Abstract

The fluctuations of measured signals are not always artifacts. A variety of active sensing circumstances where the excitation fields interact with random potentials result in measured signals that fluctuate. As inverse problems are notoriously ill posed, only stochastic information is available in these conditions. Here, we demonstrate a stochastic sensing technique where the statistically non-stationary properties of the excitation fields are purposely manipulated such that the fluctuations of the scattered intensity are enhanced and further exploited to recover information about the interaction potentials. The numerical and experimental results establish the efficacy of this new sensing approach and validate its robustness against various sources of perturbations. A range of remote sensing and biomedical applications that rely on recovering the properties of random scattering potentials and are usually limited by noise could benefit from the sturdy performance of this sensing procedure.

© 2016 Optical Society of America

1. INTRODUCTION

Measuring the properties of random systems is ubiquitous. Their characterization can be approached in different ways, but in many situations, the only meaningful depiction is in statistical terms. For instance, in active sensing, information is recovered through a stochastic analysis of the system’s output in response to controlled external stimuli. Examples include the characterization of structural dynamics [1] or individual scattering objects [2,3], to mention just a few. Nevertheless, the external stimuli can be themselves stochastic, like in the case of events of molecular origin [4].

In all these examples, the information about the targeted system is contained in the fluctuations of the measured signals. Therefore, efficient information retrieval requires augmenting such fluctuations, which is reminiscent of stochastic resonance [5]. We note that the traditional stochastic resonance approach relies on (i) some sort of system nonlinearity and (ii) an increase of the interaction energy through the external control of the level of additional noise.

An effective enhancement of the signal fluctuations can be generated not only by selecting a specific type of physical interaction, but also by appropriate manipulations of the probing stimulus. Thus, it can be argued that using stochastic probing, one may be able to enhance the signal variability. This kind of approach may also extend certain practical attributes of the sensing process, such as robustness against noisy and unpredictable environments.

2. FLUCTUATION-BASED SENSING

In this paper, we introduce a stochastic sensing method for characterizing properties of random systems. We will show that the statistical features of a scattering medium can be recovered efficiently by simply controlling the statistical realizations ξ of a stochastic probe S(x;ξ) that interrogates the targeted system V(x) via a multidimensional interaction in the domain {x}. In a nutshell, we will show that changes in the statistical parameters of a non-stationary probe S(x;ξ) lead to abrupt modifications in the signals’ statistics, which can be related to the target properties. Even though this approach is reminiscent of stochastic resonance, it does not come at the expense of increasing the energy of interaction between the probe and the targeted system and, furthermore, it does not require the presence of any nonlinearity. Moreover, because the non-stationary stochastic probe and the different possible sources of noise are statistically independent, we will demonstrate that this sensing methodology is robust against various external perturbations.

In practice, attaining a reliable signal encoded in x is often problematic because of, among other things, the limited bandwidth of a detection system and the presence of perturbations in both transferring the stimulus (perturbation in) and recovering the response from the target (perturbation out). In these circumstances, a preferable quantity to measure is the average (integrated) signal response:

I(ξ)=A1A[(S(x;ξ)+n(x;ξ))P(x;ξ)V(x)+R(x;ξ)]dx,
where P(x;ξ), R(x;ξ) and n(x;ξ) denote all possible sources of perturbations in the sensing process. Notably, this measure cancels out the distortions in the return path as long as this information channel is lossless. It is also important to realize that the integration in Eq. (1) modifies the way in which the information is represented by projecting it from the x- into the ξ- domain. This decreases the dimensionality of the problem and permits recovering stochastic information about V(x).

Without loss of generality, we will consider an optical situation in which the intensity statistics of a two-dimensional illumination field are manipulated to recover information regarding a random scattering target V(ρ;lV) characterized by the correlation length lV. In this case, the interaction is described in the two-dimensional Cartesian domain ρ, and ξ represents a simple, one-dimensional realization index. The general setting for this optical situation is illustrated in Fig. 1. We note that the concept is not limited to backscattered light; a similar operation in forward scattering or any other direction can be described by simply folding the setup around the plane Ω.

 figure: Fig. 1.

Fig. 1. Schematic procedure of sensing statistical properties of a random scattering potential V(ρ;lV) interrogated with a controlled stochastic illumination S(ρ,ξ), which is affected by the inherent source noise n(ρ;ξ), and perturbations Ui(ρ,ξ) and Uo(ρ,ξ) in the illumination and recording paths, respectively. The measured signal is the integrated intensity I(ξ).

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A generic description of the targeted medium can be advanced using the concept of the scattering potential V(ρ)=k2[n2(ρ)n02(ρ)], where k is the wave number based on the first Born approximation. This approach is commonly used in reconstructing the distributions of refractive index variations n2(ρ) [6]. In this case, the averaged response I(ξ) in Eq. (1) is, in fact, the average intensity of scattered light from the targeted scattering potential that fluctuates when the configuration ξ of the excitation changes. The proposed stochastic sensing method relies on processing these fluctuations and not the magnitude of the integrated scattered signal itself.

