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A Learning Approach to Optical Tomography
Morteza H. Shoreh, Ulugbek S. Kamilov, Ioannis N. Papadopoulos, Alexandre Goy, Cedric Vonesch, Michael Unser, Demetri Psaltis
TL;DR
Three-dimensional tomographic imaging can produce missing-cone and parasitic granular artifacts. This paper trains a neural network by adapting refractive-index voxel values to reproduce measured scattered-light amplitudes, yielding higher-quality reconstructions with those artifacts removed.
Problem
Tomographic reconstruction can contain missing-cone and parasitic granular artifacts, limiting artifact-free imaging of three-dimensional refractive-index distributions.
Method
A neural-network algorithm adapts voxelized refractive-index values by minimizing error between modeled and experimentally measured scattered-light amplitudes.
Results
The learning algorithm accurately reproduces bead shape and produces higher-quality tomographic images while efficiently removing missing-cone and parasitic granular artifacts.
Takeaways & Limitations
Neural-network optimization provides an experimentally demonstrated approach for reconstructing three-dimensional refractive-index distributions with fewer conventional tomographic artifacts.
Takeaways & Limitations
The method assumes that measured scattered light consists of photons scattered only once and does not account for reflections.
Abstract
from arXiv · showhide
We describe a method for imaging 3D objects in a tomographic configuration implemented by training an artificial neural network to reproduce the complex amplitude of the experimentally measured scattered light. The network is designed such that the voxel values of the refractive index of the 3D object are the variables that are adapted during the training process. We demonstrate the method experimentally by forming images of the 3D refractive index distribution of cells.
Experimental Setup
The experiment used a holographic tomography system that illuminated samples from multiple angles and holographically recorded the transmitted scattered light. Samples containing polystyrene beads and cells were measured under 561 nm collimated illumination at 80 angles.
- Experimental Setup: The holographic tomography system illuminated the sample from multiple angles and holographically recorded the scattered light.The optical arrangement was most similar to the system described by Choi et al.
- Experimental Setup: Samples containing polystyrene beads and cells were prepared between two glass cover slides.
- Experimental Setup: 561 nm continuous collimated-wave illumination was applied at 80 different angles.
- Experimental Setup: The amplitude and phase of the transmitted light were measured for each illumination angle.
Mirror)
The method uses holographically recorded detector measurements as training data for a computational model of the experimental setup. The model uses BPM to represent propagation through a sliced inhomogeneous sample.
- Mirror): Holographically recorded 2D detector measurements, obtained with a reference beam, form the training set for the computational model.The sample is imaged onto a 2D detector before holographic recording.
- Mirror): The network is constructed using the beam propagation method (BPM) to simulate the experimental setup.The computational model is trained using the recorded measurements.
- Mirror): The inhomogeneous medium is divided into thin slices along the propagation direction z.This slicing is applied to samples containing beads or cells.
- Mirror): Each slice applies phase modulation from local transverse index variation, followed by propagation through a homogeneous medium at the sample’s average refractive index.The propagation calculation separates local phase modulation from propagation through the homogeneous background.
Methodology
The method reconstructs a 3D refractive-index perturbation by training a layered beam propagation model against experimentally measured complex amplitudes. The adaptable voxel perturbations are optimized iteratively using gradient-based minimization with sparsity and positivity constraints.
- BPM network: The layered structure models beam propagation by summing complex amplitudes and applying phase shifts determined by the unknown 3D index perturbation Δn(x, y, z).The discretized Fresnel diffraction kernel describes propagation between sampled transverse planes, with δ and δz specifying transverse and longitudinal sampling intervals.
- BPM network: In the experiments, the network contains 420 layers, and Δn(x, y, z) is the adaptable variable representing the object.Unlike a conventional neural network, the output is linear in the input complex field amplitude but depends nonlinearly on Δn(x, y, z).
- Optimization objective: Training minimizes the mean squared difference between experimental measurements and BPM predictions for multiple illumination beams.Ek(Δˆn) denotes the current BPM prediction for beam k, while Mk(Δn) denotes the corresponding optical measurement.
- Optimization objective: The BPM is optimized by steepest descent, analogous to neural-network backpropagation, by differentiating the error with respect to each adaptable parameter.The optimization is performed iteratively.
- Constraints: A sparsity constraint enhances contrast, its weight τ is tuned by systematic search, and positivity enforces real, positive index perturbations.The positivity constraint relies on the assumption that the index perturbation is real and positive.
following steepest descent
The method updates the estimated refractive-index change in proportion to the error and its derivative. A recursively computed gradient provides the back-propagation step of the learning algorithm.
- following steepest descent: The error is defined as ϵk = ∥Ek(∆ˆn)−Mk(∆n)∥.
- following steepest descent: The change in ∆ˆn is proportional to the error and its derivative, with α as a constant.
- following steepest descent: A recursive computation of the gradient efficiently provides the back-propagation part of the learning algorithm.
Results
The learning-based BPM reconstruction produced accurate 3D refractive-index images of synthetic beads and biological cells, outperforming inverse Radon reconstruction by reducing missing-cone and diffraction artifacts. It also distinguished overlapping cells and converged to essentially the same result from constant or Radon initialization after 100 iterations.
- Initialization and convergence: After 100 iterations, constant-value and inverse-Radon initializations yielded essentially identical results.This convergence behavior was observed in the HeLa-cell experiment.
- Scattering and image quality: The BPM model allows multiple forward-scattering events and produced better-quality images than Radon reconstruction, including efficient removal of missing-cone artifacts and parasitic granular structure.Reflections were not included, although they could be incorporated without fundamentally altering the approach.
- hTERT-RPE1 cells: The learning reconstruction distinguished two aligned hTERT-RPE1 cells, whereas Radon reconstruction merged them because of z-blurring from the missing cone.The result demonstrates improved separation of stacked biological cells in three-dimensional imaging.