LocNet: deep learning-based localization on a rotating point spread function with applications to telescope imaging.

Three-dimensional (3D) point source recovery from two-dimensional (2D) data is a challenging problem with wide-ranging applications in single-molecule localization microscopy and space-debris localization telescops. Point spread function (PSF) engineering is a promising technique to solve this 3D localization problem. Specifically, we consider the problem of 3D localization of space debris from a 2D image using a rotating PSF where the depth information is encoded in the angle of rotation of a single-lobe PSF for each point source. Instead of applying a model-based optimization, we introduce a convolution neural network (CNN)-based approach to localize space debris in full 3D space automatically. A hard sample training strategy is proposed to improve the performance of CNN further. Contrary to the traditional model-based methods, our technique is efficient and outperforms the current state-of-the-art method by more than 11% in the precision rate with a comparable improvement in the recall rate.

Three-dimensional tracking using a single-spot rotating point spread function created by a multiring spiral phase plate.

Significance: Three-dimensional (3D) imaging and object tracking is critical for medical and biological research and can be achieved by multifocal imaging with diffractive optical elements (DOEs) converting depth ( z ) information into a modification of the two-dimensional image. Physical insight into DOE designs will spur this expanding field. Aim: To precisely track microscopic fluorescent objects in biological systems in 3D with a simple low-cost DOE system. Approach: We designed a multiring spiral phase plate (SPP) generating a single-spot rotating point spread function (SS-RPSF) in a microscope. Our simple, analytically transparent design process uses Bessel beams to avoid rotational ambiguities and achieve a significant depth range. The SPP was inserted into the Nomarski prism slider of a standard microscope. Performance was evaluated using fluorescent beads and in live cells expressing a fluorescent chromatin marker. Results: Bead localization precision was < 25 nm in the transverse dimensions and ≤ 70 nm along the axial dimension over an axial range of 6 µ m . Higher axial precision ( ≤ 50 nm ) was achieved over a shallower focal depth of 2.9 µ m . 3D diffusion constants of chromatin matched expected values. Conclusions: Precise 3D localization and tracking can be achieved with a SS-RPSF SPP in a standard microscope with minor modifications.

Joint 3D localization and classification of space debris using a multispectral rotating point spread function.

We consider the problem of joint three-dimensional (3D) localization and material classification of unresolved space debris using a multispectral rotating point spread function (RPSF). The use of RPSF allows one to estimate the 3D locations of point sources from their rotated images acquired by a single 2D sensor array, since the amount of rotation of each source image about its x, y location depends on its axial distance z. Using multispectral images, with one RPSF per spectral band, we are able not only to localize the 3D positions of the space debris but also classify their material composition. We propose a three-stage method for achieving joint localization and classification. In stage 1, we adopt an optimization scheme for localization in which the spectral signature of each material is assumed to be uniform, which significantly improves efficiency and yields better localization results than possible with a single spectral band. In stage 2, we estimate the spectral signature and refine the localization result via an alternating approach. We process classification in the final stage. Both Poisson noise and Gaussian noise models are considered, and the implementation of each is discussed. Numerical tests using multispectral data from NASA show the efficiency of our three-stage approach and illustrate the improvement of point source localization and spectral classification from using multiple bands over a single band.

Quantum Limited Superresolution of an Incoherent Source Pair in Three Dimensions.

The error in estimating the separation of a pair of incoherent sources from radiation emitted by them and subsequently captured by an imager is fundamentally bounded below by the inverse of the corresponding quantum Fisher information (QFI) matrix. We calculate the QFI for estimating the full three-dimensional pair separation vector, extending previous work on pair separation in one and two dimensions. We also show that the pair-separation QFI is, in fact, identical to source localization QFI, which underscores the fundamental importance of photon-state localization in determining the ultimate estimation-theoretic bound for both problems. We also propose general coherent-projection bases that can attain the QFI in two special cases. We present simulations of an approximate experimental realization of such quantum limited pair superresolution using the Zernike basis, confirming the achievability of the QFI bounds.

Spatial statistics of optical beams perturbed by aero-optical turbulence.

