1. Introduction
Microlocal analysis originated in the 1950s, and by now it is a substantial mathematical theory with many different facets and applications. One might view microlocal analysis as
a kind of “variable coefficient Fourier analysis” for solving variable coefficient PDEs; or
as a theory of pseudodifferential operators (DOs) and Fourier integral operators (FIOs); or
as a phase space (or time-frequency) approach to studying functions, operators and their singularities (wave front sets).
DOs were introduced by Kohn and Nirenberg [
1], and FIOs and wave front sets were studied systematically by Hörmander [
2]. Much of the theory up to the early 1980s is summarized in the four volume treatise of Hörmander [
3]. There are remarkable applications of microlocal analysis and related ideas in many fields of mathematics. Classical examples include spectral theory and the Atiyah-Singer index theorem, and more recent examples include scattering theory [
4], behavior of chaotic systems [
5], general relativity [
6], and inverse problems.
In this note we will describe certain classical applications of microlocal analysis in inverse problems, together with a very rough non-technical overview of relevant parts of microlocal analysis (intended for readers who may not be previously familiar with microlocal analysis). In a nutshell, here are a few typical applications:
- 1.
Computed tomography/X-ray transform—the X-ray transform is an FIO, and under certain conditions its normal operator is an elliptic DO. Microlocal analysis can be used to predict which sharp features (singularities) of the image can be reconstructed in a stable way from limited data measurements. Microlocal analysis is also a powerful tool in the study of geodesic X-ray transforms related to seismic imaging applications.
- 2.
Calderón problem/Electrical Impedance Tomography—the boundary measurement map (Dirichlet-to-Neumann map) is a DO, and the boundary values of the conductivity as well as its derivatives can be computed from the symbol of this DO.
- 3.
Gel’fand problem/seismic imaging—the boundary measurement operator (hyperbolic Dirichlet-to-Neumann map) is an FIO, and the scattering relation of the sound speed as well as certain X-ray transforms of the coefficients can be computed from the canonical relation and the symbol of this FIO.
This note is organized as follows—in
Section 2, we will motivate the theory of
DOs and discuss some of its properties without giving proofs.
Section 3 will continue with a brief introduction to wave front sets and FIOs (again with no proofs). The rest of the note is concerned with applications to inverse problems.
Section 4 considers the Radon transform in
and its normal operator, and describes what kind of information about the singularities of
f can be stably recovered from the Radon transform.
Section 5 and
Section 6 discuss the Gel’fand and Calderón problems, and prove results related to recovering X-ray transforms or boundary determination. The treatment is motivated by
DO and FIO theory, but we give direct and (in principle) elementary proofs based on quasimode constructions.
The results discussed in this note are classical. For more recent results and for further references, we refer to the surveys [
7,
8] on X-ray type transforms, survey [
9] on inverse problems for hyperbolic equations, and survey [
10] on inverse problems for elliptic equations. We also mention the article [
11] that studies inverse problems in rather general settings by using constructions like the ones in
Section 5 and
Section 6.
Notation
We will use multi-index notation. Let
be the set natural numbers. Then
consists of all
n-tuples
where the
are nonnegative integers. Such an
n-tuple
is called a
multi-index. We write
and
for
. For partial derivatives, we will write
If is a bounded domain with boundary, we denote by the set of infinitely differentiable functions in whose all derivatives extend continuously to . The space consists of functions having compact support in . The standard based Sobolev spaces are denoted by with norm , with denoting the Fourier transform. We also write . The notation means that for some uniform (with respect to the relevant parameters) constant C. In general, all coefficients, boundaries and so forth are assumed to be for ease of presentation.
2. Pseudodifferential Operators
In this note we will give a very brief idea of the different points of view to microlocal analysis mentioned in the introduction (and repeated below), as
- (1)
a kind of “variable coefficient Fourier analysis” for solving variable coefficient PDEs; or
- (2)
a theory of DOs and FIOs; or
- (3)
a phase space (or time-frequency) approach to studying functions, operators and their singularities (wave front sets).
