Abstract

Recent development of active imaging system technology in the defense and security community have driven the need for a theoretical understanding of its operation and performance in military applications such as target acquisition. In this paper, the modeling of active imaging systems, developed at the U.S. Army RDECOM CERDEC Night Vision & Electronic Sensors Directorate, is presented with particular emphasis on the impact of coherent effects such as speckle and atmospheric scintillation. Experimental results from human perception tests are in good agreement with the model results, validating the modeling of coherent effects as additional noise sources. Example trade studies on the design of a conceptual active imaging system to mitigate deleterious coherent effects are shown.

©2007 Optical Society of America

1. Introduction

Active imaging systems are being developed and marketed widely in the defense and security community as possible identification aides in reconnaissance systems [1, 2, 3]. These systems have several distinct technological and practical advantages over passive imaging systems making them attractive alternatives, or at the very least, complementary components to existing infrared imaging technology. Gating, for example, reduces camera noise, mitigates the scattering effects of obscurants, and reduces the amount of background clutter and path radiance. Proper gating can also be utilized to capture silhouette images with very high scene contrast. The use of an active illuminator boosts the signal-to-noise ratio while relaxing the sensitivity requirements in the receiver. In addition, the availability of inexpensive near infrared lasers and image intensifier tubes can make the system practical and cost efficient. One common Army operational concept and implementation of an active imaging system that has been developed under the Cost-Effective Targeting System (CETS) program, depicted in Fig. 1, involves the use of an uncooled long wave infrared imager (FLIR) for wide field of view (FOV) search and detection coupled with a laser range gated (LRG) short wave infrared (SWIR) imager for narrow FOV identification.

 figure: Fig. 1.

Fig. 1. Operational concept and implementation of an active imaging system.

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Recently, it has been proposed that the use of an active 2-D/3-D imaging system can provide significant improvements in identification ranges [1, 4]. However, these advantages can be lost due to the reflective properties of the objects viewed (speckle) and the effects of atmospheric optical turbulence on the illumination pulse (scintillation) [5, 6]. Accurate predictions of observer performance require that these new effects relevant to active imaging systems be accounted for in target acquisition models. The primary degradations are distortion of the outbound illuminating beam resulting in non-uniform illumination of the target and inbound distortion of the illuminated image. While turbulence induced distortions of the inbound image can be severe, human observers have the psychophysical ability to mitigate them [7]. The observer appears to be able to ignore large-scale distortions while small-scale distortions appear as additional blur to the image to degrade resolution [8]. Resolution degradations due to turbulence are included in current performance models by use of an atmospheric modulation transfer function (MTF) developed by Goodman [9]. Distortion of the coherent illuminating beam results in a non-uniform distribution of reflected intensities from the target, which could have a more severe effect on observer performance than distortions along the return path.

The U.S. Army RDECOM CERDEC Night Vision & Electronic Sensors Directorate (NVESD) has a validated set of models for the performance of passive imagers for target acquisition. The introduction and growing interest of active imaging systems has prompted the development of extensions to these models to account for the coherent nature of the source and the effect of atmospheric turbulence on the illuminating beam, i.e., speckle and scintillation, respectively.

In this paper, a brief overview and discussion of NVESD models for target acquisition will be given. Next, the extensions to the model to account for the active imaging capability are introduced. Radiometry is based on a radar approach. Speckle and scintillation are modeled as additional noise sources combined in quadrature with other system noise sources. A method for simulating scintillation using beam propagation is reviewed. The results for the validation experiments using human perception tests for the model extensions are then analyzed. Then, the model is used to perform trade studies of several candidate active imaging systems. Finally, conclusions regarding this model and plans for future extensions are discussed.

 figure: Fig. 2.

Fig. 2. Relationship between system CTF, limiting frequency, and supra-threshold contrast.

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2. Theory of target acquisition

The models produced by NVESD all operate from the same principles stemming from two fundamental assumptions: 1) target acquisition performance is related to the image quality and 2) image equality is related to the threshold vision of the observer through the sensor. These assumptions lead to three primary equations describing the models.

First, a contrast threshold function (CTF) that characterizes the observer and the sensor is defined. This function defines the contrasts visible to the observer using the sensor. The CTF is a function of the spatial frequency f and is predicted by the following [10],

CTFsys(f)=CTFEye(f)Hsys(f)1+κ2σ2(f)L2

where CTFsys is the degraded CTF of the observer through the sensor system, CTFeye is the CTF of the human eye [11], Hsys is the modulation transfer function (MTF) of the system, σ is the standard deviation of the perceived noise on the display, L is the average display luminance, κ is an empirically determined calibration constant. An example of the system CTF plot is shown in Fig. 2. As noted in the introduction, turbulence blur effects are accounted for in the model as an additional MTF that contributes to the overall system MTF in Eq. 1. The form for the short term turbulence MTF is [10],

Hatmospheric(f)short=e57.4aCn2Rλ13f53(1u(fλD0)13)

where f is spatial frequency, λ is the wavelength, C2 n is the refractive index structure parameter, R is the round trip path length, D 0 is the optics focal length, a = 3/8 and a = 1 for spherical and plan wave, respectively, and u = 0.5 and u = 1 for far and near field, respectively.

The perceived noise on the display is calculated using,

σ2(f)=Sn(ξ)HPost(ξ)HPer(ξ,f)2dξ

where Sn is the power spectral density of the noise source, HPost is any filter in the sensor that occurs after the point where noise is generated, and HPer is a filter describing the human perception process. HPost includes the MTFs associated with the display and the eye. The perceptual filter is a band pass filter designed to mimic the spatial frequency channels of the human visual system. In psychophysical tests, it has been shown that it is only the noise within an octave of the center frequency that interferes with the perception of a sine wave. In the model, an expression introduced by Barten [12] is used and is given by,

HPer(ξ,fc)=exp[2.2log2(ξfc)]

where ξ is spatial frequency and fc is the center frequency of the filter.

Second, an ensemble of targets having an average contrast of Ct is compared with the CTFsys using the target task performance (TTP) metric [10] given by,

TTP=floflimitCtCTFsys(f)df[cycmrad]

where Ct is the root sum square (RSS) contrast of the target and the limits of integration are determined by

CTFsys(f)flo,flimit=Ctflimit>flo

As shown in Fig. 2, The TTP metric is the area above the system CTF curve and below the target contrast, namely, the supra-threshold contrast. The TTP metric is given in units of cycles/mrad. But range performance is dictated by an effective number of cycles on the target, which is calculated by

V(R)=1000(AtR)TTP[cyc]

where At is the area of the target in square meters and R is the range to the target in meters. Equation (7) is essentially a measure of the information available to the observer, i.e., the number of cycles on target, and is a function of both the sensitivity and resolution of the observer using the sensor relative to the target.

