A theoretically exact filtered back-propagationformula for single-time wave inversion

1 Introduction

• Each camera image is a 2D wave field whose initial value is the linear projection of the 3D initial pressure, so reconstruction proceeds by 2D inversion followed by Radon inversion.Source: §1 · p. 1
• Processing camera images independently makes the method modular and fast, requiring a few FFTs, and permits incomplete images to be completed in 2D before final image formation.Source: §1 · p. 1
• The reduction to 2D assumes constant sound speed; for variable sound speed, full 3D methods are a natural alternative.Source: §1 · p. 1
• Fourier back-propagation is exact in odd dimensions but leaves a smooth bias in 2D, which the paper gives in closed form.Source: §1 · p. 1
• An explicit radial filter gives exact inversion for every cT > R, with a physical-space formula and a characterization of filters in this class.The paper compares back-propagation, filtered back-propagation, and Radon-domain band selection in accuracy, noise, and run time.Source: §1 · p. 2

2.1 Problem statement

• Single-time wave inversion recovers f from a wave field at time T, assuming constant sound speed and zero initial time derivative.For PAT, the 2D initial pressure is a projection of the 3D pressure; the stated setup assumes complete images.Source: §2.1 · p. 2

2.2 Notation

• The Radon FBP formula uses the ramp filter Λ, whose Fourier symbol is |σ|, followed by backprojection.The paper uses this notation to relate wave inversion filters to Radon-domain filtering.Source: §2.2 · pp. 2–3

2.3 Back-propagation and its error

• In 2D, back-propagation gives f_BP = f + W_2T f, and the smooth error can be substantial for short propagation times.For a homogeneous disc, the relative error is 9.1% at cT = 1.25R and 3.3% at cT = 2R.Source: §2.3 · pp. 3–4
• For odd dimensions, back-propagation is exact when cT ≥ R, whereas in 2D its error has a closed form and an explicit bound.The 2D error is smooth, so it produces a low-frequency bias rather than edge artefacts.Source: §2.3 · pp. 3–4

2.4 Exact filtered back-propagation in 2D

• For every cT > R, exact inversion follows from an even 1D filter K whose effective kernel matches the ramp kernel on (−2R, 2R).The paper characterizes all filters in this class that satisfy the exactness condition.Source: §2.4 · pp. 4, 6
• Since shifts commute with convolution, filtering the three bands of RfBP with K is equivalent to filtering the single band Rf with effective kernel ATK; consequently K*RfBP = ΛRf for |s| < R.Source: §2.4 · pp. 4–5
• When cT ≥ 2R, a truncated ramp filter yields exact reconstruction, with its correction using back-propagated data outside the source disc.For general cT > R, the paper also gives an explicit filter and expresses the reconstruction through circular means.Source: §2.4 · pp. 5–6
• For cT ≤ R, no exact filter exists within the paper’s characterized class.The stated obstruction is that the required function is not integrable near ±2cT.Source: §2.4 · p. 6

3 Numerical results

• The tests use a smoothed disc and a vessel-like tree, measuring relative L2 error on Bd(0, 0.98R).The phantom tree lies in Bd(0, 0.8R); the reported setup uses R = c = 1 and grid spacing h = 0.02.Source: §3 · pp. 6–7
• Back-propagation errors range from 0.7–12.7 % for cT/R = 3, . . . , 1.1, appearing as a smooth negative bias that agrees with the analytic error up to 1.3 · 10−4.Figure 1 compares the two error fields at cT = 1.25R using the same colour scale.Source: §3 · p. 7, Figure 1 · p. 7
• Filtered back-propagation reduces relative L2 error to at most 7.5 · 10−5, falling to 10−6 at h = 0.01, while the back-propagation error does not change.The remainder at h = 0.01 is attributed to discretisation error.Source: §3 · p. 7, Figure 1 · p. 7
• Radon-domain band selection gives 4.7 % error for the vessels, equal to the discretisation error of the Radon reconstruction.Source: §3 · p. 7
• With 2 % noise, back-propagation and filtered back-propagation amplify white noise equally, with gains of 1.41–1.44.The filter changes only low frequencies.Source: §3 · p. 7
• Filtered back-propagation takes 0.015 s per image, compared with 1.4 s for the Radon route.Source: §3 · p. 7

• Back-propagation is exact in odd dimensions but biased in two, the dimension of full-field PAT projection images.Source: §4 · p. 7
• For every cT > R, an explicit radial filter gives exact inversion using back-propagation with a modified Fourier multiplier.Source: §4 · p. 8
• Open questions include choosing among exact filters under noise, the case cT ≤ R, and using the filter as a preconditioner for variable sound speed and attenuation.Source: §4 · p. 8

• The study is theoretical and numerical, so no ethical approval was required.Source: §5 · p. 8