We develop the spatial structure function (SSF) of the phase of an optical beam propagated through an aero-optically active turbulent boundary layer at a flying aircraft, which is characterized by anisotropic spatial fluctuations that arise from pressure fluctuations and are highly non-Kolmogoroff. We use the phase SSF to calculate in closed form the spatial statistics of a monochromatic Gaussian beam transmitted through such a turbulent layer and subsequently propagated in free space, discussing in particular the correlations of its amplitude and intensity fluctuations to arbitrary orders.

Extended Taylor frozen-flow hypothesis and statistics of optical phase in aero-optics.

We present an extended Taylor frozen-flow model for the statistics of the spatiotemporal disturbances of the index of refraction of air and the phase of an optical beam propagated through the turbulent boundary and shear layers in a high-Reynolds-number flow. By incorporating rapid random fluctuations of the flow velocity about a mean convection velocity and an anisotropic spatial power spectrum for the index of refraction, we calculate both the short-delay temporal structure function and the power spectral density of these disturbances. We discuss the predicted scaling behaviors for these quantities in the context of existing experimental observations, showing specifically the agreement of these predictions with some optical phase data obtained by the Airborne Aero-Optical Laboratory.

High-numerical-aperture microscopy with a rotating point spread function.

Rotating point spread function (PSF) microscopy via spiral phase engineering can localize point sources over large focal depths in a snapshot mode. The present work gives an approximate vector-field analysis of an improved rotating PSF design that encodes both the 3D location and polarization state of a monochromatic point dipole emitter for high-numerical-aperture microscopy. By examining the angle of rotation and the spatial form of the PSF, one can jointly localize point sources and determine the polarization state of light emitted by them over a 3D field in a single snapshot. Results of numerical simulations of noisy data frames under Poisson shot noise conditions and the errors in the recovery of 3D location and dipole orientation for a single point source are discussed.

Rotating point spread function via pupil-phase engineering.

A simple approach based on the use of a properly designed pupil-phase profile can yield a 3D point-spread function (PSF) that rotates with changing defocus, while keeping its transverse shape approximately invariant over ±3-4 waves of defocus. Unlike Gauss-Laguerre mode-based approaches, it generalizes readily for encoding spherical aberration too via PSF rotation.

New error bounds for M-testing and estimation of source location with subdiffractive error.

I present new lower and upper bounds on the minimum probability of error (MPE) in Bayesian multihypothesis testing that follow from an exact integral of a version of the statistical entropy of the posterior distribution, or equivocation. I also show that these bounds are exponentially tight and thus achievable in the asymptotic limit of many conditionally independent and identically distributed measurements. I then relate the minimum mean-squared error (MMSE) and the MPE by means of certain elementary error probability integrals. In the second half of the paper, I compare the MPE and MMSE for the problem of locating a single point source with subdiffractive uncertainty. The source-strength threshold needed to achieve a desired degree of source localization seems to be far more modest than the well established threshold for the different optical super-resolution problem of disambiguating two point sources with subdiffractive separation.

Joint segmentation and reconstruction of hyperspectral data with compressed measurements.

This work describes numerical methods for the joint reconstruction and segmentation of spectral images taken by compressive sensing coded aperture snapshot spectral imagers (CASSI). In a snapshot, a CASSI captures a two-dimensional (2D) array of measurements that is an encoded representation of both spectral information and 2D spatial information of a scene, resulting in significant savings in acquisition time and data storage. The reconstruction process decodes the 2D measurements to render a three-dimensional spatio-spectral estimate of the scene and is therefore an indispensable component of the spectral imager. In this study, we seek a particular form of the compressed sensing solution that assumes spectrally homogeneous segments in the two spatial dimensions, and greatly reduces the number of unknowns, often turning the underdetermined reconstruction problem into one that is overdetermined. Numerical tests are reported on both simulated and real data representing compressed measurements.

Support-assisted optical superresolution of low-resolution image sequences: the one-dimensional problem.

We analyze the problem of optical superresolution (OSR) of a one-dimensional (1D) incoherent spatial signal from undersampled data when the support of the signal is known in advance. The present paper corrects and extends our previous work on the calculation of Fisher information (FI) and the associated Cramer-Rao lower bound (CRB) on the minimum error for estimating the signal intensity distribution and its Fourier components at spatial frequencies lying beyond the optical band edge. The faint-signal and bright-signal limits emerge from a unified noise analysis in which we include both additive noise of detection and shot noise of photon counting via an approximate Gaussian statistical distribution. For a large space-bandwidth product, we derive analytical approximations to the exact expressions for FI and CRB in the faint-signal limit and use them to argue why achieving any significant amount o unbiased bandwidth extension in the presence of noise is a uniquely challenging proposition. Unlike previous theoretical work on the subject of support-assisted bandwidth extension, our approach is not restricted to specific forms of the system transfer functions, and provides a unified analysis of both digital and optical superresolution of undersampled data.

