E(~γ(x′,v′)(t))Θl(~γ(x′,v′)(t))

×

∫

M

[
∆k(γ(x′,v′)(t))k(y2) + k̃(γ(x′,v′)(t))∆k(y2)

]
φl(γ(x′,v′)(t), w̄1)Θ(y2, ŵ1, w̄2)

× E(γ(x′,v′)(t),y2)E(y2,x)J (γ(x′,v′)(t),y2,x)m(x,ŵ2) dy2 dt

=

∞∑

l=1

IΘlE,1
[
Ψ2,1,l + Ψ2,2,l

]
(x′,v′)

where

Ψ2,1,l(z) =

∫

M

∆k(z)k(y2)ϕl(z,w̄1(z,y2))Θ(y2, ŵ1(z,y2), w̄2(y2,x))E(z,y2)E(y2,x)

× J (z,y2,x)m(x,ŵ2(y2,x)) dy2,

Ψ2,2,l(z) =

∫

M

k̃(z)∆k(y2)ϕl(z,w̄1(z,y2))Θ(y2, ŵ1(z,y2), w̄2(y2,x))E(z,y2)E(y2,x)

× J (z,y2,x)m(x,ŵ2(y2,x)) dy2.



CUBO
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Optical Tomography for Media ... 93

In a similar manner, with y1 = γ(x′,v′)(t),

∆αj(x,x
′,v′)

=

∞∑

l=1

∫ τ+(x′,v′)

0

E(~γ(x′,v′)(t)Θl(~γ(x′,v′)(t))

×

∫

M

· · ·

∫

M

( j∑

i=1

k̃(y1) · · · k̃(yi−1)∆k(yi)k(yi+1) · · ·k(yj)
)
ϕl(γ(x′,v′)(t), w̄1)

×
( j∏

i=2

Θ(yi, ŵi−1, w̄i)E(yi−1,yi)
)
E(yj,x)J (γ(x′,v′)(t),y2, . . . ,yj,x)

× m(x,ŵj) dyj · · ·dy2 dt

=

∞∑

l=1

j∑

i=1

IΘlE,1Ψj,i,l(x
′,v′).

Explicitly,

Ψj,i,l(y1) =

∫

M

· · ·

∫

M

k̃(y1) · · · k̃(yi−1)∆k(yi)k(yi+1) · · ·k(yj)ϕl(y1, w̄1)

×
( j∏

i=2

Θ(yi, ŵi−1, w̄i)E(yi−1,yi)
)
E(yj,x)J (·) m(x,ŵj) dyj · · ·dy2.

One may estimate ‖Ψj,i,l‖L2(M) (see [LM]):

‖Ψj,i,l‖L2(M) ≤ Cε(ε‖Θ‖∞(diam M)|S
n−1|)j−2‖∆k‖L2(M). (5.25)

Now, using the relation βIΘj E,1 = IΘj E,β, we have

∥∥∥I∗w,ββ
∞∑

j=2

∆αj

∥∥∥
H1(M)

≤

∞∑

j=2

j∑

i=1

∞∑

l=1

‖I∗w,βIΘj E,βΨj,i,l‖H1(M)

≤
∞∑

j=2

j∑

i=1

∞∑

l=1

C

1 + l2
‖Ψj,i,l‖L2(M)

≤

∞∑

j=2

j∑

i=1

∞∑

l=1

C′

1 + l2
ε(ε‖Θ‖∞(diam M)|S

n−1|)j−2‖∆k‖L2(M)

≤ εC′′‖∆k‖L2(M)

∞∑

j=2

j(ε‖Θ‖∞(diam M)|S
n−1|)j−2.

This follows from Proposition 5.3 and Lemma 5.3, with C depending continuously on the C3n

norm of g and Θ, and the C2 norms of σ and m as well. We see that for sufficiently small ε this

series converges (thus justifying the formal computations performed above).

Proof of theorem 5.2. Fix real analytic (m,g,σ, Θ). Given the hypothesis of Theorem 5.2 we are

ensured that both Proposition 5.2 and Proposition 5.4 hold. Combining them, we have the existence



94 Stephen McDowall CUBO
11, 5 (2009)

of a constant C, depending continuously on the C2 norms of (m,σ), and the C3n norms of (Θ,g)

such that

‖∆k‖L2(M) ≤ Cε‖∆k‖L2(M)

and so for 0 ≤ ε < C−1, we must have k = k̃.

