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A new reconstruction method in integral geometry

A general method for analytic inversion of geometric integral transforms is proposed


Introduction
There are numerous applications of integral geometry to various problem of image reconstruction, in particular to X-ray, emission, magnetic resonance, wave, acoustic, thermo and photoacoustic, scattering, Doppler tomographies as well as to radar technique and texture analysis. We propose here a general method of analytic inversion formulas for various transformations in integral geometry. Several examples for the case of integral transforms in a plane domain are given.

Curves and integrals
Let X and Σ be smooth 2-manifolds and Φ be a smooth real function defined in X × Σ such that d x Φ = 0. For any point σ ∈ Σ the set F (σ) = {x ∈ X; Φ (x, σ) = 0} is a smooth curve in X. We call Φ generating function of the curve family {F (σ) , σ ∈ Σ}.
Let dS be the Riemannian area form; we define a Funk-Radon transform generated by Φ by for continuous functions f compactly supported in X. The quotient f dS/d x Φ denotes an arbitrary 1-form q such that d x Φ ∧ q = f dS. It is defined up to a term hd x Φ where h is a continuous function. An orientation of a curve F (σ) is defined by means of the form d x Φ and the integral of q over F (σ) is uniquely defined. Let g be a Riemannian metric in a manifold X; the form d g s = g (dx) is the Riemannian line element. We have then where |∇ g a| . = g (da) is the Riemannian gradient of a function a. Suppose that the gradient factorizes through X and Σ that is |∇ g Φ (x, σ)| = m (x) µ (σ) for some positive continuous functions m in X and µ in Σ. It follows that data of the Funk-Radon transform is equivalent to data of Riemannian curve integrals . The reconstruction problem of a function f from Riemannian integrals R g f is then reduced to inversion of the operator M Φ f.

Main theorem
We assume now that Σ = R ×S 1 and a generating function is linear in the first argument: be a regular generating function in X × Σ such that the function ψ is analytic in X and the principal value integral vanishes for any x = y ∈ X. Then the reconstruction formula holds for an arbitrary function f ∈ L 2 (X) comp . The integral (2) converges in L 2 (X) loc .

Remark 1.
A more invariant form of (2) is a reconstruction of the densityt.
Here the quotient dS/D (x) depends only of the conformal class of the Riemannian metric g. Remark 2. G. Beylkin studied "the generalized Radon transform" in R n [3] which coincides with our Funk-Radon transform M Φ in a Euclidean space (see also the last Section). He constructed a Fourier integral operator which serves as parametrix for this transform. Beylkin's parametrix gives a high frequency approximation to a solution but not an exact inversion due to limitation of the method of FIO. The explicit form of the nucleus N (x, y) does not appear in Beylkin's approach but is important in our method. Note that if the nucleus does not vanish it is still a smooth function in the complement to the diagonal, since condition (ii) keeps the number of real zeros of ψ (x, ϕ) − ψ (y, ϕ) constant. Then the right-hand side of (2) is a parametrix.
Proof of Theorem. To simplify our arguments we assume that X is an open set in R 2 and g is the Euclidean metric, the area form is denoted dx. Let q be a 1-form in X such that dψ ∧ q = dx; we have Mf (λ, ϕ) = λ+ψ=0 f q. For an arbitrary x ∈ X and any function f that vanishes in a neighborhood of x we find since dλ = −dψ as λ + ψ (y, ϕ) = 0 and dλ ∧ q = −dy by definition of q. It follows that the function N is the off-diagonal nucleus of I and it vanishes since of the assumption. Therefore the nucleus I (x, y) of the operator I (see (5) is supported in the diagonal and by Lemma 3.3 below we have I (x, y) = a (x) δ x (y) for some continuous function a. To calculate this function at a point x 0 ∈ X we apply I to a density h ε that is equal to dx in ε-neighborhood X ε of x 0 and h ε = 0 in the complement: Lemma 3. 2 We have for arbitrary ϕ ∈ S 1 , x 0 ∈ X and small ε Integrating over ϕ yields the equation Proof of Lemma. Replacing ψ by ψ/ |∇ψ (x 0 , ϕ)| we can assume that |∇ψ (x 0 , ϕ)| = 1. Shift the origin to the point x 0 and suppose first that ψ (x, ϕ) is linear in x. By an orthogonal transformation we can assume that ψ (x, ϕ) = x 1 . In this case we have In the general case we can write is analytic in 3ε-neighborhood of the origin if ε is sufficiently small. We have The first integral in the right-hand side is equal to 2π and shall show the second integral is small. The field The function σ has a holomorphic continuation to the domain For any point z ∈ S we have The same equation holds in T. It follows that the chain S +T is contained in the complex algebraic curve z 2 = ε 2 . This implies that the integral of the holomorphic form σdz over S + T vanishes. This yields for an arbitrary The left-hand side tends to the second term of (4) as δ → 0. The right-hand side can be estimated from above by the sum and dS is the Euclidean area element. The area of the chain V 1 is estimated by const ε 2 since |∇v| is bounded in X ε . For we have any point z ∈ V 1 This yields This completes proofs of Lemma 3.2 and Theorem 3.1. ◮

