Source-linked AI summary
Reference-less measurement of the transmission matrix of a highly scattering material using a DMD and phase retrieval techniques
Angelique Dremeau, Antoine Liutkus, David Martina, Ori Katz, Christophe Schulke, Florent Krzakala, Sylvain Gigan, Laurent Daudet
TL;DR
The paper addresses reference-free measurement of a complex transmission matrix in a highly scattering medium using fast binary-amplitude DMD modulation and intensity-only detection. It compares four phase-retrieval algorithms on noisy experimental data and finds that variational Bayesian methods, especially prVBEM, offer precise, computationally tractable estimates when enough calibration measurements are available.
Problem
Measuring a transmission matrix typically requires holographic detection with a reference beam, creating an interferometric-stability problem, while fast modulators are often not phase-only.
Method
The study measures the complex transmission matrix using binary amplitude patterns from a DMD, intensity measurements without a reference arm, and compares Gerchberg-Saxton, PhaseCut, prGAMP, and prVBEM phase retrieval.
Results
For α ≥3, prVBEM outperforms the other algorithms in prediction correlation, reaching around 0.95, while the estimates are also validated through singular-value distributions and single- or multi-point focusing.
Takeaways & Limitations
A reference-less DMD-based transmission-matrix measurement can combine high speed, high pixel counts, and a simple, fast, robust implementation, supporting applications including focusing and imaging.
Takeaways & Limitations
The Bayesian model assumes additive centered isotropic Gaussian noise and independently uniform missing phases.
Abstract
from arXiv · showhide
This paper investigates experimental means of measuring the transmission matrix (TM) of a highly scattering medium, with the simplest optical setup. Spatial light modulation is performed by a digital micromirror device (DMD), allowing high rates and high pixel counts but only binary amplitude modulation. We used intensity measurement only, thus avoiding the need for a reference beam. Therefore, the phase of the TM has to be estimated through signal processing techniques of phase retrieval. Here, we compare four different phase retrieval principles on noisy experimental data. We validate our estimations of the TM on three criteria : quality of prediction, distribution of singular values, and quality of focusing. Results indicate that Bayesian phase retrieval algorithms with variational approaches provide a good tradeoff between the computational complexity and the precision of the estimates.
1. Introduction
The paper proposes measuring a scattering medium’s full complex transmission matrix with a binary-amplitude DMD and intensity-only detection, avoiding a reference arm. It compares phase-retrieval algorithms and evaluates the estimates through prediction, singular values, and focusing.
- 1. Introduction: DMDs provide binary amplitude modulation at more than 20 kHz, with high pixel counts and low pixel pitch.They offer a faster alternative to commonly used liquid-crystal phase modulators.
- 1. Introduction: Holographic transmission-matrix measurements require a reference beam, creating an unavoidable interferometric-stability problem.The reference may be co-propagating or supplied through an external arm.
- 1. Introduction: The proposed reference-less approach uses a binary DMD and no detection-side reference, but requires phase-retrieval processing on many calibration measurements.Four algorithms are compared on noisy experimental measurements, including computational cost and performance versus measurement count.
- 1. Introduction: The figure contrasts reference-arm holography, DMD holographic phase modulation, and the presented intensity-only measurement approach.The proposed approach is the third configuration in the comparison.
- 1. Introduction: The study evaluates transmission-matrix estimates by output prediction, singular-value distributions, and single- or multi-point light focusing.The introduction states that the measured singular-value distribution varies according to random matrix theory.
2. Experimental setup
The setup illuminates a thick white-paint scattering layer with binary DMD patterns and records the resulting speckle field on a camera. Fast pattern loading and acquisition support calibration while monitoring medium stability.
- 2. Experimental setup: A 1920×1080 DMD switches mirrors between ON and OFF directions, sending selected light toward the focusing system.OFF pixels direct light toward a beam dump, while ON pixels transmit it toward the sample.
- 2. Experimental setup: A 100 mm lens focuses DMD amplitude patterns onto an approximately 100-micrometer white-paint layer that produces a mixed speckle pattern.The DMD pixels correspond roughly to incidence angles on the sample.
- 2. Experimental setup: Random amplitude masks are loaded into the DMD driver and displayed through triggering electronics during calibration.The experiment typically uses a few times more input patterns than the number of controlled input pixels.
- 2. Experimental setup: The transmitted speckle is collected through a microscope objective and detected by a camera, with a 400×400-pixel camera subregion used for speed.The overall acquisition rate is 31 images per second.
