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Inverse Transport Theory of Photoacoustics

Guillaume Bal, Alexandre Jollivet, Vincent Jugnon

arXiv:0908.4012v1math-phmath.MP

TL;DR

Recovering optical properties from photoacoustic data is difficult because deposited thermal energy combines absorption and radiation intensity. The paper analyzes internal measurements through a singular-component decomposition of the transport measurement operator. It identifies reconstructible attenuation and scattering properties, including the Henyey-Greenstein anisotropy parameter in dimensions 2 and 3, with a stated limitation on general uniqueness.

  • Problem

    Deposited energy is the product σaI, so extracting intrinsic attenuation from it is difficult and using it as a proxy for σa can distort images when intensity is not uniform.

    Source: §1 · p. 2

  • Method

    The authors decompose the internal-measurement operator into singular components to analyze which optical coefficients can be reconstructed from multiple radiation patterns.

    Source: §1 · pp. 2–3, §2.1 · p. 4

  • Results

    The analysis reconstructs intrinsic attenuation and spatial scattering, and uniquely determines the Henyey-Greenstein anisotropy coefficient g(x) in dimensions 2 and 3 when σs(x)>0 almost everywhere.

    Source: §1 · p. 3, §2.7 · p. 12

  • Takeaways & Limitations

    The most singular component provides attenuation and spatial-scattering information, while the once-scattered term provides anisotropy information and supports stable reconstruction.

    Source: §1 · p. 3, §2.4 · p. 8

    The results do not establish uniqueness of optical-parameter reconstruction in the most general setting, although they provide stable, unique reconstructions in several settings.

    Source: §2.4 · p. 10

Abstract

from arXiv · show

We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are then used to construct the deposited thermal energy map. The latter depends on the constitutive optical parameters in a nontrivial manner. In this paper, we develop and use an inverse transport theory with internal measurements to extract information on the optical coefficients from knowledge of the deposited thermal energy map. We consider the multi-measurement setting in which many electromagnetic radiation patterns are used to probe the domain of interest. By developing an expansion of the measurement operator into singular components, we show that the spatial variations of the intrinsic attenuation and the scattering coefficients may be reconstructed. We also reconstruct coefficients describing anisotropic scattering of photons, such as the anisotropy coefficient $g(x)$ in a Henyey-Greenstein phase function model. Finally, we derive stability estimates for the reconstructions.

1. Introduction

Photoacoustic imaging pairs optical contrast with acoustic spatial resolution, but extracting optical coefficients from deposited-energy measurements remains difficult. The paper uses multiple radiation patterns and a singular-component decomposition to reconstruct selected attenuation and scattering properties with stability.

1. Introduction

  • Extracting intrinsic attenuation from deposited energy σaI remains essentially unsolved, and treating σaI as a proxy for σa can distort images when intensity is not uniform.The paper identifies this coefficient-reconstruction step as distinct from recovering the initial thermal source from acoustic data.Source: §1 · p. 2
  • The paper studies high-frequency near-infrared photon transport, modeling photon density with a transport equation that accounts for propagation, absorption, and scattering.The setting is distinguished from low-frequency thermoacoustic tomography, which the paper does not consider.Source: §1 · p. 2
  • The authors decompose the internal-measurement operator into singular components to reconstruct intrinsic attenuation σa and the spatial scattering component σs.The most singular component is associated with ballistic photons.Source: §1 · pp. 2–3
  • The second most singular term, associated with photons scattered once, partially determines scattering anisotropy; in dimensions 2 and 3, it uniquely determines the Henyey-Greenstein anisotropy coefficient g(x).The full scattering phase function cannot be reconstructed by these techniques.Source: §1 · p. 3
  • Reconstructions of the parameters that are identifiable by these methods have Hölder-type stability.When scattering is very large, the paper briefly notes a diffusion-equation regime characterized by D(x) and σa(x).Source: §1 · p. 3

2. Derivation and main results

The section derives a stationary transport model from short-pulse photoacoustic measurements and frames reconstruction of optical coefficients from deposited energy. It uses the difference between light and sound time scales to relate the acoustic inverse problem to an initial thermal source.

