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Universal bounds for imaging in scattering media
Abstract
In this work we establish universal ensemble independent bounds on the mean and variance of the mutual information and channel capacity for imaging through a complex medium. Both upper and lower bounds are derived and are solely dependent on the mean transmittance of the medium and the number of degrees of freedom N. In the asymptotic limit of large N, upper bounds on the channel capacity are shown to be well approximated by that of a bimodal channel with independent identically Bernoulli distributed transmission eigenvalues. Reflection based imaging modalities are also considered and permitted regions in the transmission-reflection information plane defined. Numerical examples drawn from the circular and DMPK random matrix ensembles are used to illustrate the validity of the derived bounds. Finally, although the mutual information and channel capacity are shown to be non-linear statistics of the transmission eigenvalues, the existence of central limit theorems is demonstrated and discussed.
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For the quantum correlations between scattered modes in the disordered media, the previous works focus mainly on the cases where the inputs are non-superposed states, for instance, products of Fock states (Ott et al. (2010)). A natural question that arises is how the superpositions affect the quantum correlations. Following this trail, the comparison between superpositions and products of Fock states is performed. It is found an interesting phenomenon that for the superposition and the corresponding product of Fock state (non-Gaussian states), their averaged quantum correlations are nearly same, whereas the distributions of their quantum correlations might be different. Therefore, superpositions may affect the distributions of the quantum correlations. In addition, to examine how the Gaussian states affect the quantum correlations, we compare the typical Gaussian states with the non-Gaussian states (superpositions and products of Fock states). It is discovered that the non-Gaussian-state input could result in the quantum correlation that is either positive or negative, depending on the number of the input modes and the number of the photons in each mode, whereas the Gaussian-state input always leads to the non-negative quantum correlation. Besides, it is demonstrated that with the increase of the disorder strength, the mean strength of the quantum correlation increases for multi-mode-state inputs (except for multi-mode-coherent-state inputs). These results may be useful to control and adjust the quantum properties of scattered modes after the quantized lights propagating through the disordered medium.
Modern digital cameras employ silicon focal plane array (FPA) image sensors featuring millions of pixels. However, it is possible to make a camera that only needs one pixel. In these cameras a spatial light modulator, placed before or after the object to be imaged, applies a time-varying pattern and synchronized intensity measurements are made with a single-pixel detector. The principle of compressed sensing then allows an image to be generated. As the approach suits a wide a variety of detector technologies, images can be collected at wavelengths outside the reach of FPA technology or at high frame rates or in three dimensions. Promising applications include the visualization of hazardous gas leaks and 3D situation awareness for autonomous vehicles.
We study theoretically the mutual information between reflected and transmitted speckle patterns produced by wave scattering from disordered media. The mutual information between the two speckle images recorded on an array of N detection points (pixels) takes the form of long-range intensity correlation loops, that we evaluate explicitly as a function of the disorder strength and the Thouless number g. Our analysis, supported by extensive numerical simulations, reveals a competing effect of cross-sample and surface spatial correlations. An optimal distance between pixels is proven to exist, that enhances the mutual information by a factor Ng compared to the single-pixel scenario.
A fundamental challenge in physics is controlling the propagation of waves in disordered media despite strong scattering from inhomogeneities. Spatial light modulators enable one to synthesize (shape) the incident wavefront, optimizing the multipath interference to achieve a specific behaviour such as focusing light to a target region. However, the extent of achievable control is not known when the target region is much larger than the wavelength and contains many speckles. Here we show that for targets containing more than g speckles, where g is the dimensionless conductance, the extent of transmission control is substantially enhanced by the long-range mesoscopic correlations among the speckles. Using a filtered random matrix ensemble appropriate for coherent diffusion in open geometries, we predict the full distributions of transmission eigenvalues as well as universal scaling laws for statistical properties, in excellent agreement with our experiment. This work provides a general framework for describing wavefront-shaping experiments in disordered systems.
We compare two recently developed multiple-frame deconvolution approaches for the reconstruction of structured illumination microscopy (SIM) data: the pattern-illuminated Fourier ptychography algorithm (piFP) and the joint Richardson–Lucy deconvolution (jRL). The quality of the images reconstructed by these methods is compared in terms of the achieved resolution improvement, noise enhancement, and inherent artifacts. Furthermore, we study the issue of object-dependent resolution improvement by considering the modulation transfer functions derived from different types of objects. The performance of the considered methods is tested in experiments and benchmarked with a commercial SIM microscope. We find that the piFP method resolves periodic and isolated structures equally well, whereas the jRL method provides significantly higher resolution for isolated objects compared to periodic ones. Images reconstructed by the piFP and jRL algorithms are comparable to the images reconstructed using the generalized Wiener filter applied in most commercial SIM microscopes. An advantage of the discussed algorithms is that they allow the reconstruction of SIM images acquired under different types of illumination, such as multi-spot or random illumination.