It should be noted that, rigorously, the intensity distribution across a detector placed at an axial distance ζ from the target is E(ρ,ζ)E*(ρ,ζ), where E(ρ,ζ)=v(ρ)ES(ρ,0)G(ρρ,ζ)dρ is the field scattered due to the stochastic illumination ES(ρ,0). In this general scattering problem, the scattering potential is characterized by a complex field scattering coefficient v(ρ) (|v(ρ)|2=V(ρ)), while the field propagation to the detector plane is described by the corresponding Green’s function, G(ρ,ζ). However, one can show that the average scattered intensity can still be modeled with Eq. (1) for any ζ, as long as the propagation of scattered light is lossless, or if the loss is the same for all the realizations ξ. The interested reader is referred to Supplement 1 for more details.

In the following proof-of-concept demonstration, we consider only static scattering potentials, but this is not a conceptual limitation. Temporal fluctuations and their evolution can be recovered using dynamic illumination with the appropriate temporal characteristics. We also note that if all the realizations ξ of the stimulus S(ρ;ξ) are known, one could, in principle, invert Eq. (1) to recover an “image” of the static target V(ρ;lV) using variants of the so-called “ghost imaging” techniques [79]. The problem at hand, however, is different, as it refers to stochastic illumination in noisy ambient conditions. In this situation, the question is: can any information about V(ρ;lV) still be recovered?

In the following, we will first outline how the statistical moments characterizing the scattering potential and the spatial and dynamic properties of the illumination relate to the fluctuations of the recorded intensity. Then, we will exemplify how this stochastic sensing formulation can be used in different practical applications. We will demonstrate the method’s robustness against perturbations in both the structure of illumination and the intensity detection.

3. STOCHASTIC OPTICAL SENSING

The stochastic Fredholm integral equation, Eq. (1), relates the unknown scattering potential V(ρ;lV) to four random processes, S(ρ;ξ), n(ρ;ξ), P(ρ;ξ), and R(ρ;ξ). These processes are considered to be statistically stationary spatially and also statistically independent of each other, which is a reasonable practical assumption. For each realization ξ, these processes are characterized by probability distributions and corresponding statistical moments with respect to variable ρ. Upon averaging over an ensemble of all realizations ξ, these functions can be treated as constants in the further analysis.

We will now examine the impact of different types of fluctuating illumination intensity. First, we will discuss the case of a δ-correlated distribution defined by S˜(ρ;ξ)S˜(ρ+ρ0;ξ)ρ=m˜S(2)δ(ρ0), with m˜S(k)=(S˜(ρ,ξ))kξ in terms of S˜(ρ,ξ)=S(ρ,ξ)S(ρ,ξ)ξ. In this case, one can use the statistical properties of the distribution σ(ρ;ξ)=(S(ρ;ξ)+n(ρ;ξ))P(ρ;ξ), effectively impinging on the scattering potential to evaluate recursively the moments as follows:

MV(k)=m˜I(k)m˜σ(k)n=0k22k!m˜R(k2n)(k2n)!(2n)!m˜σ(2n)m˜σ(k)MV(2n).
This equation defines the spatial distribution of the scattering potential. The details of the derivation are included in Supplement 1. Of course, the reconstruction of higher-order moments must be considered in the context of the overall procedural noise [1013], which is outside of the scope of this paper. In practical cases where the first two moments are sufficient, it can be shown that the Fano factors FV=MV(2)/MV(1) and FI=m˜I(2)/mI(1) [14,15], which describe the scattering potential and the detected intensity, respectively, are related by
FV=FImσ(1)mI(1)m˜σ(2)(mI(1)mR(1))m˜R(2)mσ(1)m˜σ(2)(mI(1)mR(1)).
In Eq. (3), the Fano factor measures how significant the fluctuations are with respect to their average. This measure will be used throughout this paper.

In the practical case where the correlation length lS of the illumination field is finite, then this length scale can be manipulated to infer information about the spatial properties of the scattering potential. The illumination can be thought of as the convolution S(ρ;ξ,lS)=e(ρ;lS)*C(ρ;ξ) of a positively defined elementary unit e(ρ;lS) with a spatially δ-correlated stochastic process C(ρ;ξ). In these conditions, we show in Supplement 1 that

m˜I(2)(lS)=ψ(ρ;ξ,lS,lV)ψ(ρ;ξ,lS,lV)ξdρdρ+m˜R(2),
where the function ψ(ρ;ξ,lS,lV)=σ˜(ρ;ξ,lS)V(ρ;lV) denotes the intensity fluctuations of the scattered light without any perturbation R(ρ,ξ). As is clear from Eq. (4) and in contrast to the previous case of δ-correlated illumination, finding the statistical characteristics of the scattering potential involves a nonlinear inversion. This is a typical ill-posed problem that can be solved only in certain conditions and by appealing to a priori information [13,16].