2. Derivation and main results

  • The transport model uses total attenuation σ and scattering k, with intrinsic attenuation σa=σ−σs assumed bounded below by a positive constant; the coefficients to reconstruct are inside X.The source is supported outside the bounded domain, and the scattering coefficient vanishes outside it; the domain has a C1 boundary, with convexity used for simplicity rather than required by the results.Source: §2.1 · pp. 3–4
  • Thermal energy is proportional to radiation intensity u and intrinsic absorption σa, linking the transport solution to the heat source measured acoustically.The acoustic pressure model uses known constant coupling coefficient β and sound speed cs.Source: §2.1 · p. 4
  • Because the acoustic source H is space-time dimensional while boundary pressure data have one fewer dimension, the inverse acoustic problem is ill-posed before using the light–sound time-scale separation.The model assumes constant sound speed, rescales it to one, and treats transport as much faster than acoustics.Source: §2.1 · p. 4
  • For a radiation source on a time scale much faster than acoustics, the deposited energy approaches a source supported at t=0, and the acoustic pressure converges weakly to a wave-equation solution.The derivation takes the short-pulse limit ε→0.Source: §2.1 · p. 5
  • After recovering the initial thermal source, the optical inverse problem asks which parts of σ and k can be reconstructed from H0 across multiple radiation patterns.The paper assumes H0 has been accurately reconstructed as a functional of the radiation source.Source: §2.1 · pp. 5–6

2.2. Inverse transport with internal measurements

The inverse transport problem is posed using internal measurements generated by boundary radiation. Its objective is to determine which optical coefficients can be recovered, and with what stability, from the albedo operator.

2.2. Inverse transport with internal measurements

  • The source supported outside X and vanishing scattering outside X allow the transport equation to be written as a boundary-value problem with incoming radiation conditions on Γ−.The incoming conditions φ are obtained by solving the outside-domain transport problem with known σa.Source: §2.2 · p. 6
  • The albedo operator maps incoming boundary radiation to angularly averaged internal measurements, and the inverse problem asks what can be recovered about σ and k from complete or partial knowledge of that operator.The section also asks for stability of the reconstructions.Source: §2.2 · p. 6

2.3. Albedo operator and decomposition

The albedo operator is decomposed into contributions from unscattered, once-scattered, and more highly scattered photons. Their distinct singular behavior supports extraction of information about optical coefficients from internal measurements.

2.3. Albedo operator and decomposition

  • The kernel α represents internal-energy measurements for radiation concentrated near a boundary point and direction, obtainable as a limit of localized source-and-detector experiments.The operator’s kernel accounts for propagation including all scattering orders, while its singular components can be extracted without multiple-scattering effects.Source: §2.3 · p. 7
  • A0 represents particles that undergo no scattering, A1 particles that scatter exactly once, and G2 the remaining contribution to the albedo operator.The full operator is their sum, A=A0+A1+G2.Source: §2.3 · p. 7
  • The attenuation factor E for a sequence of points is the product of segment attenuation factors and measures attenuation along the resulting broken path.The construction extends the two-point attenuation function by induction over the number of path segments.Source: §2.3 · p. 7
  • The ballistic kernel α0 is more singular than α1 and Γ2, while α1 is asymptotically larger than Γ2 near the support of α0.The α1 singularity blows up faster than Γ2 as the observation point approaches the ballistic segment.Source: §2.3 · p. 8
  • The faster α1 singularity carries optical information beyond the ballistic term and allows that information to be reconstructed stably.The once-scattered contribution therefore provides information not contained in the unscattered contribution alone.Source: §2.3 · pp. 7–8

2.4. Stability estimates

The section establishes stability for extracting singular components and optical-parameter functionals from albedo measurements. The single-scattering stability estimate uses measurements away from the ballistic component, and the paper notes that uniqueness is not guaranteed in the most general setting.

2.4. Stability estimates

  • Small albedo-measurement errors yield small errors when extracting the terms that characterize the ballistic and single-scattering components.The section presents stability estimates for useful optical-parameter functionals in terms of measurement errors.Source: §2.4 · pp. 8–9
  • The section derives stability estimates for functionals of the optical parameters from appropriate error metrics on the albedo operator.The stated results include stability of the ballistic term and a separate result for the single-scattering component.Source: §2.4 · p. 9
  • Single-scattering stability is estimated using Γ1, the albedo kernel with the ballistic part removed, rather than errors on A.The relevant singular behavior occurs near the ballistic component and cannot be captured by the L1 norm used for the ballistic estimate.Source: §2.4 · p. 9
  • The results do not guarantee uniqueness in the most general setting, but do provide stable, unique reconstructions in several settings of interest.The supplied passages do not specify the full estimates or all conditions for the single-scattering result.Source: §2.4 · p. 10

2.5. Scattering-free setting

In the scattering-free setting, the albedo operator uniquely determines intrinsic attenuation. The section also gives a stability result under positive upper and lower bounds and describes a local beam experiment that determines attenuation along its direction.