In this section we will discuss (1) and (2) in the context of
DOs. We will continue with (2) and (3) in the context of FIOs in
Section 3. The treatment is mostly formal and we will give no proofs whatsoever. A complete reference for the results in this section is ([
3], Section 18.1).
2.1. Constant Coefficient PDEs
We recall the following facts about the Fourier transform (valid for sufficiently nice functions):
If
u is a function in
, its
Fourier transform is the function
The Fourier transform converts derivatives to polynomials (this is why it is useful for solving PDEs):
A function
u can be recovered from
by the Fourier inversion formula
, where
is the
inverse Fourier transform.
As a motivating example, let us solve formally (i.e., without worrying about how to precisely justify each step) the equation
This is a constant coefficient PDE, and such equations can be studied with the help of the Fourier transform. We formally compute
The same formal argument applies to a general constant coefficient PDE
where
. Then
where
is the
symbol of
. Moreover, one has
The argument leading to (
1) gives a formal solution of
:
Thus, formally
can be solved by dividing by the symbol
on the Fourier side. Of course, to make this precise one would need to show that the division by
(which may have zeros) is somehow justified. This can indeed be done, and the basic result in this direction is the Malgrange-Ehrenpreis theorem ([
3], Theorem 7.3.10).
2.2. Variable Coefficient PDEs
We now try to use a similar idea to solve the variable coefficient PDE
where
and
for all multi-indices
. Since the coefficients
depend on
x, Fourier transforming the equation
is not immediately helpful. However, we can compute an analogue of (
2):
where
is the (full)
symbol of
.
Now, we could try to obtain a solution to
in
by dividing by the symbol
as in (
3):
Again, this is only formal since the division by needs to be justified. However, this can be done in a certain sense if A is elliptic:
Definition 1. The principal symbol (i.e., the part containing the highest order derivatives) of the differential operator is We say that A is elliptic if its principal symbol is nonvanishing for .
A basic result of microlocal analysis states that the function
with
where
is a cutoff with
in a sufficiently large neighborhood of
(so that
does not vanish outside this neighborhood), is an
approximate solution of
in the sense that
where
is one derivative smoother than
f. Applying this construction iteratively to the error term—thus to
in the first step above—it is possible to construct an approximate solution
so that
2.3. Pseudodifferential Operators
In analogy with the Formula (
4), a
pseudodifferential operator (
DO) is an operator
A of the form
where
is a
symbol with certain properties. The most standard symbol class
is defined as follows:
Definition 2. The symbol class consists of functions such that for any there is with If , the corresponding Ψ
DO is defined by (7). We denote by the set of Ψ
DOs corresponding to . Note that symbols in
behave roughly like polynomials of order
m in the
-variable. In particular, the symbols
in (
5) belong to
and the corresponding differential operators
belong to
. Moreover, if
is elliptic, then the symbol
as in (
6) belongs to
. Thus the class of
DOs is large enough to include differential operators as well as approximate inverses of elliptic operators. Also normal operators of the X-ray transform or Radon transform in
are
DOs (see
Section 4 and ([
12], Appendix A)).
Remark 1 (Homogeneous symbols)
. We saw in Section 2.1 that the elliptic operator has the inverseThe symbol is not in , since it is not smooth near 0. However, G is still considered to be a Ψ
DO. In fact, one can writewhere satisfies near 0. Now is a Ψ
DO in , since , and is smoothing in the sense that it maps any function into a function (at least if ). In general, in Ψ
DO theory smoothing operators are considered to be negligible (since at least they do not introduce new singularities), and many computations in Ψ
DO calculus are made only modulo smoothing error terms. In this sense one often views G as a Ψ
DO by identifying it with . The same kind of identification is done for operators whose symbol is homogeneous of some order m in ξ, i.e., for . More generally one can consider(step one) polyhomogeneous
symbols having the formwhere each is homogeneous of order in ξ for , and ∼ denotes asymptotic summation meaning that for any . Corresponding Ψ
DOs are called classical
DOs.
It is very important that one can compute with DOs in much the same way as with differential operators. One often says that DOs have a calculus, and in fact the DOs defined above form an algebra with respect to composition. The following theorem lists typical rules of computation (it is instructive to think first why such rules are valid for differential operators):
Theorem 1 (DO calculus).