However, information about the two-point correlation of the scattering potential can still be accessed without a priori knowledge by simply examining the fluctuations of the recorded intensity over different realizations ξ of the illumination intensity. In this case, the fluctuations of the integrated intensity become

I˜(ξ;lS)=S˜(ρ;ξ,lS)P(ρ;ξ)V(ρ;lV)dρ+mS(1)P˜(ρ;ξ)V(ρ;lV)dρ+n(ρ;ξ)P(ρ;ξ)V(ρ;lV)dρ+R˜(ρ;ξ)dρ
which consists of four zero-mean, statistically independent contributions. As a result, the second moment of the intensity fluctuations is simply the sum of the second moments of these four contributions. If the perturbations are statistically stationary, the second moments of the last three terms are constant during the measurement. Thus, the overall level of the recorded intensity fluctuations is regulated by the value of the first integral in Eq. (5), which can be adjusted by varying lS. In terms of the spatial fluctuations e˜(ρ;lS) of the elementary illumination unit, the first integral can be written as
S˜(ρ;ξ,lS)P(ρ;ξ)V(ρ;lV)dρ=C(ρ;ξ)(e˜(ρ;lS)*P(ρ;ξ)V(ρ;lV))dρ.
This means that the value of this integral is determined by randomly sampling the convolution between e˜(ρ;lS) and a perturbed projection of the scattering potential. When the characteristic lengths of this perturbation is much larger than lV, it can be shown that the maximum of the intensity fluctuations in the first term of Eq. (5) is proportional to e˜(ρ;lS)e˜(ρ;lV)dρ. The contrast of these fluctuations depends on lS and, for a given average detected intensity, can be quantified by a corresponding Fano factor, as shown before.

The procedure described above suggests a direct way to infer lV from the values of the Fano factor of the recorded intensity as a function of lS. When changing lS, it is evident based on Schwarz’s inequality that this Fano factor attains its maximum when lS=lV. The interested reader is referred to Supplement 1 for more details.

The same conclusion can be reached by analyzing this sensing process in the spatial frequency domain. It is clear that the dominant frequency components of the spectral distributions (the Fourier transform of the corresponding autocorrelation function) of the illumination intensity and the scattering potential will overlap when lV=lS. The similar extent of the spectral distributions enhances the magnitude of the intensity fluctuations [6]. Of course, one could have used a deterministic illumination to reconstruct the characteristic length of the targeted potential. However, as follows from Eq. (5), a statistical analysis makes our method more robust to perturbations. Moreover, using random illumination may also lead to faster measurements.

We also note that this procedure is not limited to a single statistical process. A superposition of uncorrelated scattering potentials with different characteristic length scales can be represented, for instance, by a non-monotonic autocorrelation function leading to a Fano spectrum having multiple maxima. The repeatability of this technique is rather high because, as mentioned before, four terms in Eq. (5) are statistically independent. As a result, the variations in the Fano spectrum are independent of the considered perturbations. The repeatability is demonstrated in the Supplement 1 for different targeted potentials in noisy environments.

4 Concept Demonstration

In the following, we present a proof-of-concept demonstration in which we will incorporate the two main requirements for a generic sensing procedure. First, when changing lS, the total incident irradiance may change. Because in some circumstances, the scattering potential could depend on the irradiance level, we impose that mS(1) remains constant throughout the procedure. Second, sometimes there is simply no a priori information about the targeted potential and, therefore, an effective sensing method must sample it uniformly and isotropically.

A. Random Scattering Potentials

In the first example, we simulate a scattering potential based on a random array of circular disks with lV=Δ and a volume fraction of 15%. Figure 2(a) shows the normalized Fano spectrum of the intensity fluctuations corresponding to illumination patterns with varying lS and a filing fraction of 10%. The elementary unit used here is spatially isotropic. It is evident that the maximum in the Fano spectrum corresponds to the situation where lS=lV. This matching condition can be used to describe the targeted potential and the illumination, and is rigorously demonstrated in the Supplement 1.

 figure: Fig. 2.

Fig. 2. Numerical calculation of the Fano spectrum corresponding to (a) random targeted potential consisting of circular disks with lV=Δ, and (b) the corresponding Fano spectrum. The maximum of the Fano spectrum is normalized to unity.

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B. Multi-Scale Scattering Potentials

In the first example, we considered a random system with only one length scale. The effectiveness of this stochastic sensing technique as outlined in Eq. (5) was also studied in more complex situations where the scattering potential has two characteristic lengths. For example, in Fig. 3, we present the normalized FI spectrum corresponding to a scattering target consisting of two rectangular areas with width 2Δ, which are separated by a distance of 3Δ. To mimic natural perturbations, the illumination was purposely corrupted by additive noise: S(ρ;ξ)+n(ρ;ξ). In addition, the scattered light was also distorted by additive and multiplicative white noise to effectively create detection conditions with an SNR of 2.5 dB. As can be seen in Fig. 3(b), even in this highly perturbed condition, the characteristic features of the target can be effectively recovered. This is the benefit of relying on statistical parameters, which makes the information distinguishable from the influence of any stationary perturbations.

 figure: Fig. 3.

Fig. 3. (a) Simulated reflective target perturbed by illumination noise and (b) the corresponding normalized Fano spectrum. The measurement conditions are similar to those in Fig. 2. (c) Targeted standard microscopy chart under noisy illumination and (d) the corresponding normalized Fano spectrum.

