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circular Radon transform, thermoacoustic tomography, injectivity
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\begin{document}
\newtheorem{theorem}{Theorem}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
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\author{Gaik Ambartsoumian and Peter Kuchment\\
Mathematics Department\\
Texas A \& M University\\
College Station, TX 77843-3368\\
kuchment@math.tamu.edu, haik@tamu.edu}
\title{On the injectivity of the circular Radon transform arising in thermoacoustic tomography}
\date{}
\maketitle
\begin{abstract}
The circular Radon transform integrates a function over the set of
all spheres with a given set of centers. The problem of
injectivity of this transform (as well as inversion formulas,
range descriptions, etc.) arises in many fields from approximation
theory to integral geometry, to inverse problems for PDEs, and
recently to newly developing types of tomography. A major
breakthrough in the $2D$ case was made several years ago in a work
by M.~Agranovsky and E.~T.~Quinto. Their techniques involved
intricate microlocal analysis and knowledge of geometry of zeros
of harmonic polynomials in the plane, which are somewhat
restrictive in more general circumstances. Since then there has
been an active search for alternative methods, especially the ones
based on simple PDE techniques. The article discusses known and
provides new results that one can obtain by methods that
essentially involve only the finite speed of propagation and
domain dependence for the wave equation.
\end{abstract}

\section{Introduction}
Most tomographic methods of medical imaging (as well as industrial
non-destructive evaluation, geological imaging, sonar, and radar)
are based on the following procedure: one sends towards a
non-transparent body some kind of a signal (acoustic or
electromagnetic wave, X-ray or visual light photons, etc.) and
measures the wave after it passes through the body. Then the
problem becomes to use the measured information to recover the
internal structure of the object of study. The common feature of
all traditional methods of tomography is that the same kinds of
physical signals are sent and are measured. Although the
development of tomography in the last half of the century has
brought many remarkable successes \cite{Natt4,Natt2001}, each of
the methods has its own shortfalls. For instance, when imaging
biological tissues, microwaves and optical imaging often provide
good contrasts between different types of tissues, but are
inferior in terms of resolution in comparison with ultrasound or
X-rays. This, in particular, is responsible for practical
impossibility of getting any good resolution in optical or
electrical impedance tomography, unless one wants to image only
skin-deep areas. On the other hand, ultrasound, while giving good
resolution, does not do a good job in terms of contrast. It is
amazing then that the idea of combining different types of
radiation for triggering the signal and for the measured signal
had to wait for such a long time to appear. By now, thermoacoustic
tomography (TAT or TCT) and its sibling photoacoustic tomography
(PAT) have already made significant advances (e.g.,
\cite{Kruger},\cite{MXW}-\cite{XWAK}), while some others are still
in a development stage. Since PAT in terms of the relevant
mathematics is identical to TAT, we will describe briefly only the
latter. In TAT, a short microwave or radiofrequency
electromagnetic pulse is sent through the biological object. At
each internal location $x$ certain energy $H(x)$ will be absorbed.
It is known (see the references above), that cancerous cells
absorb several times more MW (or RF) energy than the normal ones,
which means that significant increases of the values of $H(x)$ are
expected at tumorous locations. It is believed that this contrast
is due to the increased water and sodium content in tumors, which
is partly due to extra blood vessel growth there. The absorbed
energy, due to resulting heating, causes a thermoelastic
expansion, which in turn creates a pressure wave. This wave can be
detected by ultrasound transducers placed at the edges of the
object. Since the pulses delivered are very short, the thermal
diffusion during the experiment can be neglected. Now the former
weakness of ultrasound (low contrast) becomes an advantage.
Indeed, in many cases (e.g., for mammography) one can assume the
sound speed to be constant. Hence, the sound waves detected at any
moment $t$ of time are coming from the locations at a constant
distance (depending on time and sound speed) from the transducer.
The strength of the signal coming from a location $x$ reflects the
energy absorption $H(x)$. Thus, one effectively measures the
integrals of $H(x)$ over all spheres centered at the transducers'
locations. In other words, one needs to invert a generalized Radon
transform of $H$ (``generalized,'' since integration is done over
spheres). Exact implementation of this idea involves simple
handling of the wave equation
(\cite{Kruger},\cite{MXW}-\cite{XWAK}).

This method amazingly combines advantages of two types of
radiation used (contrast for microwaves and resolution for
ultrasound), while avoiding their deficiencies.

It is clear from the dimension considerations that it should be
sufficient to run the transducers along a curve in the case of a
$2D$ problem or a surface in $3D$. The most popular geometries of
these surfaces (curves) that have been implemented are spheres,
planes, and cylinders \cite{MXW}-\cite{YXW2}.

Let us mention some of the central problems that arise in these
studies:
\begin{itemize}
\item Uniqueness of reconstruction: is the information collected
sufficient for the unique determination of the energy deposition
function $H$?
\item Reconstruction formulas and algorithms, and stability of the reconstruction.
\item Description of the range of the transform: what conditions
should ideal data satisfy?
\item Incomplete data problems: what happens to the reconstruction
if only a part of transducers' locations can be (or are) used?
\end{itemize}
All these questions have been essentially answered for the
classical Radon transform that arises in X-ray CT, Positron
Emission Tomography (PET), and Magnetic Resonance Imaging (MRI)
\cite{Natt4,Natt2001}. However, they are much more complex and not
that well understood for the circular Radon transform that arises
in TAT.

The aim of this paper is to address the uniqueness problem. We
will survey the known results and describe some new approaches
developed originally in \cite{FPR} that significantly clarify this
question. Then we show how one can make additional progress in
this direction.

The results of this paper were presented at the special sessions
on tomography at the AMS Meetings in Binghamton, NY in October
2003 and in Lawrenceville, NJ in April 2004 and at the Inverse
problems workshop at IPAM in November 2003.

The next section contains the mathematical formulation of the
problem, its history, and the description of the beautiful
uniqueness result obtained by M.~Agranovsky and E.~T.~Quinto in
\cite{AQ}. Alternative PDE methods developed in \cite{ABK,FPR} are
also presented. The following section contains the main results of
this paper. It is followed by sections containing further remarks
and acknowledgements.