Finally, the amount of information necessary to perform a task to a given probability is empirically determined using a logistic function known as the target transform probability function (TTPF), which fits measured task performance probabilities reasonably well. The TTPF gives the probability of an observer performing a task (detection, recognition, identification) as a function of both the effective cycles on target, which is a function of range, and a predetermined set of criteria for that task. The TTPF is given by,

Ptask(R)=(V(R)V50(task))E1+(V(R)V50(task))E

where V 50 is the number of effective cycles needed to perform a given task at 50% probability, E is an exponent determined by fitting (8) to perception experiments. The value of E derived from previous experiments was found to be

E=1.51+0.24(V(R)V50)

where V 50 is equal to 2.7, 14.5, and 18.8 for the task of target detection, recognition, and identification, respectively, calibrated using the NVESD 12 target long-wave infrared (LWIR) set [10]. Note that these values are given only as examples. Task difficulty calibration values will be different for LRG SWIR images.

 figure: Fig. 3.

Fig. 3. Illustration of the radiometric laser model.

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3. Model extensions for active imaging systems

3.1. Radiometry

The radiometry model, based on a radar approach, is illustrated in Fig. 3 and is shown below. Assuming perfect target reflection from a Lambertian target that is unresolved with respect to the transmitted beam size but resolved with respect to the receiver, the power at the receiver for a monostatic LRG system is [13],

PRx=(PTxπR2(tan(ϕlas2))2)(Adetf2)(1π)(πD024)τatm2τRx

where PTx is the transmitted power, R is the range to the target,ϕlas is the laser beam divergence, Adet is the area of the detector pixel, f is the focal length, Do is the aperture diameter,τatm is the one-way atmospheric transmission, and τRx is the optics transmission. Details of the radiometry and how contrast is calculated for the model are in the unpublished NVLRG manual.

3.2. Speckle

Coherent illumination of targets and background with rough surfaces (compared to a wavelength) produces speckle in the imagery. Speckle is a fine grain, high contrast spatial noise. Figure 4 shows speckle from a field measurement using an LRG imager. As such, speckle impacts the observer’s ability to perform discrimination tasks such as identification of the target [14]. Blur and noise are major factors in sensor performance and must be included in performance models. Hence, the effects of speckle are most easily modeled as another noise source [7]. The speckle noise term is multiplicative (rather than additive) because the brightness of the speckle pattern is governed by the average intensity in the local area of the scene.

Size of the noise (autocorrelation, power spectral density) and contrast are key parameters for noise, including speckle. Speckle results from averaging electric fields (phasors) present within each imager point spread function (PSF). That is, the speckle is produced in the imager itself as each imager pixel sums the phasor fields from each point within the PSF footprint at the target range. For well designed imagers, the PSF of the optics is essentially the diffraction limited spot size. Thus, one input to the LRG model is the aperture size of the optics. This, along with the center wavelength, determines the image plane speckle size. The detectors then sample the PSF at the image plane. Each detector provides an image pixel. The detector pitch compared to the PSF size then determines the speckle size in pixels in the image itself.

 figure: Fig. 4.

Fig. 4. Speckle from a target board at 1 kilometer as imaged by a laser range gated imager under conditions with virtually no turbulence.

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The expression for the speckle noise variance used in the model is,

σspeckle2(f)=1NSspeckle(ξ)HPost(ξ)HDet(ξ)HPer(ξ,f)2dξ

where Sspeckle is the power spectral density of the speckle, N is the number of independent speckle samples averaged during the formation of the displayed image, HPost is the postfilter MTF, HDet is the detector MTF, and HPer is the perceptual filter MTF. Sspeckle is given by [9],

Sspeckle(f)=LHOptics(f)0HOptics(ξ)dξ

where HOptics is the optics aperture MTF and L is the average display luminance.

For many LRG imagers, the design provides only a few detectors per PSF. This results in an almost white power spectral density for the speckle noise. Speckle contrast is defined as,

Cspeckle=σII

where σI and 〈I〉 are the standard deviation and mean of the intensity. For some situations the speckle contrast is high (Cspeckle → 1) and can represent a significant impediment to imager performance.

Some of the physical factors reducing contrast are spectral averaging (short coherence length), tilted surfaces with short coherence lengths and temporal (frame) averaging. Lasers used for illumination are not highly coherent. Their wide spectral bandwidths result in illumination with a spread of wavelengths. The sensor integrates over its spectral response, resulting in an averaging over the wavelengths present in the laser illumination. A tilted surface illuminated by the short coherence length laser also tends to reduce speckle contrast [5, 15]. Halford shows that the contrast reduction can be modeled simply. The combination of tilt and short coherence length create “cells” within the imager pixel. Summing the fields over these cells averages independent speckle patterns, resulting in significant reduction in contrast. The number of independent cells within a pixel is readily determined from the geometry and is,

K=imagerpixelsizelc/sinα

where lc is the coherence length of the laser and α is the tilt of the surface relative to the normal longitudinal direction from the target to the source, i.e., α=0 for normal incidence. The contrast reduction is then the reciprocal of the square root of K.

 figure: Fig. 5.

Fig. 5. Propagation through turbulence.

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Frame averaging also reduces the speckle contrast by adding independent (in time) speckle noise patterns. Most LRG imagers allow frame averaging. Consequently the active imager model uses the number of frames averaged to set the speckle contrast. Each frame is assumed to have a contrast of 1, providing a worst case effect of speckle noise. The final image after averaging has a contrast of 1 divided by the square root of the number of frames averaged together.