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In Figs. 3(c) and 3(d), we present the results of an experiment where a reflective target consisting of 2 μm wide strips and a 10 μm center-to-center distance was sequentially illuminated with 200 random realizations for each lS (in this case, ξ represents time). The illumination was generated using a liquid crystal spatial light modulator with pitch size of 15 μm and 512pixels×512pixels. The backscattered light was integrated on a single photodetector with peak quantum efficiency of 0.38. The measurements were conducted in noisy conditions characterized by signal-to-noise ratio (SNR) of 1.8 dB, and more details of the experimental setup are included in Supplement 1. As evident, both scale lengths characterizing this target are clearly identified in the experimental Fano spectrum.

In these examples, the correlation function characterizing the targeted potential is non-monotonic. For observing higher-order peaks in the Fano spectrum, the lS range of variation must be large enough to cover the spatial domain of interest. As a result, the level of variations in the first term of Eq. (5) is reduced, which determines the decrease in the overall level of fluctuations, as can be seen in the results summarized in Figs. 3(b) and 3(d).

C. Disturbed Illumination

So far, we have shown that perturbations that can be modeled as statistically stationary fluctuations do not degrade the performance of this stochastic sensing procedure. However, perturbations in the illumination path that can disturb the illumination structure may affect the results. In other words, the performance may be influenced by introducing uncertainties in the value of lS, our only tuning parameter. This case is illustrated in Fig. 4, where the size of the elementary function e(ρ;lS) changes randomly according to a Gaussian distribution with a standard deviations of Δ/3.

 figure: Fig. 4.

Fig. 4. (a) Normalized Fano spectrum for an unperturbed condition (-) and for the case of lS distributed Gaussianly with a variance of Δ/3 (-.-). (b) Correlation length lσ of the effective illumination (red shadow) as a function of the correlation length lS of the unperturbed illumination (black solid line).

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As suggested in the inset of Fig. 4(b), this is a dramatic variation for the value of lS. The Fano spectra corresponding to a scattering potential consisting of two rectangular objects characterized by length scales of 2Δ and 5Δ demonstrate that, even though the maxima broaden because of the uncertainty in lS, the characteristic length scales can still be clearly identified.

D. Non-Sparse Scattering Potentials

The examples illustrated so far prove the efficacy of this stochastic sensing procedure. They were, however, conducted on rather sparse scattering potentials. In the following, we will demonstrate that this is not a conceptual limitation. We will now consider targets consisting of densely packed and randomly shaped objects having slightly different statistical parameters. In general, the low sparsity complicates the detection of such subtle changes, which can be easily masked by experimental noise. To ensure realistic conditions, we considered measurement circumstances where noise was added to both the scattering potential and the illumination pattern to create an overall SNR of 2 dB.

We considered potentials with different spatial symmetries, as shown in Figs. 5(a) and 5(b). Note that, in the asymmetric case of Fig. 5(b), some of the objects are more elongated and, due to the asymmetry in the density-density correlation function, they generate two additional characteristic length scales. As shown in Fig. 5(c), this asymmetry is reflected in the Fano spectrum of the recorded intensity. The inset shows the ratio between the two spectra, FI,Asym./FI,Sym and, as can be seen, for larger values of lS associated with larger linear dimensions of the asymmetric objects, the difference between them increases because of an increased contribution of larger-sized objects. The autocorrelation functions of these two targets are compared in Figure (S4) of Supplement 1, where an additional discussion is provided.

 figure: Fig. 5.

Fig. 5. Simulated reflective targets consisting of randomly shaped objects having, on average, symmetric (a) and asymmetric (b) correlation functions. (c) Normalized Fano spectra corresponding to the symmetric (continuous line) and asymmetric (dashed line) targets, respectively. The inset shows the ratio of the two spectra as a function of the correlation length lS of the illumination pattern.

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Clearly, only an averaged characterization is meaningful for the asymmetry of the targets illustrated in Fig. 5. When the overall asymmetry increases, the corresponding Fano spectrum tends to develop maxima corresponding to the additional scales defined by the asymmetry and, as a result, the spectrum displays a longer tail. As clearly seen in the inset, this departure from the symmetric signature evolves nonlinearly because of the way the variation of the first term in Eq. (5) reduces when the characteristic scale lS increases under conditions of constant total reflectivity.

E. Weak Scattering Targets

Up to this point, we have shown that the stochastic optical sensing operates efficiently in harshly perturbed conditions and for different levels of target sparsity. Also important for a sensing method is its ability to recover information from targets with different scattering potential strengths. To examine this property experimentally, we examined two different targets with very different backscattering reflectivities. First, we examined the high-contrast element five of the group 5 in the positive standard USAF 1951 chart with lV=9.8μm. This is close to the case already shown in Fig. 3(c). As a second example, we used a significantly less reflective target, an H2c9 cell placed in an aqueous medium (DMEM+10%FPS+1×p/s). The cell has an almost symmetrical shape with a characteristic diameter of Δ10μm, similar to the length scale characterizing the USAF chart. Similar illumination conditions were used, and the typical results of these two experiments are illustrated Fig. 6. We note that during the 0.25 second illumination, the cumulative intensity of light used for the cells’ characterization was eight orders of magnitudes smaller than the sunlight intensity. As clearly seen, the stochastic sensing technique works quite well even in the case of such very low-contrast targets.

 figure: Fig. 6.