\section{Formulation of the problem and known results}

The discussion of the previous section motivates the study of the
following ``circular'' Radon transform. Let $f(x)$ be a continuous
function on $\Bbb{R}^n$, $n\ge 2$.

\begin{definition}\label{D:circular}The circular Radon transform of $f$ is defined
as
$$
Rf(p,r)=\int_{|y-p|=r}f(y)d\sigma(y),
$$
where $d\sigma(y)$ is the surface area on the sphere $|y-p|=r$
centered at $p \in \Bbb{R}^n$.
\end{definition}

In this definition we do not restrict the set of centers $p$ or
radii $r$. It is clear, however, that this mapping is
overdetermined, since the dimension of pairs $(p,r)$ is $n+1$,
while the function $f$ depends on $n$ variables only. This (as
well as the tomographic motivation of the previous section)
suggests to restrict the set of centers to a set (hypersurface) $S
\subset \Bbb{R}^n$, while not imposing any restrictions on the
radii. This restricted transform will be denoted by $R_S$:
$$
R_Sf(p,r)=Rf(p,r)|_{p \in S}.
$$

\begin{definition}
The transform $R$ is said to be injective on a set $S$ ($S$ is a
{\bf set of injectivity}) if for any $f\in C_c(R^n)$ the condition
$Rf(p,r)=0$  for all $r\in \Bbb{R}$ and all $p\in S$ implies
$f\equiv0$.

In other words, $S$ is a set of injectivity, if the mapping $R_S$
is injective on $C_c(\Bbb{R}^n)$.
\end{definition}

One can wonder why we impose the condition of compactness of
support on $f$, since it does not seem to be all that natural for
the circular Radon transform . The answer is that the situation
can be significantly different without compactness of support (or
at least some decay) condition \cite{ABK,AQ}. Besides, the problem
becomes significantly harder in this case. Fortunately,
tomographic problems usually yield compactly supported functions
(such as the energy deposition function $H$ of the previous
section).

One now arrives to the

\begin{problem}\label{P:injec}
Describe all sets of injectivity for the circular Radon transform
$R$ on $C_c(\Bbb{R}^n)$.
\end{problem}

In other words, we are looking for a description of those sets of
positions of transducers that enable one to recover uniquely the
energy deposition function.

This problem has been around in different guises for quite a
while. For instance, it is formulated (in a more general setting
of integration over level sets of polynomials) in the recent book
\cite{Leon} by L.~Ehrenpreis, preliminary drafts of which have
been circulating among experts for years. The same question in its
dual form was posed in terms of approximation theory and studied
by V.~Lin and A.~Pincus \cite{LP1,LP2}:
\begin{problem}\label{P:LinPinc}
Describe sets $\Gamma\subset \Bbb{R}^n$, $n\ge2$, such that the
system of shifted radial functions (spherical waves)
$$
\psi(|x-a|),\;a\in\Gamma,\;\; \psi\; is\; a\; function\; of\;
one\; variable,
$$
is complete in $C(\Bbb{R}^n)$ in the topology of uniform
convergence on compacta.
\end{problem}

As it will be clear a little bit later, this problem is equivalent
to the description of possible nodal sets of oscillating infinite
membranes. The paper \cite{AQ} contains a survey of some other
problems that lead to the injectivity question for $R_S$.

There might have been other sources of this problem, unknown to
the authors.

As it turns out, the injectivity sets $S$ are more common than the
non-injectivity ones. This means that one should aim for a
description of those ``bad'' non-injectivity sets (i.e. sets of
transducers' positions from which one cannot recover the energy
deposition function).

The first important observations concerning non-injectivity sets
were made by V.~Lin and A.~Pincus \cite{LP1,LP2}. We will present
these following the notations and formulations of \cite{AQ}.

First of all, convolution with radial mollifiers easily shows that
one can assume that the function $f$ to which the transform is
applied is arbitrarily smooth (e.g., \cite{AQ}). So, we will not
bother with smoothness conditions, assuming that the function is
at least continuous.

Let us associate with each $f\in C(\Bbb{R}^n)$ that decays at
infinity faster than any power of $|x|$ the set
$$
S[f]=\{x\in \Bbb{R}^n|\;Rf(x,r)=0\;\forall r\in \Bbb{R}_+\}.
$$
We also introduce the infinite family of polynomials $Q_k$ of
degree $deg Q_k\le2k$:
$$
Q_k(x)=Q_k[f](x)=r^{2k}*f=\int_{\Bbb{R}^n}\|x-\xi\|^{2k}f(\xi)d\xi,
\;\;r^2=x_1^2+\dots+x_n^2.$$
For any polynomial $Q$ with real
coefficients, we denote by $V[Q]$ the real algebraic variety
$$
V[Q]=\{x\in \Bbb{R}^n|\;\; Q(x)=0\}.
$$

\begin{lemma}\label{L:S(f)} (V.~Lin and A.~Pincus \cite{LP1,LP2})
$S[f]=\bigcap_{k=0}^\infty V[Q_k]$.
\end{lemma}
\begin{lemma}\label{L:harm}
Let $f\in C_c(\Bbb{R}^n)$. Then $f\equiv0$ if and only if
$Q_k[f]\equiv0$ for all $k=0,1,\dots$ . If $f$ is not identically
zero, and $P=Q_{k_{0}}[f]$ is the minimal degree nontrivial
polynomial among $Q_k$, then $P$ is harmonic.
\end{lemma}
The harmonicity in Lemma \ref{L:harm} was discovered by N.~Zobin
\cite{Zob}.

These statements imply in particular that if $R$ is not injective
on $S$, then $S$ is the zero set of a harmonic polynomial.
Therefore we get a sufficient condition for injectivity:

\begin{corollary}\label{C:harmonic_zero}
Any set $S\subset \Bbb{R}^n$ of uniqueness for the harmonic
polynomials is a set of injectivity for the transform $R$.
\end{corollary}
In particular, this implies
\begin{corollary}\label{C:boundary} If $U \subset \Bbb{R}^n$ is
any bounded domain, then $S=\partial U$ is a injectivity set of
$R$.
\end{corollary}
We will see later a different proof of this fact that does not use
harmonicity.

