Text from the interactive explanation

Local reconstruction of coefficients in quantitative photoacoustic tomography

Bao, Gang, Karamehmedovic, Mirza, Triki, Faouzi

Quantitative photoacoustic tomography

Recovering what tissue is made of — only in the thin layer where the measurements can be trusted.

Measuring the heat signal from each hidden vessel…

Illustrative Monte-Carlo simulation. Each streak is a packet of near-infrared light taking a random walk through tissue; the amber glow is the heat it leaves behind; cyan rings are the ultrasound that this heat sends to the detector elements on the edges. The dashed band by the lit edge marks a thin layer where light is still strong. All three vessels absorb light equally strongly — only their depth differs.

Light goes in. Sound comes out.

Hearing light inside the body

In photoacoustic tomography , pulses of harmless near-infrared light are sent into tissue. Wherever light is absorbed the tissue warms very slightly, expands for an instant and sends out an ultrasound wave — and detectors on the skin record it. From those sounds, a first step reconstructs a map of where light energy was absorbed: the heat map.

That map is not yet the answer. What we want is what the tissue is made of: how strongly each spot absorbs light and how strongly it scatters it. Recovering those two numbers from the heat map is the second step — the subject of the paper explored here, by Gang Bao, Mirza Karamehmedović and Faouzi Triki.

You can already see the catch above: light fades fast with depth. Experiments put the imaging depth of photoacoustic imaging at typically only a few centimetres, and the deep signals are weak and noisy. Yet most existing reconstruction methods solve equations that tie every point to every other, so they need measurements throughout the entire region — including the noisy depths — and the whole image can suffer, even when the structures of interest are shallow.

The paper's questionCan we work out what tissue is made of locally — only where the light is still strong?

01 · Two hidden numbers

Every spot of tissue keeps two secrets

Absorption (written µa) says how readily tissue soaks up light and turns it into heat. Scattering (µs) says how often light is knocked into a new direction. Light obeys the radiative transport equation: it travels in every direction at once, losing energy to absorption and being redirected by scattering.

Fire photons down this strip of tissue and change the two numbers. More absorption turns more light into heat. More scattering sends more of it wandering off sideways or back — and the far end goes dark either way.

Firing photons…

Coefficients are per strip length; scattering here is forward-leaning (anisotropy 0.9, see chapter 2).

The heat rule

The heat map does not show absorption directly. At every point it records a product:

heat = absorption × light arriving

So a strongly absorbing spot sitting in the dark can glow exactly as brightly as a weak absorber in full light. Try it.

The absorbed energy density is the absorption coefficient times the light summed over all directions d:

Here u(x, d) is the light travelling through x in direction d, the solution of the radiative transport equation. In practice the measured data are Υ·H, where Υ (the Grüneisen coefficient, how efficiently heat becomes sound) is assumed known in the paper.

Spots A and B glow equally on the heat map. Which one absorbs light more strongly?

That is the knot this paper unties. The heat map tangles absorption together with the light field — and the light reaching any point depends on the absorption and scattering of everything it passed through on the way. Working backwards from heat to µa and µs is called an inverse problem.

To untangle it, we first need to know how light changes direction.

02 · How light bends

Mostly forwards, with a nudge

When a photon scatters in tissue, it usually carries on roughly the way it was going, deflected by a small angle. The odds of each new direction are described by the Henyey–Greenstein law , which needs just one number: the anisotropy η (eta), between −1 and 1.

At η = 0, every new direction is equally likely. As η approaches 1, scattering becomes more and more forward-peaked and the beam stays narrower. In biological tissue, scattering is typically strongly forward-directed: η lies between 0 and 1.

Because gentle nudges hardly change the light's course, many forward scatterings act like a few "real" ones. That is captured by the effective scattering µs′ = (1 − η) µs, which shrinks to zero as η approaches 1.

In two dimensions the Henyey–Greenstein phase function gives the probability density of scattering from direction d′ into d:

The photons in this instrument are drawn exactly from this law (in 2D it is the "wrapped Cauchy" distribution). Its average deflection cosine is exactly η.