Digital superresolution and the generalized sampling theorem.

The technique of reconstructing a higher-resolution (HR) image of size MLxML by digitally processing LxL subpixel-shifted lower-resolution (LR) copies of it, each of size MxM, has now become well established. This particular digital superresolution problem is analyzed from the standpoint of the generalized sampling theorem. It is shown both theoretically and by computer simulation that the choice of regularly spaced subpixel shifts for the LR images tends to maximize the robustness and minimize the error of reconstruction of the HR image. In practice, since subpixel-level control of LR image shifts may be nearly impossible to achieve, however, a more likely scenario, which is also discussed, is one involving random subpixel shifts. It is shown that without reasonably tight bounds on the range of random shifts, the reconstruction is likely to fail in the presence of even small amounts of noise unless either reliable prior information or additional data are available.

Fisher-information-based analysis of a phase-diversity-speckle imaging system.

The question of just how much to defocus the second image in the conventional two-channel phase-diversity speckle imaging technique may be addressed in a number of ways. Fisher information furnishes a useful metric for optimizing the choice of defocus as a functional of the object class, operating conditions, and the imaging task. Approximate closed-form expressions for the Fisher information relative to object parameters, rather than the pupil phase, are derived and discussed for phase-diversity-speckle imaging under conditions of strong turbulence and additive Gaussian noise. As an application of our general information-theoretic approach, the optimization of defocus when the imaging task is to estimate the midfrequency power spectrum of the object is discussed.

Information-optimized phase diversity speckle imaging.

Fisher-information-based optimization of a conventional phase diversity speckle imaging system is presented. I demonstrate how Fisher information can be used to determine the optimum value of defocus in the diversity channel for estimating the mid-frequency integrated object power spectrum. This approach is likely to be useful in optimizing the design and performance of a general imaging system.

Light scattering from a randomly occupied optical lattice. I. Born approximation.

A theoretical study of the scattering of light from a randomly occupied optical lattice of resonant atoms is presented to reveal both the characteristics of the lattice and the properties of light scattered from the lattice. In the first-order Born approximation we discuss here, a number of interesting effects are established, including sideband Stokes scattering, a finite angular coherence of the scattered light, and spectral line narrowing. Specifically, the degree of angular coherence of the scattered light is calculated, and it is shown that such coherence is strongly influenced by the regularity and size of the underlying lattice structure. The previously observed phenomenon of the sideband spectral line narrowing is also explained in terms of the localization of atoms in the trapping potential wells. Important information about the lattice can thus be recovered by analyzing the scattered light in the Born approximation.

Light scattering from a randomly occupied optical lattice. II. The multiple scattering problem.

In this paper, we study the problem of multiple scattering of light from a randomly occupied optical lattice, thereby extending the first-order Born analysis of the previous paper. A full multiple-scattering analysis is essential to a complete understanding of the nature of light propagation inside a medium. Our calculations show that the incident wave, when resonant with the atomic medium, is rapidly extinguished due to multiple scattering. The decay constant depends critically on the incident wavelength, the lattice constant, the average number density of atoms, and their polarizability. Both the Bragg scattering amplitudes and directions are modified as a result of multiple scattering. Because of the random site occupation of an otherwise regular lattice structure, a coherent enhancement of the scattering cross section is also predicted to occur along a discrete set of directions that are related to the strictly backward direction by reciprocal lattice vectors.

Statistical-information-based performance criteria for Richardson-Lucy image deblurring.

Iterative image deconvolution algorithms generally lack objective criteria for deciding when to terminate iterations, often relying on ad hoc metrics for determining optimal performance. A statistical-information-based analysis of the popular Richardson-Lucy iterative deblurring algorithm is presented after clarification of the detailed nature of noise amplification and resolution recovery as the algorithm iterates. Monitoring the information content of the reconstructed image furnishes an alternative criterion for assessing and stopping such an iterative algorithm. It is straightforward to implement prior knowledge and other conditioning tools in this statistical approach.