So, what are possible non-injectivity sets? It is rather obvious
that any hyperplane $S$ is such a set. Indeed, for any function
$f$ that is odd with respect to $S$ one obviously gets $R_S f
\equiv 0$. There are other options as well. In order to describe
them in $2D$, let us first introduce the following definition.
\begin{definition}\label{D:Coxeter}
For any $N\in \Bbb{N}$ denote by $\Sigma_N$ the Coxeter system of
$N$ lines $L_0, \dots, L_{n-1}$:
$$
L_k=\{te^{i\pi k/n}| -\infty<t<\infty\}.
$$
\end{definition}
In other words, $\Sigma_N$ is a ``cross'' of $N$ lines passing
through the origin and forming equal angles $\pi/N$. It is rather
easy to construct a non-zero compactly supported function that is
simultaneously odd with respect to all lines of a given Coxeter
set (look at the Fourier series expansions with respect to the
polar angle). Hence, $\Sigma_N$ is a non-injectivity set as well.
It is clear that applying any rigid motion $\omega$ one preserves
non-injectivity property. Thus, all sets $\omega \Sigma_N$ are
non-injectivity sets. It was conjectured by V.~Lin and A.~Pincus
that these are the only non-injectivity sets for compactly
supported functions on the plane. It was discovered in \cite{AQ}
that one can also add any finite set $F$ preserving
non-injectivity. And that is all that can happen. M.~Agranovsky
and E.~Quinto presented the following complete solution in $2D$
for the (equivalent) Problems \ref{P:injec}, \ref{P:LinPinc} in
\cite{AQ}:
\begin{theorem}\label{T:AQ}\cite{AQ}
The following condition is necessary and sufficient for a set
$S\subset \Bbb{R}^2$ to be a set of injectivity for the circular
Radon transform on $C_c(\Bbb{R}^2)$:

$S$ is not contained in any set of the form
$\omega(\Sigma_N)\bigcup F$, where $\omega$ is a rigid motion in
the plane and $F$ is a finite set.
\end{theorem}
In other words, only Coxeter ``crosses'' plus finite sets are
non-injectivity sets for the circular Radon transform $R$ {\bf
considered on compactly supported functions}.

Before we go any further, we would like to state the (unproven)
conjecture that describes non-injectivity sets in higher
dimensions (still relative to the class of compactly supported
functions).

\begin{conjecture}\label{C:conj}\cite{AQ}
The following condition is necessary and sufficient for $S$ to be
a set of injectivity for the circular Radon transform on
$C_c(\Bbb{R}^n)$:

$S$ is not contained in any set of the form $\omega(\Sigma)\bigcup
F$, where $\omega$ is a rigid motion of $\Bbb{R}^n$, $\Sigma$ is
the zero set of a {\bf homogeneous} harmonic polynomial, and $F$
is an algebraic subset in $\Bbb{R}^n$ of co-dimension at least
$2$.
\end{conjecture}
The reader easily notices that for $n=2$ this boils down to
Theorem \ref{T:AQ}.

The beautiful proof of Theorem \ref{T:AQ} by M. Agranovsky and E.
Quinto is built upon three kinds of tools: Lemmas \ref{L:S(f)} and
\ref{L:harm}, microlocal analysis (Fourier Integral Operators
technique) that in particular guarantees existence of certain
analytic wave front sets at the boundary of the support of a
function located on one side of a smooth surface (8.5.6 in
\cite{Hor}), and known geometric structure of level sets of
harmonic polynomials in $2D$ (e.g., \cite{Flatto}). These methods,
unfortunately, restrict applicability of the proof. The microlocal
tool used works at an edge of the support and hence is not
applicable for non-compactly-supported functions, no matter how
fast they decay. On the other hand, the geometry of level sets of
harmonic polynomials does not work well in dimensions higher than
2, and definitely not on more general Riemannian manifolds (e.g.,
on the hyperbolic plane). Hence, the quest has been active for
alternative approaches since \cite{AQ} has appeared.

One can guess what such approaches could possibly be, by looking
at alternative reformulations of the problem (which there are
plenty \cite{AQ}). We have already seen the one in terms of
approximation theory, but unfortunately this has not lead to
significant advances yet. There is, however, another revealing
reformulation \cite{AQ,Kuch} that stems from known relations
between spherical integrals and the wave equation (e.g., \cite{CH,
John}). Namely, consider the initial value problem for the wave
equation in $\Bbb{R}^n$:
\begin{equation}
u_{tt}-\triangle u=0, \;\; x\in \Bbb{R}^n, t\in \Bbb{R}
\label{E:wave1}
\end{equation}
\begin{equation}
u|_{t=0}=0, \;\; u_t|_{t=0}=f.\label{E:wave2}
\end{equation}
Then
$$
u(x,t)=\frac{1}{(n-2)!}\frac{\partial^{n-2}}{\partial
t^{n-2}}\int_0^t r(t^2-r^2)^{(n-3)/2}(Rf)(x,r)dr, \;\;t\ge0.
$$
Hence, it is not hard to show \cite{AQ} that the original problem
is equivalent to the problem of recovering $u_t(x,0)$ from the
value of $u(x,t)$ on subsets of $S\times(-\infty,\infty)$.
\begin{lemma}\label{L:wave}\cite{AQ,Kuch} A set $S$ is a
non-injectivity set for $C_c(\Bbb{R}^n)$ if and only if there
exists a non-zero compactly supported continuous function $f$ such
that the solution $u(x,t)$ of the problem
(\ref{E:wave1})-(\ref{E:wave2}) vanishes for any $x \in S$ and any
$t$.
\end{lemma}
Hence, non-injectivity sets are exactly the nodal sets of
oscillating free infinite membranes. In other words, injectivity
sets are those that observing the motion of the membrane over $S$
gives complete information about the motion of the whole membrane.
Analogous relation holds also for solutions of the heat equation
\cite{AQ}.