Proof of Lemma 5.3. In a fixed coordinate system for M we define a smooth bijection ΩxM → S
n−1

and define Θ̃ ∈ C∞(ΩM × Sn−1)

Θ̃(x,v′,θ) = Θ(x,v′,v(θ)).

For f ∈ Hk, the space of spherical harmonics of order k, the Laplacian ∆S on S
n−1 is ∆Sf =

−k(k+n−2)f. Denote by Zxk (θ) the so-called zonal harmonics for which f(x) = 〈f,Z
x
k (θ)〉L2(Sn−1)

for all f ∈ Hk. Then one has (see, for example, [F]),

dim(Hk) = dk ≤ cn(k
n−2

+ 1)

‖Zxk‖L2(Sn−1) = c
′
n

√
dk for all x ∈ S

n−1.

Let {ψ̃kl}
dk
l=1 ∈ L

∞
(S
n−1

) be an L2(Sn−1) orthonormal basis for Hk (so Z
x
k =

∑dk
l=1 ψ̃kl(θ)ψ̃kl(x)).

Define ψkl := ‖ψ̃kl‖
−1
L∞(Sn−1)

ψ̃kl. Then for each (x,v
′
) ∈ ΩM,

Θ̃(x,v′,θ) =

∞∑

k=0

dk∑

l

Θkl(x,v
′
)ψkl(θ)

with

Θkl(x,v
′
) = ‖ψ̃kl‖L∞(Sn−1)

∫

Sn−1

Θ̃(x,v′,θ)ψ̃kl(θ) dθ.

For each θ,

|ψ̃kl(θ)| = |〈ψ̃kl,Z
θ
k〉| ≤ ‖ψ̃kl‖L2(Sn−1)‖Z

θ
k‖L2(Sn−1) = c

′
n

√
dk.

Next, since ψ̃kl ∈ Hk, for any N ∈ N and k ≥ 1,

|Θkl(x,v
′
)| = ‖ψ̃kl‖L∞(Sn−1)

∣∣∣
∫

Sn−1

Θ̃(x,v′,θ)ψ̃kl(θ) dθ
∣∣∣

= ‖ψ̃kl‖L∞(Sn−1)

∣∣∣
∫

Sn−1

Θ̃(x,v′,θ)
(∆S)

Nψ̃kl(θ)

(−k(k + n − 2))N
dθ

∣∣∣

≤
c′n
√
dk

(k(k + n − 2))N

∣∣∣
∫

Sn−1

ψ̃kl(∆S)
N

Θ̃(x,v′,θ) dθ
∣∣∣

≤
c′′nk

(n−2)/2

(k(k + n − 2))N
‖(∆S)

N
Θ̃(x,v′, ·)‖L2(Sn−1),



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Optical Tomography for Media ... 95

where c′′n depends only on the dimension n. Thus for sufficiently large N (in fact N ≥ 5n/2), there

is cn,N such that

|Θkl(x,v
′
)| ≤

cn,N
1 + k2n

‖(∆S)
N

Θ̃(x,v′, ·)‖L2(Sn−1).

Now renumber the collection of coefficient functions and define new basis functions as follows: with

j = d0 + d1 + · · · + dk−1 + l, set

Θj(x,v
′
) = Θkl(x,v

′
), and ϕj(x,v) := ψkl(θx(v)).

Then

Θ(x,v′,v) =

∞∑

j=1

Θj(x,v
′
)ϕj(x,v).

Now

j ≤

k∑

m=0

dm ≤ cn(k + 1)(k
n−2

+ 1) ≤ ĉnk
n

so

|Θj(x,v
′
)| = |Θkl(x,v

′
)| ≤

cn,N
1 + k2n

‖(∆S)
N

Θ̃(x,v′, ·)‖L2(Sn−1)

≤
c̃n,N

1 + j2
‖(∆S)

N
Θ̃(x,v′, ·)‖L2(Sn−1).

If one applies the same decomposition to ∂αx Θ(x,v
′,v), α ∈ {0, 1, 2}n, one finds that the coefficients

are nothing more than ∂αx Θj(x,v
′
) and these satisfy

|∂αx Θj(x,v
′
)| ≤

c̃n,N
1 + j2

‖(∆S)
N∂αx Θ̃(x,v

′, ·)‖L2(Sn−1) ≤
C

1 + j2
.

Note that C depends on 2N derivatives of Θ̃ in the last variable, and hence 2N derivatives of g

due to the change of variables introduced earlier. This proves the claim of the lemma.

Received: December, 2008. Revised: April, 2009.

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