2.5. Scattering-free setting

  • When k ≡ 0, the albedo operator uniquely determines σa(x, v) throughout X and for every direction v.The attenuation is recovered from the ballistic measurement by differentiating the negative logarithm of the relevant expression.Source: §2.5 · p. 10
  • When k ≡ 0 and σa is bounded above and below by positive constants, the section states a stability result for σa.The supplied passage does not include the stability inequality itself.Source: §2.5 · p. 10
  • The uniqueness result is local: a beam in direction v′ passing through x determines σa(x, v′) from measurements along that line.The paper describes this beam experiment as an asymptotic transport solution.Source: §2.5 · p. 10

2.6. Reconstruction of the spatial optical parameters

With scattering present, the section describes local reconstruction of both intrinsic attenuation and total attenuation, then establishes uniqueness and stability under additional assumptions. It also reports stronger-norm stability for σ when the coefficients satisfy a priori regularity bounds.

2.6. Reconstruction of the spatial optical parameters

  • The theorem's assumptions include knowing σ in a δ0-vicinity of the boundary; the paper says this hypothesis is not very restrictive in practice.The constant in the stability estimate also depends on δ0.Source: §2.6 · pp. 10–11
  • Under the stated assumptions, σa and σ are uniquely determined by A, with stability in L1(X) for σa and W^-1,1(X) for σ.The estimate bounds these errors by the L1-to-L1 operator error between the albedo operators; the constant depends on δ0 and uniform bounds.Source: §2.6 · p. 11
  • With k ≠ 0, the ballistic component and the Theorem 2.3 estimate allow local unique reconstruction of σa(x, v′) and σ(x, v′).The local experiment sends a beam in direction v′ through x and uses measurements along its path.Source: §2.6 · pp. 10–11
  • A priori W^r,p regularity bounds on σ yield stability estimates in stronger norms for σ.The stated corollary assumes p > 1, r > -1, and gives estimates for -1 ≤ s ≤ r.Source: §2.6 · p. 11

2.7. Application to Henyey-Greenstein kernels

For Henyey–Greenstein scattering in dimensions 2 and 3, the anisotropy parameter g(x) is uniquely recoverable when σs(x) > 0. Stability depends on the parameter range, while unrestricted scattering kernels remain only partially determined.

2.7. Application to Henyey-Greenstein kernels

  • For an unrestricted scattering kernel, the measurements provide only 3n−3 dimensions of information about its 3n−2-dimensional form.In dimension 2, the recovered information is k(x,v′,v′) + k(x,v′,−v′); in dimensions n ≥ 3, it is an angular integral of the kernel.Source: §2.7 · p. 11
  • For the Henyey–Greenstein phase function in dimensions 2 and 3, g(x) is uniquely determined when σs(x) > 0.The argument uses the phase function's one-parameter representation and the recovered quantities σg(x) and σs(x).Source: §2.7 · p. 12
  • Stability estimates for g(x) hold under positive lower bounds on σs and W1,∞ control of h(g).The estimate constant also depends on δ0, σ0, σs,0, M, and the smaller of the W1,∞ norms of h(g) and h(ĝ).Source: §2.7 · p. 13
  • Reconstruction sensitivity worsens as g approaches 0, but reconstructions are very accurate when g is close to 1.The stated stability constant diverges as the maximum norm of g and ĝ tends to 0, and tends to 0 as their minimum norm tends to 1.Source: §2.7 · p. 13
  • Under corresponding bounds on g, the stability estimate can instead control differences in g², g, or 1/(1−g).The stated range conditions determine which of these error measures applies.Source: §2.7 · p. 13

2.8. Reconstructions in the diffusive regime.

In the diffusive regime, measurements are H(x) = σa(x)I(x), with intensity obtained from a diffusion equation. Attenuation can be recovered from one measurement if D is known, but joint recovery of D and σa requires multiple measurements.