Lemma \ref{L:wave} suggests to try to understand the geometry of
non-injectivity sets in terms of wave propagation. The first
example of such a consideration was the original proof \cite{Kuch}
of Corollary \ref{C:boundary} that did not use harmonicity (which
was not known at the time). The proof went essentially as follows
(one can easily make what is written below precise). Let
$S=\partial U$ be a non-injectivity (and hence nodal for wave
equation) set, where $U$ is a bounded domain. Then one can adopt
two different points of view. On one hand, the membrane is free
and hence the energy of the initial compactly supported
perturbation must move away. Thus, its portion inside $U$ should
decay to zero. On the other hand, one can think  that $S$ is a
fixed boundary (since it does not move anyway) and hence the
energy inside must stay constant. This contradiction allows one to
conclude that in fact $f=0$. The same PDE idea, with many more
technical details, was implemented in \cite{ABK} to prove the
following statement:
\begin{theorem}\label{T:ABK}\cite{ABK}If $U$
is a bounded domain in $\Bbb{R}^n$, then $S=\partial U$ is an
injectivity set for $R$ in the space $L^q(\Bbb{R}^n)$ if $q \leq
2n/(n-1)$. This statement fails when $q > 2n/(n-1)$, in which case
spheres fail to be injectivity sets.
\end{theorem}
The rough idea of what is going on in this theorem can be
explained as follows: when $q > 2n/(n-1)$, the function $f$ in
(\ref{E:wave1})-(\ref{E:wave2}) has a long tail, and so there is
enough energy at infinity to constantly come and replace the
energy leaving the domain $U$. This can keep the set $S$ nodal for
all times. On the other hand, for $q \leq 2n/(n-1)$ there is not
enough energy at infinity to keep the balance.

In spite of these limited results, it still remained unclear what
distinguishes in terms of wave propagation the ``bad'' flat lines
$S$ in Theorem \ref{T:AQ} that can be nodal for all times, from
any truly curved $S$ that according to this theorem cannot stay
nodal. An approach to this question was found in the recent paper
\cite{FPR} by D.~Finch, Rakesh, and S.~Patch, where in particular
some parts of the injectivity results due to \cite{AQ} were
re-proven by simple PDE means without using microlocal tools and
harmonicity.

\begin{theorem}\label{T:Finch}\cite{FPR}
Suppose $D$ is a bounded, open, subset of $\Bbb{R}^n$, $n\ge2$,
with a strictly convex smooth boundary $S$. Let $\Gamma$ be any
relatively open subset of $S$. If $f$ is a smooth function on
$\Bbb{R}^n$, supported in $\bar{D}$, and $(Rf)(p,r)=0$ for all
$p\in\Gamma$ and all $r$, then $f=0$.

Equivalently, if $u$ is the solution of the initial value problem
(1), (2) and $u(p,t)=0$ for all $p\in\Gamma$ and all $t$, then
$f=0$.
\end{theorem}

One can find out easily that the statement of this theorem follows
immediately from a microlocal statement in
\cite{AQ}\footnote{Results of \cite{AQ} make the situation
described in Theorem \ref{T:Finch} impossible, since the support
of $f$ lies on one side of a tangent plane to $\Gamma$. See also
Theorem \ref{tangent}} The significance of this theorem, however,
lies not in its statement, but rather in the proof provided in
\cite{FPR} (that paper contains other important results as well,
which we do not touch here).

The following two standard statements about the wave equation were
the basis of the proof of the Theorem \ref{T:Finch} in \cite{FPR}.
The first one concerns the unique continuation for the time-like
Cauchy problem for the wave equation, while the second states
finiteness of speed of propagation.

\begin{proposition}\label{P:uniqueness_cont}\cite{FPR}
Let $B_\epsilon(p)=\{x\in \Bbb{R}^n \, |\, |x-p|<\epsilon\}$. If
$u$ is a distribution and satisfies (\ref{E:wave1}) and $u$ is
zero on $B_\epsilon(p)\times(-T,T)$ for some $\epsilon>0$, and
$p\in \Bbb{R}^n$, then u is zero on
$$
\{(x,t):|x-p|+|t|<T\},
$$
and in particular on
$$
\{(x,0):|x-p|<T\}.
$$
\end{proposition}

Let now $D$ be a bounded, open subset of $\Bbb{R}^n$ with a smooth
boundary $S$. For points $p,q$ outside $D$, let $d(p,q)$ denote
the infimum of the lengths of all the piecewise $C^1$ paths in
$\Bbb{R}^n\setminus D$ joining $p$ to $q$. Then $d(p,q)$ is a
metric on $\Bbb{R}^n\setminus D$. For any point $p$ in
$\Bbb{R}^n\setminus D $ and any positive number $r$, define
$E_r(p)$ to be the ball of radius $r$ and center at $p$ in
$\Bbb{R}^n\setminus D$ with respect to this metric, i.e.
$$
E_r(p)=\{x\in \Bbb{R}^n\setminus D:d(x,p)<r\}.
$$
The second statement is about the domain of dependence of
solutions of the wave equation in an exterior domain. The result
claims that the value of the solution of the wave equation in an
exterior domain, at a point $(x,s)$, affects the values of the
solution at the point $(y,t)$ only if $d(x,y)\le t-s$.

\begin{proposition}\label{P:domain_dep}\cite{FPR}
Suppose $D$ is a bounded, open, connected subset of $\Bbb{R}^n$,
with a smooth boundary $S$. Let $u$ be a smooth solution of the
exterior problem
$$
u_{tt}-\triangle u=0, \;\;x\in \Bbb{R}^n\setminus D,\;\;t\in R
$$
$$
u=h\;\; on \;\; S\times R.
$$
Suppose $p$ is not in $D$, and $t_0<t_1$ are real numbers. If
$u(.,t_0)$ and $u_t(.,t_0)$ are zero on $E_{t_1-t_0}(p)$ and $h$
is zero on
$$
\{(x,t):x\in S, t_0\le t\le t_1, d(x,p)\le t_1-t \},
$$
then $u(p,t)$ and $u_t(p,t)$ are zero for all $t\in [t_0,t_1)$.
\end{proposition}

Now the proof of Theorem \ref{T:Finch} roughly boils down to the
following consideration: the signal from $f$ can reach a point
near $\Gamma$ that is outside $D$ at a time dictated by the
``curvilinear'' distance $d$, while on the other hand the membrane
is free, so the signals can propagate along straight paths and so
should have arrived earlier. This contradiction proves the
theorem.