- (a)
(Principal symbol) There is a one-to-one correspondence between operators in and (full) symbols in , and each operator has a well defined principal symbol . The principal symbol may be computed by testing A against highly oscillatory functions (this is valid if A is a classical Ψ
DO): - (b)
(Composition) If and , then and ;
- (c)
(Sobolev mapping properties) Each is a bounded operator for any ;
- (d)
(Elliptic operators have approximate inverses) If is elliptic, there is so that and where , i.e., are smoothing (they map any function to for any t, hence also to by Sobolev embedding).
The above properties are valid in the standard DO calculus in . However, motivated by different applications, DOs have been considered in various other settings. Each of these settings comes with an associated calculus whose rules of computation are similar but adapted to the situation at hand (for instance, one may need extra conditions for compositions to be well defined). Examples of different settings for DOs include
open sets in
(local setting) ([
3], Section 18.1);
compact manifolds without boundary, possibly acting on sections of vector bundles ([
3], Section 18.1);
compact manifolds with boundary (transmission condition/Boutet de Monvel calculus) [
13];
non-compact manifolds (e.g., Melrose scattering calculus) [
14];
operators with a small or large parameter (semiclassical calculus) [
15]; and
operators with real-analytic coefficients (analytic microlocal analysis) [
16,
17].
3. Wave Front Sets and Fourier Integral Operators
For a reference to wave front sets, see Reference ([
3], Chapter 8). Sobolev wave front sets are considered in Reference ([
3], Section 18.1). FIOs are discussed in Reference ([
3], Chapter 25). We mention that FIO type methods were independently developed by Maslov [
18].
3.1. The Role of Singularities
We first discuss the singular support of u, which consists of those points such that u is not a smooth function in any neighborhood of . We also consider the Sobolev singular support, which also measures the “strength” of the singularity (in the Sobolev scale).
Definition 3 (Singular support)
. We say that a function or distribution u is (resp.
) near
if there is with near such that is in (resp. in ). We define Example 1. Let be bounded domains with boundary in so that for , and definewhere are constants, and is the characteristic function of . Thensince for , butsince u is not near any boundary point. Thus in this case the singularities of u are exactly at the points where u has a jump discontinuity, and their strength is precisely . Knowing the singularities of u can already be useful in applications. For instance, if u represents some internal medium properties in medical imaging, the singularities of u could determine the location of interfaces between different tissues. On the other hand, if u represents an image, then the singularities in some sense determine the “sharp features” of the image. Next we discuss the
wave front set which is a more refined notion of a singularity. For example, if
is the characteristic function of a bounded strictly convex
domain
D and if
, one could think that
f is in some sense smooth in tangential directions at
(since
f restricted to a tangent hyperplane is identically zero, except possibly at
), but that
f is not smooth in normal directions at
since in these directions there is a jump. The wave front set is a subset of
, the cotangent space with the zero section removed:
Definition 4 (Wave front set)
. Let u be a distribution in . We say that u is (microlocally) (resp.
) near
if there exist with near and so that near and ψ is homogeneous of degree 0, such that(resp. ). The wave front set (resp. wave front set ) consists of those points where u is not microlocally (resp. ). Example 2. The wave front set of the function u in Example 1 iswhere is the conormal bundle of , The wave front set describes singularities more precisely than the singular support, since one always has
where
is the projection to
x-space.
It is an important fact that applying a DO to a function or distribution never creates new singularities:
Theorem 2 (Pseudolocal/microlocal property of
DOs)
. Any has the pseudolocal propertyand the microlocal property Elliptic operators are those that completely preserve singularities:
Theorem 3 (Elliptic regularity)
. Let be elliptic. Then, for any u,Thus any solution u of is singular precisely at those points where f is singular. There are corresponding statements for Sobolev singularities.