3.3. Scintillation

Optical turbulence is caused by refractive index variations in the air induced by mixing of air at different temperature [6]. The variations in refractive index occur at all scales, as governed by a refractive index power spectrum [16] as notionally illustrated in Fig. 5. The most significant spatial spectral region is the inertial subrange beginning at the inner scale (lo) and extending to the outer scale (L 0). Within this range the turbulence strength is characterized by a single quantity, C2 n, the refractive index structure parameter. Refractive index variations at scales within the inertial subrange cause most of the observed effects in imagery. In the absence of atmospheric turbulence, the observed fields at the target due to an illuminating laser beam will have a high degree of spatial coherence. When propagated through a turbulent medium, the fields become spatially random due to phase decorrelation upon propagation through pockets of randomly varying refractive index. Phase decorrelation then leads to focused and defocused regions in the propagating wavefront creating the amplitude fluctuations termed scintillation. The high frequency dissipation subrange just above the inertial range has important implications for scintillation effects. Statistically, the scintillation is often assumed log normally distributed, but this is only strictly the case in weak scintillation. As turbulence increases, the scintillation distribution passes through intermediate phases before assuming an exponential distribution at asymptotically high turbulence levels. The statistical character of the scintillated field is determined by propagation range, strength of the turbulence (C2 n), inner scale length, and wavelength. In general, the field will possess multiple scales of correlation in amplitude and phase corresponding to various observed scales in the turbulent medium. These scales are observed in the variations in the intensity of illumination of the target.

When the random incident field resulting from the beam propagating through a turbulent atmosphere interacts with a target having a random rough surface, the surface imposes further spatial incoherence. This process is independent of the atmosphere and acts at scales much smaller than the atmosphere. The resulting scattered field is the random incident field modulated by the random rough surface. The modulation at the surface is mostly with respect to phase and to a much lesser degree with respect to magnitude. The phase variations caused by the surface are dominant in the scattered field and together with the aperture function of the receiver, are responsible for the observed speckle in active imagery with coherent sources. Propagation back to the sensor (the inbound path) involves another pass through the random atmosphere. However, now the wave is spatially incoherent due to reflection from a rough surface. As a result, the large-scale coherent effects imposed on the outbound path cannot produce the same large-scale intensity variations seen on the inbound path. The field is essentially spatially decorrelated to a scale smaller than that of the atmosphere and hence further decorrelation at larger scales has little effect. The image can still suffer refractive distortions due to the turbulence and some of these can be severe [8]. However, it has been shown that human observers are able to mitigate the effects of large-scale distortions [7].

 figure: Fig. 7.

Fig. 7. Example imagery showing scintillation patterns varying with C2 n. The images were taken at measured C2 n values of (Left) 1.22E-12, (Center) 1.41E-13, and (Right) 1.11E-14 m2/3.

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The effects of the atmosphere on the outbound beam can be demonstrated experimentally. In the Spring of 2003, an experiment was conducted using a laser and a gated camera. A schematic of the experiment is shown in Fig. 6. Two panels were set up at a range of 2 km. One panel was painted white and the other black. A laser operating at 1.57 μm was used to illuminate the panels. The terrain between the laser and the panels was flat sandy soil. A gated camera was set up approximately 30 m from the panels.

Gating of the down range camera was accomplished by generating a delayed trigger pulse from the received signal of an avalanche photodiode pointed toward the laser. As the incoming pulse from the laser was received an appropriate delay was added and the camera was gated. This images from the camera give an estimate of the effects of turbulence on the outbound beam without the confounding influence of a second propagation through the 2 km path. Example imagery from this experiment is shown in Fig. 7. Note how the beam patterns change with decreasing values of C2 n.

Because of the statistical nature of turbulence, characterizing it devolves to the characterization of a random process. Random patterns resulting from turbulence are defined by two aspects, the intensity contrast and the scale sizes. An autocovariance function captures the random nature of the shapes of the intensity variations. Characteristic sizes present in the intensity variation correspond to hill shaped features in the autocovariance function. The e-1 points provide estimates of the scale sizes. The autocovariance function is an average measurement of these characteristic sizes.

With turbulence, the laser-illuminated target acquires the scintillation pattern of the laser pulse. This scintillation has a large contrast and two characteristic scale sizes under strong turbulence conditions. Dunphy and Kerr [17] and Gracheva, et al. [18] and others present experimental evidence supporting this. Theoretical models for the strong turbulence regime show the two scale sizes and model the contrast as well [6, 19].

Both of the scale sizes are functions of the strength of the turbulence. However, under strong turbulence one scale size is significantly smaller than the other. Hence, the terms small scale and large scale are used to denote the sizes. Other terms for these scales are spatial coherence radius (small scale) and scattering disk radius (large scale). A product of random processes models the two scale random process for the intensity[20]. The autocovariance of the product, expressed in terms 9f the log irradiance covariances for the small and large scale terms, is,

B(ρ)=exp[Blnx(ρ)+Blny(ρ)]1,

where BI(ρ) is the autocovariance function for the intensity for two points on the wavefront of the outbound beam with a transverse separation of ρ = √x 2 + y 2. Blnx and Blny, represent the contributions for the large scale term and the small scale term respectively.

Blnx and Blny are obtained from a heuristic model developed in the frequency domain. An effective atmospheric spectral density is obtained from a modified Kolmogorov spectrum that uses two filter functions to model the small scale and large scale contributions. This model yields Blnx and Blny,

Blnx(ρ)=K010η116eη/ηxJ0(ρξkηL){1cos[ηξ(1ξ)]},dηdξ
Blnx(ρ)=K010(η+ηy)116J0(ρξkηL){1cos[ηξ(1ξ)]},dηdξ
K=2.65β02

where β2 0 is the Rytov variance for a spherical wave given by,

β02=0.4σ12
σ12=1.23Cn2k76L116

where ηx and ηy are given by,

ηx=8.561+0.19σ1125
ηy=9(1+0.23σ1125)

and where J0 is the Bessel function of the first kind of order zero, k is the wave number and L is the range.

The autocovariance function is obtained by using numerical integration to obtain Blnx and Blny, which determine the autocovariance, BI(ρ). The autocovariance defines the effective noise in an imaging system due to scintillation. The numerical procedure is to carry out the numerical integration for each value from an array of values for ρ. The Fourier transform of this quantity yields the power spectral density which is used in the model, viz.,

σscintillation2(f)=Sscintillation(ξ)HPost(ξ)HPer(ξ,f)2dξ.

3.4. Caveats On Noise Extensions

Note that the model currently accounts for realistic sensor blur and noise effects as well as atmospheric blurring effect due to turbulence and display and human vision characteristics. The above extensions to the current model, i.e., speckle and scintillation noise effects are both first order extensions and are treated as separate and independent effects. Speckle is modeled as being fully developed and is a worst case effect. The scintillation model does not account for inner scale effects; future versions will include a first level modification rescaling the peak value according to an inner scale dependent formula and a second level that includes inner scale as another autocovariance transverse length scale. Furthermore, inbound scintillation effects are ignored as discussed above. Finally, interaction between turbulence blur and speckle and the interaction between return path scintillation and speckle, which can affect the overall noise variances, are not presently included. Future extensions to the model will address each of these effects, as well as account for partial coherence, target tilt, and frame averaging of independent speckle events to more realistically treat speckle.

