[Paper Review] Remarks on the general Funk-Radon transform and thermoacoustic tomography

Victor Palamodov|ArXiv.org|Jan 7, 2007

Photoacoustic and Ultrasonic Imaging8 references16 citations

TL;DR

This paper establishes theoretical foundations for the general Funk-Radon transform in the context of thermoacoustic tomography, providing two-sided estimates, range descriptions, and approximation theorems for the kernel of the dual operator. It proves injectivity of the transformed operator for regularized data and adapts the Kaczmarz method for stable inversion, offering a convergence-proof iterative algorithm for solving the inverse problem under general geometric conditions on the incidence manifold.

ABSTRACT

We study properties of the general integral transform defined for a family of hypersurfaces in a smooth manifold. Estimates of Sobolev norms, range conditions and approximation theorem for the kernel of the integral transform are stated. Applications to the spherical mean transform that appears in thermo/opto/photoacoustic tomography are discussed.

Motivation & Objective

• To establish two-sided estimates and describe the range of the general Funk-Radon transform on manifolds with incidence structures.
• To characterize the kernel of the dual Funk-Radon operator using approximation theorems.
• To prove injectivity of the regularized transform operator $ M^*\varepsilon M $ for $ \alpha > 1/2 $, ensuring uniqueness in inverse problems.
• To adapt the Kaczmarz method for stable iterative inversion of the general Funk-Radon operator in thermoacoustic tomography.

Proposed method

• The paper uses a double fibration framework with a smooth incidence manifold $ F \subset X \times \Sigma $ defined by a non-vanishing differential $ dI \neq 0 $, satisfying rank and transversality conditions.
• It defines the Funk transform $ Mf(\sigma) = \int_{F(\sigma)} \frac{f}{d_xI} $, where $ f $ is a density with compact support, and interprets the result as a hypersurface integral.
• The dual operator $ M^\circ $ is analyzed via geometric and analytic conditions, particularly the non-degeneracy of the Hessian-like determinant $ \det \Phi \neq 0 $, ensuring local diffeomorphism properties.
• The Kaczmarz-type iterative method is constructed using the operator $ Q = I - \omega M^* R^{-1} M $, where $ R = -MM^* + \theta I $, ensuring convergence for $ 0 < \omega < 2 $.
• The method relies on the self-adjointness and invertibility of $ R $, and the contraction property $ \|Qg\| < \|g\| $ for $ M g \neq 0 $, guaranteeing convergence to the solution.
• Theoretical convergence is proven via strong convergence of the sequence $ \{f^k\} $ to the true solution $ f $, under injectivity and range conditions.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions is the general Funk-Radon transform injective?
• RQ2 What is the precise range of the Funk-Radon transform, and how can its kernel be approximated?
• RQ3 How can the Kaczmarz method be adapted to ensure convergence for the general Funk-Radon operator in thermoacoustic tomography?
• RQ4 What regularity conditions on the data ensure stability and uniqueness in the inverse problem?

Key findings

• The operator $ M^*\varepsilon M $ is injective for any $ \delta > 0 $ and all $ \alpha > 1/2 $, ensuring uniqueness in the regularized inverse problem.
• The inequality $ \|Mf\| \geq c \|f\| $ holds for $ \alpha > 1/2 $, providing a two-sided estimate for the transform.
• The Kaczmarz-type iteration $ f^{k+1} = f^k + \omega M^* R^{-1}(\varphi - Mf^k) $ converges strongly to the solution $ f $ for $ 0 < \omega < 2 $.
• The operator $ R = -MM^* + \theta I $ is self-adjoint, positive, and invertible, enabling stable inversion via the iterative scheme.
• The method ensures $ \|f^{k+1} - f\| < \|f^k - f\| $, with convergence to the true solution $ f $, provided $ M $ is injective and $ \varphi $ is in the range.
• The framework generalizes to arbitrary compact manifolds $ K $ and $ \Lambda $ with incidence structures satisfying conditions (i) and (ii), including thermoacoustic tomography settings.