Proof. First note that by Theorem 2,
Conversely, since
is elliptic, by Theorem 1(d) there is
so that
Since
L is smoothing,
, which implies that
modulo
. Thus it follows that
Thus
. The claim for singular supports follows by (
9). □
3.2. Fourier Integral Operators
We have seen in
Section 2.3 that the class of pseudodifferential operators includes approximate inverses of elliptic operators. In order to handle approximate inverses for hyperbolic and transport equations, it is required to work with a larger class of operators.
Motivation 1. Consider the initial value problem for the wave equation, This is again a constant coefficient PDE, and we will solve this formally by taking the Fourier transform in space, After taking Fourier transforms in space, the above equation becomes For each fixed this is an ODE in t, and the solution is Taking inverse Fourier transforms in space, we obtain Generalizing (
10), we can consider operators of the form
where
is a symbol (for instance in
), and
is a real valued phase function. Such operators are examples of
Fourier integral operators (more precisely, FIOs whose canonical relation is locally the graph of a canonical transformation, see ([
3], Section 25.3)). For
DOs the phase function is always
, but for FIOs the phase function can be quite general, though it is usually required to be homogeneous of degree 1 in
, and to satisfy the non-degeneracy condition
.
We will not go into precise definitions, but only remark that the class of FIOs includes pseudodifferential operators as well as approximate inverses of hyperbolic and transport operators (or more generally real principal type operators). There is a calculus for FIOs, analogous to the pseudodifferential calculus, under certain conditions in various settings. An important property of FIOs is that they, unlike pseudodifferential operators, can move singularities. This aspect will be discussed next.
3.3. Propagation of Singularities
Example 3. Let be fixed, and consider the operators from (10), Using FIO theory, since the phase functions are , it follows thatwhere is the canonical transformation
(i.e., diffeomorphism of that preserves the symplectic structure) given by This means that the FIO takes a singularity of the initial data f and moves it along the line through x in direction to . In fact one has equality in (12) since has inverse and has inverse . Thus singularities of solutions of the wave equation propagate along straight lines with constant speed one. Remark 2. In general, any FIO has an associated canonical relation
that describes what the FIO does to singularities. The canonical relation of the FIO A defined in (11) is (see ([3], Section 25.3))and A moves singularities according to the rulewhere Using these formulas, it is easy to check that the canonical relation of in Example 3 is the graph of in the sense thatand one indeed has . There is a far reaching extension of Example 3, which shows that the singularities of a solution of propagate along certain curves in phase space (so called null bicharacteristic curves) as long as P has real valued principal symbol.
Theorem 4 (Propagation of singularities, ([
3], Theorem 26.1.1))
. Let have real principal symbol that is homogeneous of degree m in ξ. Ifthen is contained in the characteristic set . Moreover, if , then the whole null bicharacteristic curve through is in as long as it remains in . Here is the solution of the Hamilton equations Example 4. We compute the null bicharacteristic curves for the wave operator . The principal symbol of P is The characteristic set iswhich consists of light-like
cotangent vectors on . The equations for the null bicharacteristic curves are Thus, if , then the null bicharacteristic curve through is The result of Example 3 may thus be interpreted so that singularities of solutions of the wave equation propagate along null bicharacteristic curves for the wave operator.
4. The Radon Transform in the Plane
In this section we outline some applications of microlocal analysis to the study of the Radon transform in the plane. Similar ideas apply to X-ray and Radon transforms in higher dimensions and Riemannian manifolds as well. The microlocal approach to Radon transforms was introduced by Guillemin [
19]. We refer to [
8,
12] and references therein for a more detailed treatment of the material in this section.
4.1. Basic Properties of the Radon Transform
The X-ray transform of a function f in encodes the integrals of f over all straight lines, whereas the Radon transform encodes the integrals of f over -dimensional planes. We will focus on the case , where the two transforms coincide. There are many ways to parametrize the set of lines in . We will parametrize lines by their direction vector and distance s from the origin.
Definition 5. If , the Radon transform
of f is the function Here is the vector in obtained by rotating ω counterclockwise by .
There is a well-known relation between and the Fourier transform . We denote by the Fourier transform of with respect to s.
Proof. Parametrizing
by
, we have
This result gives the first proof of injectivity of the Radon transform:
Corollary 1. If is such that , then .