3.5. Simulating scintillation

A key goal in determining the effects of optical turbulence on system performance is the ability of trained observers to discriminate targets in the presence of turbulent image degradation. To simulate these effects for active imaging systems, a first phase was to develop two-dimensional templates of scintillation effects on active illumination pulses.

These templates of instantiations of turbulence effects were generated using a wide-angle scintillated beam model [21]. To generate such instantiations the phase screen propagation method [22] is typically used. However, the standard phase screen approach is only tailored for focused and collimated beam calculations. Illumination pulses represent relatively highly divergent beams. Assuming the focal plane for such beams is the system exit aperture [23], high resolution calculations are required initially, and a large domain at the target illumination plane. Modeling such beams using a single transverse wave array is computationally very expensive, and unnecessary. The beam can be more easily handled via subdivision into a series of low divergence sub-beams using the decomposition model:

p0(r)=Apexp(r2w02)=exp(θ2θ02)n0(rθ)dθ

where p 0(r⃗) is the original Gaussian beam intensity pattern in the exit aperture. This wave is decomposed into sub-beams n 0(r⃗,θ⃗) each centered in the exit aperture, but directed into a unique off-axis path according to launch direction θ⃗. Dimension z designates the downrange axis. Vector r⃗ = (x,y) indicates positions in the transverse plane. In the paraxial approximation a unit vector denoting the centroid of the direction of propagation of a sub-beam is written Ω⃗ ≈ (θx,θy,1). This is the small angle approximation [24]. In this approximation the limits of the angular integral are extended to ±∞ because the exponential envelope will attenuate all radiation at angles large compared to θ 0. In this approximation, sin(θ) θdθdϕxy.

The sub-beams in the exit aperture are characterized as,

n0(r,θ)=Anexp(r2v02)exp(ikrθ)

leading to the relation, w0=v0v02k2θ024. That is, due to beam forming, the full wave will always be narrower than each sub-beam in the exit aperture. Conversely, at range z,w 2 z = v 2 z + θ 2 0 z 2. The sub-beams thus produce an overlapping applique of illumination at the target range. In the propagation algorithm, the integral is replaced by a summation of individual sub-beams weighted by the ΔΩ of the interval between adjacent beams. Since each sub-beam is initially wider than the original full beam, the computational cost per sub-beam is considerably reduced.

Once a given sub-beam is designated, the propagation procedure is approximately the same as for the standard phase screen method: the optical path is divided into a series of segments. For each segment the turbulence properties are characterized by the refractive index structure parameter Cn2[m23], and inner and outer scales of turbulence, lo and Lo, respectively. Within each segment spatial wave propagation is modeled as a diffraction limited calculation in a Fourier space. The wavefront incident at the slab boundary is first propagated halfway through the slab. There, the random effects of turbulence are applied through multiplication by the position-dependent phase factor, exp [(r⃗)], using,

ϕ(r)=(kc/n0)δτ(r)

to represent a phase perturbation based on propagation time fluctuation δτ of a ray passing through the complete slab. Here, k = 2π/λ is the monochromatic wavenumber, c is velocity of light in vacuum, and n 0 is the mean background refractive index. After passing through the phase screen the wavefront is propagated through the remainder of the slab using another Fourier diffraction calculation.

The phase screen approach is based on the concept that the phase perturbations acting on the wave will be small over the length of a single path segment. The segment must also be long enough to ensure successive phase screens are decorrelated. The key to generating a screen is the randomized δτ model, which is obtained as the inverse transform of a randomized spectrum. This spectrum is based on an analysis of the crossing time autocovariance function: 〈δτ(s1)δτ(s2)〉 [25]. The results of this analysis are based on the refractive index spectrum Φn(κ), slab thickness Δ, and phase screen dimensions X and Y. The randomized crossing times are then evaluated via an inverse Fast-Fourier Transform of the array,

Tlm=(2π)3ΔXYc2GlmΦn(2πl/X,2πm/Y,0)

where Glm is a complex Gaussian noise spectrum [22].

The refractive index spectrum used includes both inner and outer scale effects, represented by,

Φn(κx,κy,κz)=Φn(κ)=Φn(κ)=ΦI(κ,lo)+ΦX(κ,Lo)ΦK(κ)

where ΦK(κ)=0.033Cn2κ113 is the Kolmogorov spectrum function of spatial frequency κ[m-1]. ΦI is the inner scale modified spectrum such that ΦI → ΦK as κ → 0; ΦX is the outer scale modified spectrum such that ΦXΦK as κ → ∞. The combined forms model the complete spectrum. The inner scale term is modeled after Hill’s [16] bump spectrum using,

ΦI(κ,lo)=0.033Cn2κ113F(κlo/1.412)
F(x)=2.2J(x/7.5)1.2J(x/2.5)

based on Belen’kii’s [26] J(x)=x116K116(x)/[256Γ(116)].

The outer scale portion [27] is based on a reanalysis of von-Karman [28] and using the Kaimal et al. [29] one-dimensional spectrum of temperature. The form derived was

ΦX(κ,Lo)=0.033Cn2i=1,2aiX(κ,i)
 figure: Fig. 8.

Fig. 8. Scintillated beam profiles generated by simulation: no scintillation, C2 n=1E-14,C2 n=3.16E-14, and C2 n=1E-13.

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where the i are related to the integral scale, and X(κ,L)=κ2L173/(1+κ2L2)176. A suitable fit to Kaimal et al.’s [29] 1-D spectral forms was found using a 1 = 8.2, a 2 = -7.2, 1 = 2.07Lo, 2 = 2.48Lo. This form replaces the standard, though misnamed, von-Karman spectrum, since von-Karman’s [28] original theory recommended a spectrum equal to zero at zero frequency. To avoid confusion with the currently named von-Karman spectrum, the term Kaimal spectrum appears appropriate.

With the establishment of an appropriate spectrum, phase screens of any dimension are possible. Our approach utilizes screens with a maximum dimension of 200 m to avoid loss of low frequency fluctuational energy. However, with such a low frequency limit, a single screen cannot provide sufficient small scale resolution. Therefore, a method of building cascaded phase screens was developed. The model developed uses 4 screens of increasing resolution in each path segment. The resolution of the finest screen is 0.19 mm. Subsequent higher frequency screens periodically repeat within the domain of the lowest resolution screen. Higher frequency screens increase spatial resolution by a factor of 16 above the previous screen. To ensure independence, the appropriate low frequency region of each higher frequency screen must be set to zero.