The last two propositions can be proven under less stringent
conditions on the domains and functions, but as it will be seen in
the next section, this is in fact not needed.

\section{Further injectivity results by PDE means}\label{S:main}

We will show now how simple tools similar to the Propositions
\ref{P:uniqueness_cont} and \ref{P:domain_dep}, namely finite
speed of propagation and domain of dependence for the wave
equation allow one to obtain stronger results concerning
injectivity sets.

Let us start with some initial remarks that will narrow the cases
we need to consider. First of all, according to Lemma
\ref{L:S(f)}, any non-injectivity set $S$ is contained in an
algebraic surface that is also a non-injectivity set. So, one can
as well assume from the start algebraicity of $S$. Since it is
known \cite{AQ} that algebraic surfaces of co-dimension higher
than 1 are automatically non-injectivity sets, we can restrict our
attention to algebraic hypersurfaces in $\Bbb{R}^n$ only. The
reader should understand that anything that is not algebraic (or
rather does not fit into an algebraic surface) is automatically an
injectivity set. So, confining ourselves to the case of algebraic
hypersurfaces solely we do not lose any generality. One can also
assume irreducibility of that surface, if this helps. When needed,
one can also exclude the case of {\bf closed hypersurfaces}, since
according to Lemma \ref{L:harm} those are all injectivity sets,
albeit this is mostly not needed in what follows. Notice that
these reductions so far do not require us to confine ourselves to
the class of compactly supported functions, since they work as
well in the class of continuous functions decaying at infinity
faster than any power of $|x|$. Thus, we do not impose at this
stage compactness of support restriction, although it will appear
in some of the results of this section.

Our goal now is to exclude many pairs $(S,f)$, where $S$ is an
algebraic surface and $f$ is a function as possible candidates for
satisfying the non-injectivity condition $R_Sf=0$. Notice that
Theorem \ref{T:Finch} does exactly that when $S$ contains an open
part of the boundary of a strictly convex domain where $f$ is
supported. Theorem \ref{T:AQ}, on the other hand excludes all
compactly supported $f$'s in $\Bbb{R}^2$, unless
$S=\omega\Sigma_N$. Similarly, Theorem \ref{T:ABK} excludes
boundaries $S$ of bounded domains when $f$ is in an appropriate
space $L_p(\Bbb{R}^n)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $S$ be an irreducible algebraic hypersurface that splits
$\Bbb{R}^n$ into parts $H^+$ and $H^-$ (for instance, if $P(x)$ is
the irreducible polynomial defining $S$, we can take
$H^+=\{x\,|\,P(x)>0\}$). One can define the interior metric in
$H^+$ as follows:
\begin{equation}\label{E:metric}
d^+(p,q)=inf\{\mbox{length of}\, \gamma\},
\end{equation}
where the infimum is taken over all $C^1$-curves $\gamma$ in $H^+$
joining points $p,q \in H^+$. This metric extends naturally to
$S$. A similar metric $d^-$ is defined on $H^- \cup S$.

\begin{theorem}\label{T:halves}
Let $S$ be as above and $f\in C(\Bbb{R}^n)$ be such that $R_Sf=0$.
Let also $x\in H^+\cup S$. Then
\begin{equation}\label{E:eqhalves_ineq}
\begin{array}{c}
  dist(x, \mbox{supp } f \cap H^+)=dist^+(x, \mbox{supp } f \cap H^+)\\
  \leq dist(x, \mbox{supp } f \cap H^-),
\end{array}
\end{equation}
where distances $dist^\pm$ are computed with respect to the
metrics $d^\pm$, while $dist$ is computed with respect to the
Euclidean metric in $\Bbb{R}^n$.

A similar statement holds for $x\in H^-\cap S$.

In particular, for  $x\in S$
\begin{equation}\label{E:eqhalves_eq}
\begin{array}{c}
  dist(x, \mbox{supp } f \cap H^+)=dist^+(x, \mbox{supp } f \cap H^+)\\
  = dist(x, \mbox{supp } f \cap H^-)=dist^-(x, \mbox{supp } f \cap
  H^-).
\end{array}
\end{equation}
\end{theorem}
\begin{remark}
\begin{itemize}
\item Notice that the theorem does not require
the function $f$ to be compactly supported. It in fact does not
require any control at infinity.
\item The assumed algebraicity of $S$ (although it comes for free when $f$
decays faster than any power of $|x|$) is also not truly
necessary. The reader can easily figure out from the proof that
any piecewise-smooth (or even Lipschitz) hypersurface suffices.
\end{itemize}
\end{remark}
\noindent {\bf Proof of the theorem.} Let us prove first the
equality

\begin{equation}\label{E:onehalf}
   dist(x, \mbox{supp } f \cap H^+)=dist^+(x, \mbox{supp } f \cap
   H^+),
\end{equation}
which would also prove the similar one with $d^-$ and $H^-$.

We need a simple auxiliary statement, where we use the notation
$B_r(p)$ for the ball of radius $r$ centered at $p$.
\begin{lemma}\label{L:smooth}
For any compact $K$, $\epsilon>0$, and a point $p\in H^+\cup S$
there exists a smooth function $d_\epsilon^+(q)$ in a neighborhood
of $K\cap (H^+\cup S)$ such that $|\nabla d_\epsilon^+(q)|\leq1$
and $|d^+(p,q)-d_\epsilon^+(q)|<\epsilon$ for any $q\in K\cap
(H^+\cup S)$.
\end{lemma}
This lemma removes the function theory difficulties related to
Proposition \ref{P:domain_dep} faced in \cite{FPR}.

{\bf Proof of the lemma.} Consider for a small positive $\delta$
the enlarged set $H^+_\delta=\{x\,|\,P(x)>-\delta\}\supset H^+$.
Introduce the distance $D^+_\delta$ in $H^+_\delta$ the same way
$d^+$ was introduced in $H^+$. Then  $D^+_\delta$ on $H^+$ does
not exceed $d^+$ and for small $\delta$ can be made as close as
necessary to $d^+$ on any compact $K \subset H^+\cup S$.
Mollifying $D^+_\delta$ with a mollifier of a small support, one
obtains the required function $d^+_\epsilon$. Indeed, closeness of
the two functions is clear. Since $D^+_\delta$ has gradient of
length not exceeding $1$ a.e., we get the needed estimate on the
gradient of $d^+_\epsilon$ on $K$. The lemma is proven.

