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Deep Learning Diffuse Optical Tomography

Jaejun Yoo, Sohail Sabir, Duchang Heo, Kee Hyun Kim, Abdul Wahab, Yoonseok Choi, Seul-I Lee, Eun Young Chae, Hak Hee Kim, Young Min Bae, Young-wook Choi, Seungryong Cho, Jong Chul Ye

arXiv:1712.00912v2cs.CVcs.AIcs.LGstat.ML

TL;DR

Diffuse optical tomography is difficult because photon scattering makes reconstruction nonlinear and ill-posed. This paper designs a deep network around the Lippman-Schwinger equation and trains it on simulated data. The authors report improved reconstructions on simulation and real data, including phantom and animal experiments, while noting that the method requires homogeneous-background flux measurements.

  • Problem

    DOT reconstruction is nonlinear and ill-posed, and conventional methods can depend on assumptions such as small optical perturbations or matched boundary conditions.

    Source: §I · p. 1, A. Lippman-Schwinger integral equation · p. 2

  • Method

    The authors design a deep network to learn Lippman-Schwinger inversion and train it using numerical data.

    Source: §VI · p. 10

  • Results

    The learned network improved reconstruction results over existing approaches in simulation and real-data experiments, accurately reconstructing anomalies without iteration or linear approximation.

    Source: §VI · p. 10

  • Takeaways & Limitations

    The network trained only on simulation data successfully reconstructed sparse phantom and extended mouse targets without regularizers.

    Source: §V · p. 8

    The method requires flux measurements from a homogeneous background, which may be difficult to obtain in clinical applications such as breast imaging.

    Source: §V · p. 10

Abstract

from arXiv · show

Diffuse optical tomography (DOT) has been investigated as an alternative imaging modality for breast cancer detection thanks to its excellent contrast to hemoglobin oxidization level. However, due to the complicated non-linear photon scattering physics and ill-posedness, the conventional reconstruction algorithms are sensitive to imaging parameters such as boundary conditions. To address this, here we propose a novel deep learning approach that learns non-linear photon scattering physics and obtains an accurate three dimensional (3D) distribution of optical anomalies. In contrast to the traditional black-box deep learning approaches, our deep network is designed to invert the Lippman-Schwinger integral equation using the recent mathematical theory of deep convolutional framelets. As an example of clinical relevance, we applied the method to our prototype DOT system. We show that our deep neural network, trained with only simulation data, can accurately recover the location of anomalies within biomimetic phantoms and live animals without the use of an exogenous contrast agent.

I. INTRODUCTION

DOT inversion is difficult because photon scattering produces a nonlinear, ill-posed problem, while conventional methods often rely on full measurements or accurately known boundary conditions. The paper models scattered fluence with the Lippman-Schwinger equation and proposes directly recovering optical anomalies with a neural network.

I. INTRODUCTION

  • DOT is severely nonlinear and ill-posed, and classical reconstruction often requires entire measurements or perfectly known boundary conditions that are usually impractical.The inverse problem is to recover PDE constitutive parameters from boundary measurements after photons scatter through tissue.Source: §I · p. 1
  • Prior machine-learning inverse-scattering methods did not directly invert the three-dimensional distribution of anomalies.The proposed network instead uses a Lippman-Schwinger-based design, with a fully connected layer followed by a CNN encoder-decoder structure.Source: §I · pp. 1–2

A. Lippman-Schwinger integral equation

  • The diffusion model describes fluence using absorption and diffusion coefficients, with boundary conditions, and the paper focuses on absorption changes while treating diffusion as known.The study's DOT system targets absorption changes associated with hemoglobin concentration.Source: A. Lippman-Schwinger integral equation · p. 2
  • The Lippman-Schwinger equation relates scattered fluence to the absorption perturbation and total fluence, retaining nonlinear dependence on the unknown perturbation.Born or Rytov approximations replace total fluence with background fluence under a small-perturbation assumption, which breaks down for significant abnormalities.Source: A. Lippman-Schwinger integral equation · p. 2
  • The paper extends joint sparse-support recovery by using a neural network to recover optical anomalies directly from scattered measurements.The inverse problem is severely ill-posed because diffusive waves dissipate and measurements are fewer than unknowns.Source: A. Lippman-Schwinger integral equation · p. 2

B. Neural network for inverting Lippman-Schwinger equation

The network uses convolutional-framelet theory to translate Lippman-Schwinger inversion into a learned architecture. It combines an approximate inverse mapping with 3D encoder-decoder filtering, while retaining practical limits on parameter identifiability and hyperparameter selection.

