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Physics-based Learned Design: Optimized Coded-Illumination for Quantitative Phase Imaging

Michael R. Kellman, Emrah Bostan, Nicole Repina, Laura Waller

arXiv:1808.03571v3eess.SPcs.CV

TL;DR

Traditional experimental-design analyses use linear measurement and reconstruction models, while coded-illumination phase retrieval involves nonlinear measurement formation and reconstruction. This work learns illumination patterns by unrolling a reconstruction algorithm with the system physics included, and reports that two-measurement designs achieve quality similar to Fourier Ptychography using 69 measurements.

  • Problem

    Linearized experimental-design analyses are restricted to linear reconstruction and do not necessarily improve accuracy with nonlinear iterative phase recovery.

    Source: §I · p. 1

  • Method

    The approach unrolls a nonlinear reconstruction algorithm and incorporates system physics into learning coded-illumination patterns.

    Source: §I · pp. 1–2

  • Results

    Two-measurement learned designs achieve phase-reconstruction quality similar to Fourier Ptychography with 69 measurements and better than qDPC with four measurements.

    Source: §V · p. 7

  • Takeaways & Limitations

    The learned designs achieve high-quality experimental reconstructions without retraining, while supporting improved acquisition temporal resolution and real-time processing.

    Source: §V · p. 8, §VII · p. 8

Abstract

from arXiv · show

Coded-illumination can enable quantitative phase microscopy of transparent samples with minimal hardware requirements. Intensity images are captured with different source patterns and a non-linear phase retrieval optimization reconstructs the image. The non-linear nature of the processing makes optimizing the illumination pattern designs complicated. Traditional techniques for experimental design (e.g. condition number optimization, spectral analysis) consider only linear measurement formation models and linear reconstructions. Deep neural networks (DNNs) can efficiently represent the non-linear process and can be optimized over via training in an end-to-end framework. However, DNNs typically require a large amount of training examples and parameters to properly learn the phase retrieval process, without making use of the known physical models. Here, we aim to use both our knowledge of the physics and the power of machine learning together. We develop a new data-driven approach to optimizing coded-illumination patterns for a LED array microscope for a given phase reconstruction algorithm. Our method incorporates both the physics of the measurement scheme and the non-linearity of the reconstruction algorithm into the design problem. This enables efficient parameterization, which allows us to use only a small number of training examples to learn designs that generalize well in the experimental setting without retraining. We show experimental results for both a well-characterized phase target and mouse fibroblast cells using coded-illumination patterns optimized for a sparsity-based phase reconstruction algorithm. Our learned design results using 2 measurements demonstrate similar accuracy to Fourier Ptychography with 69 measurements.

I. INTRODUCTION

Coded-illumination uses multiple LED-array patterns to capture intensity measurements for computational phase reconstruction. This work learns illumination designs by incorporating system physics and nonlinear reconstruction into an unrolled network, using few parameters and training examples.

I. INTRODUCTION

  • Coded-illumination uses asymmetric source patterns and multiple measurements to retrieve 2D phase; qDPC uses four rotated half-circle patterns with a partially coherent linearized model.Performance depends on how phase information is encoded by illumination and decoded by phase recovery.Source: §I · p. 1
  • Linearized design analyses are restricted to linear reconstructions and do not necessarily improve accuracy with nonlinear iterative phase recovery.The exact measurement model is nonlinear, so optimal source design cannot be predicted without knowing the sample phase a priori.Source: §I · p. 1
  • Earlier data-driven approaches use CNNs to learn measurement formation and phase reconstruction together, requiring tens to hundreds of thousands of parameters and many training examples.The approach here aims to use known system physics rather than asking a CNN to learn both stages from data.Source: §I · p. 1
  • The method unrolls a generic nonlinear reconstruction algorithm into layers of specified measurement-formation and sparse-regularization operations.These operations replace standard network operations such as generic convolutions.Source: §I · p. 1
  • Each layer uses only a few LED-brightness variables, allowing design learning with fewer than 100 simulated training examples.The learned designs are reported to generalize to experimental biological samples.Source: §I · p. 2

II. QUANTITATIVE PHASE IMAGING

Quantitative phase imaging recovers sample phase from intensity measurements. The section models the nonlinear measurement process and describes approximations and representations used to form phase measurements from single- and multiple-LED illumination.