Fig. 6. Average over 100 Fano spectra corresponding to (a) element five of group 5 of USAF 1951 positive resolution chart, and (b) a live H2c9 cell placed in an aqueous medium.

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Moreover, the repeatability of the procedure is rather high, and the interested reader can find an ample discussion in Supplement 1.

5. CONCLUSION

Fluctuations are not always artifacts. In this paper, we proposed and demonstrated a stochastic sensing technique that relies on enhancing the fluctuations measured in a scattering experiment under structured illumination. Using the dispersion of the integrated scattered intensities’ fluctuations, we showed that the characteristic lengths of the interaction potential can be consistently recovered. We have also demonstrated the efficiency of this method in extremely perturbed conditions, either in illumination or detection. In addition, we proved that this method is efficient over a large range of practical conditions. We successfully characterized targets with both very low and very high scattering coefficients.

Our stochastic sensing procedure can be regarded as a type of “noise spectroscopy,” where abrupt changes in the non-stationary properties of a signal are controlled by adjusting the stochastic properties of the excitation field. Varying the correlation length of the illumination can be exploited to recover structural information about the interaction potentials.

We also present a model that essentially relies on the fact that, even for a Gaussian distribution of a random scattering potential, the scattered intensity is not necessarily Gaussian. By tailoring the illumination, one can therefore enforce non-Gaussian scattering processes, leading to measurable departures from the central limit theorem [1720]. Being non-universal, these deviations are measurable signatures of the specific properties of the scattering potential.

Finally, our experiments demonstrate that, in addition to being robust, this stochastic sensing approach is capable of providing spatial measurements with low errors in highly noisy conditions (SNR below 2 dB). This sensing method is not limited to the optical situations illustrated here and it should be appealing for a range of biomedical applications [21,22], material sciences [23,24], and atmospheric characterization [25,26]. Finally, although the examples presented here addressed static scattering potentials, a similar stochastic sensing approach can be extended to characterizing dynamic phenomena [24,27].

Funding

Air Force Office of Scientific Research (AFOSR) (FA95501010190); National Institute of General Medical Sciences (NIGMS) (1R21GM10794201A1).

Acknowledgment

We thank Dr. Lucia Cilenti for providing the H2c9 cells.

 

See Supplement 1 for supporting content.

REFERENCES

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21. N. Brenner and Y. Shokef, “Nonequilibrium statistical mechanics of dividing cell populations,” Phys. Rev. Lett. 99, 138102 (2007). [CrossRef]  

22. C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004). [CrossRef]  

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References

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  1. B. J. Berne and R. Pecora, Dynamic Light Scattering: with Applications to Chemistry, Biology, and Physics (Courier Corporation, 2000).
  2. S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative anisotropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
    [Crossref]
  3. D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
    [Crossref]
  4. H. Bayley and P. S. Cremer, “Stochastic sensors inspired by biology,” Nature 413, 226–230 (2001).
    [Crossref]
  5. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
    [Crossref]
  6. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
    [Crossref]
  7. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
    [Crossref]
  8. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
    [Crossref]
  9. J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
    [Crossref]
  10. J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
    [Crossref]
  11. D. H. Lenschow, V. Wulfmeyer, and C. Senff, “Measuring second-through fourth-order moments in noisy data,” J. Atmos. Ocean. Technol. 17, 1330–1347 (2000).
    [Crossref]
  12. A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
    [Crossref]
  13. W. J. Padgett and C. P. Tsokos, “Existence of a solution of a stochastic integral equation in turbulence theory,” J. Math. Phys. 12, 210–212 (1971).
    [Crossref]
  14. U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26–29 (1947).
    [Crossref]
  15. A. Bousselham, H. H. Barrett, V. Bora, and K. Shah, “Photoelectron anticorrelations and sub-Poisson statistics in scintillation detectors,” Nucl. Instrum. Methods Phys. Res. A 620, 359–362 (2010).
    [Crossref]
  16. K. Balachandran, K. Sumathy, and H. H. Kuo, “Existence of solutions of general nonlinear stochastic volterra fredholm integral equations,” Stoch. Anal. Appl. 23, 827–851 (2005).
    [Crossref]
  17. E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
    [Crossref]
  18. R. Barakat, “Direct derivation of intensity and phase statistics of speckle produced by a weak scatterer from the random sinusoid model,” J. Opt. Soc. Am. 71, 86–90 (1981).
    [Crossref]
  19. E. Jakeman and R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
    [Crossref]
  20. K. Kanazawa, T. Sagawa, and H. Hayakawa, “Stochastic energetics for non-Gaussian processes,” Phys. Rev. Lett. 108, 210601 (2012).
    [Crossref]
  21. N. Brenner and Y. Shokef, “Nonequilibrium statistical mechanics of dividing cell populations,” Phys. Rev. Lett. 99, 138102 (2007).
    [Crossref]
  22. C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
    [Crossref]
  23. M. Parrinello and A. Rahman, “Crystal structure and pair potentials: a molecular-dynamics study,” Phys. Rev. Lett. 45, 1196–1199 (1980).
    [Crossref]
  24. A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
    [Crossref]
  25. D. H. Lenschow and B. B. Stankov, “Length scales in the convective boundary layer,” J. Atmos. Sci. 43, 1198–1209 (1986).
    [Crossref]
  26. C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
    [Crossref]
  27. S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
    [Crossref]