A 2 km path length was chosen, and C2 n values ranging over 3 orders of magnitude in half decade increments were computed. 10 sample instantiations of turbulence effects were generated as illumination templates at each turbulence strength. Examples of the processes are shown in Fig. 8.

4. Model component validation

4.1. Speckle noise

Imagery with speckle noise was produced using an LRG simulation developed under a previous research effort [14]. Simulated images were produced under four conditions. The first was incoherent (spatial and temporal incoherence). Under this condition, a pristine visible band image was blurred by an incoherent sensor point spread function. No sensor noise was added. The second condition was a single shot LRG-SWIR mode where temporal coherence was maintained and the spatial phase was randomized. The third condition was a two-pulsed average. The fourth condition was an eight-pulsed average. This imagery was processed by first doing pixel replication and taking a square root operator. Then a random phase was applied to each pixel. After convolution with the coherent point spread function of the optics, the squared magnitude of the image was convolved with the remaining incoherent point spread functions of the simulated system. This process produced fully developed speckle in the images and was repeated to simulate the effect of range, i.e., 1,3,5,7.5,10, and 15 km, for each of the four conditions. The standard NVESD 12 target set was used in the perception experiment. An example of the simulated speckle imagery at 5 km range for the four conditions is shown in Fig. 9(a). The incoherent image was taken as a baseline for imagery comparison and image averages were obtained by adding independent speckle images. It can be seen that the speckle from the single-shot image can degrade the identification performance significantly.

 figure: Fig. 9.

Fig. 9. 9(a) Speckled images: incoherent, 1 shot, 2 shot average, and 8 shot average. 9(b) Comparison of model predictions and experimental results. The TTPF explains approximately 94% of the variance in the measured data.

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Fifteen soldiers were trained to identify targets with 95% proficiency prior to participating in the experiment. The experimental cells were randomized to vary the level of target identification difficulty. The images were displayed on high-resolution grayscale monitors. The monitors have Gaussian MTFs with equivalent spot sizes of 0.306 mm horizontally and 0.237 mm vertically. There were approximately 70.9 pixels per centimeter on the displays. The images were displayed with an average luminance of 5 fL and were viewed from a nominal distance of 15 inches. After correcting for chance guesses and mistakes, the average probability of ID for each cell was calculated and the results plotted against the model predictions (TTPF). As shown in Fig. 9(b), the experimental results and the model predictions are in good agreement with the model explaining 94% of the observed variance in the data [7].

 figure: Fig. 10.

Fig. 10. 10(a) Scintillated images: unscintillated, C2 n=1E-14, C2 n=3.16E-14, and C2 n=1E-13. 10(b) Comparison of model predictions and experimental results. The TTPF explains approximately 89% of the variance in the measured data.

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4.2. Scintillation noise

Imagery with scintillation noise was produced by multiplying the simulated scintillation masks in Fig. 8, corresponding to the unscintillated, C2 n=1E-14, C2 n=3.16E-14, and C2 n=1E-13 scintillated conditions, with the NVESD 12 target visible set [21]. The effect of range was added to each of the four conditions by processing the images with a prescribed blur, which varied from 5 to 17.5 pixels. An example of the simulated scintillated imagery at 10 pixel blur is shown in Fig. 10(a).

 figure: Fig. 11.

Fig. 11. 11(a) Trade study comparing aperture size vs. C2 n on range performance. 11(b) Effect of frame averaging vs. turbulence strength on range performance.

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In a manner similar to the speckle perception experiment, six soldiers and two civilians were trained and participated in the scintillation noise human perception experiment. After correcting for chance guesses and mistakes, the average probability of ID for each cell was calculated and the results plotted against the model predictions (TTPF). As shown in Fig. 10(b), the experimental results and the model predictions are in good agreement with the model explaining 89% of the observed variance in the data [30].

5. Model use: Trade Studies

One of the practical applications of the NVLRG performance model is in the implementation of trade studies. Trade studies are necessary in order to assess the model-predicted performance for various environmental and system conditions, such as turbulence strength and aperture size, respectively. Given the strong impact of turbulence on the performance of active imaging systems, a trade study was designed to see the effect of aperture size and frame averaging on mitigating/limiting the impact of turbulence on range performance. The results of the trade study are shown in Fig. 11.

The modeled active system had the following sensor parameters: an 8 mJ, 20 ns, 1 Hz pulsed illuminator at λ=1.57μm;a 640×480 electron bombardedCCD (EBCCD) detector with 12 μm pixel size; an F#8, 1 m focal length, 5 inch centrally obscured aperture with 0.44×0.33 FOV. As shown in Fig. 11(a), the aperture size can improve the ID range performance by a factor of 2 in benign conditions while showing minimal effect in moderate-to-strong turbulence conditions. All curves converge to the same result. The same result is true in the use of frame averaging to mitigate the effect of turbulence (Fig. 11(b)).

These results show that turbulence-induced effects present in active imaging systems employing a laser illuminator can severely limit range performance. Mitigation of these deleterious effects, therefore, must be explored. Optimized system design through the use of a larger aperture or simple frame averaging show some improvement. A change in tactical concept of operations (CONOPS), i.e., operating in the low turbulence night conditions, may also help. However, a recent study by Weiss-Wrana [31] showed that the frequency of occurrence of C2 n is quite high in the range between 1E-14–1E-13. As shown in Fig. 11, in this range, the dependence on aperture size is a minimal effect.

6. Conclusion

In this paper, an approach to modeling the performance of active imaging systems has been introduced. This approach leverages physically accurate models of speckle, scintillation, and psychophysically accurate models of target acquisition. The agreement obtained between the model and perception experiments for each noise component is good and indicate that the effects of speckle and scintillation on the illuminating beam are major factors in degrading performance of active imagers. Future work on full validation of the end-to-end model will require extensive field and perception tests. The trade studies performed analyzing the effect of aperture size and frame averaging versus turbulence strength show that one can, by proper system design, mitigate, or at least, limit the impact of these turbulence effects on range performance. This model should be a useful tool in the future design of active imaging systems. The U.S. Army RDECOM CERDEC NVESD NVLRG model will be publicly released in 2007. Subsequent releases will include the effect of partially coherent sources, image dancing, turbulence-induced speckle, speckle/scintillation interaction, inner scale effects to the scintillation noise model, and calibrated task difficulties. Further extensions to the model will include imaging simulations to visually show active imaging effects on a representative set of targets.