Lemma 3.3 The integral transform
is a continuous operator L 2 (X) comp → L 2 (X) loc .

Proof of Lemma. The condition (i) can be written in the form
x 1 , x 2 are coordinates in X. According to [4], Theorem 25.3.1 the map M Φ : L 2 (X) comp → L 2 (Σ) loc is a Fourier integral operator of order −1/2 since the corresponding Lagrange manifold is locally a graph of a canonical transformation (see details in [10]; the bundles Ω 1/2 (X) and Ω 1/2 (Σ) are trivial in our case). The image of this operator is contained in where π X and π Σ are projections of F to X and Σ respectively. The projection π X is obviously proper. The integral transform can be written by means of a Fourier integral operator with the phase function Φ. The corresponding Lagrange manifold coincides with that of M Φ with source and target spaces interchanged. The order of S equals 1/2 due to the factor |t|. It follows that the composition I = SM Φ is well defined as a PDO of order 0. ◮

Integrals of rational trigonometric functions
We study now the condition N (x, y) = 0. A function Any trigonometric polynomial is 2π-periodic and is well-defined and holomorphic in the cylinder C/2πZ. It always has 2k zeros in the cylinder. If a polynomial is real the number of real zeros is even.
Lemma 4.1 Let t (ϕ) , s (ϕ) be real trigonometric polynomials such that deg s < n deg t for a natural n and all zeros of t are real and simple. Then Proof. The function r (ζ) = s (ζ) t −n (ζ) is meromorphic for ζ = ϕ + iτ ∈ C/2πZ and has no nonreal poles since of the assumption. Fix a small positive ε we choose a real continuous function λ = τ (ϕ) defined on the circle 0 ≤ ϕ ≤ 2π that vanishes except for ε-neighborhood of the zero set of t and in ε-neighborhood of each zero α of t. We have t (α + iτ (α) 0) = t (α) + i0 for any zero α and since t (ϕ + iτ ) = 0 for τ = 0 and the function st −n tends to zero at infinity. ◮ The following formula can be useful for calculation of D (x) : where the sum is taken over k zeros of t with positive (or negative) imaginary part.
For a proof we apply the Residue theorem for the form s (ϕ) dϕ/t (ϕ) in the upper half-cylinder C + /2πZ.

Radon's and Funk's reconstructions
We apply Theorem 3.1 to recover some known and unknown formulas for regular curve families.
We apply (9) to the function h (x) = x 0 f (x) and rearrange this formula as follows where dσ = cos θdθdϕ is the area form in the hemisphere S + . This formula coincides with original Funk's result [1] after a partial integration.

Geodesic transform in Lobachevski plane
Take a generating function Φ (x; λ, ϕ) = λ + ψ, ψ = −2 |x| 2 + 1 −1 x, e (ϕ) , −1 < λ < 1 in the unit disc X = D. The curves F (λ, ϕ) are geodesics in the Poincaré model of the Lobachevski plane and d g s = 2 1 − |x| 2 −1 ds is the hyperbolic metric. We have where ds, d g s means the Euclidean and hyperbolic line element respectively. Further since the integral in the right-hand side is equal to 2π 1 − |y| 2 −1/2 , y = 2 |x| 1 − |x| 2 −1 (e.g. Lemma 4.2). By Theorem 3.1 this yields We apply this equation to the function f * (x) = 4 1 − |x| 4 −1 f (x) and get where R f (λ, ϕ) = f d g s and the integral is taken over a geodesic F (λ, ϕ) in Lobachevski plane. This formula is equivalent to Helgason's theorem [6]. Applying a map we transform the Poincaré model to the Lorentz model in the "upper" sheet hyperboloid with the metric induced from the Euclidean metric in E 3 . For any central plane P in E 3 the hyperbola γ = P ∩ Q + is a geodesic curve in Q + . Vice versa, the image in Q + of an arbitrary geodesic circle F (λ, ϕ) = {λ = 2 1 + |x| 2 −1 x, e (ϕ) } is contained in the plane P = {y; λy 0 − e (ϕ) ,y ′ = 0}. This plane is orthogonal to a vector ξ .

