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High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images
Shwetadwip Chowdhury, Michael Chen, Regina Eckert, David Ren, Fan Wu, Nicole Repina, Laura Waller
TL;DR
Standard ODT is sensitive to interferometric instabilities and typically limited to weakly scattering samples. This work uses intensity-only measurements and a multi-slice beam-propagation model to reconstruct high-resolution 3D RI in multiple-scattering biological samples, including a whole C. elegans worm.
Problem
Standard ODT relies on interferometric systems and reconstruction frameworks typically limited to weakly scattering samples, excluding multiple-scattering biological specimens.
Method
The method uses a multi-slice beam-propagation model with intensity-only measurements from different illumination angles to reconstruct 3D refractive-index distributions.
Results
The method reconstructed 3D RI with high fidelity in 3T3 fibroblasts, C. elegans embryos, and a whole C. elegans worm, including high-resolution NA>1.0 imaging of multiple-scattering samples.
Takeaways & Limitations
Intensity-only computational imaging extends high-resolution 3D RI microscopy to dense cell clusters and multicellular organisms without requiring samples to be weakly scattering.
Takeaways & Limitations
The approach requires higher computational resources than standard ODT, and more acquisitions or higher reconstruction resolution further slow computation.
Abstract
from arXiv · showhide
Optical diffraction tomography (ODT) reconstructs a samples volumetric refractive index (RI) to create high-contrast, quantitative 3D visualizations of biological samples. However, standard implementations of ODT use interferometric systems, and so are sensitive to phase instabilities, complex mechanical design, and coherent noise. Furthermore, their reconstruction framework is typically limited to weakly-scattering samples, and thus excludes a whole class of multiple-scattering samples. Here, we implement a new 3D RI microscopy technique that utilizes a computational multi-slice beam propagation method to invert the optical scattering process and reconstruct high-resolution (NA>1.0) 3D RI distributions of multiple-scattering samples. The method acquires intensity-only measurements from different illumination angles, and then solves a non-linear optimization problem to recover the sample 3D RI distribution. We experimentally demonstrate reconstruction of samples with varying amounts of multiple scattering: a 3T3 fibroblast cell, a cluster of C. elegans embryos, and a whole C. elegans worm, with lateral and axial resolutions of 250 nm and 900 nm, respectively.
I. Introduction
The paper develops intensity-only 3D RI microscopy using a multi-slice beam-propagation model to address high-NA reconstruction of multiple-scattering samples. MSBP models propagation through thin layers, while the exit field captures accumulated diffraction and multiple-scattering information.
- Motivation: Fluorescent imaging requires exogenous labels, whereas ODT reconstructs 3D refractive-index distributions from intrinsic optical variation.ODT therefore avoids some drawbacks associated with fluorescent imaging, including photobleaching, slow acquisition, and low signal-to-noise.
- Motivation: Earlier intensity-based MSBP reconstruction used a 2D gradient-descent simplification that accumulated error as axial sampling density increased.The paper presents a new MSBP-based technique using intensity-only measurements for 3D RI reconstruction of multiple-scattering biological samples.
- Multi-slice beam propagation: MSBP approximates a 3D object as thin layers and models light propagation through sequential layer-to-layer electric-field propagation.Each layer has a complex transmittance determined by its refractive index and the surrounding medium.
- Multi-slice beam propagation: The exit field y_N(r) accumulates diffraction and multiple-scattering effects during propagation and contains information about the sample’s 3D structure.The image-plane field is formed by pupil filtering and propagation from the sample volume, followed by intensity measurement.
B. Inverse problem formulation
The inverse problem estimates a sample’s 3D RI from intensity measurements acquired under multiple illumination angles. Reconstruction minimizes the discrepancy between measured amplitudes and amplitudes predicted by the nonlinear forward model.
- Measurements: Multiple measurements are acquired at varying illumination angles, with each incident planar field represented by its wave-vector.The corresponding measurements are intensity images indexed by illumination angle.
- Objective: The reconstruction estimates 3D RI by least-squares minimization of differences between measured amplitudes and forward-model predictions.Measured amplitude is the square root of intensity, while the forward model predicts the camera-plane electric field.
- Objective: The 3D RI variable n(r_3D) maps the 2D position and layer index to the refractive index of each layer.The nonlinear operator predicts the measured 2D electric field from the 3D RI and an incident illumination field.