Forward scattering keeps light moving inwards. But every step costs some of it. How fast does it fade?

03 · The fade — and a guarantee

Light dies away exponentially

Here photons enter a uniform block of tissue (the same illustrative medium as the opening scene), and the plot keeps count of the heat they leave at each depth. The vertical axis is logarithmic: every grid line is ten times less than the one above. A straight line on such an axis means exponential decay — the signal is divided by the same factor for every extra step of depth.

Real detectors have a noise floor. Move it, and see where the heat signal sinks into the noise. Beyond that depth, the measurement says almost nothing about the tissue — and a method that needs data from everywhere must still use it.

Counting photons…

A guarantee, with a condition

Is this fading guaranteed in general, or just what one simulation happens to show? The authors prove (their Theorem 2.1 ) that light inside the tissue decays exponentially with depth — provided scattering is forward-peaked enough, that is, η is above some threshold.

The proof hinges on a number t0, built from the shape of the domain, the weakest absorption and the strongest scattering. If t0 < 1, the guarantee holds. The curve on the left is recomputed in this page from the paper's definition, for the paper's test medium (a unit square, background absorption 0.8, scattering up to 0.5). It crosses 1 near η ≈ 0.52; the authors run their tests at η = 0.6.

Below the crossing the proof simply gives no guarantee. That does not mean the light stops fading — only that this argument cannot promise it.

τ₋(x, d) is the distance from x back to the boundary along −d, i.e. how deep x lies along the light's path. When t0 < 1, the scattering operator is a contraction, the light u is bounded by the illumination g times a factor that decays like e−(min µa)·depth.

If deep light is faint, stay shallow. But how shallow — and what do we gain by it?

04 · The shortcut at the edge

Near the surface, light is simple

Close to where it enters, light has not travelled far enough to do anything complicated. The paper's Theorem 3.1 shows that within a thin layer of depth ρ0, the light can be written as three plain terms:

light ≈ straight from the source − lost on the way + scattered in

— with an error that grows only like depth squared. The leading term identifies absorption; the next correction captures scattering.

A curve and its straight-line start are nearly indistinguishable close to the start (shaded) and drift apart further out — the gap grows like the square of the distance. The exponential fade e−∫µ is the curve; the paper's shortcut 1 − ∫µ is the line.

On the right, the paper's central ray: the light computed carefully (straight-through plus once-scattered) against the shortcut, as the ray goes deeper. Drag the cursor. Then make the bumps of absorption stronger and watch the trustworthy layer shrink.

For x inside the layer, with x′ the entry point and t the distance along direction d:

where µ = µa + µs. It holds for depths up to

With the paper's values (µ up to 2.5, illumination peak 1) this gives ρ0 = 0.096 — about a tenth of the square's width. The "computed" light in this instrument keeps the exact exponential fade and adds once-scattered light; it is an accurate reference near the entry, which is where the comparison matters.

Trust layer ρ0: computing…

A formula simple enough to run backwards. Time to untangle.

05 · Untangling

The local reconstruction lab

Divide the heat by the three-term light formula, expand once more ("linearise"), and the two unknowns appear linearly: a plain system of equations, one per pixel and per illumination, stacked over several beams. Solving it gives µa and µs — but only inside a small region where the light, within depth ρ0, is strong.

This lab re-creates the paper's simulated test in your browser: a unit square lit at the middle of its left edge by three narrow beams; three small hidden bumps of absorption (peak 2.0 on a background of 0.8) and three of scattering (peak 0.5); anisotropy η = 0.6. The reconstruction window on the map is everything the solver ever touches.

Scattering is the harder unknown: it enters the equations only through an integral, never directly.

Use only the leading term: heat divided by the light straight from the source.

The error jumps to around 15%, growing with depth. The leading term forgets the light lost on the way, so it underestimates absorption more and more the deeper it looks. The correction terms are what make 2–3% possible.

Keep only the π/8 illumination and run the full method.

Absorption is still recovered (with a somewhat larger error), but in this re-creation the scattering map comes back essentially empty: from one viewpoint, scattering can hide behind absorption. With two or three beams the scattering blob reappears in the right place.