3. Estimating the TM with intensity-only measurements and binary inputs
The TM is estimated from binary DMD inputs and intensity-only outputs by treating reconstruction as a complex phase-retrieval problem. Four algorithms are compared experimentally, with prVBEM offering strong prediction performance at relatively low computational cost.
- Problem formulation: The calibration problem estimates a complex-valued transmission matrix from known binary DMD inputs and measurements containing only transmitted-wave moduli.The moduli are obtained as square roots of camera intensity measurements, and each complex-valued TM column is estimated from the input-output data.
- Phase retrieval: Phase retrieval is non-convex and difficult because the observations provide magnitudes without phases.The paper reviews alternating-projection, convex-relaxation, and Bayesian approaches to this problem.
- Bayesian variational approximations: The Bayesian prVBEM method uses variational Bayes expectation-maximization to approximate the posterior over the complex signal and hidden phase variables.The model introduces hidden observation phases and additive Gaussian acquisition noise, then applies a mean-field approximation.
- Experiments: The experiments compare Gerchberg-Saxton, PhaseCut, prGAMP, and prVBEM using a cross-validation-like setup with N = 900 DMD mirrors, M = 40000 camera pixels, and P = 6000 patterns.Each TM row is learned from p = αN calibration measurements with α ranging from 1 to 6, then evaluated on the remaining measurements.
- Results: For α ≥3, prVBEM outperforms the other algorithms with a normalized cross-correlation around 0.95, while its computational cost is closely followed only by prGAMP.PhaseCut has a prohibitive running time in the stated application context, whereas Gerchberg-Saxton is relatively slow but performs well.
- Results: prGAMP achieves good correlation but retains a high MSE independently of α because it recovers the solution only up to a multiplicative factor.The paper notes that this scaling ambiguity does not affect the reported focusing experiment but may limit more complex tasks.
4. Focusing with the DMD
The paper uses a Bayesian Mean-Field inversion adapted to binary DMD inputs to focus light on single or multiple target points. Focusing quality depends strongly on the number of calibration measurements, with Mean-Field generally outperforming phase conjugation in single-target tests.
- Mean-Field-based inversion: A Bayesian Mean-Field approach adapts the inversion procedure to the DMD’s binary inputs for computing focusing patterns.The model uses Bernoulli inputs and a marginalized MAP estimation with a Bayesian Mean-Field approximation.
- Experimental setup: The focusing experiments use binary DMD configurations optimized for one to four target points, with intensity enhancement measured over 100 trials.The enhancement factor compares target intensity with average background intensity.
- Single target: For single-target focusing, enhancement rises sharply from α = 2 to α = 3 and then continues increasing less significantly for α ≥3.Here p = αN calibration measurements are used to estimate the transmission matrix.
- Single target: The Mean-Field method tends to exceed phase conjugation in mean and maximum enhancement, although the difference is not always statistically significant.The comparison uses box plots across calibration conditions.
- Multiple targets: In multiple-target focusing, increasing target count or reducing calibration measurements decreases enhancement and increases missed detections.With two targets, missed detections approach 40% at α = 2 but remain around 10% at α = 3.
5. Conclusion
The paper demonstrates reference-less estimation of a strongly scattering material’s full complex transmission matrix using real-valued inputs and outputs. Bayesian variational phase retrieval, especially VBEM, provides precise, computationally tractable, scalable estimates when calibration data are sufficient, validated by prediction, singular values, and focusing.
- Reference-less TM estimation: The full complex-valued transmission matrix can be estimated up to a global phase factor on each row using real-valued inputs and outputs.The setup uses binary DMD amplitude modulation and intensity measurements from a CCD camera without a reference arm.
- Bayesian phase retrieval: VBEM provides precise transmission-matrix estimates with tractable computational complexity and scalability for large signals when sufficiently many calibration signals are available.The method uses Bayesian phase retrieval with a variational approach based on mean-field updates.
- Experimental validation: Experimental validation covers output prediction, singular-value distributions, and single- or multi-point light focusing.These criteria test both the estimated matrix and its use for controlling light through the scattering material.
6. Appendix: Focusing with a Mean-Field based algorithm
The appendix describes VBEM as an iterative mean-field procedure for estimating latent factors, followed by an approximation of the posterior and binary decisions by thresholding.
- VBEM updates: VBEM iteratively updates the factors of a mean-field approximation for the specified model.The appendix presents update equations specialized to models (9)–(10).
- Posterior approximation: The posterior approximation uses modified Bessel functions of the first kind of orders 0 and 1.These functions are denoted I0 and I1 in the update expressions.
- Binary estimation: The estimated binary variable is set to 1 when q(x_i = 1) > 0.5 and to 0 otherwise.Thus, the final decision is obtained through a simple thresholding operation.