C. Reconstruction framework
The reconstruction iteratively alternates forward propagation, residual and back-propagation updates, illumination-angle diversity, and 3D total-variation regularization. Iterations continue until the cost function levels out.
- Iterative reconstruction: Forward-model residuals drive back-propagation and gradient-descent updates through every layer of the reconstruction volume.The back-propagation recursion proceeds from layer N toward layer 1, with α controlling the gradient-descent step size.
- Iterative reconstruction: The algorithm initializes an N-layer reconstruction volume with the surrounding-medium refractive index and begins iterative optimization.Each iteration randomly selects an illumination angle and computes the corresponding layer fields and predicted intensity.
- Illumination diversity: Each illumination angle incrementally refines the layer RI estimates, which are consolidated into a single 3D RI volume after one complete angular pass.Angular diversity supplies successive measurements for refinement.
- Regularization: 3D total-variation regularization stabilizes convergence against camera noise, coherent source noise, and optical aberrations unaccounted for by the forward model.The regularization strength is controlled by β, and the regularized volume becomes the current iterative RI estimate.
- Convergence: Iterations repeat until the iterative cost function levels out with respect to iteration index d.This criterion identifies convergence of the reconstruction process.
A. Optical system design
The optical system uses a fiber-coupled LED, angle scanning, Fourier-plane NA control, and a camera to acquire intensity measurements. Detection NA is limited to 1.1 to reduce aberrations while targeting submicrometer theoretical resolution.
- Optical hardware: A 532 nm LED is fiber-coupled into a 50 µm-core multimode fiber before transmission imaging onto a 20-megapixel CMOS sensor.The fiber increases spatial coherence while avoiding speckle and other coherent artifacts associated with highly coherent illumination.
- Optical hardware: An adjustable Fourier-plane iris tunes the system NA, which is limited to NA = 1.1 to avoid aberrations.For weakly scattering samples, this corresponds to theoretical lateral and axial resolutions of 240 nm and 890 nm, respectively.
B. Data acquisition
The system acquires intensity images under programmable illumination angles and uses postacquisition self-calibration to correct inaccurate hardware-reported angles. A 120-angle spiral scan produces the measurements used for reconstruction.
- Illumination scanning: 120 illumination angles are scanned along a spiral trajectory that fills the pupil.Each scan point triggers the sensor for image acquisition.
- Measured data: The system records raw intensity acquisitions and their associated Fourier-transform amplitudes under varying illumination angles.
- Angle calibration: Hardware-reported illumination angles lack sufficient accuracy for satisfactory 3D RI reconstruction because of system imperfections, sample-induced changes, and misalignments.
- Angle calibration: Algorithmic self-calibration estimates illumination angles from Fourier-transform circle centers and compares them with kinematic-mirror outputs.
IV. Experimental results
The method reconstructs 3D refractive-index distributions across calibration objects, a fibroblast, embryo clusters, and whole worms, including samples exhibiting multiple scattering. Reconstructions agree with known bead properties and reveal cellular and embryonic structure in three dimensions.
- Experimental scope: The experiments span calibration objects, a weakly scattering fibroblast, and increasingly multiple-scattering C. elegans embryos and whole worm.
- Polystyrene microspheres: 3D RI reconstruction of two 3 um polystyrene microspheres captures their spherical geometry and agrees with expected RI values.The spheres have n=1.598 and are immersed in oil with n=1.552.
- Fibroblast cells: The 3T3 fibroblast reconstruction provides lateral cross-sections at z=0.0, 1.05, and 2.10 um, plus a 3D rendering of cell morphology.Both grayscale and RGB views are used for RI visualization.
- C. elegans embryos: C. elegans embryos show Fourier spatial-frequency content outside the circular regions expected for weak scattering, providing evidence of multiple scattering.
- C. elegans embryos: Embryo reconstructions resolve individual cellular compartments, globular heterogeneous structure, eggshells, and RI values from 1.33 to above 1.37.The reconstruction includes lateral and axial views at z=-2.6, 0, and +2.6 um.
D. Whole Caenorhabditis elegans worms
The method reconstructs whole adult C. elegans worms by stitching 14 individually reconstructed patches and resolves major anatomical structures across multiple regions. Quantitative RI cross-sections reveal relatively consistent bulk-tissue RI, heterogeneous fertilized eggs, and highly variable scattering around lipid droplets.