Let us now return to the proof of (\ref{E:onehalf}). Since
$d^+(p,q)\geq |p-q|$, it is sufficient to prove that the left hand
side expression cannot be strictly smaller than the one on the
right. Assume the opposite, that
\begin{equation}\label{E:hal_opposite}
dist(x, \mbox{supp } f \cap H^+)=d_1<d_2=dist^+(x, \mbox{supp } f
\cap
   H^+).
\end{equation}
Pick a smaller segment $[d_3,d_4]\subset (d_1,d_2)$. Then, by
continuity, for any point $p$ in a small ball $B \subset H^+$ near
$x$ (not necessarily containing $x$, which happens when $x \in S$)
one has
\begin{equation}\label{E:hal_opposite}
dist(p, \mbox{supp } f \cap H^+)\leq d_3<d_4\leq dist^+(p,
\mbox{supp } f \cap
   H^+).
\end{equation}
For such a point $p$, let us choose $\epsilon < (d_4-d_3)/2$, the
compact $K$ which is the ball with respect to $d^+$ centered at
$p$ and of radius $2d_4$, and a function $d^+_\epsilon(q)$ from
the previous lemma. Then for any point $q\in K\cap \mbox{supp } f
\cap H^+$ we have
$$
d^+_\epsilon(q)> (d_3+d_4)/2>d_3.
$$
Consider the volume $V$ in the space-time region
$H^+\times\Bbb{R}$ bounded by the space-like surfaces $t=0$ and
$t=(d_3+d_4)/2-d^+_\epsilon(p)$ and the ``vertical'' boundary
$S\times \Bbb{R}$. Consider the solution $u(x,t)$ of the wave
equation problem (\ref{E:wave1})-(\ref{E:wave2}) with the initial
velocity $f$. Then, by construction, this solution and its time
derivative are equal to zero at the lower boundary $t=0$ and on
the lateral boundary $S$. Hence, the standard domain of dependence
argument (see, e.g., Section 2.7, Ch. 1 in \cite{BJS}) we conclude
that $u=0$ in $V$. In particular, $u(p,t)=0$ for all $p\in B$ and
$|t|\leq (d_3+d_4)/2-\epsilon$. Notice that
$(d_3+d_4)/2-\epsilon>d_3$. Now applying Proposition
\ref{P:uniqueness_cont} to the wave equation in the whole space,
we conclude that
\begin{equation}\label{E:wholespace}
dist(p,\mbox{supp }f )>d_3,
\end{equation}
and hence
\begin{equation}\label{E:almostthere}
  dist(p,\mbox{supp }f \cap H^+)>d_3,
\end{equation}
which is a contradiction. This proves (\ref{E:onehalf}). It is now
sufficient to prove
\begin{equation}\label{E:equals}
  dist(x,\mbox{supp }f \cap H^+)\leq dist(x,\mbox{supp }f \cap H^-).
\end{equation}
This in fact is an immediate consequence of (\ref{E:wholespace}).
Alternatively, we can repeat the same consideration as above in a
simplified version. Namely, suppose that
\begin{equation}\label{E:nonequalhalves}
  dist(x,\mbox{supp }f \cap H^+)> d_2>d_1> dist(x,\mbox{supp }f \cap H^-)
\end{equation}
for a point $x$, and hence for all points $p$ in a small ball in
$H^+$. Consider the volume $V$ in the space-time region
$H^+\times\Bbb{R}$ bounded by the space-like surfaces $t=0$ and
$t=d_2-|x-p|$ ($p$ fixed in the small ball) and the boundary
$S\times \Bbb{R}$. Consider the solution $u(x,t)$ of the wave
equation problem (\ref{E:wave1})-(\ref{E:wave2}) with the initial
velocity $f$. Then, by construction, this solution and its time
derivative are equal to zero at the lower boundary $t=0$ and on
the lateral boundary $S$. Hence, by the same standard domain of
dependence argument (see, e.g., Section 2.7, Ch. 1 in \cite{BJS})
we conclude that $u=0$ in $V$. In particular, $u(p,t)=0$ for all
$p\in B$ and $|t|\leq d_2$. Now applying Proposition
\ref{P:uniqueness_cont} to the wave equation in the whole space,
we conclude that
$$
dist(p,\mbox{supp }f )>d_2,
$$
and hence
\begin{equation}
  dist(p,\mbox{supp }f \cap H^-)>d_2,
\end{equation}
which is a contradiction. \noindent {\bf Q.E.D.}

We will now show several corollaries that can be extracted from
Theorem \ref{T:halves}.

Let  $S\subset \Bbb{R}^n$. For any points $p,q \in \Bbb{R}^n-S$ we
define the distance $d_S(p,q)$ as the infimum of lengths of $C^1$
paths in $\Bbb{R}^n-S$ connecting these points. Clearly
$d_S(p,q)\geq |p-q|$. Using this metric, we can define the
corresponding distances $dist_S$ from points to sets.

\begin{theorem}\label{T:piece}
Let a set $S\subset \Bbb{R}^n$ and a non-zero function $f \in
C(\Bbb{R}^n)$ decaying at infinity faster than any power of $|x|$
be such that $R_Sf=0$. Then for any point $p\in \Bbb{R}^n-S$
\begin{equation}\label{E:piece}
  dist_S(p,\mbox{supp} f)=dist(p,\mbox{supp} f).
\end{equation}
\end{theorem}

