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On a singular value decomposition of the normal Radon transform operator acting on 3D 2-tensor fields

We consider the problem of reconstructing the potential part of a symmetric 2-tensor field defined in a unit ball by its known values of the normal Radon transform. To solve the problem, a singular value decomposition of the normal Radon transform operator is constructed. The basis functions and fields using the Jacobi, Gegenbauer polynomials and spherical harmonics are constructed.


Introduction
The classical integral geometry operator acting on the vector and the 2-tensor fields is the ray transform [1]. In the two-dimensional case, for the full reconstruction of the vector field, it is necessary to know values of two ray transforms (the longitudinal and transverse ray transforms) [2], since each of them has a nonzero kernel. In order to completely restore the symmetric 2-tensor field in R 2 , the values of three ray transforms must be known (for more details see [3], [4]). In the three-dimensional case, only the solenoidal part of the vector or the symmetric 2-tensor field can be reconstructed by the longitudinal ray transform (see, for example, [5], [6]). To restore the potential part of the vector and the tensor fields, we need to have data of another type. One of the operators, which allows reconstructing the potential part of a vector and a symmetric 2-tensor field, is the normal Radon transform. We mention the report [7], in which the inversion of the Radon transform of a symmetric 2-tensor field in the space R 3 and, in particular, the normal Radon transform, is investigated.
In this paper, we construct a singular value decomposition of the normal Radon transform operator acting on the three-dimensional symmetric 2-tensor fields. The singular value decompositions of the operators of the Radon transform [8]- [10] and the ray transform [11] acting on the scalar fields in R 3 are well known. At the same time, singular value decompositions of the integral operators acting on the vector and the symmetric 2-tensor fields have been obtained relatively recently. In particular, in the space R 2 , the singular value decompositions of the ray transforms of the vector [12] and tensor fields [13], [14] were constructed. In the space R 3 , the singular value decompositions were constructed for the operators acting on the vector fields: for the ray transform operator [15] and for the normal Radon transform operator [16], [17]. We note the papers devoted to the development and research of algorithms for the numerical solution of the vector and the tensor tomography problems in R 2 and R 3 using the above singular value decompositions [18]- [20]. The paper [21] proposes an algorithm for the numerical solution of the 2-tensor tomography problem for reconstructing the three-dimensional potential symmetric 2-tensor field by its known values of the normal Radon transform. The algorithm is based on the method of truncated singular value decomposition. It should be noted that [21] deals with the numerical solution of the three-dimensional tomography problem. At the same time, the theoretical justification of the algorithm proposed is not fully made. Namely, the orthogonality in the main space was verified by using the Wolfram Mathematica 9 software only for basis fields of degrees not exceeding 50. In the present paper, this statement is theoretically justified for the basis fields of an arbitrary degree. Thus, a singular value decomposition of the normal Radon transform operator acting on a three-dimensional symmetric 2-tensor field is constructed.

