Source-linked AI summary
Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing
Yoav Shechtman, Yonina C. Eldar, Alexander Szameit, Mordechai Segev
TL;DR
Sub-wavelength recovery under partially incoherent illumination is difficult because measurements depend quadratically on the unknown object. The paper introduces quadratic compressed sensing and theoretically demonstrates sparse sub-wavelength reconstruction from blurred intensity measurements.
Problem
Partially incoherent sub-wavelength imaging yields non-convex quadratic measurements that existing linear sparse-recovery methods do not directly address.
Method
The method finds a sparse solution to quadratic imaging equations by minimizing the rank of a positive semidefinite matrix under physical constraints.
Results
Theoretical reconstructions recover sub-wavelength features from blurred intensity measurements using only the object’s real-space sparsity prior.
Takeaways & Limitations
Quadratic compressed sensing extends sparse correlation-recovery ideas to partially incoherent imaging and can accommodate white-light microscopy when the source spectrum is known.
Takeaways & Limitations
The low-pass optical system causes inherent information loss and non-uniqueness, so successful recovery requires a sparse input model.
Abstract
from arXiv · showhide
We demonstrate that sub-wavelength optical images borne on partially-spatially-incoherent light can be recovered, from their far-field or from the blurred image, given the prior knowledge that the image is sparse, and only that. The reconstruction method relies on the recently demonstrated sparsity-based sub-wavelength imaging. However, for partially-spatially-incoherent light, the relation between the measurements and the image is quadratic, yielding non-convex measurement equations that do not conform to previously used techniques. Consequently, we demonstrate new algorithmic methodology, referred to as quadratic compressed sensing, which can be applied to a range of other problems involving information recovery from partial correlation measurements, including when the correlation function has local dependencies. Specifically for microscopy, this method can be readily extended to white light microscopes with the additional knowledge of the light source spectrum.
Introduction
The introduction frames sub-wavelength imaging as overcoming Abbe’s diffraction limit using sparsity, then presents quadratic compressed sensing for partially incoherent light. The method also extends to known-spectrum white-light microscopy and broader correlation-recovery problems.
- Abbe’s limit traditionally sets the smallest resolvable feature at about λ/2, leaving sub-wavelength features blurred and unresolvable.
- Earlier bandwidth-extrapolation approaches could recover evanescent-wave information only through far-field measurements at extremely high precision.
- Prior sparsity-based bandwidth extrapolation overcame the diffraction limit by selecting the sparsest image consistent with measurements.
- For partially incoherent illumination, sparse real-space objects produce non-linear quadratic measurements rather than the linear measurements available in fully coherent or incoherent extremes.
- The proposed reconstruction applies to known-spectrum broad-band illumination, enabling extension to white-light microscopes and other partial-correlation information-recovery problems.
Propagation of images borne on partially-spatially-coherent light
Partially spatially coherent illumination produces blurred images whose sub-wavelength information cannot be recovered reliably using fully coherent or incoherent models. The proposed approach uses real-space sparsity to formulate and solve the resulting non-convex quadratic recovery problem, while phase recovery remains limited by finite transverse coherence.
- Limitations: Partial spatial coherence limits phase recovery because points separated by more than the transverse correlation distance Lc are not phase-correlated.Even a hypothetically delta-function impulse response cannot remove this limitation.
- Forward model: The measured image intensity is determined by the coherent impulse response and the mutual intensity immediately after the object through a quadratic propagation relation.The mutual intensity incorporates correlations in the partially spatially incoherent input field.
- Motivation: Partially incoherent images differ substantially from coherent and incoherent images for closely spaced, sub-wavelength features, requiring a model specific to partial spatial coherence.All coherence regimes produce blurred images, but the coherence effect is strongest when peaks are close together.
- Quadratic compressed sensing: After discretization, recovery becomes a non-convex optimization problem with quadratic measurement-consistency and energy constraints.The formulation promotes a solution matrix that is sparse in rows and within each row.
- Sparse recovery: Because low-pass propagation makes the inverse problem non-unique under noise, the method assumes that the input field is sparse in real space.The goal is to find the sparsest input field consistent with the measured image.
2. Solve
The method solves the quadratic reconstruction problem by iteratively identifying an off-support, then extracting the final object from the dominant singular component. Numerically, sparsity-based reconstruction recovers sub-wavelength sparse features from blurred intensity measurements, outperforming reconstruction without the sparsity prior.
- Algorithm: The algorithm repeats solution and thresholding stages, enlarging the off-support when diagonal entries satisfy the threshold criterion, until no further solution exists.The last successful iterate is retained; if it is not sufficiently close to rank 1, iteration continues with the previous successful off-support.
- Algorithm: The final estimate ˆa is obtained from the singular value decomposition of the resulting X using the column associated with the largest singular value.The reconstruction is formed as ˆa = U_1√s_1.
- Imaging model: Each transfer matrix M_u is constructed from the coherent impulse response and the mutual-intensity matrix, with element-wise multiplication by a Toeplitz representation of the latter.For Gaussian mutual intensity, each row of B is a shifted Gaussian.
- Imaging model: The coherent impulse response imposes diffraction-limited smearing, so every measurement is a weighted sum over a region of the object rather than a point value.Its width is bounded below by approximately λ/2, while coherence controls mixing between more distant object values.
- Numerical results: A four-spike object separated by 0.5λ under partially incoherent illumination was reconstructed practically identically when the sparsity prior was used, whereas the non-sparse reconstruction could not retrieve the object.The mutual-intensity Gaussian had FWHM 0.55λ, and both reconstructions remained consistent with the measurements.
- Numerical results: The sparsity-based method has lower reconstruction error than the non-sparsity method, with the advantage especially clear at low noise levels.The noise study averaged each plotted point over 50 reconstructions; performance improved for both methods as noise decreased.