Ackowledgements

The authors acknowledge Dr. Ron Driggers and Dr. Joe Reynolds of the Night Vision and Electronic Sensors Directorate for useful discussions in support of this research. We also thank the reviewers for very helpful comments and suggestions.

References and links

01. A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol.4024 of Proc. SPIE , pp.180–191 (2000). [CrossRef]  

02. O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol.3707 of Proc. SPIE, pp.432–448 (1999). [CrossRef]  

03. J. Busck, “Underwater 3-D optical imaging with a gated viewing laser radar,” Opt. Eng. 44,116,001 (2005). [CrossRef]  

04. P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng. 45,034,301 (2006). [CrossRef]  

05. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).

06. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Belling-ham, WA, 2001). [CrossRef]  

07. E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol.5407 of Proc. SPIE , pp.201–210 (2004). [CrossRef]  

08. K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001). [CrossRef]  

09. J. W. Goodman, Statistical Optics (Wiley Interscience, New York, NY, 2000).

10. R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004). [CrossRef]  

11. P. G. J. Barten, “Evaluation of subjective image quality with the square-root integral method,” J. Opt. Soc. Am. A 7,2024–2031 (1990). [CrossRef]  

12. P. G. J. Barten, Contrast sensitivity of the human eye and its effects on image quality (SPIE Press Monograph, PM72, 1999). [CrossRef]  

13. W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).

14. R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003). [CrossRef]  

15. C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007). [CrossRef]  

16. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech.88,541–562 (1978).

17. J. R. Dunphy and J. Kerr, “Scintillation measurements for large integrated path turbulence,” J. Opt. Soc. Am. 63,981–986 (1973). [CrossRef]  

18. M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

19. V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).

20. L. C. Andrews and R. L. Phillips, “I-K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2,160–163 (1985). [CrossRef]  

21. D. H. Tofsted and S. G. O’Brien, “Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems,” in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement, and W. R. Reynolds, eds., vol.5432 of Proc. SPIE , pp.160–171 (2004). [CrossRef]  

22. J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976). [CrossRef]  

23. A. E. Siegman, Lasers (Univ. Sci. Books, Mill Valley, CA, 1986).

24. W. G. Tam and A. Zardecki, “Multiple scattering corrections to the Beer-Lambert Law. I: Open Detector,” Appl. Opt. 21,2405–2412 (1980). [CrossRef]  

25. D. H. Tofsted, “Turbulence Simulation: On Phase and Deflector Screen Generation,” Tech. rep., U.S. Army Res. Lab. (2001).

26. M. S. Belen’kii, “Effect of the inner scale of turbulence on the atmospheric modulation transfer function,” J. Opt. Soc. Am. A 13,1078–1082 (1996). [CrossRef]  

27. D. H. Tofsted, “Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum,” Tech. rep., U.S. Army Res. Lab. (2000).

28. T. von Karman, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. U.S. 34,530–539 (1948). [CrossRef]  

29. J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972). [CrossRef]  

30. E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol.5987 of Proc. SPIE , pp.598,703–1–598,703–11 (2005).

31. K. Weiss-Wrana, “Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions,” in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., vol.5891 of Proc. SPIE , pp.58,910D–1–58,910D–12 (2005).

References

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  1. A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol. 4024 of Proc. SPIE, pp.180–191 (2000).
    [Crossref]
  2. O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
    [Crossref]
  3. J. Busck, “Underwater 3-D optical imaging with a gated viewing laser radar,” Opt. Eng. 44,116,001 (2005).
    [Crossref]
  4. P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng. 45,034,301 (2006).
    [Crossref]
  5. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).
  6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Belling-ham, WA, 2001).
    [Crossref]
  7. E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol. 5407 of Proc. SPIE, pp.201–210 (2004).
    [Crossref]
  8. K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
    [Crossref]
  9. J. W. Goodman, Statistical Optics (Wiley Interscience, New York, NY, 2000).
  10. R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004).
    [Crossref]
  11. P. G. J. Barten, “Evaluation of subjective image quality with the square-root integral method,” J. Opt. Soc. Am. A 7,2024–2031 (1990).
    [Crossref]
  12. P. G. J. Barten, Contrast sensitivity of the human eye and its effects on image quality (SPIE Press Monograph, PM72, 1999).
    [Crossref]
  13. W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).
  14. R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
    [Crossref]
  15. C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007).
    [Crossref]
  16. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88,541–562 (1978).
  17. J. R. Dunphy and J. Kerr, “Scintillation measurements for large integrated path turbulence,” J. Opt. Soc. Am. 63,981–986 (1973).
    [Crossref]
  18. M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).
  19. V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).
  20. L. C. Andrews and R. L. Phillips, “I-K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2,160–163 (1985).
    [Crossref]
  21. D. H. Tofsted and S. G. O’Brien, “Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems,” in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement, and W. R. Reynolds, eds., vol. 5432 of Proc. SPIE, pp.160–171 (2004).
    [Crossref]
  22. J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976).
    [Crossref]
  23. A. E. Siegman, Lasers (Univ. Sci. Books, Mill Valley, CA, 1986).
  24. W. G. Tam and A. Zardecki, “Multiple scattering corrections to the Beer-Lambert Law. I: Open Detector,” Appl. Opt. 21,2405–2412 (1980).
    [Crossref]
  25. D. H. Tofsted, “Turbulence Simulation: On Phase and Deflector Screen Generation,” Tech. rep., U.S. Army Res. Lab. (2001).
  26. M. S. Belen’kii, “Effect of the inner scale of turbulence on the atmospheric modulation transfer function,” J. Opt. Soc. Am. A 13,1078–1082 (1996).
    [Crossref]
  27. D. H. Tofsted, “Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum,” Tech. rep., U.S. Army Res. Lab. (2000).
  28. T. von Karman, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. U.S. 34,530–539 (1948).
    [Crossref]
  29. J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
    [Crossref]
  30. E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).
  31. K. Weiss-Wrana, “Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions,” in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., vol. 5891 of Proc. SPIE, pp.58,910D–1–58,910D–12 (2005).

2006 (1)

P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng. 45,034,301 (2006).
[Crossref]

2005 (3)

J. Busck, “Underwater 3-D optical imaging with a gated viewing laser radar,” Opt. Eng. 44,116,001 (2005).
[Crossref]

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).

K. Weiss-Wrana, “Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions,” in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., vol. 5891 of Proc. SPIE, pp.58,910D–1–58,910D–12 (2005).

