Holographic particle localization under multiple scattering Download: 531次
1 Introduction
Three-dimensional (3-D) particle-localization using in-line holography is fundamental to many applications, such as biological sample characterization,1,2 flow cytometry,3,4 fluid mechanics,5,6 and optical measurement.7
Multiple scattering induces a nonlinear relation between the permittivity contrast and the scattered field, making it difficult to invert.13 Many algorithms have been proposed to solve the inverse multiple scattering problem and demonstrated improved performance over single-scattering methods, such as iterative Born series,14
To calculate multiple scattering, we build our model based on the Born series expansion.13 To make it computationally efficient, we take a multislice approximation by discretizing the 3-D object volume into a series of two-dimensional (2-D) thin axial slices. At each slice, each object voxel takes a uniform refractive index value. Between neighboring slices, the uniform background medium is assumed. By adjusting the voxel size and interslice distance, our model allows us to flexibly trade computational complexity for model accuracy. At the limit when the voxel size equals the interslice distance, our discretization reduces to the existing approaches in Refs. 1416" target="_self" style="display: inline;">–
To compute multiple scattering, we introduce a 3-D-to-3-D operator to efficiently evaluate the internal scattered fields within the volume. The computational framework discretizes the 3-D object as a set of 2-D slices, and multiple scattering is modeled as recursive propagation among them. Starting from the initial field, each subsequent recursion estimates the next higher-order scattering term within the object volume. This process can be carried out up to an arbitrary order until the field converges to a steady state. To evaluate the convergence, we adapt a metric derived from the residual error of the internal field.19,25 Next, we devise a 3-D-to-2-D operator that computes the external scattered fields by propagating the multiply scattered internal 3-D field to the 2-D sensor plane. Finally, the intensity measured by the hologram is the interference between the scattered and the unscattered fields. This further complicates the model by introducing the “twin-image” problem.34 If only single scattering is considered, our model reduces to linear compressive holography.11 As a result of multiple scattering computation, the hologram encodes information about the high-angle scattering within the volume, which is otherwise ignored in single scattering-based methods. We show that this extra information leads to better recovery of the scatterers, in particular at larger depths.
To solve the inverse scattering problem, we derive an optimization procedure that iteratively minimizes the data-fidelity term measuring the difference between the estimated and measured holograms and imposes a sparsity-promoting regularization on the object. The overall structure of the algorithm follows the proximal-gradient method.35 The key ingredient is the gradient computation of the data-fidelity term. Conveniently, our recursive forward model leads to a similarly structured recursive gradient computation. Further exploiting the convolution structure in the scattering operators, the algorithm is implemented using efficient FFT-based computations.
Distinct from prior Born-series-based models,14
An important feature of our multislice-based framework is that the 3-D object can be flexibly estimated with any desired number of axial slices, as set by the targeted resolution. In particular, we show that it is possible to use much fewer axial slices to achieve high localization accuracy while still exploiting the extra information contained in the multiple scattering. This allows us to handle much larger scale problems with reduced computational cost as compared to existing techniques that are often limited by fine sampling requirements.
Single scattering-based methods tend to underestimate the refractive index contrast. This underestimation can be mitigated by incorporating multiple scattering.17,25,37,38 We show this effect using our multislice-based approach in single-shot in-line holography and demonstrate improved particle localization and axial resolution under multiple scattering.
Next, we demonstrate the localization accuracy of our method by imaging 3-D distributed particles in water at various densities in both simulation and experiment. To facilitate quantitative comparison of different methods, we use a classification framework and use the receiver operating characteristic (ROC) curve to determine each method’s best performance. At low particle density, our multiple-scattering model converges to the single scattering solution as expected since the information is dominated by the first-order scattering. At high particle density, our model largely improves the accuracy since multiple-scattering becomes more significant. We observe that the localization accuracy is highly depth-dependent. Following the classification framework, we use the Dice coefficient39 to quantify the localization result slice-by-slice. We show that our multiple-scattering model provides greater improvement at larger depths.
2 Theory and Method
2.1 Forward Model
Consider the imaging geometry in
Fig. 1. In-line holography with multiple scattering. (a) A plane-wave is incident on a 3-D object containing distributed scatterers. The field undergoes multiple scattering within the volume and then propagates to the image plane. A hologram is recorded, which is then used to estimate the unknown scatterers’ distribution. (b) An inline holography setup is used that consists of a collimated laser for illumination and a 4F system for magnification. (c) The raw data are a single hologram. (d) The reconstruction implements a nonlinear inverse multiple scattering algorithm.40 (e) The output estimates the 3-D distribution of the scatterers.
