Abstract
This paper is concerned with a general condition for the solvability of integral geometry problems along the plane curves of given curvatures. As two important results, the solvabilities of integral geometry problems along the family of circles with fixed radius and along the family of circles of varying radius centered on a fixed circle are given. By using some extension of the class of unknown functions, the proofs are based on the solvabilities of equivalent inverse problems for transportlike equation.
MSC: 35R30, 53C65, 65N30.
Keywords:
integral geometry problem; inverse problem; Galerkin method; transportlike equation1 Introduction
The problems of integral geometry are to determine a function, given (weighted) integrals
of this function over a family of manifolds, and there has been significant progress
in the classical Radon problem when manifolds are hyperplanes and the weight function
is unity, there are interesting results in the plane case when a family of curves
is regular or in the case of a family of straight lines with arbitrary regular attenuation
[[1], Chapter 7]. It is assumed that the basis of the integral geometry problems is the
Radon transform [2]. The Radon transform R integrates a function f on
(see [[3], Chapter 2]).
The problems of integral geometry have important applications in imaging and provide the mathematical background of tomography, where the main goal is to recover the interior structure of a nontransparent object using external measurements. The object under investigation is exposed to radiation at different angles, and the radiation parameters are measured at the points of observation. The basic problem in computerized tomography is the reconstruction of a function from its line or plane integrals, and there are many applications related with computerized tomography: medical imaging, geophysics, diagnostic radiology, astronomy, seismology, radar and many other fields (see, e.g., [4]).
From the applied point of view, the importance of integral geometry problem over a family of straight lines in the plane is indicated in [5], where the problem models Xrays, and applicable to the problems of radiology and radiotherapy. Because of their many practical applications, a considerable attention has been devoted to other family of curves in the plane as well as straight lines. Invertibility of the Radon transforms on some families of curves in the plane is given with explicit inversion formulas via circular harmonic decomposition in [6] and for the explicit inversion formulas of the attenuated Radon transform, see, e.g., [7,8]. Note that the circle is the simplest nontrivial curve in the plane next to the straight line, and the representation of a function by its circular Radon transform also arises in applications. In [9], invertibility of the Radon transforms over all translations of a circle of fixed radius and circles of varying radius centered on a fixed circle is considered, where the proofs require microlocal analysis of the Radon transforms and a microlocal Holmgren theorem. In [10], some existence and uniqueness results on recovering a function from its circular Radon transform with partial data are presented and the relations to applications in medical imaging are described. There are several other ways related to the selection of a family of curves, such as circles of varying radius centered on a straight line or a fixed curve, circles passing through a fixed point, along paths that are not on the zero sets of harmonic polynomials, circular arcs having a chord of fixed length rotating around its middle point etc., which are meaningful in applications on thermoacoustic and photoacoustic tomography, synthetic aperture radar, Compton scattering tomography, ultrasound tomography etc. (see, e.g., [1012] and the references therein).
In fact, since the seminal work of Radon [2], the various integral geometry problems with numerous applications have been considered in several important aspects which are not mentioned here, but for a comprehensive list, see, e.g., [1316] and the references therein. Furthermore, the problems of integral geometry and inverse problems for transport equations are interrelated and the latter are also of great importance in theory and applications; see, e.g., [1720] and for the derivation and applications of transport equations, see, e.g., [2123].
In this paper, a general condition for the solvability of integral geometry problems along plane curves of given curvatures is presented and its relation with some previous results is indicated. Moreover, as two important results, the solvabilities of integral geometry problems along the family of circles with fixed radius and along the family of circles of varying radius centered on a fixed circle are given. Since the curvature of a circle is defined to be the reciprocal of the radius of the circle, in the former case the curvature is a constant, while in the latter one the curvature depends on the point and the direction. To investigate the solvability of the integral geometry problem (IGP) given in Section 2.1, which is overdetermined since the underlying operator of the IGP is compact and its inverse operator is unbounded (see Section 2.2), it is reduced to an equivalent overdetermined inverse problem for a transportlike equation, and then, with the use of a similar method which was proposed in [24] (see also [[13], Chapter 1]), on using some extension of the class of unknown functions, this inverse problem is replaced by a determined one. Thus, the solvability of IGP is proved via the solvability of an inverse problem for a transportlike equation. The above mentioned method on the solvability was also previously utilized in [2527] for IGP along some family of plane curves of given curvatures and straight lines, in [28] for IGP along geodesics and in [29] for IGP along the family of curves whose curvatures are given by the Christoffel symbols. Here, the presented general condition for the solvability covers those of [2528], in the manner indicated by Remark 1 in Section 3.1. Moreover, these previous solvability conditions do not hold for IGP along the above given two families of circles, and this is the main importance and motivation of this study.