2004 (3)

D. H. Tofsted and S. G. O’Brien, “Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems,” in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement, and W. R. Reynolds, eds., vol. 5432 of Proc. SPIE, pp.160–171 (2004).
[Crossref]

E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol. 5407 of Proc. SPIE, pp.201–210 (2004).
[Crossref]

R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004).
[Crossref]

2003 (1)

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

2001 (1)

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

2000 (2)

A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol. 4024 of Proc. SPIE, pp.180–191 (2000).
[Crossref]

D. H. Tofsted, “Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum,” Tech. rep., U.S. Army Res. Lab. (2000).

1999 (1)

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

1996 (1)

1990 (1)

1985 (1)

1980 (1)

1978 (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88,541–562 (1978).

1976 (1)

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976).
[Crossref]

1973 (1)

1972 (1)

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
[Crossref]

1948 (1)

T. von Karman, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. U.S. 34,530–539 (1948).
[Crossref]

Andersson, P.

P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng. 45,034,301 (2006).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, “I-K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2,160–163 (1985).
[Crossref]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Belling-ham, WA, 2001).
[Crossref]

Banakh, V. A.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).

Barnard, K. J.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

Barten, P. G. J.

P. G. J. Barten, “Evaluation of subjective image quality with the square-root integral method,” J. Opt. Soc. Am. A 7,2024–2031 (1990).
[Crossref]

P. G. J. Barten, Contrast sensitivity of the human eye and its effects on image quality (SPIE Press Monograph, PM72, 1999).
[Crossref]

Belen’kii, M. S.

Bolander, G.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

Bowman, T.

A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol. 4024 of Proc. SPIE, pp.180–191 (2000).
[Crossref]

Busck, J.

J. Busck, “Underwater 3-D optical imaging with a gated viewing laser radar,” Opt. Eng. 44,116,001 (2005).
[Crossref]

Cote, O. R.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
[Crossref]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).

Devitt, N.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

Driggers, R.

R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004).
[Crossref]

Driggers, R. G.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007).
[Crossref]

Driggers, R. N.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

Dunphy, J. R.

Espinola, R. L.

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).

Feit, M. J.

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976).
[Crossref]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley Interscience, New York, NY, 2000).

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

Groenwall, C. A.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

Halford, C.

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

Halford, C. E.

E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol. 5407 of Proc. SPIE, pp.201–210 (2004).
[Crossref]

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007).
[Crossref]

Hill, R. J.

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88,541–562 (1978).

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Belling-ham, WA, 2001).
[Crossref]

Izumi, Y.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
[Crossref]

Jacobs, E.

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).

R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004).
[Crossref]

Jacobs, E. L.

E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol. 5407 of Proc. SPIE, pp.201–210 (2004).
[Crossref]

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007).
[Crossref]

Kaimal, J. C.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
[Crossref]

Kasharov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

Kerr, J.

Klager, G.

A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol. 4024 of Proc. SPIE, pp.180–191 (2000).
[Crossref]

Kopeika, N. S.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

Krapels, K.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

Letalick, D.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

Milton, A. F.

A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol. 4024 of Proc. SPIE, pp.180–191 (2000).
[Crossref]

Mironov, V. L.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976).
[Crossref]

O’Brien, S. G.

D. H. Tofsted and S. G. O’Brien, “Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems,” in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement, and W. R. Reynolds, eds., vol. 5432 of Proc. SPIE, pp.160–171 (2004).
[Crossref]

Olsson, H.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, “I-K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2,160–163 (1985).
[Crossref]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Belling-ham, WA, 2001).
[Crossref]

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

Robinson, A. L.

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (Univ. Sci. Books, Mill Valley, CA, 1986).

Steinvall, O. K.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

Tam, W. G.

Tofsted, D.

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).

Tofsted, D. H.

D. H. Tofsted and S. G. O’Brien, “Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems,” in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement, and W. R. Reynolds, eds., vol. 5432 of Proc. SPIE, pp.160–171 (2004).
[Crossref]

D. H. Tofsted, “Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum,” Tech. rep., U.S. Army Res. Lab. (2000).

D. H. Tofsted, “Turbulence Simulation: On Phase and Deflector Screen Generation,” Tech. rep., U.S. Army Res. Lab. (2001).

Vollmerhausen, R.

R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004).
[Crossref]

Vollmerhausen, R. H.

E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol. 5407 of Proc. SPIE, pp.201–210 (2004).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

von Karman, T.

T. von Karman, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. U.S. 34,530–539 (1948).
[Crossref]

Weiss-Wrana, K.

K. Weiss-Wrana, “Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions,” in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., vol. 5891 of Proc. SPIE, pp.58,910D–1–58,910D–12 (2005).

Wolfe, W.

W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).

Wyngaard, J. C.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
[Crossref]

Zardecki, A.

Zissis, G.

W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).

Appl. Opt. (1)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time-dependent propagation of high-energy laser-beams through atmosphere,” Appl. Phys. 10,129–160 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (5)

R. Vollmerhausen, E. Jacobs, and R. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43,2806–2818 (2004).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, “Impact of speckle on laser range-gated shortwave infrared imaging system target identification performance,” Opt. Eng. 42,738–746 (2003).
[Crossref]

J. Busck, “Underwater 3-D optical imaging with a gated viewing laser radar,” Opt. Eng. 44,116,001 (2005).
[Crossref]

P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng. 45,034,301 (2006).
[Crossref]

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, “Atmospheric turbulence modulation transfer function for infrared target acquisition modeling,” Opt. Eng. 40,1906–1913 (2001).
[Crossref]

Proc. Natl. Acad. Sci. U.S. (1)

T. von Karman, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. U.S. 34,530–539 (1948).
[Crossref]

Proc. SPIE (5)

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, “Beam scintillation effects on identification performance with active imaging systems,” in Electro-Optical and Infrared Systems: Technology and Applications II, R. G. Driggers and D. A. Huckridge, eds., vol. 5987 of Proc. SPIE, pp.598,703–1–598,703–11 (2005).

K. Weiss-Wrana, “Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions,” in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., vol. 5891 of Proc. SPIE, pp.58,910D–1–58,910D–12 (2005).