The background-removed hologram thus represents the real component of the scattered field at the measurement plane and is given as
Equations (3) and (4) can be discretized to get the following recursive forward model:
We consider the 3-D volume to be a set of discrete 2-D slices along the longitudinal axis.
Fig. 2. Illustration of the 3-D internal scattered field operator in Eq. (6). (a) Each object slice is first voxelwise multiplied by the lower order scattered field ; it is then propagated to every other slice within the volume. (b) This computed scattered-field is added to the incident-field to obtain the next higher-order Born-field . This process is recursively applied to compute the multiply scattered field within the volume.
An important numerical treatment to
2.2 Inverse Problem
To estimate the object
The minimization in Eq. (7) is implemented via the proximal-gradient method,46 in which the
Similar to the forward model, the gradient computation is also a
3 Results
We test our model on both simulations and experiments. In our experiment, the inline holography setup uses a linearly polarized HeNe laser (632.8 nm, 500:1 polarization ratio, Thorlabs HNL210L) that is collimated for illumination [
Importantly, Eq. (6) requires computation of high-angle multiply scattered field propagating within the volume; thus the internal field needs to be sampled at the Nyquist rate
For large-scale simulation, we model the system parameters to approximately match the physical setup. On such a scale, rigorous solutions such as FDTD are computationally prohibitive, and sample complexity makes analytical solutions such as Mie theory nontrivial. We first study the effect of multiple scattering on simulated holograms using 3-D SEAGLE,25 which is an accurate forward model that incorporates multiple scattering, including scattering within each particle. It is based on a rigorous optimization procedure that solves the Lippmann Schwinger equation. We further simulate the hologram at high particle densities using our model with a sufficiently high scattering order, e.g.,
The Boston University Shared Computing Cluster (SCC) was used for all computations. The average times of computing one iteration for the single- and multiple-scattering models on a
3.1 Effect of Multiple Scattering in Small-Scale Inversion: A Multislice-Based Approach
It has been shown that in the presence of strong multiple scattering, the single-scattering models underestimate the permittivity contrast.17,25,37 Here we validate our model on a small-scale simulation and make similar observation by showing that the underestimation is mitigated as multiple scattering is incorporated in the inversion.
The utility of our multislice-based computational approach is also demonstrated, in which the number of axial slices can be arbitrarily chosen in the inverse reconstruction. Effectively, we approximate the 3-D object with a fixed number of slices, such that the computation is tractable when expanding to large-scale problems.
We simulate a volume of
Fig. 3. Small-scale multiple-scattering inversion. (a) An accurate 3-D forward model is used to simulate the hologram. (b) Multislice 3-D reconstruction is performed from a single simulated measurement using our method. The number of slices in the inverse reconstruction can be flexibly chosen. (c) Full 3-D inversion is performed by reconstructing all axial slices in the original object using our method. The multiple-scattering method outperforms the single-scattering method by providing both more accurate permittivity contrast estimation and improved optical sectioning. (d) Our multislice approach enables 3-D reconstruction using a much reduced number of slices while still maintaining the benefit of incorporating multiple scattering. Reconstruction using only three slices is compared to demonstrate the improved localization capability by our method.
An inline hologram is simulated at
In order to test the utility of our multislice-based approach, we perform reconstruction for two cases. In the first case, we reconstruct all 37 slices for the object [
In the second case, we estimate the object using only three slices to perform the inverse scattering reconstruction [
3.2 Large-Scale Inversion of Multiple-Scattering: Simulation
Next, we demonstrate the inversion of multiple scattering from single-shot measurement in large-scale. For this purpose, we design a simulation that involves estimating the concentration of particles in a suspension from its inline hologram. We show that our multiple-scattering model improves the accuracy in estimating the particle density, particularly at larger depths.
The simulated volume is
As a measure of particle density, we consider the geometric cross-section
In Sec.
Fig. 4. Effect of particle density on the scattered intensity term contribution in the hologram. (a) Contribution is negligible compared to the hologram for low particle densities and becomes gradually important as the particle density increases. (b) The ratio between the total intensity of the hologram and the terms for all values of tested in the simulation. For , the total intensity of the hologram is at least an order of magnitude larger than the term.