II. QUANTITATIVE PHASE IMAGING

  • The section introduces quantitative phase imaging as recovering a sample’s complex transmittance from several coded-illumination measurements.The recovery optimization compares measured intensities with measurements predicted from the current phase estimate.Source: §II · p. 2

A. System Modelling

  • A thin sample is modeled by a complex transmission function whose phase depends on refractive-index change, wavelength, and sample thickness.The measured intensity is a nonlinear function of this transmission function, formed through illumination and the microscope point spread function.Source: A. System Modelling · p. 2
  • For closely index-matched biological samples, a weak-object approximation linearizes measurement formation, enabling phase recovery by linear deconvolution with weak-object transfer functions.This approximation relies on a small scatter-scatter term and, for unstained samples, weak absorption.Source: A. System Modelling · p. 2
  • With mutually spatially incoherent LEDs, a multi-LED measurement and its phase transfer function are weighted sums of the corresponding single-LED quantities.The weights are LED brightness values, c_w.Source: A. System Modelling · p. 3
  • The model represents Fourier-space measurements as a system operator applied to phase, with the operator formed from the phase transfer function.For each coded measurement, single-LED weights combine the single-LED transfer functions into a corresponding multi-LED transfer function.Source: A. System Modelling · p. 3

B. Phase Recovery

Phase recovery is posed as a regularized linear inverse problem and solved iteratively with accelerated proximal gradient descent.

B. Phase Recovery

  • Accelerated proximal gradient descent solves the regularized phase-recovery inverse problem through acceleration, gradient, and proximal updates.The proximal operator corresponds to the user-chosen regularizer.Source: B. Phase Recovery · p. 3

III. PHYSICS-BASED LEARNED DESIGN

The section introduces learning coded-illumination designs for a specified phase-reconstruction algorithm and training set. It also notes that measurement images are flattened by background subtraction.

  • The proposed contribution is learning coded-illumination designs for a given reconstruction algorithm and training set.Source: §III · p. 3
  • In practice, y denotes the flattened image after background energy is removed by background subtraction.Source: §III · p. 3
  • The phase-recovery procedure is identified as accelerated proximal gradient descent (APGD).Source: §III · p. 3

A. Unrolled Physics-based Network

The method unrolls reconstruction iterations into a physics-based network whose learned parameters control LED brightness. Its learning objective fits designs to phase-reconstruction error while imposing feasibility constraints.

A. Unrolled Physics-based Network

B. Learning Objective

  • The learning objective minimizes training-data phase-reconstruction error over LED configurations subject to physical-feasibility and non-degeneracy constraints.Source: B. Learning Objective · p. 3

A. Unrolled Physics-based Network

  • The network takes intensity measurements parameterized by coded-illumination designs and produces a phase reconstruction evaluated against ground truth using a user-chosen loss.Source: Figure 2 · p. 4

B. Learning Objective

  • Non-negativity enforces nonnegative LED brightness, scale normalization makes each design's weights sum to one, and geometric constraints exclude conjugate-symmetric LED pairs within a measurement.The geometric constraint confines each source pattern to one major semicircle set.Source: B. Learning Objective · p. 4
  • Accelerated projected gradient descent updates the designs using gradients, then projects them onto non-negativity, geometric, and scale constraints.Source: B. Learning Objective · p. 4

C. Gradient Update

The design gradient is accumulated through the unrolled reconstruction network, accounting for both measurement and WOTF dependence. Acceleration requires backpropagating through two successive iterations.

  • The design-parameter gradient receives contributions at every network layer through both measurement terms and phase WOTF terms.Source: C. Gradient Update · p. 4
  • Because accelerated updates combine the previous two estimates, error backpropagation requires contributions from two successive layers.Source: C. Gradient Update · p. 4
  • The method combines layerwise error backpropagation with gradients of the loss with respect to the design parameters; general-purpose automatic differentiation can also perform the updates.Source: C. Gradient Update · pp. 4–5

IV. RESULTS

The results section specifies the reconstruction settings used for learned designs. It uses total-variation-based sparsity with a proximal operator implemented by a parallel proximal method.