2015 (1)

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

2012 (2)

K. Kanazawa, T. Sagawa, and H. Hayakawa, “Stochastic energetics for non-Gaussian processes,” Phys. Rev. Lett. 108, 210601 (2012).
[Crossref]

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

2010 (1)

A. Bousselham, H. H. Barrett, V. Bora, and K. Shah, “Photoelectron anticorrelations and sub-Poisson statistics in scintillation detectors,” Nucl. Instrum. Methods Phys. Res. A 620, 359–362 (2010).
[Crossref]

2008 (4)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative anisotropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
[Crossref]

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
[Crossref]

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

2007 (2)

A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
[Crossref]

N. Brenner and Y. Shokef, “Nonequilibrium statistical mechanics of dividing cell populations,” Phys. Rev. Lett. 99, 138102 (2007).
[Crossref]

2005 (1)

K. Balachandran, K. Sumathy, and H. H. Kuo, “Existence of solutions of general nonlinear stochastic volterra fredholm integral equations,” Stoch. Anal. Appl. 23, 827–851 (2005).
[Crossref]

2004 (1)

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

2001 (1)

H. Bayley and P. S. Cremer, “Stochastic sensors inspired by biology,” Nature 413, 226–230 (2001).
[Crossref]

2000 (1)

D. H. Lenschow, V. Wulfmeyer, and C. Senff, “Measuring second-through fourth-order moments in noisy data,” J. Atmos. Ocean. Technol. 17, 1330–1347 (2000).
[Crossref]

1998 (3)

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
[Crossref]

C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
[Crossref]

1995 (1)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

1991 (1)

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[Crossref]

1988 (1)

E. Jakeman and R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[Crossref]

1986 (1)

D. H. Lenschow and B. B. Stankov, “Length scales in the convective boundary layer,” J. Atmos. Sci. 43, 1198–1209 (1986).
[Crossref]

1981 (1)

1980 (1)

M. Parrinello and A. Rahman, “Crystal structure and pair potentials: a molecular-dynamics study,” Phys. Rev. Lett. 45, 1196–1199 (1980).
[Crossref]

1978 (1)

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[Crossref]

1971 (1)

W. J. Padgett and C. P. Tsokos, “Existence of a solution of a stochastic integral equation in turbulence theory,” J. Math. Phys. 12, 210–212 (1971).
[Crossref]

1947 (1)

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26–29 (1947).
[Crossref]

Abate, A. R.

A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
[Crossref]

Alexander, S.

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

Balachandran, K.

K. Balachandran, K. Sumathy, and H. H. Kuo, “Existence of solutions of general nonlinear stochastic volterra fredholm integral equations,” Stoch. Anal. Appl. 23, 827–851 (2005).
[Crossref]

Barakat, R.

Barrett, H. H.

A. Bousselham, H. H. Barrett, V. Bora, and K. Shah, “Photoelectron anticorrelations and sub-Poisson statistics in scintillation detectors,” Nucl. Instrum. Methods Phys. Res. A 620, 359–362 (2010).
[Crossref]

Bayley, H.

H. Bayley and P. S. Cremer, “Stochastic sensors inspired by biology,” Nature 413, 226–230 (2001).
[Crossref]

Behrendt, A.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering: with Applications to Chemistry, Biology, and Physics (Courier Corporation, 2000).

Bora, V.

A. Bousselham, H. H. Barrett, V. Bora, and K. Shah, “Photoelectron anticorrelations and sub-Poisson statistics in scintillation detectors,” Nucl. Instrum. Methods Phys. Res. A 620, 359–362 (2010).
[Crossref]

Bousselham, A.

A. Bousselham, H. H. Barrett, V. Bora, and K. Shah, “Photoelectron anticorrelations and sub-Poisson statistics in scintillation detectors,” Nucl. Instrum. Methods Phys. Res. A 620, 359–362 (2010).
[Crossref]

Boyd, R. W.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

Brasseur, J. G.

C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
[Crossref]

Brenner, N.

N. Brenner and Y. Shokef, “Nonequilibrium statistical mechanics of dividing cell populations,” Phys. Rev. Lett. 99, 138102 (2007).
[Crossref]

Cremer, P. S.

H. Bayley and P. S. Cremer, “Stochastic sensors inspired by biology,” Nature 413, 226–230 (2001).
[Crossref]

Dogariu, A.

S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative anisotropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
[Crossref]

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
[Crossref]

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
[Crossref]

Durian, D. J.

A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
[Crossref]

Eichinger, M.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Fano, U.