{\bf Proof.} Assume that (\ref{E:piece}) does not hold, i.e.
$$
dist_S(p,\mbox{supp} f)>dist(p,\mbox{supp} f).
$$
As it has been mentioned before, under the conditions of the
theorem, we can assume $S$ to be a part of an algebraic surface
$\Sigma$ for which $R_\Sigma f=0$. Since $d_S$ does not react on
presence of lower dimensional algebraic manifolds, we can assume
$S$ to be a part of an algebraic hypersurface $\Sigma$ dividing
the space into parts $H^\pm$. Then, in notations of the previous
theorem, we have
\begin{equation}
dist^\pm(p,\mbox{supp } f\cap H^\pm) \geq dist_S(p,\mbox{supp}
f)>dist(p,\mbox{supp} f).
\end{equation}
Then Theorem \ref{T:halves} shows that this is impossible.
\noindent {\bf Q.E.D.}

\begin{corollary}\label{C:perpendicular_ray}
Let $f$ be continuous and decaying faster than any power and
$S\subset \Bbb{R}^n$ be an algebraic hypersurface such that
$R_Sf=0$. Let $L$ be any hyperplane such that $L\cap \mbox{supp
}f\neq \emptyset$ and such that $\mbox{supp }f$ lies on one side
of $L$. Let $x\in L\cap \mbox{supp }f$ and $r_x$ be the open ray
starting at $x$, perpendicular to $L$, and going into the
direction opposite to the support of $f$. Then either $r_x \subset
S$, or $r_x$ does not intersect $S$ (albeit tangency is allowed).
\end{corollary}
{\bf Proof.} Assuming otherwise, take any point $p$ on $r_x$ after
the first intersection with $S$. The only point closest to $p$ on
$\mbox{supp }f$ is $x$, and the segment connecting the two points
intersects $S$. Then obviously $dist_S(p,\mbox{supp
}f)>|p-x|=dist(p,\mbox{supp }f)$. Now the previous theorem
finishes the job. {\bf Q.E.D.}

Let us formulate another example of a geometric constraint on
pairs $S,\,f$ such that $R_Sf=0$.
\begin{theorem}\label{tangent}
Let $S\subset \Bbb{R}^n$ be a relatively open piece of a smooth
hypersurface and $f$ be a continuous function decaying faster than
any power of $|x|$ such that $R_Sf=0$. If there is a point $p_0\in
S$ such that the support of $f$ lies on one side of the tangent
plane $T_{p_0}S$ to $S$ at $p_0$, then $f=0$.\footnote{This
implies, in particular, Theorem \ref{T:Finch}.}
\end{theorem}
{\bf Proof of the theorem .} Let us denote by $K_p(\mbox{supp }f)$
the convex cone with the vertex $p$ consisting of all the rays
starting at $p$ and passing through the convex hall of the support
of $f$. Then $K_{p_0}(\mbox{supp }f)$, due to the condition of the
theorem, lies on one side of $T_{p_0}S$. Let us pull the point
$p_0$ to the other side of the tangent plane along the normal to a
nearby position $p$. Then for $p$ sufficiently close to $p_0$ all
rays of the cone $K_{p}(\mbox{supp }f)$ will intersect $S$. This
means in particular, that for this point $p$ we have $dist_S(p,
\mbox{supp }f)>dist(p, \mbox{supp }f)$. According to Theorem
\ref{T:piece}, this implies that $f=0$. {\bf Q.E.D.}

\begin{corollary}\label{C:intersect}
Let $S\subset \Bbb{R}^n$ be an algebraic hypersurface and $f$ be a
continuous function decaying at infinity faster than any power. If
$R_Sf=0$, then every tangent plane to $S$ intersects the convex
hull of the support of $f$.
\end{corollary}

The above results present significant restrictions on the geometry
of the non-injectivity sets $S$ and corresponding functions $f$ in
the kernel of $R_S$. One can draw from them more specific
conclusions about these sets.

\begin{proposition}\label{P:asymptotes}
Let $S\subset \Bbb{R}^2$ be an algebraic curve such that $R_Sf=0$
for some non-zero compactly supported continuous function $f$.
Then $S$ has no compact components, and each its component has
asymptotes at infinity.
\end{proposition}
{\bf Proof.} Corollary \ref{C:boundary} excludes bounded
components. So, we can think that $S$ is an irreducible unbounded
algebraic curve. Existence of its asymptotes can be shown as
follows. Let us take a point $p\in S$ and send it to one of the
infinite ends of $S$. According to Corollary \ref{C:intersect},
every tangent line $T_pS$ intersects the convex hull of the
support of $f$, which is a fixed compact in $\Bbb{R}^2$. This
makes this set of lines on the plane compact. Hence, we can choose
a sequence of points $p_j$ such that the lines $T_{p_j}S$ converge
to a line $T$ in the natural topology of the space of lines (e.g.,
one can use normal coordinates of lines to introduce such
topology). This line $T$ is in fact the required asymptote.
Indeed, let us choose the coordinate system where $T$ is the
$x$-axis. Then the slopes of the sequence $T_{p_j}S$ converge to
zero. Due to algebraicity, for a tail of this sequence, the
convergence is monotonic, and in particular holds for all $p\in S$
far in the tail of $S$. Let us for instance assume that these
slopes are negative. Then the tail of $S$ is the graph of a
monotonically decreasing positive function. This means that $S$
has a horizontal asymptote. This asymptote must be the $x$-axis
$T$, otherwise the $y$-intercepts of $T_{p_j}S$ would not converge
to zero, which would contradict the convergence of $T_{p_j}S$ to
$T$. {\bf Q.E.D.}