Proof. If , then by Theorem 5 and consequently . □
To obtain a different inversion method, and for later purposes, we will consider the adjoint of
R. The formal adjoint of
R is the
backprojection operator . The formula for
is obtained as follows: if
,
one has
The following result shows that the normal operator is a classical DO of order in , and also gives an inversion formula.
Theorem 6 (Normal operator)
. One hasand f can be recovered from by the formula Remark 3. Above we have written, for , The notation is also used.
Proof. The proof is based on computing
using the Parseval identity, Fourier slice theorem, symmetry and polar coordinates:
The same argument, based on computing
instead of
, leads to the famous
filtered backprojection (FBP) inversion formula:
where
. This formula is efficient to implement and gives good reconstructions when one has complete X-ray data and relatively small noise, and hence FBP (together with its variants) has been commonly used in X-ray CT scanners.
However, if one is mainly interested in the singularities (i.e., jumps or sharp features) of the image, it is possible to use the even simpler
backprojection method: just apply the backprojection operator
to the data
. Since
is an elliptic
DO, Theorem 3 guarantees that the singularities are recovered:
Moreover, since
is a
DO of order
, hence smoothing of order 1, one expects that
gives a slightly blurred version of
f where the main singularities should still be visible. The ellipticity of the normal operator is also important in the analysis of statistical methods for recovering
f from
[
20].
4.2. Visible Singularities
There are various imaging situations where complete X-ray data (i.e., the function for all s and ) is not available. This is the case for limited angle tomography (e.g., in luggage scanners at airports, or dental applications), region of interest tomography, or exterior data tomography. In such cases explicit inversion formulas such as FBP are usually not available, but microlocal analysis (for related normal operators or FIOs) still provides a powerful paradigm for predicting which singularities can be recovered stably from the measurements.
We will try to explain this paradigm a little bit more, starting with an example:
Example 5. Let f be the characteristic function of the unit disc , i.e., if and for . Then f is singular precisely on the unit circle (in normal directions). We have Thus is singular precisely at those points with , which correspond to those lines that are tangent to the unit circle.
There is a similar relation between the singularities of f and in general, and this is explained by microlocal analysis:
Theorem 7. The operator R is an elliptic FIO of order . There is a precise relationship between the singularities of f and singularities of .
We will not spell out the precise relationship here, but only give some consequences. It will be useful to think of the Radon transform as defined on the set of (non-oriented) lines in . If is an open subset of lines in , we consider the Radon transform restricted to lines in . Recovering f (or some properties of f) from is a limited data tomography problem. Examples:
If , then is called exterior data.
If and , then is called limited angle data.
It is known that any
is uniquely determined by exterior data (Helgason support theorem ([
21], Theorem 2.6)), and any
is uniquely determined by limited angle data (Fourier slice and Paley-Wiener theorems). However, both inverse problems are very unstable (inversion is not Lipschitz continuous in any Sobolev norms, but one has conditional logarithmic stability).
Definition 6. A singularity at is called visible from if the line through in direction is in .
One has the following dichotomy:
If is visible from , then from the singularities of one can determine for any whether or not . If uniquely determines f, one expects the reconstruction of visible singularities to be stable.
If is not visible from , then this singularity is smoothed out in the measurement . Even if would determine f uniquely, the inversion is not Lipschitz stable in any Sobolev norms.
5. Gel’fand Problem
Seismic imaging gives rise to various inverse problems related to determining interior properties, for example, oil deposits or deep structure, of the Earth. Often this is done by using acoustic or elastic waves. We will consider the following problem, also known as the
inverse boundary spectral problem (see the monograph [
22]):
Gel’fand problem: Is it possible to determine the interior structure of Earth by controlling acoustic waves and measuring vibrations at the surface?
In seismic imaging one often tries to recover an unknown sound speed. However, in this presentation we consider the simpler case where the sound speed is constant (equal to one) and one attempts to recover an unknown potential at each point , where is a ball in .