Move the absorption bumps to new places inside the region, then reconstruct.

The method doesn't know where the bumps are: wherever they sit inside the region, the full method still lands within a few percent. Drag them yourself to check.

Keep the first three terms of the heat expansion, H ≈ µa(m0 + m1 + m2), where m0 is the light arriving straight from the source, m1 (negative) the light lost on the way and m2 the light scattered in. Solving for µa and expanding once:

m1 and m2 depend linearly on µa and µs along the incoming rays, so this becomes a linear system with integral operators T1, T2:

stacked over the illuminations g1, …, gM. Absorption enters through the identity I (directly, pixel by pixel); scattering only through T2 — which is why it is harder to recover. In the paper's test, µa is solved pixelwise on the region R, and µs as a sum of Gaussian blobs on an 11 × 31 lattice, by least squares with small smoothing terms and bounds:

This lab uses a coarser grid than the paper (cells of 0.005), noise-free data generated with the same asymptotic model (as in the paper), and its own regularisation weights and iterative solver, so its numbers are close to — but not identical with — the published ones.

So what, exactly, did the authors establish?

06 · What the paper shows

Two theorems and a simulated test

If scattering is forward-peaked enough (η above a threshold), light decays exponentially with depth. In the test medium the condition t0 < 1 holds above η ≈ 0.52.

Near the entry point, light ≈ straight − lost + scattered in, with an error that grows like depth squared. The same holds for the heat map.

For the test's values, the expansion is valid to a depth of about a tenth of the domain width — and it shrinks when absorption or scattering get stronger.

Pointwise relative error in µa on the order of 2%, solving only in a region next to the entry point.

µa enters through the identity term, µs only through an integral operator, so scattering is recovered less accurately. Linearising also stabilised the solve: it beat the nonlinear optimisation of the original asymptotic model.

Because the unknowns live only in the small local region, the inverse solve took about 5 seconds on a standard desktop. For scale, simulating the full light field on a 100 × 100 × 64 grid (used to check the shortcut) took about 11 minutes.

Keep in mind: all numbers come from computer simulations, not measurements. The reconstruction used noise-free data generated with the paper's asymptotic model, and the Grüneisen coefficient — how efficiently absorbed heat becomes sound — is assumed known.

07 · Back to the opening

Reading the layer where light still speaks

The inset is the absorption map from your latest lab run (the paper's test window, scaled up); the rest is the illustrative opening simulation.

Here is the opening scene again. Deep inside, the light is still faint and the sound it makes is weak and noisy. But the thin layer beside the entry point now holds a recovered map of absorption — the one you just reconstructed.

That is the paper's answer to its question. To read the shallow region of interest, you do not need to trust the deep, noisy data at all. Close to the surface, light follows a simple formula with a known error; with it, the tangle of absorption and light becomes a straightforward set of linear equations, solved only where it matters. In the authors' simulations, that recovered absorption to about 2%, with an inverse solve of about five seconds — a simpler, local reconstruction in the region of interest.

Source & credits

Gang Bao, Mirza Karamehmedović and Faouzi Triki, “Local reconstruction of coefficients in quantitative photoacoustic tomography”, arXiv preprint arXiv:2606.15222 [math.AP], 2026. DOI 10.48550/arXiv.2606.15222 · arxiv.org/abs/2606.15222

Imaging-depth limit as cited by the paper: S. Liu, T. Wang, X. Zheng, Y. Zhu and C. Tian, “On the imaging depth limit of photoacoustic tomography in the visible and first near-infrared windows”, Optics Express 32(4):5460–5480, 2024.

All visuals are original simulations drawn for this page; no figures from the paper are reproduced. The lab re-creates the paper's Section 5 test in your browser, with our own discretisation and regularisation choices. The opening scene and the Monte-Carlo scenes in chapters 1–3 are illustrative. The t0 curve and the central-ray profiles are computed in the page from the paper's definitions.

Typefaces: Instrument Serif and Instrument Sans (The Instrument Project Authors) and IBM Plex Mono (IBM Corp.), all under the SIL Open Font License 1.1.