  • The learned design is trained for a given reconstruction algorithm and training set.Source: §IV · p. 5
  • Reconstruction uses 40 iterations and step size α = 0.2, with total-variation-based sparsity.The iteration count is fixed to set CPU time.Source: §IV · p. 5
  • The TV weight τ = 1e−3 trades off TV cost against data consistency, and its proximal operator is implemented in closed form via a parallel proximal method.Source: §IV · p. 5

A. Learning

The learned designs improve simulated phase reconstruction over previously proposed illumination schemes, including when using fewer measurements than traditional qDPC. The designs were trained on a small simulated dataset.

A. Learning

  • Training used 100 simulated examples: 90 for training and 10 for testing.Each example contained a 95 × 95-pixel phase region and 69 simulated single-LED measurements.Source: A. Learning · p. 5
  • The learned designs were evaluated with four, three, and two measurements, while traditional qDPC uses four to cover frequency space.Table I reports mean and standard deviation PSNR for test examples across illumination schemes and measurement counts.Source: A. Learning · p. 5, Table I · p. 6
  • Learned designs significantly improve reconstruction over the compared schemes, recovering high- and low-frequency phase more accurately.The comparison includes traditional qDPC, annular illumination, condition-number optimization, and A-optimal design.Source: A. Learning · p. 5, Figure 4 · p. 6

B. Experimental Validation

On both the USAF phase target and 3T3 mouse fibroblast cells, learned-design reconstructions matched Fourier Ptychography (FP) more closely than traditional qDPC. The comparisons include learned designs using four, three, or two measurements against FP using 69.

B. Experimental Validation

  • The experiments used an LED-array microscope and compared reconstructions with pupil-corrected FP using 69 single-LED measurements at equivalent resolution.The samples were a USAF phase target and fixed 3T3 mouse fibroblast cells.Source: B. Experimental Validation · p. 6
  • For the USAF target, learned-design phase reconstructions remained similar to FP as measurements fell from four to two, while qDPC quality degraded.The cross-sections show learned-design phase closer to FP than qDPC.Source: B. Experimental Validation · p. 7, Figure 5 · p. 7
  • For 3T3 mouse fibroblast cells, learned-design reconstructions matched FP well, while qDPC consistently underestimated phase.Figure 6 compares the methods at four, three, and two measurements against FP using 69 measurements.Source: B. Experimental Validation · p. 7, Figure 6 · p. 7

V. DISCUSSION

The discussion presents two-measurement learned designs as comparable in quality to 69-measurement Fourier Ptychography and better than four-measurement qDPC. It also emphasizes efficient training and reconstruction within a chosen iteration budget.

V. DISCUSSION

  • Two-measurement learned designs achieved phase quality similar to 69-measurement Fourier Ptychography and better than four-measurement qDPC.The authors report a 2× temporal-resolution improvement over qDPC and superior testing performance to previously proposed coded-illumination designs.Source: §V · p. 7
  • Learned reconstructions accurately recovered both low- and high-frequency information.The discussion describes the reconstructions as visually close to the ground-truth phase.Source: §V · pp. 7–8
  • Few weights per measurement allow efficient training on a small simulated dataset, with experimental quality similar to simulation and no retraining.The authors note that large experimental microscopy datasets may be difficult to obtain.Source: §V · p. 8
  • Designs are trained for a specified number of reconstruction iterations, incorporating a compute-time budget into the design process.The authors say this produces high-quality reconstructions within reasonable compute time; restricting qDPC iterations can limit convergence and cause low-frequency artifacts.Source: §V · p. 8

VI. OUTLOOK

The outlook describes possible use of the framework in other physics-based nonlinear reconstruction problems and states conditions for applying it. The conclusion emphasizes accurate experimental phase reconstruction without retraining.

VI. OUTLOOK

  • The framework could be applied to MRI undersampling patterns or fluorescence-microscopy point-spread functions with suitable model parameterization and physical constraints.The outlook names these as examples of other experimental-design applications.Source: §VI · p. 8
  • Applying the method requires differentiable reconstruction updates; a closed-form proximal operator is practically important to avoid backpropagating through many additional iterations.The paper uses anisotropic-TV regularization with a proximal operator approximated in closed form.Source: §VI · p. 8

VII. CONCLUSION

  • The learned coded-illumination designs improve acquisition temporal resolution and enable real-time processing while maintaining high accuracy experimentally without retraining.The conclusion presents the work as a general framework incorporating known physics and nonlinearities of regularized reconstruction.Source: §VII · p. 8
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