[Paper Review] A family of inversion formulas in Thermoacoustic Tomography

Linh V. Nguyen|ArXiv.org|Feb 15, 2009

Photoacoustic and Ultrasonic Imaging17 references4 citations

TL;DR

This paper presents a unified family of closed-form inversion formulas for thermoacoustic tomography (TAT) with constant sound speed, applicable in both time- and frequency-domain formulations. By leveraging Hankel functions and intertwining operators, the method generalizes and unifies existing filtered backprojection-type formulas, including those by Finch, Patch, Rakesh, Kunyansky, and Xu-Wang, while resolving ambiguities in their equivalence across dimensions and data types.

ABSTRACT

We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both time-domain and frequency-domain versions. As special cases, they imply most of the previously known filtered backprojection type formulas.

Motivation & Objective

• To resolve unresolved questions about the equivalence and relationships between previously derived inversion formulas in thermoacoustic tomography.
• To unify existing closed-form inversion formulas—particularly those of Finch-Patch-Rakesh, Kunyansky, and Xu-Wang—into a single coherent framework.
• To extend the applicability of inversion formulas to both odd and even dimensions, including finite-time data reconstructions.
• To establish connections between time-domain and frequency-domain inversion formulas through Fourier and Hankel transforms.
• To derive inversion formulas from Neumann-type boundary data, extending prior work limited to Dirichlet data.

Proposed method

• Derives a family of inversion formulas using the Fourier transform of the even extension of thermoacoustic data, denoted $\hat{g}_0(y,\lambda)$, in the frequency domain.
• Introduces a kernel function $G(s,\lambda) = \frac{i}{4}\left(\frac{\lambda}{2\pi s}\right)^{\frac{n-2}{2}}H^{(1)}_{\frac{n-2}{2}}(\lambda s)$, based on Hankel functions of the first kind.
• Applies an intertwining operator $\mathcal{W}$ that maps the wave equation into the Darboux equation, enabling transformation to spherical means operators.
• Uses the spherical Radon transform $\mathcal{R}_S$ and spherical means operator $\mathcal{M}_S$ to relate measured data to the initial pressure distribution.
• Employs symmetry properties of the kernel $K(x,z,\lambda)$ and duality between $\mathcal{W}$ and its adjoint $\mathcal{W}^*$ to derive inversion identities.
• Applies integration by parts and Fourier-Bessel transform inversion techniques to recover the initial function $f(x)$ from data $g$.

Experimental results

Research questions

• RQ1 What is the relationship between the finite-time and infinite-time inversion formulas for even-dimensional thermoacoustic tomography?
• RQ2 How can existing inversion formulas—such as those by Kunyansky and Xu-Wang—be unified under a single mathematical framework?
• RQ3 Can the same family of formulas recover $f(x)$ from Neumann-type boundary data (normal derivative of pressure) as well as Dirichlet data?
• RQ4 What is the role of Hankel functions and the intertwining operator $\mathcal{W}$ in constructing a unified inversion formula across all dimensions?
• RQ5 Under what conditions is the operator $\mathcal{W}\frac{d}{ds}\mathcal{W}^*$ equivalent to the identity, and how does this relate to inversion?

Key findings

• The proposed family of inversion formulas includes all previously known filtered backprojection-type formulas as special cases, including those by Finch, Patch, Rakesh (odd dimensions), Kunyansky (all dimensions), and Xu-Wang (n=3).
• For even dimensions, the method provides finite-time inversion formulas that are equivalent to the infinite-time formulas of Finch, Haltmeier, and Rakesh under appropriate conditions.
• The kernel $G(s,\lambda)$ ensures that the inversion formula is valid for all dimensions $n \geq 2$, with smoothness properties depending on parity: $C^\infty$ for odd $n$, and $C^\infty$ away from zero with logarithmic singularity at zero for even $n$.
• The intertwining operator $\mathcal{W}$ transforms the second derivative into a Bessel-type operator, enabling the derivation of inversion formulas via spherical means and Fourier-Bessel transforms.
• The symmetry of the kernel $K(x,z,\lambda)$ implies that $\int_{\mathbb{R}} K(x,z,\lambda) d\lambda = \int_{\mathbb{R}} K(z,x,\lambda) d\lambda$, which underpins the duality used in the inversion.
• An inversion formula is derived for Neumann data: $f(x) = 2\int_S \mathcal{W}(\partial_{\nu_y}u)(y, |x-y|) d\sigma(y)$, extending prior results to general $n$ and Neumann-type measurements.