Photoacoustic geometries
Elliptical source curve. D. Finch and Rakesh [8] gave a formula of type (2) for reconstruction for the family of spheres centered on a sphere. Another reconstruction formula was proposed by L. Kunyansky [9]; it looks different however can be reduced to a form close to (2) after one-fold integration. We show that a reconstruction can be done for ellipse and more general source curves by means of Theorem 3.1. F. Natterer applied quite different method for reconstruction from the family of spheres centered in an ellipsoid [12]. Take first a generating function Φ = λ + ψ where ψ (x, ϕ) = |x − e (ϕ)| 2 , e (ϕ) = (cos ϕe 1 , sin ϕe 2 ) , where e 1 , e 2 are arbitrary positive numbers. The source curve s = e (ϕ) , ϕ ∈ S 1 is an ellipse with half-axes e 1 and e 2 .
Proof. We have where θ = arg (y − x) and z e . = (z 1 e 1 ) 2 + (z 2 e 2 ) 2 1/2 . This norm and the dual norm z * e = (z 1 /e 1 ) 2 + (z 2 /e 2 ) 2 1/2 fulfil the inequality since y + x * e < 2 for x, y belonging to the support of f . This implies that the trigonometric polynomial (12) has only simple real zeros. By Lemma 4.1 N (x, y) = 0 for arbitrary x = y ∈ E. We have |∇ψ| = 2 |x − e (ϕ)| and and our statement follows. ◮ A similar reconstruction from spherical means in R n can be done by means of Theorem 11.1.

Isofocal hyperbolas and parabolas
Hyperbolas. The equation λ = |x| − εx 1 , ε > 1 defines a fold of the hyperbola with a focus at the origin. The function Φ (x; λ, ϕ) = λ+ψ (x, ϕ) , ψ (x, ϕ) = ε x, e (ϕ) − |x| generates the family of all one-fold hyperbolae with focuses in the origin. The first order trigonometric polynomial ψ (x, ϕ) − ψ (y, ϕ) = ε x − y, e (ϕ) − |x| + |y| has two real zeros if x = y. since ||x| − |y|| < ε |x − y| . By Lemma 4.1 the nucleus N (x, y) vanishes for y = x. We have |∇ψ| 2 = 1 + ε 2 − 2ε |x| −1 x, e and Parabolas. A parabola with a focus at the origin can be given by 2px 1 + x 2 2 − p 2 = 0 where p is a positive parameter. Set λ = p 1/2 and write this equation in the form ( This function generates all parabolas with focus at the origin. For any x = y ∈ R 2 the trigonometric polynomial ψ (x, ϕ) − ψ (y, ϕ) is of order 1 and zero constant term. Therefore it has two real roots and by Lemma 4.1 the function N (x, y) vanishes if y = x. We have Corollary 9.2 For any L 2 -function f with compact support in R 2 \{0} a reconstruction is given by In [5] Fourier coefficients of f are reconstructed from Mf by Cormack's method.

Higher dimensions
Let Φ be a generating function defined in X × Σ where X ⊂ R n is a smooth manifold of dimension n with a Riemannian metric g, Σ = R × S n−1 and Φ (x; λ, ω) = λ + ψ (x, ω), λ ∈ R, ω ∈ S n−1 . We say that a generating function Φ is regular if a n + 1 × n + 1 determinant like (6) does not vanish in F = {Φ = 0} and conjugated points are absent, see [10] for details. The Funk-Radon transform is defined by where dV means the Riemannian volume form and F (λ, ϕ) = {x : Φ (x; λ, ϕ) = 0} . The result of Sec. 3 is generalized as follows. Define a function N (x, y) = Re S n−1 dω (ψ (x, ω) − ψ (y, ω) ± i0) n , x = y where dω the Euclidean volume form in S n−1 and D (x) = 1 |S n−1 | S n−1 dω |∇ g ψ (x, ω)| n Theorem 11.1 If a regular generating function Φ is real analytic in X and N (x, y) = 0 for any x = y ∈ X then a reconstruction of type (2) exists for an arbitrary function f ∈ L 2 (X) with compact support. For even n it reads The integrals converge in L 2 (X) loc .
A proof will be given elsewhere.