For the series expansion approach used in our model, it is important to evaluate its convergence. In
Fig. 5. Validation of our multiple-scattering method on large-scale simulation. (a) Convergence properties of the forward model are studied under varying particle densities. Higher-order scattering is generally required for convergence when the object is strongly scattering. In most cases studied, two orders of scattered field sufficiently capture the majority of the contribution. (b) For higher refractive index contrast ( ), multiple-scattering performs similarly to single-scattering for low concentration ( ), and better than single-scattering for . Reconstruction fails for very high concentration ( ), i.e., when the SNR drops below an empirically chosen value of 1 dB. The error in the predicted versus the ground truth particle concentrations also shows a similar trend. (c) For lower contrast ( ), multiple scattering contributions are negligible and both methods give similar performance. (d) A 3-D rendering depicting localized particles is shown for and . Both methods have similar performance for slices close to the image plane, but our multiple-scattering model performs better at increased depths.
Next, we evaluate the reconstruction accuracy by measuring the signal-to-noise ratio (SNR)
For higher contrast (
For lower index contrast (
The depth-dependent performance is highlighted in
Fig. 6. Reconstruction performance as a function of depth. (a) Segmentation maps of reconstructed slices (zoomed-in regions) at different depths (true positive, white; true negative, black; false positive, green; false negative, pink). For object slices close to the hologram, both multiple and single scattering methods provide high accuracy. At larger depths, the accuracy deteriorates for both methods. Our multiple-scattering method performs notably better at larger depths for higher particle densities. (b) The slicewise Dice coefficient plotted as a function of slice depth also indicates that the multiple-scattering model provides improved segmentation accuracy, especially at greater depth. (c) The particle localization accuracy is quantified using the ROC curve. The curves corresponding to the multiple-scattering solutions consistently have larger areas underneath, indicating better localization accuracy as compared to the single-scattering method in all cases studied.
3.3 Large-Scale Experimental Validation
Finally, we demonstrate our method on a set of large-scale experiments. We reconstruct over 100 million object voxels (
We prepare polystyrene microsphere suspensions, ranging from dense to sparse concentrations via successive dilution, with corresponding
First, we evaluate the results based on the optimization convergence cost [
Fig. 7. Experimental validation of our method in large-scale. (a) The multiple-scattering model converges to a lower cost than the single-scattering model for all concentrations indicating better fit to the cost function. (b) The reconstructed particle density follows a trend similar to the simulation where multiple-scattering performs better than the single-scattering method for ; both methods fail for . (c) As increases, the hologram gradually resembles speckle patterns, as quantified by the CR.
Next, we assess the estimated particle density. Our multiple-scattering model consistently performs better than the single-scattering model for
Evidently, the recorded holograms gradually resemble speckle patterns as the particle density increases [
Finally, we closely examine the 3-D reconstructions for
Fig. 8. A 3-D visualization of the localized particles under different concentrations from our experiment and their lateral cross sections at different depths. For low density, both multiple- and single-scattering methods perform similarly. For high density, the underestimation of particles from the single-scattering method is clearly visible, especially at increased depth. Our multiple-scattering model mitigates the underestimation as it accounts for the intercoupling between particles whose strength increases as the depth. The traditional BPM is effective for low density but completely fails for high density and the reconstruction resembles speckles throughout the volume.
4 Conclusion
We have presented a new computational framework for utilizing multiple scattering in in-line holography for large-scale 3-D particle localization. Our model recursively computes both forward and backward multiple scattering in a computationally efficient manner. Both simulations and experiments demonstrate the significance of modeling multiple scattering in alleviating depth-dependent artifacts and improving the 3-D localization accuracy compared to traditional methods. Our method may open up new opportunities for large-scale imaging applications utilizing multiple scattering.
Our model is currently limited by the convergence regime of the classical Born series expansion, preventing its application to particle density higher than 0.1 geometric cross-section. Recent work on convergent Born series expansion52 provides a promising avenue to extend our model to higher scattering scenarios.
The multislice structure proposed in our model provides a flexible framework for trading computational cost for model accuracy. Still, higher-order scattering calculation necessitates longer computational times, which is less appealing for applications requiring real-time reconstructions. To facilitate rapid volumetric estimation without sacrificing accuracy, recent machine learning-based inverse scattering approaches53
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Article Outline
Waleed Tahir, Ulugbek S. Kamilov, Lei Tian. Holographic particle localization under multiple scattering[J]. Advanced Photonics, 2019, 1(3): 036003.