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26–29 (1947).
[Crossref]

Fink, C.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Friedl, P.

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

Gammaitoni, L.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Geissler, E. K.

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

Glotzer, S. C.

A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
[Crossref]

Haefner, D.

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
[Crossref]

S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative anisotropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
[Crossref]

Hammann, E.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

Hänggi, P.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Hayakawa, H.

K. Kanazawa, T. Sagawa, and H. Hayakawa, “Stochastic energetics for non-Gaussian processes,” Phys. Rev. Lett. 108, 210601 (2012).
[Crossref]

Hirschberg, M.

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

Jakeman, E.

E. Jakeman and R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[Crossref]

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[Crossref]

Jung, P.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Kanazawa, K.

K. Kanazawa, T. Sagawa, and H. Hayakawa, “Stochastic energetics for non-Gaussian processes,” Phys. Rev. Lett. 108, 210601 (2012).
[Crossref]

Kauczor, H. U.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Keys, A. S.

A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
[Crossref]

Khanna, S.

C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
[Crossref]

Koehl, G. E.

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

Kuo, H. H.

K. Balachandran, K. Sumathy, and H. H. Kuo, “Existence of solutions of general nonlinear stochastic volterra fredholm integral equations,” Stoch. Anal. Appl. 23, 827–851 (2005).
[Crossref]

Lenschow, D. H.

D. H. Lenschow, V. Wulfmeyer, and C. Senff, “Measuring second-through fourth-order moments in noisy data,” J. Atmos. Ocean. Technol. 17, 1330–1347 (2000).
[Crossref]

D. H. Lenschow and B. B. Stankov, “Length scales in the convective boundary layer,” J. Atmos. Sci. 43, 1198–1209 (1986).
[Crossref]

Ley, S.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Marchesoni, F.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).
[Crossref]

Mendel, J. M.

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[Crossref]

Muppa, S. K.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

Padgett, W. J.

W. J. Padgett and C. P. Tsokos, “Existence of a solution of a stochastic integral equation in turbulence theory,” J. Math. Phys. 12, 210–212 (1971).
[Crossref]

Pal, S.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

Parrinello, M.

M. Parrinello and A. Rahman, “Crystal structure and pair potentials: a molecular-dynamics study,” Phys. Rev. Lett. 45, 1196–1199 (1980).
[Crossref]

Pecora, R.

B. J. Berne and R. Pecora, Dynamic Light Scattering: with Applications to Chemistry, Biology, and Physics (Courier Corporation, 2000).

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Plathow, C.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Puderbach, M.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Pusey, P. N.

E. Jakeman and P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978).
[Crossref]

Rahman, A.

M. Parrinello and A. Rahman, “Crystal structure and pair potentials: a molecular-dynamics study,” Phys. Rev. Lett. 45, 1196–1199 (1980).
[Crossref]

Sagawa, T.

K. Kanazawa, T. Sagawa, and H. Hayakawa, “Stochastic energetics for non-Gaussian processes,” Phys. Rev. Lett. 108, 210601 (2012).
[Crossref]

Schmähl, A.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Senff, C.

D. H. Lenschow, V. Wulfmeyer, and C. Senff, “Measuring second-through fourth-order moments in noisy data,” J. Atmos. Ocean. Technol. 17, 1330–1347 (2000).
[Crossref]

Sergienko, A. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Shah, K.

A. Bousselham, H. H. Barrett, V. Bora, and K. Shah, “Photoelectron anticorrelations and sub-Poisson statistics in scintillation detectors,” Nucl. Instrum. Methods Phys. Res. A 620, 359–362 (2010).
[Crossref]

Shapiro, J. H.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11, 949–993 (2012).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Shokef, Y.

N. Brenner and Y. Shokef, “Nonequilibrium statistical mechanics of dividing cell populations,” Phys. Rev. Lett. 99, 138102 (2007).
[Crossref]

Stankov, B. B.

D. H. Lenschow and B. B. Stankov, “Length scales in the convective boundary layer,” J. Atmos. Sci. 43, 1198–1209 (1986).
[Crossref]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Sukhov, S.

S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative anisotropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
[Crossref]

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
[Crossref]

Sumathy, K.

K. Balachandran, K. Sumathy, and H. H. Kuo, “Existence of solutions of general nonlinear stochastic volterra fredholm integral equations,” Stoch. Anal. Appl. 23, 827–851 (2005).
[Crossref]

Tong, C.

C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
[Crossref]

Tough, R. J. A.

E. Jakeman and R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[Crossref]

Tsokos, C. P.

W. J. Padgett and C. P. Tsokos, “Existence of a solution of a stochastic integral equation in turbulence theory,” J. Math. Phys. 12, 210–212 (1971).
[Crossref]

Wolf, E.

Wulfmeyer, V.

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

D. H. Lenschow, V. Wulfmeyer, and C. Senff, “Measuring second-through fourth-order moments in noisy data,” J. Atmos. Ocean. Technol. 17, 1330–1347 (2000).
[Crossref]

Wyngaard, J. C.

C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
[Crossref]

Zuna, I.