Consider the free wave operator
We assume that the medium is at rest at time
and that we take measurements until time
. If we prescribe the amplitude of the wave to be
on
, this leads to a solution
u of the wave equation
Given any
, there is a unique solution
(see ([
23], Theorem 7 in §7.2.3)). We assume that we can measure the normal derivative
, where
and
is the outer unit normal to
. Doing such measurements for many different functions
f, the ideal boundary measurements are encoded by the hyperbolic Dirichlet-to-Neumann map (DN map for short)
The Gel’fand problem for this model amounts to recovering
from the knowledge of the map
. We will prove the following result due to [
24].
Theorem 8 (Recovering the X-ray transform)
. Let and assume that . Ifthen and satisfywhenever γ is a maximal line segment in with length . It is natural that the region where one can recover information depends on
T. By finite propagation speed the map
is unaffected if one changes
q outside the set
Indeed, if
u and
solve (
13) for potentials
q and
with the same Dirichlet data
f, and if
in
, then
solves
where
vanishes in
and in
. Moreover,
on
and
. By finite speed of propagation
. This proves that
.
For T large enough, one can recover everything:
Corollary 2. If , then implies .
Proof. If
, then by Theorem 8 one has
for any maximal line segment
in
. Thus
and
have the same X-ray transform in
. This transform is injective by Corollary 1 when
. Tiling
by two-planes gives injectivity when
. Thus
. □
Theorem 8 could be proved based on the following facts, see e.g., [
25]:
The map is an FIO of order 1 on .
The X-ray transform of q can be read off from the symbol of (more precisely, from the principal symbol of ).
We will give an elementary proof that is based on testing
against highly oscillatory boundary data (compare with (
8)).
The first step is an integral identity.
Lemma 1 (Integral identity)
. Assume that . For any , one haswhere solves (13) with and , and solves an analogous problem with vanishing Cauchy data on : Proof. We first compute the adjoint of the DN map: one has
where
with
v solving
so that
and
on
. To prove this, we let
u be the solution of (
13) and integrate by parts:
Now, if
and
are as stated, the computation above gives
and
The result follows by subtracting these two identities. □
The second step is to construct special solutions to the wave equation that concentrate near curves
where
is a line segment. These curves are projections to the
variables of null bicharacteristic curves for □ (see Example 4). These solutions are closely related to Theorem 4 concerning propagation of singularities. In fact, similar methods can be used to prove that Theorem 4 is sharp in the sense that there are approximate solutions whose wave front set is precisely on a given null bicharacteristic curve ([
3], Theorem 26.1.5). One can also go in the other direction and use suitable concentrating solutions to prove Theorem 4, see Reference [
26].
The proof is based on a standard geometrical optics/WKB quasimode construction.
Proposition 1 (Concentrating solutions)
. Assume that , and let be a maximal line segment in with . For any there is a solution of in with on , such that for any one hasMoreover, if , there is a solution of in with on , such that for any one has At this point it is easy to prove the main result:
Proof (Proof of Theorem 8). (Proof of Theorem 8)
. Using the assumption
and Lemma 1, we have
for any solutions
of
in
so that
on
, and
on
.
Let
be a maximal unit speed line segment in
with
, and let
be the solution constructed in Proposition 1 for the potential
with
on
. Moreover, let
be the solution constructed in the end of Proposition 1 for the potential
with
on
. Taking the limit as
in (17) and using (16) with
, we obtain that
Thus the integrals of and over maximal line segments of length in are the same. □
Proof of Proposition 1. Let be a maximal unit speed line segment in with , and let be the unit speed line so that for . Write and , so that and . After a translation and rotation, we may assume that and .
We first construct an approximate solution
for the operator
, having the form
where
is a real phase function, and
a is an amplitude supported near the curve
. Note that
Using a similar expression for
, we compute
We would like to have
. To this end, we first choose
so that the
term in (
18) vanishes. This will be true if
solves the
eikonal equationThere are many possible solutions, but we make the simple choice
With this choice, (
18) becomes
where
L is the constant vector field
It is convenient to consider new coordinates
in
, where
Then
L corresponds to
in the sense that
where
corresponds to
F in the new coordinates:
We next look for the amplitude
a in the form
Inserting this to (
18) and equating like powers of
, we get
We would like the last expression to be
. This will hold if
and
satisfy the
transport equationsLet
be supported near 0, and choose
We will later choose
to depend on
. Next we choose
These functions satisfy (22), and they vanish unless
w is small (i.e.,
is close to
t). Then (21) becomes
where
Using the Cauchy-Schwarz inequality, one can check that
uniformly over
. This concludes the construction of the approximate solution
.