Definitions
We use the following notations: For functions we use the notations f and g. Denote the potentials by φ(x), ψ(x), . . . We give some definitions for an arbitrary rank m of tensor fields, but only m = 0 (functions), m = 1 (vector fields) and m = 2 (symmetric 2-tensor fields) are used. A set of symmetric m-tensor We use the spaces of the square-integrable symmetric m-tensor fields L 2 (S m (B)). The inner product in the space L 2 (S m (B)) is defined by The Sobolev spaces of the symmetric m-tensor fields are denoted by H k (S m (B)). The spaces of the symmetric 2-tensor fields from H k (S m (B)) that vanish at the boundary of the domain together with all their derivatives up to (k − 1)th order are denoted by H k 0 (S m (B)). Also, we use the weighted L 2 -space L 2 (Z, ρ), where the weight function ρ(s) > 0 is defined on Z. The inner product of the functions f and g in L 2 (Z, ρ) is given by We use the following differential operators: 1) The inner derivation operator d : H k (S m (B)) → H k−1 (S m+1 (B)) acts on the potential ψ and the vector field v as follows: 2) The curl operator curl : H k (S 1 (B)) → H k−1 (S 1 (B)) acts on the vector field w by the rule 3) The divergence operator div : H k (S m+1 (B)) → H k−1 (S m (B)) acts on the symmetric m-tensor field w by the formula We need to recall that the symmetric m-tensor It is well known [1] that every symmetric m-tensor field v ∈ L 2 (S m (B)) can be uniquely represented as the sum In [21], it is shown that every symmetric 2-tensor field can be uniquely decomposed as the The plane P ξ,s ∈ R 3 is defined by the normal equation ξ, x − s = 0 for x = (x 1 , x 2 , x 3 ) and ξ = (ξ 1 , ξ 2 , ξ 3 ), |ξ| = 1. Here |s| is a distance from the plane P ξ,s to the origin and ξ is the normal vector of the plane.
The Radon transform Rf : The normal Radon transform R ⊥ 2 : L 2 (S 2 (B)) → L 2 (Z, ρ) acts on the symmetric 2-tensor field u(x) by the rule Further, we formulate the properties of the normal Radon transform R ⊥ 2 (for details see [21]). Statement 1. For any function ψ ∈ H 2 0 (B) the equality holds. Statement 2. The kernel of the operator of the normal Radon transform R ⊥ 2 consists of any linear combinations of the two types of symmetric 2-tensor fields: 1) solenoidal symmetric 2-tensor fields w such that . From the Statements it follows that by the known values of the normal Radon transform of a symmetric 2-tensor field, only its potential part of the form d 2 ψ, ψ ∈ H 2 (B), can be restored. The purpose of this paper is to solve the problem of reconstruction of the potential 2-tensor fields of the form d 2 φ, φ ∈ H 2 0 (B), by the known values of the normal Radon transform For this a singular value decomposition of the normal Radon transform operator R ⊥ 2 is constructed. The basis functions and fields using the Jacobi, Gegenbauer polynomials and spherical harmonics are constructed.
Recall the definitions and some properties of the orthogonal polynomials.
The Jacobi polynomial P (p,q) n (t) of degree n with the indices (p, q) and t ∈ [0, 1] is defined by the formula where C k n is the binomial coefficient. On the segment [0, 1] the Jacobi polynomials are orthogonal with the weight t q−1 (1 − t) p−q , i.e. we have where Γ(α) is the gamma function and δ nm is the Kronecker symbol. The first and the second derivatives of the Jacobi polynomial P (p,q) n are calculated as The Gegenbauer polynomial C (µ) n (t) of degree n with the index µ is given by where [ · ] denotes the integer part of the number. The Gegenbauer polynomials are orthogonal on [−1, 1] with the weight (1 − t 2 ) µ−1/2 . In other words we have The Legendre polynomial L k (t) of degree k is a particular case of the Gegenbauer polynomial: . These polynomials are orthogonal on [−1, 1], and the norm can be calculated by the formula The associated Legendre polynomial L kl (t) of degree k with the integer index l = 0, . . . , k is defined via the Legendre polynomial The spherical function Y kl of order k with the integer index l = −k, . . . , k is defined by the rule Y kl (θ, ϕ) = L k|l| (cos θ) · cos lϕ, l ≥ 0, sin |l|ϕ, l < 0.
The spherical functions are orthogonal on the unit sphere, and the norm is calculated by the formula The harmonic polynomial H kl (x) of degree k with the integer index l = −k, . . . , k in the spherical coordinate system has the form H kl (r, θ, ϕ) = r k Y kl (θ, ϕ).

The singular value decomposition of the normal Radon transform operator
The basis potential symmetric 2-tensor fields in L 2 (S 2 (B)) are constructed using the method of potentials. Namely, we choose a basis system of functions in H 2 0 (B) and generate a 2-tensor potential basis in L 2 (S 2 (B)) applying the operator d 2 .
In [21], the work on the construction of a singular value decomposition of the normal Radon transform operator R ⊥ 2 was begun. Namely, 1) the images [R ⊥ 2 T kln ](s, ξ) of the fields T kln (x) were calculated; 2) the orthogonality of the functions [R ⊥ 2 T kln ](s, ξ) in the space L 2 (Z, (1−s 2 ) −1 ) was proved; 3) the orthogonality of the fields T kln (x) in the space L 2 (S 2 (B)) for 2n + k + 2 50 using Wolfram Mathematica 9 software was numerically verified.
The numbers G kln (s, θ, ϕ), for a potential symmetric 2-tensor field w = d 2 φ, φ ∈ H 2 0 (B) and a value of the inverse operator can be calculated by the formula
Lemma 2. The values of the basis fields components can be calculated by the formulas T kln ij (r, θ, ϕ) = A ij r k+2 F n + B ij r k F n + C ij r k−2 F n , i, j = 1, 2, 3, where A ij , B ij , C ij are defined in Lemma 1 and F n = (1 − r 2 ) 2 P (k+3.5, k+1.5) n (r 2 ). This fact can be proved by rearranging the terms in the representation from Lemma 1. The proof of the orthogonality of basis fields in this case consists in using in Lemma 2, direct calculation on the integrals by the usage of the orthogonality of the spherical functions Y kl .
The third step. We need to calculate the norm of the basis fields: (T kln ) 2 ij (r, θ, ϕ)   r 2 sin θdθ dϕ dr.
In other words, we have proved, that T kln 2 = 1.
Theorem 1 is proven.