C. Plathow, C. Fink, S. Ley, M. Puderbach, M. Eichinger, I. Zuna, A. Schmähl, and H. U. Kauczor, “Radiotherapy and oncolog,” Radiotherapy Oncology 73, 349–354 (2004).
[Crossref]

Adv. Phys. (1)

E. Jakeman and R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[Crossref]

Atmos. Chem. Phys. (1)

A. Behrendt, V. Wulfmeyer, E. Hammann, S. K. Muppa, and S. Pal, “Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar,” Atmos. Chem. Phys. 15, 5485–5500 (2015).
[Crossref]

Histochem. Cell Biol. (1)

S. Alexander, G. E. Koehl, M. Hirschberg, E. K. Geissler, and P. Friedl, “Dynamic imaging of cancer growth and invasion: a modified skin-fold chamber model,” Histochem. Cell Biol. 130, 1147–1154 (2008).
[Crossref]

J. Atmos. Ocean. Technol. (1)

D. H. Lenschow, V. Wulfmeyer, and C. Senff, “Measuring second-through fourth-order moments in noisy data,” J. Atmos. Ocean. Technol. 17, 1330–1347 (2000).
[Crossref]

J. Atmos. Sci. (2)

D. H. Lenschow and B. B. Stankov, “Length scales in the convective boundary layer,” J. Atmos. Sci. 43, 1198–1209 (1986).
[Crossref]

C. Tong, J. C. Wyngaard, S. Khanna, and J. G. Brasseur, “Resolvable-and subgrid-scale measurement in the atmospheric surface layer: Technique and issues,” J. Atmos. Sci. 55, 3114–3126 (1998).
[Crossref]

J. Math. Phys. (1)

W. J. Padgett and C. P. Tsokos, “Existence of a solution of a stochastic integral equation in turbulence theory,” J. Math. Phys. 12, 210–212 (1971).
[Crossref]

J. Opt. Soc. Am. (1)

Nat. Phys. (1)

A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian, “Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material,” Nat. Phys. 3, 260–264 (2007).
[Crossref]

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Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1. Schematic procedure of sensing statistical properties of a random scattering potential V(ρ;lV) interrogated with a controlled stochastic illumination S(ρ,ξ), which is affected by the inherent source noise n(ρ;ξ), and perturbations Ui(ρ,ξ) and Uo(ρ,ξ) in the illumination and recording paths, respectively. The measured signal is the integrated intensity I(ξ).
Fig. 2.
Fig. 2. Numerical calculation of the Fano spectrum corresponding to (a) random targeted potential consisting of circular disks with lV=Δ, and (b) the corresponding Fano spectrum. The maximum of the Fano spectrum is normalized to unity.
Fig. 3.
Fig. 3. (a) Simulated reflective target perturbed by illumination noise and (b) the corresponding normalized Fano spectrum. The measurement conditions are similar to those in Fig. 2. (c) Targeted standard microscopy chart under noisy illumination and (d) the corresponding normalized Fano spectrum.
Fig. 4.
Fig. 4. (a) Normalized Fano spectrum for an unperturbed condition (-) and for the case of lS distributed Gaussianly with a variance of Δ/3 (-.-). (b) Correlation length lσ of the effective illumination (red shadow) as a function of the correlation length lS of the unperturbed illumination (black solid line).
Fig. 5.
Fig. 5. Simulated reflective targets consisting of randomly shaped objects having, on average, symmetric (a) and asymmetric (b) correlation functions. (c) Normalized Fano spectra corresponding to the symmetric (continuous line) and asymmetric (dashed line) targets, respectively. The inset shows the ratio of the two spectra as a function of the correlation length lS of the illumination pattern.
Fig. 6.
Fig. 6. Average over 100 Fano spectra corresponding to (a) element five of group 5 of USAF 1951 positive resolution chart, and (b) a live H2c9 cell placed in an aqueous medium.

Equations (6)

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I(ξ)=A1A[(S(x;ξ)+n(x;ξ))P(x;ξ)V(x)+R(x;ξ)]dx,
MV(k)=m˜I(k)m˜σ(k)n=0k22k!m˜R(k2n)(k2n)!(2n)!m˜σ(2n)m˜σ(k)MV(2n).
FV=FImσ(1)mI(1)m˜σ(2)(mI(1)mR(1))m˜R(2)mσ(1)m˜σ(2)(mI(1)mR(1)).
m˜I(2)(lS)=ψ(ρ;ξ,lS,lV)ψ(ρ;ξ,lS,lV)ξdρdρ+m˜R(2),
I˜(ξ;lS)=S˜(ρ;ξ,lS)P(ρ;ξ)V(ρ;lV)dρ+mS(1)P˜(ρ;ξ)V(ρ;lV)dρ+n(ρ;ξ)P(ρ;ξ)V(ρ;lV)dρ+R˜(ρ;ξ)dρ
S˜(ρ;ξ,lS)P(ρ;ξ)V(ρ;lV)dρ=C(ρ;ξ)(e˜(ρ;lS)*P(ρ;ξ)V(ρ;lV))dρ.

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