We next find an exact solution
of (
13) having the form
where
r is a correction term. Note that for
t close to 0,
is supported near
and hence
on
. Note also that
. Thus
u will solve (
13) for
if
r solves
By the wellposedness of this problem ([
23], Theorem 5 in §7.2.3), there is a unique solution
r with
We now fix the choice of
so that (15) will hold. Let
satisfy
near 0 and
, and choose
where
Since
, the integral in (
15) has the form
Using that
is compactly supported in
, we have
by changing variables as in (20). Finally, changing
to
and
w to
and letting
(so
) yields
by the normalization
and the fact that
. This proves (15).
It remains to prove (16). Since , we have on , and we may alternatively arrange that r solves (23) with on instead of . We can do such a construction for the potential instead of q. Since and are independent of q, the same argument as above proves (16). □
6. Calderón Problem: Boundary Determination
Electrical Impedance Tomography (EIT) is an imaging method with potential applications in medical imaging and nondestructive testing. The method is based on the following important inverse problem.
Calderón problem: Is it possible to determine the electrical conductivity of a medium by making voltage and current measurements on its boundary?
The treatment in this section follows [
27].
Let us begin by recalling the mathematical model of EIT. The purpose is to determine the electrical conductivity at each point , where represents the body which is imaged (in practice ). We assume that is a bounded open set with boundary, and that is positive.
Under the assumption of no sources or sinks of current in
, a voltage potential
f at the boundary
induces a voltage potential
u in
, which solves the Dirichlet problem for the conductivity equation,
Since
is positive, the equation is uniformly elliptic, and there is a unique solution
for any boundary value
. One can define the Dirichlet-to-Neumann map (DN map) as
Here is the outer unit normal to and is the normal derivative of u. Physically, is the current flowing through the boundary.
The Calderón problem (also called the inverse conductivity problem) is to determine the conductivity function from the knowledge of the map . That is, if the measured current is known for all boundary voltages , one would like to determine the conductivity .
We will prove the following theorem.
Theorem 9 (Boundary determination)
. Let be positive. Ifthen the Taylor series of and coincide at any point of . This result was proved in Reference [
28], and it in particular implies that any real-analytic conductivity is uniquely determined by the DN map. The argument extends to piecewise real-analytic conductivities. A different proof was given in [
29], based on two facts:
Remark 4. The above argument is based on studying the singularities of the integral kernel of the DN map, and it only determines the Taylor series of the conductivity at the boundary. The values of the conductivity in the interior are encoded in the part of the kernel, and different methods (based on complex geometrical optics solutions) are required for interior determination.
Let us start with a simple example:
Example 6 (DN map in half space is a
DO)
. Let , so . We wish to compute the DN map for the Laplace equation (i.e., ) in Ω.
ConsiderWriting and taking Fourier transforms in gives Solving this ODE for fixed and choosing the solution that decays for gives We may now compute the DN map: Thus the DN map on the boundary is just corresponding to the Fourier multiplier . This shows that at least in this simple case, the DN map is an elliptic ΨDO of order 1.
We will now prove Theorem 9 by an argument that avoids showing that the DN map is a DO, but is rather based on directly testing the DN map against oscillatory boundary data. The first step is a basic integral identity (sometimes called Alessandrini identity) for the DN map.
Lemma 2 (Integral identity)
. Let . If , thenwhere solves in Ω with . Proof. We first observe that the DN map is symmetric: if
is positive and if
solves
in
with
, then an integration by parts shows that
The result follows by subtracting the above two identities. □
Next we show that if
is a boundary point, there is an approximate solution of the conductivity equation that concentrates near
, has highly oscillatory boundary data, and decays exponentially in the interior. As a simple example, the solution of
that decays for
is given by
, which concentrates near
and decays exponentially when
if
is large. Roughly, this means that the solution of a Laplace type equation with highly oscillatory boundary data concentrates near the boundary. Note also that in a region like
, the function
u is harmonic and concentrates near the origin.