The concept of invariance of information capacity is discussed and applied to the resolution of an optical system. Methods of obtaining superresolution in microscopy are discussed, and scanning microscopy has many distinct advantages for such applications.
Coherent wave propagation in disordered media gives rise to many fascinating phenomena as diverse as universal conductance
fluctuations in mesoscopic metals and speckle patterns in light scattering. Here, the theory of electromagnetic wave propagation
in diffusive media is combined with information theory to show how interference affects the information transmission rate
between antenna arrays. Nontrivial dependencies of the information capacity on the nature of the antenna arrays are found,
such as the dimensionality of the arrays and their direction with respect to the local scattering medium. This approach provides
a physical picture for understanding the importance of scattering in the transfer of information through wireless communications.
The optical microscope is a powerful instrument for observing cellular events. Recently, the increased use of microscopy in
quantitative biological research, including single molecule microscopy, has generated significant interest in determining
the performance limits of an optical microscope. Here, we formulate this problem in the context of a parameter estimation
approach in which the acquired imaging data is modeled as a spatio-temporal stochastic process. We derive formulations of
the Fisher information matrix for models that allow both stationary and moving objects. The effects of background signal,
detector size, pixelation and noise sources are also considered. Further, formulations are given that allow the study of defocused
objects. Applications are discussed for the special case of the estimation of the location of objects, especially single molecules.
Specific emphasis is placed on the derivation of conditions that guarantee block diagonal or diagonal Fisher information matrices.
We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain
class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity
of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for
several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d.
random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are
strongly dependent even if the entries of matrices are independent.
Rather than decreasing efficiency, scattering can actually increase the information transfer rate for cell phones and other wireless microwave communication devices. Mesoscopic physics helps explain how.
We experimentally measure the monochromatic transmission matrix (TM) of an
optical multiple scattering medium using a spatial light modulator together
with a phase-shifting interferometry measurement method. The TM contains all
information needed to shape the scattered output field at will or to detect an
image through the medium. We confront theory and experiment for these
applications and we study the effect of noise on the reconstruction method. We
also extracted from the TM informations about the statistical properties of the
medium and the light transport whitin it. In particular, we are able to isolate
the contributions of the Memory Effect (ME) and measure its attenuation length.
We introduce a method to experimentally measure the monochromatic transmission matrix of a complex medium in optics. This method is based on a spatial phase modulator together with a full-field interferometric measurement on a camera. We determine the transmission matrix of a thick random scattering sample. We show that this matrix exhibits statistical properties in good agreement with random matrix theory and allows light focusing and imaging through the random medium. This method might give important insight into the mesoscopic properties of a complex medium.
Wigner statistics for correlations of matrix eigenvalues are shown to be a property of any matrix ensemble with a density of levels and probability distributions for matrix elements that are smooth. This justifies the universality of level correlations in generic quantum systems, while suggesting that level widths and other eigenvector-dependent statistics are system dependent.
The newly emerging field of wave front shaping in complex media has recently seen enormous progress. The driving force behind these advances has been the experimental accessibility of the information stored in the scattering matrix of a disordered medium, which can nowadays routinely be exploited to focus light as well as to image or to transmit information even across highly turbid scattering samples. We will provide an overview of these new techniques, of their experimental implementations as well as of the underlying theoretical concepts following from mesoscopic scattering theory. In particular, we will highlight the intimate connections between quantum transport phenomena and the scattering of light fields in disordered media, which can both be described by the same theoretical concepts. We also put particular emphasis on how the above topics relate to application-oriented research fields such as optical imaging, sensing and communication.
Coherent imaging and communication through or within heavily scattering random media has been considered impossible due to the randomization of the information contained in the scattered electromagnetic field. We report a remarkable result based on speckle correlations over incident field position that demonstrates that the field incident on a heavily scattering random medium can be obtained using a method that is not restricted to weak scatter and is, in principle, independent of the thickness of the scattering medium. Natural motion can be exploited, and the approach can be extended to other geometries. The near-infrared optical results presented indicate that the approach is applicable to other frequency regimes, as well as other wave types. This work presents opportunities to enhance communication channel capacity in the large source and detector number regime, for a new method to view binary stars from Earth, and in biomedical applications.