In Section 2, IGP and its reduction to the equivalent inverse problem for a transportlike equation, the method to overcome the difficulty on investigating the solvability arising from overdeterminacy of these problems are presented and some definitions and notations which will be used throughout the paper are introduced. Section 3 is devoted to the statements of main results, and finally in Section 4, proofs of the results are given.
The investigation of approximate solutions of the concerned integral geometry problems is beyond the scope of this paper, but similar procedures as in [26] can be carried out by using the Galerkin method or the finite difference method.
2 Statement of the problem and overdeterminacy
2.1 Statement of the problem
Let D be a bounded domain in
IGPDetermine a function
Suppose that
where
Investigating the uniqueness of a solution of a problem of integral geometry by reducing it to the equivalent inverse problem for a differential equation was first carried out in [30]. Similar reduction is demonstrated for IGP formulated below.
Differentiating (1) in the direction ν at x, we obtain the following transportlike equation:
From (1), u is 2πperiodic with respect to φ, and since the integrals of λ along the curves of
(see [30] and [[13], p.11]). So, we have the following inverse problem.
Problem 1 Determine a pair of functions
2.2 Overdeterminacy
Generally, in the theory of integral geometry, reconstruction of a function of n variables from a function of
Let us propose the procedure for establishing the solvability of IGP. Assume that
the unknown function λ in IGP depends not only upon the space variables x, but also upon the direction φ in some special manner, i.e., consider
(1) The IGP or Problem 1 with the function
(2) The sufficiently smooth functions λ depending only on x satisfy this equation.
Suppose that a differential equation for
In general, the equation with the properties (1) and (2) for the same problem is not uniquely defined. Hence, the class of unknown functions λ extends so that IGP for this class becomes a determined problem and all sufficiently smooth functions in x belong to it. On using some extension of the class of functions λ, the overdetermined Problem 1 is replaced by a determined one (Problem 2 in Section 3.1).
The above method of solvability of the IGP or Problem 1 leads to the Dirichlettype
problem with conditions (3) for the thirdorder equation of the form
2.3 Definitions and notations
In this section, some notations are given based on [13]. Let
Let
The proof of Theorem 1 involves energylike estimates and the Galerkin method (see,
e.g., [[34], Chapter 5, Section 2.3], [[35], Chapter 7]), and therefore some class of functions are introduced below. In
where
Let
Let
where
The existence of the function g in the expression of
Let
3 Statements of results
3.1 Solvability of IGP along plane curves
Since Problem 1 is overdetermined, as indicated in Section 2.2, we consider the following determined problem.
Problem 2 Determine a pair of functions
provided that
In (4), it is assumed that the unknown function λ depends also on φ and the condition
If
Problem 3 Determine a pair of functions
provided that
The existence, uniqueness and the stability of the solution of Problem 3 are given by the following theorem.
Theorem 1If
for all
holds, where
Remark 1 In fact, without being aware of (6), the function g, with the appropriate choices of it, was used previously in [25,26,28]. The convenience of (6) with those of previous solvability results is indicated below.
(i) In [26], when the curvature K is sufficiently smooth and 2πperiodic, the condition for the solvability is
(ii) In [25], when
(iii) In [28], when
Note that in none of the above cases, the solvability conditions hold for the IGP
along the family of circles of varying radius centered on a fixed circle and the family
of curves of constant curvatures, i.e., the family of circles with fixed radius where the curvature is a nonzero constant
or the family of straight lines where the curvature is zero. The former cases are
investigated in Section 3.2 below and the latter case is considered in [27], where the term
3.2 Solvability of IGP along the family of circles
Since Theorem 1 was given for IGP along a regular family of plane curves for the general case, in this section the given results on the solvabilities of IGP along the family of circles depend on finding an appropriate function g satisfying (6) for the given curvature K and the domain D under the assumptions of Theorem 1.
3.2.1 Solvability of IGP along the family of circles with fixed radius
Let
The solvability of IGP along the curves of
Lemma 1Let us definegon
If
Remark 2 The above choice of g is not unique, and since the curvature
for all
3.2.2 Solvability of IGP along the family of circles of varying radius centered on a fixed circle
Let
and hence the curvature of the elements of the family
on
The solvability of IGP along the curves of
Lemma 2If the functiongis defined on
then (6) holds forKdefined in (11).
4 Proof of results
Proof of Theorem 1 First, we will prove the uniqueness of the solution
Since
and
If (6) holds, then the quadratic form
Indeed, we can estimate the terms
where
Moreover, whenever (6) holds, for sufficiently close value of ε to 1, there exists an
where
Since the domain D is bounded,
where
Thus, since
Since
which implies that
Now we will prove that there exists a solution
Determineudefined in Ω that satisfies
where
The solution u of problem (17)(18) will be approximated by
construction of which is based on finding the vector
where the system of functions
We must show that the solution of system (19) exists and is unique for any
has a nonzero solution
where
and since the quadratic form
Now we estimate the solution
Since
for
It can be verified that for sufficiently large
where the constant C is independent of N. This implies that
holds. Since
Since the linear span of
for every
holds and, by using
Now it remains to show that
for any
Moreover, from (19) we have
thus
Proof of Lemma 1 Since
to prove that condition (6) holds, we only need to show that g satisfies
By taking into account that
given in (8) defined on
and the proof is complete. □
Proof of Lemma 2 For
we have
and
Since
given in (12) defined on
Hence, to prove that condition (6) holds, we only need to show that
holds. If we take into account again that
The proof is complete. □
Competing interests
The author declares that he has no competing interests.
References