2.8. Reconstructions in the diffusive regime.

  • The diffusion approximation applies when the mean free path is small and intrinsic attenuation is small.For isotropic scattering, the diffusion coefficient is D(x) = 1/(nσs(x)).Source: §2.8 · p. 13
  • When D(x) is known, one measurement determines σa(x) by dividing H(x) by the intensity obtained from the diffusion equation.If the boundary illumination φ is nontrivial and nonnegative, the intensity is positive in X; if φ is bounded below by a positive constant, the reconstruction is unique and stable.Source: §2.8 · pp. 13–14
  • When both D(x) and σa(x) are unknown, at least two measurements are necessary.The paper states that this joint-reconstruction problem will be analyzed elsewhere.Source: §2.8 · p. 14

3. Transport equation and estimates

This section develops the transport equation framework and estimates, establishing well-posedness and decomposing the albedo operator before deriving kernel formulas and boundedness estimates. It also completes proofs of estimates (3.14) and (3.15) and establishes existence and uniqueness of the bounded function Φ used to define Γₙ₊₁.

Transport equation and estimates

  • The boundary value problem is well-posed under the stated bounded-coefficient and positive-absorption assumptions, with I − K invertible and the albedo operator defined on L1(Γ−, dξ).The solution is represented through the albedo operator and distributional kernels associated with iterates of K and their composition with J.Source: §3 · pp. 14–15
  • Explicit distributional-kernel expressions are derived for αm when m ≥ 2 and γm when m ≥ 3.The section also gives formulas for αm and γm in Lemma 3.1.Source: §3 · p. 15
  • The section establishes boundedness estimates for αm and γn+1, which are used with the kernel formulas in proving the stated theorem results.The discussion identifies Lemmas 3.2 and 3.3 as the boundedness results used in the proof of Theorem 2.1.Source: §3 · pp. 15–16

3. Transport equation and estimates

  • The proof concludes that estimates (3.14) and (3.15) hold.Source: §3 · pp. 25–26
  • A unique Φ ∈ L∞(X × Γ−) exists, and it is used to define Γ_{n+1}(x, x′, v′) = |ν(x′) · v′|Φ(x, x′, v′).Source: §3 · p. 26

4. Derivation of stability estimates

The stability proofs isolate ballistic and single-scattering contributions to the measurement operator. They yield coefficient estimates while showing how boundary assumptions and optical depth constrain reconstruction.

4.1. Estimates for ballistic part

  • The ballistic-part estimates control σ and σa through the measurement-operator difference, with constants depending on coefficient bounds and boundary information.The proof uses localized boundary test functions to isolate line contributions; its stability constant for σ is of order e^(M diam(X)), so reconstruction deteriorates as optical depth increases.Source: §4.1 · pp. 26, 27, 28, 30
  • The ballistic stability argument assumes σ is known in a δ0-vicinity of the boundary because the relevant line integral is not directly captured by the internal measurements.When boundary measurements capture the X-ray transform of σ, the text says δ0 can be set to 0.Source: §4.1 · p. 30

4.2. Estimates for single scattering

  • For the single-scattering contribution, the proof establishes that h(g) is strictly increasing on (0, 1).This monotonicity supports recovering g from the single-scattering information in the Henyey–Greenstein case.Source: §4.2 · p. 33
  • The single-scattering stability proof combines estimates for σg with the ballistic bounds for σa and σ.The cited estimates use the coefficient bounds and boundary agreement assumptions stated in the proof.Source: §4.2 · pp. 33–34

Appendix A.

Appendix A derives explicit expressions for the kernels β₂, β₃, and βₘ by induction, then proves Lemma 3.1. It establishes integral representations for the operators using changes of variables.

Appendix A.

  • The appendix derives explicit expressions for β₂ and β₃, then gives the kernel βₘ by induction.These kernel calculations precede the proof of Lemma 3.1.Source: Appendix A · p. 34
  • For m ≥ 2, the derivation obtains the boundary-integral representation (3.7) involving αₘ.The proof treats m = 2 and then m ≥ 3, using changes of variables to reach the representation.Source: Appendix A · pp. 35–36
  • For m ≥ 3, the appendix first establishes the interior integral representation (3.8) involving γₘ, then uses it in proving (3.7).The derivation uses changes of variables in the operator expressions.Source: Appendix A · pp. 34–36
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