Imaging with optical resolution through and inside complex samples is a
difficult challenge with important applications in many fields. The fundamental
problem is that inhomogeneous samples, such as biological tissues, randomly
scatter and diffuse light, impeding conventional image formation. Despite many
advancements, no current method enables to noninvasively image in real-time
using diffused light. Here, we show that owing to the memory-effect for speckle
correlations, a single image of the scattered light, captured with a standard
high-resolution camera, encodes all the information that is required to image
through the medium or around a corner. We experimentally demonstrate
single-shot imaging through scattering media and around corners using
incoherent light and various samples, from white paint to dynamic biological
samples. Our lensless technique is simple, does not require laser sources,
wavefront-shaping, nor time-gated detection, and is realized here using a
camera-phone. It has the potential to enable imaging in currently inaccessible
scenarios.
We present an analytic random matrix theory for the effect of incomplete channel control on the measured statistical properties of the scattering matrix of a disordered multiple-scattering medium. When the fraction of the controlled input channels, m_{1}, and output channels, m_{2}, is decreased from unity, the density of the transmission eigenvalues is shown to evolve from the bimodal distribution describing coherent diffusion, to the distribution characteristic of uncorrelated Gaussian random matrices, with a rapid loss of access to the open eigenchannels. The loss of correlation is also reflected in an increase in the information capacity per channel of the medium. Our results have strong implications for optical and microwave experiments on diffusive scattering media.
In the formation of an optical image, each surface element of the object gives rise to a more or less blurred distribution in the image surface, of total brightness proportional to that of the object element. The image is the sum of these distributions in the appropriate sense: when the object is coherently lit, the image is built up by adding their complex amplitudes; when the object elements are regarded as incoherent it is the intensities which are added. In both cases the image can be expressed as the convolution of the object with a spread function which characterizes the optical system. In systems for which the spread function does not change appreciably from one part of the field to another, the Fourier transform of the image is obtained to a sufficient approximation on multiplying the Fourier transform of the object with that of the spread function. More generally, this holds for any part of the field of a non-isoplanatic system over which the changes in the form of the spread function are small enough to be disregarded; we call such an area an 'isoplanatism-patch'. Working over such an area, an optical system can be regarded as a linear filter in which the Fourier components of the object reappear in the image multiplied by 'transmission factors'. These factors, first considered by Duffieux, depend on the aperture and aberrations of the system, and in Section 2 they are evaluated in terms of an ikonal function. The qualities required of an optical image are so varied that an assessment valid over the whole range of practical applications seems out of the question. Two extreme cases are considered in the present paper. In the first of these it is assumed that the aim of an optical design is to produce an image which is directly similar to the object. This is appropriate when no process of image interpretation or reconstruction is envisaged. In the second case, the aim is to produce an image containing the greatest possible amount of information about the object, without regard to the complexity of the interpretation processes which may be needed to extract it. For the first case, a criterion of image fidelity is proposed in Section 2.4 which gives a numerical measure of the resemblance of image to object in terms of the transmission factors of the optical system. In the second case, assessment is based on the information content of the image in Shannon's sense. This depends not only on the transmission factors of the system but also on the statistical properties of the presumed object set and of the unpredictable fluctuations which necessarily disturb observation; the analysis is carried through in Section 3. In Section 4 the assessment of optical images is discussed in terms of these two criteria.
We discuss the phenomenon of chaotic scattering and its application in the study of transmission of electrons in mesoscopic devices as well as the transmission of microwaves through junctions. We show that the fact that the ray optics (classical dynamics) is chaotic, implies fluctuations in the observed transmission coefficients, whose statistics is determined by the theory of random matrices. We also show how the classical distribution functions which reflect the chaotic nature of the classical dynamics, determine the dependence of the correlations observed in the fluctuating transmission coefficients on external parameters. The time domain properties of chaotic scattering systems are also examined, and are shown to depend on the chaotic nature of the classical dynamics, together with a wave mechanical enhancement in time reversal invariant systems. Finally, we study the role of absorption and discuss its effects on the transmission fluctuations and their statistics.
New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational symmetry. The probability‐distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one‐dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group‐theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper.
Non-invasive optical imaging techniques, such as optical coherence tomography, are essential diagnostic tools in many disciplines, from the life sciences to nanotechnology. However, present methods are not able to image through opaque layers that scatter all the incident light. Even a very thin layer of a scattering material can appear opaque and hide any objects behind it. Although great progress has been made recently with methods such as ghost imaging and wavefront shaping, present procedures are still invasive because they require either a detector or a nonlinear material to be placed behind the scattering layer. Here we report an optical method that allows non-invasive imaging of a fluorescent object that is completely hidden behind an opaque scattering layer. We illuminate the object with laser light that has passed through the scattering layer. We scan the angle of incidence of the laser beam and detect the total fluorescence of the object from the front. From the detected signal, we obtain the image of the hidden object using an iterative algorithm. As a proof of concept, we retrieve a detailed image of a fluorescent object, comparable in size (50 micrometres) to a typical human cell, hidden 6 millimetres behind an opaque optical diffuser, and an image of a complex biological sample enclosed between two opaque screens. This approach to non-invasive imaging through strongly scattering media can be generalized to other contrast mechanisms and geometries.