The next statement proves the Agranovsky-Quinto Theorem \ref{T:AQ}
in the particular case of functions of convex support.
\begin{proposition}\label{P:convex2D}
Let $S\subset \Bbb{R}^2$ and $f\in C_c(\Bbb{R}^2)$ be such that
the support of $f$ contains the boundary of its convex hull (in
particular, the support itself can be convex), $f\neq 0$, and
$R_Sf=0$. Then $S\subset \omega \Sigma _N \cup F$ in notations of
Theorem \ref{T:AQ}.
\end{proposition}
{\bf Proof.} First of all, up to a finite set, we can assume that
$S$ is an algebraic curve. Corollary \ref{C:perpendicular_ray}
says that all normal exterior rays to the boundary of the support
are either inside $S$, or do not intersect $S$. By the normal we
mean here the normal to any supporting plane to the boundary (in
particular, at non-smooth points of the boundary one might have
the whole cone of such rays). This immediately implies that
outside the support the set $S$ consists of rays. Algebraicity and
absence of bounded components means that $S$ is the union of
several lines $L_j$ intersecting the support. It is known that any
straight line $L$ is a non-injectivity set, but the only functions
annihilated by $R_L$ are the ones odd with respect to $L$ (e.g.,
\cite{AQ,CH,John}). Hence, $f$ is odd with respect to all lines
$L_j$. In particular, every of these lines passes through the
center of mass of the support of $f$. Hence, lines $L_j$ form a
``cross''\footnote{One can prove that all these lines pass through
a joint point also in a different manner. Indeed, due to oddness
of $f$, each line is a symmetry axis for the support of $f$. Then,
considering the group generated by reflections through these
lines, one can easily conclude that if they did not pass through a
joint point, then the support of $f$ must have been non-compact.}.
It remains now to show that the angles between the lines are
commensurate with $\pi$. This can also be shown in several
different ways. For instance, this follows immediately from
existence of a {\bf harmonic} polynomial vanishing on $S$. Another
simple option is to notice that if this is not the case, there is
no non-zero function that is odd simultaneously with respect to
all the lines. {\bf Q.E.D.}

Exactly the same consideration as above shows that in higher
dimensions at least the following statement is correct:
\begin{proposition}\label{P:convex2D}
Let $S\subset \Bbb{R}^n$ and $f\in C_c(\Bbb{R}^n)$ be such that
the support of $f$ contains the boundary of its convex hull (e.g.,
the support itself is convex) , $f\neq 0$, and $R_Sf=0$. Then $S$
is spanned by lines.
\end{proposition}
\begin{remark}
If we could also show that all these lines pass through the same
point, then this would immediately imply, as in the previous
proof, the validity of Conjecture \ref{C:conj} for this case of
convex support. We, however, have not succeeded in proving this
yet. As we were notified by M.~Agranovsky, he and E.~T.~Quinto had
made some recent further progress for this case, albeit using
microlocal tools.
\end{remark}
\section{Additional remarks}
\begin{enumerate}
\item M.~Agranovsky and E.~T.~Quinto have written several other
papers besides \cite{AQ} devoted to the problem considered here.
They consider some partial cases (e.g., distributions $f$
supported on a finite set) and variations of the problem (e.g., in
bounded domains rather than the whole space). See
\cite{A,AQ2,AQ3,AVZ} for details.

\item One of our goals was to obtain the complete Theorem \ref{T:AQ},
the main result of \cite{AQ} by simple PDE tools, avoiding using
the geometry of zeros of harmonic polynomials and microlocal
analysis (or at least one of those). Although we have not
completely succeeded in this yet, the results presented (e.g.,
Proposition \ref{P:asymptotes}) are moving in this direction.

\item The PDE methods presented here in principle bear a potential
for considering non-compactly-supported functions. In order to
achieve this, one needs to have qualitative versions of statements
like Proposition \ref{P:domain_dep} and Theorem \ref{T:piece},
where instead of just noticing whether a wave has come to certain
point at a certain moment (which was our only tool), one observes
the amount of energy it carries.

\item In this paper one of the motivations for studying the injectivity
problem was the thermoacoustic tomography. One wonders then
whether considerations of $2D$ problems bear any relevance for
TAT. In fact, they do. If either the scanned sample is very thin,
or the transducers are collimated in such a way that they register
the signals only coming parallel to a given plane, one arrives to
a $2D$ problem.

\item Most of our results can be generalized to some Riemannian
manifolds, in particular to the hyperbolic plane (where the
natural analog of Theorem \ref{T:AQ} has not been proven yet). We
plan to address these issues elsewhere.

\item A closer inspection of the results of Section \ref{S:main} shows
that most results have their local versions, where it is not
required that the whole transform $R_S$ of a function is equal to
zero, but rather only for radii up to a certain value. One can see
an example of a local uniqueness theorem for the circular
transform in \cite{LQ}. We hope to address this issue elsewhere.

\item As J.~Boman notified us during the April 2004 AMS meeting in
Lawrenceville, he jointly with J.~Sjostrand, being unaware of our
work, had recently independently obtained some results analogous
to some of those presented here (e.g., to Theorem \ref{T:piece}).

\item We have not touched the problem of finding explicit
inversion formulas for the circular transforms. Such formulas are
known for the spherical, planar, and cylindrical sets of centers
\cite{And, Den, Faw, FPR, Nil, Pal, MXW, YXW1, YXW2}. They come in
two kinds: the ones involving expansions into special functions,
and the ones of backprojection type. Exact backprojection type
formulas are known for the planar geometry \cite{Den,Pal} and
recently for the spherical geometry in odd dimensions \cite{FPR}
if the function to be reconstructed is supported inside the sphere
of transducers.

Another problem deserving attention is finding the ranges of
transforms $R_S$. Such knowledge could be used, for instance, to
replenish missing data. Some necessary range conditions have been
recently obtained in \cite{Patch} for spherical location of
transducers.

An important problem in tomography of reconstruction with
incomplete data was treated in \cite{LQ,XWAK} based on an earlier
work by E.~T.~Quinto in \cite{Q1993b}.

%\item Algebraicity vs weaker conditions

\item An important integral geometric technique of the so called
$\kappa$-operator has been developed in I.~Gelfand's school (e.g.,
\cite{GGG1,GGG2}). It has been applied recently to the problems of
the circular Radon transform (see \cite{Gi}, the last chapter of
\cite{GGG2}, and references therein), albeit applicability of this
method to the problems of the kind we consider in this paper is
not completely clear yet.
\end{enumerate}
\section{Acknowledgements}
The authors express their gratitude to M.~Agranovsky, J.~Boman,
E.~Chappa, L.~Ehrenpreis, D.~Finch, S.~Patch, E.~T.~Quinto,
L.~Wang, M.~Xu, Y.~Xu, and N.~Zobin for information about their
work and discussions.

This research was partly based upon work supported by the NSF
under Grants DMS 0296150, 9971674, and 0072248. The authors thank
the NSF for this support. Any opinions, findings, and conclusions
or recommendations expressed in this paper are those of the
authors and do not necessarily reflect the views of the National
Science Foundation.

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