B. Neural network for inverting Lippman-Schwinger equation

  • The formulation represents absorption perturbation as a voxel vector and uses convolution filters whose tap size determines their length.The derivation assumes a one-dimensional perturbation for notation, with extension to 3D described as straightforward.Source: B. Neural network for inverting… · pp. 2–3
  • Convolutional framelets yield an encoder-decoder convolution structure by unfolding a high-dimensional Hankel-matrix decomposition into signal space.The filter-channel count is tied to the Hankel matrix rank.Source: B. Neural network for inverting… · p. 3
  • The framelet construction uses paired encoder and decoder filters, and the inversion operator T is substituted into the encoder-decoder representation.The coefficient matrix at the decoder can be replaced by a denoised version.Source: B. Neural network for inverting… · p. 3
  • The proposed network uses a fully connected layer to approximate the inverse operator, paired 3D convolutional encoder-decoder layers, and an intermediate filtering layer.Training learns the mapping and convolutional filters from data; Fig. 1 illustrates the architecture.Source: B. Neural network for inverting… · p. 3, Figure 1 · p. 3
  • Parameter estimation may be non-unique because of scale ambiguity, and filter counts, tap sizes, nonlinearities, and intermediate steps require trial and error.The learned convolution filters encode a manifold structure that makes the method less affected by deviations from the analytic diffusion model.Source: B. Neural network for inverting… · p. 3

III. METHODS

The study used prototype frequency-domain DOT systems, including a multichannel setup for phantom measurements and a single-channel setup for preclinical scans. Their configurations differ chiefly in the number of sources and detectors.

III. METHODS

  • The multichannel system uses three wavelengths, 64 source positions, and 40 simultaneously read avalanche photodiodes, with all measurements acquired in about 30 seconds.A 70 MHz RF signal modulates the sources, and the system records amplitude and phase.Source: §III · p. 4
  • The preclinical single-channel system uses one source fiber and one detector on a motorized probe stage, with scanning positions and sequence set by the operator.It otherwise follows the multichannel system configuration and uses one optical switch to select three wavelengths.Source: §III · p. 4

B. Phantoms and in vivo data

The evaluation used controlled biomimetic and breast-mimetic phantoms with known inhomogeneity locations, as well as tumor-bearing mice. The phantom configurations included cavities or blood-filled structures intended to mimic tumor-like targets.

B. Phantoms and in vivo data

  • Biomimetic and breast-mimetic phantoms with known inhomogeneity locations were used for controlled real experiments.The biomimetic phantom contained a 20 mm diameter, 15 mm high cylindrical cavity filled with an optical inclusion.Source: B. Phantoms and in vivo data · p. 4
  • The breast-mimetic phantom used three blood-filled polymer-tube knots of approximately 20 mm, 10 mm, and 5 mm diameter in a chamber filled with pig lard.The blood was from a rat abdominal aorta, and the medium coagulated at room temperature before imaging.Source: B. Phantoms and in vivo data · p. 4
  • In vivo tests used MC38 colon cancer cells injected subcutaneously into the right flank of C57BL/6 mice, which were scanned under anesthesia.The mice were placed in a chamber filled with a water-and-milk mixture; the experiments had institutional animal-care approval.Source: B. Phantoms and in vivo data · p. 4

C. Data preprocessing

The preprocessing filters source–detector pairs by signal usability, reducing the measurements and fully connected layer size. FEM mesh specifications are provided for test-data generation.

C. Data preprocessing

  • ρ < 51 mm marks the source–detector separations with signals above the noise floor, and only those pairs were used.The authors report that this filtering enhanced SNR and reduced the number of parameters learned by the fully connected layer.Source: C. Data preprocessing · p. 4
  • 2560 to 466 measurement pairs reduced fully connected layer parameters from 137,625,600 to 25,105,920 in the numerical phantom configuration.The architecture specifications list #MEAS = 466 for numerical and breast-mimetic configurations.Source: C. Data preprocessing · p. 4, Table I · p. 5
  • Table II specifies the FEM mesh used to generate test data.Source: Table II · p. 5

D. Neural network training

The network is trained on simulated, phantom-specific data with preprocessing, normalization, and regularization choices described. The section also specifies the conventional DOT algorithms used for comparison.