We consider a Rayleigh model [Foschini and Gans, Wirel. Pers. Commun. 6, 311 (1998)] for the Shannon capacity characterising wireless information transfer in a network of antennae. In this model, the capacity C is a linear statistic of the channel matrix of the network. We use this fact to calculate the variance [C] of the Shannon capacity. We find that the expected value ãCã of the Shannon capacity is typical, that is the coefficient of variation is small. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
This paper reviews scattering theory required for analysis of light reflected by planetary atmospheres. Section 1 defines the radiative quantities which are observed. Section 2 demonstrates the dependence of single-scattered radiation on the physical properties of the scatterers. Section 3 describes several methods to compute the effects of multiple scattering on the reflected light.
This paper is motivated by the need for fundamental understanding of ultimate limits of bandwidth efficient delivery of higher bit-rates in digital wireless communications and to also begin to look into how these limits might be approached. We examine exploitation of multi-element array (MEA) technology, that is processing the spatial dimension (not just the time dimension) to improve wireless capacities in certain applications. Specifically, we present some basic information theory results that promise great advantages of using MEAs in wireless LANs and building to building wireless communication links. We explore the important case when the channel characteristic is not available at the transmitter but the receiver knows (tracks) the characteristic which is subject to Rayleigh fading. Fixing the overall transmitted power, we express the capacity offered by MEA technology and we see how the capacity scales with increasing SNR for a large but practical number, n, of antenna elements at both transmitter and receiver. We investigate the case of independent Rayleigh faded paths between antenna elements and find that with high probability extraordinary capacity is available. Compared to the baseline n = 1 case, which by Shannon’s classical formula scales as one more bit/cycle for every 3 dB of signal-to-noise ratio (SNR) increase, remarkably with MEAs, the scaling is almost like n more bits/cycle for each 3 dB increase in SNR. To illustrate how great this capacity is, even for small n, take the cases n = 2, 4 and 16 at an average received SNR of 21 dB. For over 99%
We present a theory of multichannel disordered conductors by directly
studying the statistical distribution of the transfer matrix for the
full system. The theory is based on the general properties of the
scattering system: flux conservation, time-reversal invariance, and the
appropriate combination requirement when two wires are put together. The
distribution associated with systems of very small length is then
selected on the basis of a maximum-entropy criterion; a fixed value is
assumed for the diffusion coefficient that characterizes the evolution
of the distribution as the length increases. We obtain a diffusion
equation for the probability distribution and compute the average of a
few relevant quantities.
Optical imaging relies on the ability to illuminate an object, collect and analyse the light it scatters or transmits. Propagation through complex media such as biological tissues was so far believed to degrade the attainable depth, as well as the resolution for imaging, because of multiple scattering. This is why such media are usually considered opaque. Recently, we demonstrated that it is possible to measure the complex mesoscopic optical transmission channels that allow light to traverse through such an opaque medium. Here, we show that we can optimally exploit those channels to coherently transmit and recover an arbitrary image with a high fidelity, independently of the complexity of the propagation.
Any linear statistic defined on a random-matrix ensemble is shown to be Gaussian distributed. This supports the prediction of weak-disorder perturbation theory in the diffusive, metallic limit for the distribution of conductance, since conductance is a linear statistic on the ensemble of transfer matrices.
We show that all the known quantum-mechanical interference effects characteristic of disordered conductors in the metallic limit can be obtained from a ``maximum-entropy model,'' based on a transfer-matrix formulation, which is independent of any particular form of the disordered microscopic Hamiltonian. In particular, we have derived the weak-localization effect and the associated backscattering peak, as well as the universal conductance fluctuations and the associated long-range correlations in transmission probabilities. We find precise quantitative agreement with microscopic Green-function calculations evaluated in the quasi-one-dimensional limit. We define two random-matrix ensembles characterizing systems with and without time-reversal symmetry (the analogs of the well-known orthogonal and unitary ensembles), and show that, within the model, breaking of time-reversal symmetry has the expected effect on these phenomena. The model has not been shown to yield behavior characteristic of the two- or three-dimensional limits, which appear to be outside its current range of validity.