Isakov, V: Inverse Problems for Partial Differential Equations, Springer, New York (2006)

Radon, J: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipz., Math.Nat.wiss. Kl.. 69, 262–277 (1917)

Natterer, F, Wübbeling, F: Mathematical Methods in Image Reconstruction, SIAM, Philadelphia (2001)

Natterer, F: The Mathematics of Computerized Tomography, Vieweg+Teubner, Wiesbaden (1986)

Cormack, AM: Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys.. 34, 2722–2727 (1963). Publisher Full Text

Cormack, AM: The Radon transform on a family of curves in the plane. Proc. Am. Math. Soc.. 83, 325–330 (1981). Publisher Full Text

Natterer, F: Inversion of attenuated Radon transform. Inverse Probl.. 17, 113–119 (2001). Publisher Full Text

Novikov, RG: An inversion formula for the attenuated Xray transformation. Ark. Mat.. 40(1), 145–167 (2002). Publisher Full Text

Quinto, ET: Radon transforms on curves in the plane. Tomography, Impedance Imaging and Integral Geometry, pp. 231–244. Am. Math. Soc., Providence (1994)

Ambartsoumian, G, GouiaZarrad, R, Lewis, M: Inversion of the circular Radon transform on an annulus. Inverse Probl.. 26, (2010) Article ID 105015

Nguyen, MK, Truong, TT: Inversion of a new circulararc Radon transform for Compton scattering tomography. Inverse Probl.. 26, (2010) Article ID 065005

Rigaud, G, Nguyen, MK, Louis, AK: Novel numerical inversions of two circulararc Radon transforms in Compton scattering tomography. Inverse Probl. Sci. Eng.. 20(6), 809–839 (2012). Publisher Full Text

Amirov, AK: Integral Geometry and Inverse Problems for Kinetic Equations, VSP, Utrecht (2001)

Gelfand, IM, Gindikin, SG, Graev, MI: Selected Topics in Integral Geometry, Am. Math. Soc., Providence (2003)

Romanov, VG: Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer, Berlin (1974)

Sharafutdinov, VA: Integral Geometry of Tensor Fields, VSP, Utrecht (1994)

Arridge, SR: Optical tomography in medical imaging. Inverse Probl.. 15, R41–R93 (1999). Publisher Full Text

Bal, G: Inverse transport theory and applications. Inverse Probl.. 25, (2009) Article ID 053001

Stefanov, P: Inverse problems in transport theory. In: Uhlmann G (ed.) Inside Out: Inverse Problems and Applications, Cambridge University Press, Cambridge (2003)

Tamasan, A: An inverse boundary value problem in twodimensional transport. Inverse Probl.. 18, 209–219 (2002). Publisher Full Text

Anikonov, DS, Kovtanyuk, AE, Prokhorov, IV: Transport Equation and Tomography, VSP, Utrecht (2002)

Case, KM, Zweifel, PF: Linear Transport Theory, AddisonWesley, Reading (1967)

Klibanov, MV, Yamamoto, M: Exact controllability for the time dependent transport equation. SIAM J. Control Optim.. 46(6), 2071–2095 (2007). Publisher Full Text

Amirov, AK: Existence and uniqueness theorems for the solution of an inverse problem for the transport equation. Sib. Math. J.. 27, 785–800 (1986)

Amirov, A, Yildiz, M, Ustaoglu, Z: Solvability of a problem of integral geometry via an inverse problem for a transportlike equation and a numerical method. Inverse Probl.. 25, (2009) Article ID 095002

Ustaoglu, Z, Heydarov, B, Amirov, S: On the solvability and approximate solution of a two dimensional coefficient inverse problem for a transportlike equation. Inverse Probl.. 26, (2010) Article ID 115019

Amirov, A, Ustaoglu, Z, Heydarov, B: Solvability of a two dimensional coefficient inverse problem for transport equation and a numerical method. Transp. Theory Stat. Phys.. 40(1), 1–22 (2011). Publisher Full Text

Golgeleyen, I: An integral geometry problem along geodesics and a computational approach. An. Univ. “Ovidius” Constanţa, Ser. Mat.. 18(2), 91–112 (2010). PubMed Abstract

Golgeleyen, I: An inverse problem for a generalized transport equation in polar coordinates and numerical applications. Inverse Probl.. 29, (2013) Article ID 095006

Lavrent’ev, MM, Anikonov, YE: A certain class of problems in integral geometry. Sov. Math. Dokl.. 8, 1240–1241 (1967)

Klibanov, MV, Timonov, A: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht (2004)

Courant, R, Hilbert, D: Methods of Mathematical Physics. Partial Differential Equations, Interscience, New York (1962)

Lavrent’ev, MM, Romanov, VG, Shishatskii, SP: IllPosed Problems of Mathematical Physics and Analysis, Am. Math. Soc., Providence (1986)

Mikhailov, VP: Partial Differential Equations, Mir, Moscow (1978)

Evans, LC: Partial Differential Equations, Am. Math. Soc., Providence (1998)

Lions, JL, Magenes, E: Nonhomogeneous Boundary Value Problems and Applications, Springer, London (1972)