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Inversion of spherical means and the wave equation in even dimensions
D. Finch, M. Haltmeier, Rakesh
TL;DR
The paper addresses recovery of functions supported in a ball from boundary-centered spherical means and recovery of wave-equation initial data from boundary traces in even dimensions. It develops filtered back-projection and related trace-based inversion formulas, including a two-dimensional finite-time reconstruction result and a discrete algorithm with second-order accuracy for exact data.
Problem
Recovering functions and wave-equation initial data from spherical means or boundary traces is important for thermoacoustic and photoacoustic tomography, but even-dimensional inversion required new formulas.
Method
The paper develops filtered back-projection formulas for spherical means and connects boundary wave traces to spherical means and trace identities.
Results
In two dimensions, the wave-trace inversion reconstructs supported data using only t ∈[0, 2R0], and the discrete exact-data FBP algorithm has second-order accuracy.
Takeaways & Limitations
The results provide even-dimensional reconstruction formulas for spherical-mean and free-space wave-equation data, including a finite-data algorithm for practical measurements.
Takeaways & Limitations
The absolute convergence of one integral requires continuous G with some decay as t→∞, for example G(p,t)=O(1/t^α) for α>0.
Abstract
from arXiv · showhide
We establish inversion formulas of the so called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form (f,0) (or (0,g)) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided the initial data has support in the ball.
1 Introduction and Statement of Results
The paper develops filtered back-projection inversion formulas for spherical means in even dimensions and relates these formulas to recovering wave-equation initial data from boundary traces. It also addresses practical reconstruction from finite measurements.
- Motivation and applications: The motivating applications are thermoacoustic and photoacoustic tomography, where acoustic measurements are used to recover absorbed energy distributions.The paper relates these measurements to spherical means and wave propagation from impulsively induced initial pressure.
- Spherical mean inversion: The paper establishes a pair of filtered back-projection inversion formulas for spherical means in even dimensions.The results are first stated and proved for circular means in two dimensions, then extended to higher even dimensions.
- Wave-equation inversion: The wave-equation approach obtains spherical means from the solution through an Abel-type equation before applying spherical-mean inversion.The paper also derives trace identities connecting inner products of initial data with weighted boundary-trace inner products.
- Wave-equation inversion: In two dimensions, a function supported in the disk can be recovered from wave-equation boundary data using only times t ∈[0, 2R0].The formula applies to Wf, the time derivative of the boundary-restricted wave solution, despite the unbounded support of the full traces in time.
2 Spherical Means
The section establishes inversion formulas for spherical means in even dimensions, first in two dimensions and then by reduction to higher dimensions. The proofs use an elementary integral identity, harmonic decomposition, and density arguments.
- The proofs begin with an elementary integral identity that is used to establish the two-dimensional inversion theorem.
- In two dimensions, a logarithmic boundary integral represents the fundamental solution of the Laplacian and underpins inversion from circular means.
- The higher-dimensional theorem is derived from the two-dimensional case using spherical harmonics and reduction identities for the Euler-Poisson-Darboux equation.
- The operator identities are first verified on data formed from radial functions and spherical harmonics, then extended by orthogonality, linearity, and density in L2.
- The resulting identities show that the relevant operators act as constant multiples of the identity, yielding the inversion formulas for functions supported in the ball.
3 The Wave Equation
The section develops two types of wave-equation inversion results from boundary traces. One recovers circular means through an Abel-type equation, while the higher-dimensional formulas follow from trace identities and continuity arguments.
- The first wave-equation inversion result is a corollary of the earlier circular-mean formula.
- Boundary traces recover the circular means by inverting an Abel-type equation.
- The adjoint-based derivation requires continuous boundary data with small decay at infinity, such as O(1/t^α) for some α > 0.
- For smooth compactly supported initial data, integration by parts and vanishing boundary terms establish the two-dimensional inversion identities.
- The trace identities extend to higher even dimensions, where the formulas hold pointwise for smooth data after first being obtained in the L2 sense.
4 Numerical results
The paper develops discrete filtered back-projection algorithms for spherical-mean and wave-equation data, using finite-difference, interpolation, and quadrature approximations. The algorithms achieve second-order accuracy for exact data, require O(N^3) operations under comparable discretizations, and produce good reconstructions in noisy numerical tests.
- Algorithm construction: The discrete FBP algorithm approximates the continuous reconstruction formula by replacing the differential, integral, and back-projection operators with finite-dimensional versions.The data are sampled as F_k,m = (M f)(p_k, r_m), and reconstruction uses f_i ≈ (B I D F)_i.
- Algorithm construction: The differential operator is discretized with symmetric finite differences, while the integral operator uses piecewise linear interpolation of the radial data.Both approximations are second order in the radial step h_r.
- Algorithm construction: The back-projection is approximated with the trapezoidal rule and piecewise linear interpolation, setting reconstructed values to zero outside the reconstruction domain.The resulting discretization error is bounded through the approximation errors of the back-projection, integral, and differential operators.
- Accuracy and cost: The derived FBP algorithm has second-order accuracy for exact data.This follows from the second-order approximations of the component operators and boundedness of the integral and back-projection operators.
- Numerical tests: Numerical tests reconstruct a phantom containing characteristic functions and a Gaussian kernel from spherical-mean and wave-equation data with added noise.The experiments use N = N_ϕ = N_r = 300, with 5% uniformly distributed noise for M f and 10% noise for W f; the implementations show good results.
- Accuracy and cost: Algorithm 1 requires O(N^3) operations when N is comparable to the radial, angular, and reconstruction resolutions.Its numerical effort matches that of the classical FBP algorithm used in x-ray CT.