- Whole-worm reconstruction: 14 individually reconstructed patches were stitched together because the imaging objective’s field of view did not cover the whole worm.The final reconstruction contained 1914⨯10408⨯118 voxels.
- Anatomical structures: Three ROIs identify the pharynx-bulb, intestinal cavity, intestinal lumen, fertilized eggs, and distal and proximal gonads.These structures are highlighted in the head, reproductive, and tail regions.
- Quantitative RI: Bulk pharynx and gonad tissue has relatively consistent RI ~1.35, whereas fertilized eggs range from 1.335 to 1.35.The fertilized-egg heterogeneity is corroborated by DIC imaging.
- Quantitative RI: Lipid droplets can reach RI values as high as 1.40 and are often surrounded by low-RI features near 1.335.The intestine and proximal gonads show the most variable scattering because they contain high densities of lipid droplets.
- Practical trade-off: The computational approach provides simple, cost-effective hardware but requires substantially greater computation than standard ODT.More acquisitions or higher reconstruction resolution further slow computation, motivating GPU or cloud-computing acceleration.
6. Conclusion
The study introduces intensity-based MSBP reconstruction for high-resolution 3D RI imaging of multiple-scattering biological samples. Demonstrations span cells, embryo clusters, and a whole worm, while the authors note increased computational demands compared with standard ODT.
- Conclusion: The method reconstructs high-resolution 3D refractive index in multiply-scattering samples using a multi-slice beam-propagation model.It uses intensity-only acquisitions and does not require samples to be weakly scattering.
- Conclusion: The system uses simpler and cheaper optical hardware than standard interferometric ODT while accepting higher computational requirements.The supplement and acknowledgments describe supporting experimental protocols and contributors.
- Conclusion: 3T3 fibroblasts, C. elegans embryos, and a whole C. elegans worm were reconstructed, extending demonstrations from single cells to multicellular organisms.The embryo cluster represents multiple scattering, while the whole worm demonstrates applicability to an organism-scale sample.
- Conclusion: The authors identify the work as the first demonstration of high-resolution (NA>1.0) intensity-based 3D RI reconstruction of multiple-scattering biological samples.The simple optical system can also be adapted to existing fluorescent microscopes for 3D multimodal imaging.
A. Preparation of 3T3 fibroblast cells
The section describes intensity-spectrum behavior under angular illumination, the MSBP reconstruction context, and stitching separate worm volumes into a larger RI volume.
- Fourier-spectrum behavior: Two symmetrically positioned Fourier-space circles characterize intensity measurements from weakly scattering samples under angular illumination.Their center-to-center separation is set by twice the illumination wave-vector magnitude.
- Fourier-spectrum behavior: The electric-field spectrum is formed by pupil filtering shifted by the illumination wave-vector, while the intensity spectrum is the Fourier transform of the squared field magnitude.The pupil radius is NA/λ for the imaging system.
- Volume synthesis: Fourteen overlapping 1200⨯1200⨯100-voxel VOIs were synthesized into a 1914⨯10408⨯118-voxel, 2.3-gigavoxel worm volume.Registration determined translations, zero-padding aligned the VOIs, and weighted masks were used in overlap regions.
- Volume synthesis: Weighted addition across registered VOIs avoided the edge artifacts produced by standard averaging.The masks used normalized weighted averages within overlap regions before combining the aligned volumes.
3. Reconstruction comparison between 1st Born and multi-slice scattering models
The comparison shows that 1st Born and MSBP reconstructions agree for a weakly scattering fibroblast but diverge for multiple-scattering embryos and worms.
- Model comparison: The 1st Born and MSBP reconstructions use the same raw data, so their differences arise from the computational scattering model.The comparison includes a 3T3 fibroblast, C. elegans embryo, and C. elegans worm.
- Sample-dependent behavior: The fibroblast’s two brightfield Fourier circles indicate weak scattering, and its 1st Born and MSBP RI reconstructions match well.The embryo shows signal outside the two-circle region, consistent with multiple scattering.
- Sample-dependent behavior: The worm’s nearly absent brightfield signal marks it as more strongly multiple scattering than the embryo.Its 1st Born reconstruction therefore exhibits greater high-pass filtering than the embryo’s reconstruction.
- Sample-dependent behavior: The 1st Born model cannot reconstruct lower spatial frequencies in the multiple-scattering embryo and worm, effectively high-pass filtering their RI content.The degree of high-pass filtering increases with the amount of multiple scattering.