Recently, a macroscopic theory of N-channel disordered condeuctors showed that the statistical distribution of the transfer matrix for a system of length L evolves with L according to a diffusion equation in N dimensions. It is proved here that the recently observed universal conductance fluctuations in normal metals at very low temperatures are a rigorous consequence of that diffusion equation, in the regime in which L≫ (mean free path) and N≫1. The value found for the fluctuation coincides with the one obtained from elaborate microscopic calculations.
Wireless communications are a fundamental part of modern information infrastructure. But wireless bandwidth is costly, prompting a close examination of the data channels available using electromagnetic waves. Classically, radio communications have relied on one channel per frequency, although it is well understood that the two polarization states of planar waves allow two distinct information channels; techniques such as 'polarization diversity' already take advantage of this. Recent work has shown that environments with scattering, such as urban areas or indoors, also possess independent spatial channels that can be used to enhance capacity greatly. In either case, the relevant signal processing techniques come under the heading of 'multiple-input/multiple-output' communications, because multiple antennae are required to access the polarization or spatial channels. Here we show that, in a scattering environment, an extra factor of three in channel capacity can be obtained, relative to the conventional limit using dual-polarized radio signals. The extra capacity arises because there are six distinguishable electric and magnetic states of polarization at a given point, rather than two as is usually assumed.
We have compared different image restoration approaches for fluorescence microscopy. The most widely used algorithms were classified with a Bayesian theory according to the assumed noise model and the type of regularization imposed. We considered both Gaussian and Poisson models for the noise in combination with Tikhonov regularization, entropy regularization, Good's roughness and without regularization (maximum likelihood estimation). Simulations of fluorescence confocal imaging were used to examine the different noise models and regularization approaches using the mean squared error criterion. The assumption of a Gaussian noise model yielded only slightly higher errors than the Poisson model. Good's roughness was the best choice for the regularization. Furthermore, we compared simulated confocal and wide-field data. In general, restored confocal data are superior to restored wide-field data, but given sufficient higher signal level for the wide-field data the restoration result may rival confocal data in quality. Finally, a visual comparison of experimental confocal and wide-field data is presented.
We consider the information content h of a scalar multiple-scattered, diffuse wave field psi(r) and the information capacity C of a communication channel that employs diffuse waves to transfer the information through a disordered medium. Both h and C are shown to be directly related to the mesoscopic correlations between the values of psi(r) at different positions r in space, arising due to the coherent nature of the wave. For the particular case of a communication channel between two identical linear arrays of n>1 equally spaced transmitters or receivers (receiver spacing a), we show that the average capacity proportional, variant n and obtain explicit analytic expressions for /n in the limit of n--> infinity and kl--> infinity, where k=2pi/lambda, lambda is the wavelength, and l is the mean free path. Modification of the above results in the case of finite but large n and kl is discussed as well. If the signal to noise ratio S/N exceeds some critical value (S/N)(c), /n is a nonmonotonic function of a, exhibiting maxima at ka=mpi (m=1,2, em leader ). For smaller S/N, ka=mpi correspond to local minima, while the absolute maximum of /n is reached at some ka approximately (S/N)(1/2)<pi. We define the maximum average information capacity (max) as maximized over the receiver spacing a and the optimal normalized receiver spacing (ka)(opt) as the spacing maximizing . Both (max)/n and (ka)(opt) scale as (S/N)(1/2) for S/N<(S/N)(c), while (ka)(opt)=mpi and (max)/n approximately log(S/N) for S/N>(S/N)(c).
This is a review of the statistical properties of the scattering matrix of a mesoscopic system. Two geometries are contrasted: A quantum dot and a disordered wire. The quantum dot is a confined region with a chaotic classical dynamics, which is coupled to two electron reservoirs via point contacts. The disordered wire also connects two reservoirs, either directly or via a point contact or tunnel barrier. One of the two reservoirs may be in the superconducting state, in which case conduction involves Andreev reflection at the interface with the superconductor. In the case of the quantum dot, the distribution of the scattering matrix is given by either Dyson's circular ensemble for ballistic point contacts or the Poisson kernel for point contacts containing a tunnel barrier. In the case of the disordered wire, the distribution of the scattering matrix is obtained from the Dorokhov-Mello-Pereyra-Kumar equation, which is a one-dimensional scaling equation. The equivalence is discussed with the nonlinear \sigma model, which is a supersymmetric field theory of localization. The distribution of scattering matrices is applied to a variety of physical phenomena, including universal conductance fluctuations, weak localization, Coulomb blockade, sub-Poissonian shot noise, reflectionless tunneling into a superconductor, and giant conductance oscillations in a Josephson junction.
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