D. Neural network training

  • Input measurements are centered and scaled, while nonzero label voxels are weighted and then rescaled at inference.The input cloud is centered on the origin and divided by its maximum value; the label weighting is based on the ratio of total voxels to nonzero voxels.Source: D. Neural network training · p. 5
  • 1,500 simulated examples contain up to three spherical heterogeneities with radii of 2–13 mm and optical properties two to five times background values.Training data were generated with FEM using phantom-specific meshes and source–detector configurations; separate data sets were generated for different phantom sizes and configurations.Source: D. Neural network training · p. 5
  • The network maps preprocessed multi-static measurements through a fully connected layer and convolutional layers to a 3D optical-property distribution.The final convolution uses ReLU to ensure positive output values.Source: D. Neural network training · p. 5
  • Training used 1,000 examples, batch size 64, Adam at learning rate 0.0001, and early stopping after 10 epochs without validation-loss improvement.The authors added Gaussian noise with σ = 0.2 and used dropout with p = 0.7; training took about 380 seconds.Source: D. Neural network training · p. 5

E. Baseline algorithm for comparison

  • The baseline comparison used distorted Rytov with modified Levenberg–Marquardt and penalized least-squares algorithms, including ℓp penalties with p = 1 and p = 2.The algorithms were implemented using TOAST and NIRFAST; the regularization constant was selected by trial and error.Source: E. Baseline algorithm for comparison · p. 6
  • Fig. 3 shows numerical-phantom reconstructions from a network trained on that phantom geometry, with ground-truth anomaly locations marked on the 3D views.Source: Figure 3 · p. 6

IV. RESULTS

The proposed network is more robust than conventional methods when test boundary conditions differ from training assumptions. Across biomimetic phantoms and mouse experiments, it recovers anomaly locations while conventional reconstructions show artifacts or underestimated support.

IV. RESULTS

  • Conventional-method performance showed no significant differences across the tested hyperparameter range; the study mainly used λ = 10 for LM and sparse recovery.The quantitative comparison used box plots of numerical-phantom results.Source: §IV · p. 6, Figure 4 · p. 7
  • For the vertically oriented cylindrical biomimetic inclusion, both LM and the proposed network recovered its location, with the proposed method more accurate; sparse recovery underestimated its support.The inclusion's high contrast was also visible in the DBT image.Source: §IV · p. 6, Figure 5 · p. 7
  • Under boundary-condition mismatch, the proposed method outperformed conventional methods in Pearson correlation and SSIM, while its RMSE varied little as conventional RMSE increased.The comparison evaluated RMSE, Pearson correlation, SSIM, and CNR on 38 randomly selected numerical-phantom images.Source: §IV · p. 6, Figure 4 · p. 7
  • In the breast-mimetic phantom, the proposed method recovered inclusion locations accurately, while conventional methods produced artifacts near the sensor plane amid strong background noise.The phantom's inhomogeneous background and blood-filled thin tubes add challenges beyond homogeneous-background and known-boundary assumptions.Source: §IV · pp. 6–7, Figure 5 · p. 7
  • The method recovered inclusions without post-processing and estimated their relative sizes, although these size results were qualitative because ground truth was unavailable.The passage notes that the method recovered inclusions only, even without post-processing.Source: §IV · p. 7
  • In mouse experiments, the network showed high δµ along the spine in the normal mouse and around the right-thigh tumor, unlike conventional reconstructions' thigh or out-of-body artifacts.Measurements were acquired with and without the mouse in a water/milk mixture at a 1000:50 ratio.Source: §IV · pp. 7–8, Figure 6 · p. 8, Figure 7 · p. 8

V. DISCUSSION

The network reconstructs anomaly locations in simulation and real data, including targets unlike its sparse training examples, without case-by-case parameter tuning. Its performance depends on the convolutional stages and remains bounded by hardware constraints and the need for homogeneous-background flux measurements.