Proposition 2. (Concentrating approximate solutions) Let be positive, let , let be a unit tangent vector to at , and let be supported near . Let also . For any there exists having the formsuch thatand as Moreover, if is positive and is the corresponding approximate solution constructed for , then for any and one hasfor some . We can now give the proof of the boundary determination result.
Proof of Theorem 9. Using the assumption that
together with the integral identity in Lemma 2, we have that
whenever
solves
in
.
Let
, let
be a unit tangent vector to
at
, and let
satisfy
near
. We use Proposition 2 to construct functions
so that
We obtain exact solutions
of
by setting
where the correction terms
are the unique solutions of
By standard energy estimates ([
23], Section 6.2) and by (27), the solutions
satisfy
We now insert the solutions
into (26). Using (28) and (27), it follows that
as
. Letting
, the formula (25) yields
In particular, .
We will prove by induction that
The case
was proved above (here we may vary
slightly). We make the induction hypothesis that (30) holds for
. Let
be boundary normal coordinates so that
corresponds to 0, and
near
corresponds to
. The induction hypothesis states that
Considering the Taylor expansion of
with respect to
gives that
for some smooth function
f with
. Inserting this formula in (29), we obtain that
Now
in boundary normal coordinates. Assuming that
N was chosen larger than
k, we may take the limit as
and use (25) to obtain that
This shows that for near 0, which concludes the induction. □
It remains to prove Proposition 2, which constructs approximate solutions (also called quasimodes) concentrating near a boundary point. This is a typical geometrical optics/WKB type construction for quasimodes with complex phase. The proof is elementary, although a bit long. The argument is simplified slightly by using the Borel summation lemma, which is used frequently in microlocal analysis in various different forms.
Lemma 3 (Borel summation, ([
3] Theorem 1.2.6))
. Let for . There exists such that Proof of Proposition 2. We will first carry out the proof in the case where and is flat near 0, i.e., for some (the general case will be considered in the end of the proof). We also assume where .
We look for
v in the form
Write
. The principal symbol of
P is
Since
, we compute
We want to choose
and
b so that
. Looking at the
term in (32), we first choose
so that
We additionally want that
and
(this will imply that
). In fact, using (31) we can just choose
and then
in
.
We next look for
b in the form
Since
, (32) implies that
We will choose the functions
so that
We will additionally arrange that
and that each
is compactly supported so that
for some fixed
.
To find
, we prescribe
,
… successively and use the Borel summation lemma to construct
with this Taylor series at
. We first set
. Writing
, we observe that
Thus, in order to have
we must have
We prescribe
to have the above value (which depends on the already prescribed quantity
). Next we compute
where
Q depends on the already prescribed quantities
and
. We thus set
which ensures that
. Continuing in this way and using Borel summation, we obtain a function
so that
to infinite order at
. The other equations in (35) are solved in a similar way, which gives the required functions
. In the construction, we may arrange so that (36) and (37) are valid.
If
and
are chosen in the above way, then (34) implies that
where each
vanishes to infinite order at
and is compactly supported in
. Thus, for any
there is
so that
in
, and consequently
Since
in
we have
Choosing
and computing the integrals over
, we get that
It is also easy to compute that
Thus, choosing , we have proved all the claims except (25).
To show (25), we observe that
Using a similar formula for
(where
is independent of the conductivity), we have
Now
and
where
, and similarly for
. Hence
We can change variables
and use dominated convergence to take the limit as
. The limit is
where
.
The proof is complete in the case when
and
is flat near 0. In the general case, we choose boundary normal coordinates
so that
corresponds to 0 and
near
locally corresponds to
. The equation
in the new coordinates becomes an equation
where
A is a smooth positive matrix only depending on the geometry of
near
. The construction of
v now proceeds in a similar way as above, except that the equation (33) for the phase function
can only be solved to infinite order on
instead of solving it globally in
. □