DeepLight: A Sobolev-trained Image-to-Image Surrogate Model for Light Transport in Tissue

TL;DR

Optoacoustic imaging relies on inverting light transport to recover tissue absorption, a challenging inverse problem that benefits from accurate forward models. The authors introduce DeepLight, a UNet-based image-to-image surrogate that predicts absorbed energy $E$ through $E=F\,\mu_a$, and train it with a Sobolev loss that includes a directional-derivative term to align network derivatives with the physical operator. They deploy a biased stochastic approximation of directional derivatives and a nonlinear scaling function $\sigma_{a,c}$ to handle high dynamic range, using Monte Carlo data from ID and OOD tissue generators and MoCA for forward and derivative computations. Results show consistent improvements in forward accuracy and derivative fidelity for both ID and OOD samples, with substantial gains in deeper tissue regions, underscoring the potential of Sobolev-trained surrogates to accelerate and stabilize inverse reconstructions in optoacoustic tomography.

Abstract

In optoacoustic imaging, recovering the absorption coefficients of tissue by inverting the light transport remains a challenging problem. Improvements in solving this problem can greatly benefit the clinical value of optoacoustic imaging. Existing variational inversion methods require an accurate and differentiable model of this light transport. As neural surrogate models allow fast and differentiable simulations of complex physical processes, they are considered promising candidates to be used in solving such inverse problems. However, there are in general no guarantees that the derivatives of these surrogate models accurately match those of the underlying physical operator. As accurate derivatives are central to solving inverse problems, errors in the model derivative can considerably hinder high fidelity reconstructions. To overcome this limitation, we present a surrogate model for light transport in tissue that uses Sobolev training to improve the accuracy of the model derivatives. Additionally, the form of Sobolev training we used is suitable for high-dimensional models in general. Our results demonstrate that Sobolev training for a light transport surrogate model not only improves derivative accuracy but also reduces generalization error for in-distribution and out-of-distribution samples. These improvements promise to considerably enhance the utility of the surrogate model in downstream tasks, especially in solving inverse problems.

Figures (5)

• Figure 1: (a) Illustration of the optoacoustic measurement setup. A probe head with an illumination source is placed on the patient’s skin. Short pulses of laser light excite acoustic pressure waves, which can be recorded by acoustic detectors. (b) Schematic depiction of the data generation and training procedure. Synthetically generated absorption and scattering coefficients ($\mu_a$, $\mu_s^\prime$), and randomly sampled derivative directions ($v_a$, $v_s$) are used as input for the Monte Carlo ray tracer MoCA to compute the absorbed energy ($E$) as well as the corresponding directional derivative ($\nabla_v E$). The training loss ($L$) of the DeepLight network includes a term that penalizes errors in $E$ ($\ell_0$) and a term for the derivative accuracy ($\ell_1$) scaled by the hyper-parameter $\alpha$. (c) Sketch of the network architecture. An adapted UNet architecture was implemented. Notably, after each downsampling step, a downsampled copy of the input was concatenated to the inputs of the next layer. BN: Batch Norm, Conv: Convolution, Avg.: Average, Conv Transp.: Transpose convolution
• Figure 2: (a) Nonlinear scaling function $\sigma$, (b) its derivative, and (c-h) their effects on absorbed energy ($E$) and directional derivative ($\nabla_v E$) in synthetic tissue samples compared to linear and logarithmic scaling. The function is plotted for $a=1$ and two cutoff-scale values $c$, which are also marked as vertical lines in panel (b). The derivatives approach a value of 1 for input values larger than $a$, corresponding to linear scaling. Between $c$ and $a$, logarithmic scaling is observed. For inputs with magnitude below $c$, the derivative flattens out again, approaching a constant derivative when inputs are close to 0. Absorbed energy with (c) linear, (d) $\sigma$ and (e) logarithmic scaling. Directional derivatives with (f) linear, (g) $\sigma$ and (h) symmetric-logarithmic scaling.
• Figure 3: Normalized dataset samples. Panels (a-f) show representative samples of absorption and reduced scattering coefficients ($\mu_a$, $\mu_s^\prime$), as well as the corresponding absorbed energies ($E$) generated by the two in-distribution (ID$_1$ and ID$_2$) generators. $E$ was scaled by $\sigma$ as defined in \ref{['eq:nl']}. Panels (g-i) show $\mu_a$, $\mu_s^\prime$ and $\sigma(E)$ for the out-of-distribution (OOD) generator. In Panels (j-l), samples of the change vectors in absorption ($v_a$), scattering ($v_s^\prime$) and corresponding scaled directional derivative ($\sigma(\nabla_v E)$) from the ID$_1$ generator are shown. Panels (m-o) and (p-r) similarly depict $v_a$, $v_s^\prime$ and $\sigma(\nabla_v E)$ for the ID$_2$ generator and the OOD generator.
• Figure 4: Quantitative comparison of prediction accuracy over the test sets. For each sample, we computed the $L^2$ norm of the prediction error for the scaled absorbed energy and its derivative using the baseline and DeepLight model. To compare errors between test sets, we rescaled the errors by the average $L^2$ norm of the corresponding test set. (a-c) Error distribution for the in-distribution (ID) samples using the ID$_1$ and ID$_2$ generators and the relative gain of the DeepLight model compared to the baseline model. (d) and (e) Error distribution and gain for the out-of-distribution (OOD) samples. (f-i) Average operator and derivative error with respect to pixel depth relative to the skin for both models and the relative gain observed for the ID and OOD samples, respectively. Rel.: relative.
• Figure 5: Qualitative samples of predicted absorbed energy $E$ and directional derivatives $\nabla_v E$ with baseline prediction error and DeepLight prediction error. The sample with the median relative gain of the DeepLight model $\Phi_D$ compared to the baseline model $\Phi_B$ is shown. The color range of the difference plots is constrained to the 99.9$^\mathrm{th}$ percentile of the pixel wise errors. (a-c) shows a representative in-distribution sample, (d-f) an out-of-distribution sample. (g-i) and (j-l) show samples of the ID$_1$ and ID$_2$ derivative generatos. (m-o) shows OOD derivative samples. RMSE: root mean squared error