V. DISCUSSION

  • The network found accurate inclusion positions without the spurious locations seen in conventional reconstructions, and inferred directly from measurements without per-case tuning or extra preprocessing.The reported examples include the breast-mimetic phantom and the top and middle rows of Fig. 6.Source: §V · p. 8
  • Despite never seeing real data during training or validation, the network recovered both sparse phantom targets and extended animal targets without regularizers.The authors interpret this as evidence that it learns a general inverse function rather than memorizing training examples.Source: §V · p. 8
  • Direct learning from measurement data, rather than conventional reconstructed images, was reported to generalize better to unseen conditions such as boundary-condition mismatch.The conventional initial guess is described as sensitive to boundary conditions and regularization parameters when the real and numerical Green's kernels mismatch.Source: §V · p. 9
  • At least one convolution layer with one filter was needed for accurate anomaly locations; removing the convolutional layers severely degraded performance and prevented training.The fully connected layer produced a noisy reconstruction, refined by subsequent convolutional layers; paired encoder-decoder filters performed better than a single convolution layer.Source: §V · p. 9, Figure 8 · p. 9
  • Fiber-based detection uses few fibers and requires a liquid-filled tank; coupling, increased scattering, and reduced detected photons can constrain reconstruction.The authors caution that the algorithm may also suffer from these hardware limitations, while noting it can be modified for different systems.Source: §V · p. 9
  • The authors state that the approach may address large absorption changes and propagation-model deviations, but not the need for homogeneous-background flux measurements.They note that obtaining this background measurement can be difficult in clinical settings such as breast-cancer imaging.Source: §V · pp. 9–10

VI. CONCLUSION

The paper presents a data-driven approach to DOT inverse scattering, designed around the Lippman-Schwinger equation. Trained only on numerical data, it improves reconstructions in simulation and real experiments, with rapid end-to-end processing.

VI. CONCLUSION

  • The proposed network learns DOT's nonlinear inverse-scattering physics through a design based on the Lippman-Schwinger equation, rather than treating reconstruction as black-box image denoising.The network was trained only on numerical data and improved reconstruction results in simulation and real-data experiments without iterative procedures or linear approximation.Source: §VI · p. 10
  • The learned network accurately reconstructs optical anomalies and processes new data end-to-end in a few hundred milliseconds, a speed the authors identify as useful for dynamic imaging applications.The reported scope includes both simulation and real-data experiments.Source: §VI · p. 10

APPENDIX A MEASUREMENT DATA CALIBRATION

The appendix describes calibration of real and simulated measurements and choices made to keep one network architecture usable across datasets. Its numerical-phantom studies report limited sensitivity to added denoising layers and filter-channel count.

APPENDIX A MEASUREMENT DATA CALIBRATION

  • Simulation amplitudes were scaled and shifted to match real-data peak magnitudes, while the noisy measurements motivated robust training with additive noise and dropout.Bulk optical properties were estimated by fitting experimental data to a diffusion-equation model under a uniform-medium assumption.Source: Appendix A · p. 11

APPENDIX B HYPER-PARAMETER SELECTION

  • For consistency across phantom and animal experiments, the same fixed-layer architecture was used, changing only the number of channels.The authors state that exhaustive architecture search for every experimental condition is difficult.Source: Appendix B · p. 11

A. Number of intermediate convolutional layers for denoising

  • In the numerical-phantom network, the fully connected layer accounted for 25,105,920 of 25,219,968 parameters (99.55%), limiting convolutional layers because of physical GPU memory constraints.Source: A. Number of intermediate… · p. 11

B. Choice of the number of filter channel r

  • Changing the filter-channel count produced marginal or negligible performance variation on the numerical phantom, whose smooth spherical inclusions could be fitted with low-rank filters.The authors attribute this insensitivity to the simplicity of the simulated abnormalities.Source: B. Choice of the number of filter… · pp. 11–12

C. MATLAB code for CNR calculation

The appendix defines a contrast-to-noise calculation and reports tests of signal-to-noise robustness and the network's layer roles. Performance varied only marginally across tested noise levels, while convolutional layers refined the fully connected layer's rough output.

C. MATLAB code for CNR calculation

  • CNR is calculated from the ROI and background means and variances, weighted by their respective area fractions.The provided code defines the ROI from nonzero ground-truth entries and the background from zero entries.Source: C. MATLAB code for CNR calculation · p. 12

D. Dependency on SNR

  • Performance variations were marginal across tested SNR values, although the test measurements had added white Gaussian noise and training used clean measurements only.The noise robustness experiment varied SNR in decibels.Source: D. Dependency on SNR · p. 12, Table V · p. 13

E. Role of the fully connected layer

  • The fully connected layer produced a noisy rough solution by inverting the forward operator; subsequent convolution layers refined it into a clean output.At least one convolution layer with a single filter was needed to recover accurate anomaly locations.Source: E. Role of the fully connected layer · p. 12, Figure 11 · p. 12

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