D. H. Tofsted and S. G. O’Brien, “Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems,” in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement, and W. R. Reynolds, eds., vol. 5432 of Proc. SPIE, pp.160–171 (2004).
[Crossref]

A. F. Milton, G. Klager, and T. Bowman, “Low cost sensors for UGVs,” in Unmanned Ground Vehicle Technology II, G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., vol. 4024 of Proc. SPIE, pp.180–191 (2000).
[Crossref]

E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, “Modeling active imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XV, G. C. Holst, ed., vol. 5407 of Proc. SPIE, pp.201–210 (2004).
[Crossref]

Proc. SPIE, (1)

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman and C. Werner, eds., vol. 3707 of Proc. SPIE, pp.432–448 (1999).
[Crossref]

Q. J. Roy. Met. Soc. (1)

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, “Spectral characteristics of surface-layer turbulence,” Q. J. Roy. Met. Soc. 98,563–589 (1972).
[Crossref]

Tech. rep., U.S. Army Res. Lab. (1)

D. H. Tofsted, “Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum,” Tech. rep., U.S. Army Res. Lab. (2000).

Other (11)

D. H. Tofsted, “Turbulence Simulation: On Phase and Deflector Screen Generation,” Tech. rep., U.S. Army Res. Lab. (2001).

A. E. Siegman, Lasers (Univ. Sci. Books, Mill Valley, CA, 1986).

J. W. Goodman, Statistical Optics (Wiley Interscience, New York, NY, 2000).

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Belling-ham, WA, 2001).
[Crossref]

P. G. J. Barten, Contrast sensitivity of the human eye and its effects on image quality (SPIE Press Monograph, PM72, 1999).
[Crossref]

W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, “Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model,” submitted to Opt. Eng. (2007).
[Crossref]

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88,541–562 (1978).

M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).

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Figures (11)

Fig. 1.
Fig. 1. Operational concept and implementation of an active imaging system.
Fig. 2.
Fig. 2. Relationship between system CTF, limiting frequency, and supra-threshold contrast.
Fig. 3.
Fig. 3. Illustration of the radiometric laser model.
Fig. 4.
Fig. 4. Speckle from a target board at 1 kilometer as imaged by a laser range gated imager under conditions with virtually no turbulence.
Fig. 5.
Fig. 5. Propagation through turbulence.
Fig. 7.
Fig. 7. Example imagery showing scintillation patterns varying with C2 n . The images were taken at measured C2 n values of (Left) 1.22E-12, (Center) 1.41E-13, and (Right) 1.11E-14 m2/3.
Fig. 8.
Fig. 8. Scintillated beam profiles generated by simulation: no scintillation, C2 n =1E-14,C2 n =3.16E-14, and C2 n =1E-13.
Fig. 9.
Fig. 9. 9(a) Speckled images: incoherent, 1 shot, 2 shot average, and 8 shot average. 9(b) Comparison of model predictions and experimental results. The TTPF explains approximately 94% of the variance in the measured data.
Fig. 10.
Fig. 10. 10(a) Scintillated images: unscintillated, C2 n =1E-14, C2 n =3.16E-14, and C2 n =1E-13. 10(b) Comparison of model predictions and experimental results. The TTPF explains approximately 89% of the variance in the measured data.
Fig. 11.
Fig. 11. 11(a) Trade study comparing aperture size vs. C2 n on range performance. 11(b) Effect of frame averaging vs. turbulence strength on range performance.

Equations (31)

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CTF sys ( f ) = CTF Eye ( f ) H sys ( f ) 1 + κ 2 σ 2 ( f ) L 2
H atmospheric ( f ) short = e 57.4 a C n 2 R λ 1 3 f 5 3 ( 1 u ( f λ D 0 ) 1 3 )
σ 2 ( f ) = S n ( ξ ) H Post ( ξ ) H Per ( ξ , f ) 2 d ξ
H Per ( ξ , f c ) = exp [ 2.2 log 2 ( ξ f c ) ]
TTP = f lo f limit C t CTF sys ( f ) d f [ cyc mrad ]
CTF sys ( f ) f lo , f limit = C t f limit > f lo
V ( R ) = 1000 ( A t R ) TTP [ cyc ]
P task ( R ) = ( V ( R ) V 50 ( task ) ) E 1 + ( V ( R ) V 50 ( task ) ) E
E = 1.51 + 0.24 ( V ( R ) V 50 )
P R x = ( P T x π R 2 ( tan ( ϕ las 2 ) ) 2 ) ( A det f 2 ) ( 1 π ) ( π D 0 2 4 ) τ atm 2 τ R x
σ speckle 2 ( f ) = 1 N S speckle ( ξ ) H Post ( ξ ) H Det ( ξ ) H Per ( ξ , f ) 2 d ξ
S speckle ( f ) = L H Optics ( f ) 0 H Optics ( ξ ) d ξ
C speckle = σ I I
K = imager pixel size l c / sin α
B ( ρ ) = exp [ B ln x ( ρ ) + B ln y ( ρ ) ] 1 ,
B ln x ( ρ ) = K 0 1 0 η 11 6 e η / η x J 0 ( ρ ξ k η L ) { 1 cos [ η ξ ( 1 ξ ) ] } , d η d ξ
B ln x ( ρ ) = K 0 1 0 ( η + η y ) 11 6 J 0 ( ρ ξ k η L ) { 1 cos [ η ξ ( 1 ξ ) ] } , d η d ξ
K = 2.65 β 0 2
β 0 2 = 0.4 σ 1 2
σ 1 2 = 1.23 C n 2 k 7 6 L 11 6
η x = 8.56 1 + 0.19 σ 1 12 5
η y = 9 ( 1 + 0.23 σ 1 12 5 )
σ scintillation 2 ( f ) = S scintillation ( ξ ) H Post ( ξ ) H Per ( ξ , f ) 2 d ξ .
p 0 ( r ) = A p exp ( r 2 w 0 2 ) = exp ( θ 2 θ 0 2 ) n 0 ( r θ ) d θ
n 0 ( r , θ ) = A n exp ( r 2 v 0 2 ) exp ( ik r θ )
ϕ ( r ) = ( kc / n 0 ) δτ ( r )
T lm = ( 2 π ) 3 Δ XYc 2 G lm Φ n ( 2 π l / X , 2 π m / Y , 0 )
Φ n ( κ x , κ y , κ z ) = Φ n ( κ ) = Φ n ( κ ) = Φ I ( κ , l o ) + Φ X ( κ , L o ) Φ K ( κ )
Φ I ( κ , l o ) = 0.033 C n 2 κ 11 3 F ( κ l o / 1.412 )
F ( x ) = 2.2 J ( x / 7.5 ) 1.2 J ( x / 2.5 )
Φ X ( κ , L o ) = 0.033 C n 2 i = 1,2 a i X ( κ , i )

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