Abstract

This document provides a correction to the derivation of the distributions of the scattered fields presented in Optica 7, 63 (2020) [CrossRef]   and discusses the implications this correction has on the rest of the paper.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. SUMMARY OF CORRECTION

Our original derivation, presented in Section 3 of Ref. [1], modeled scattering off of the virtual source, the hidden object, and the virtual detector as modulating the incident field by a uniformly distributed random phasor and a constant equal to the square root of the albedo of the scattering surface. [Eq. (5) should have read $ {E_{{O_{\rm out}}}}({x_0}) = {E_{{O_{\rm in}}}}({x_O}){r^{1/2}}({x_O}) $].

In essence, this treats the scattering at position $ x $ as if the field is interacting with just a single scatterer. However, for any meaningful discretization, the field interacts with multiple dephased scatterers at each position $ x $ and the resulting scattered fields are summed together. Thus, the scattering process is more accurately modeled by multiplying the incident field by a complex random variable drawn from a circular complex Gaussian distribution [2].

In Section 2 we present the corrected derivation of the field distributions. The main difference is that the fields emitted from the hidden object and the virtual detector follow circular complex double Gaussian distributions [3]. That is, we have speckled speckle. Additionally, we relax the far-field propagation restriction by observing that the mathematical principles underlying correlography can be extended to accommodate Fresnel propagation [4].

In Section 3 we discuss the implications these changes have on our results.

2. CORRECTED DERIVATION OF FIELDS

We assume that the virtual source surface is illuminated by a collimated beam at normal incidence so that

$${E_{V{S_{\rm in}}}}({x_{VS}}) = 1.$$

The incident field interacts with multiple scatterers on the optically rough virtual source. This causes the scattered field to follow a circular complex Gaussian distribution,

$${E_{V{S_{\rm out}}}}({x_{VS}}) \sim CN(0,\sigma _{{E_{V{S_{\rm out}}}}}^2),$$
where $ {E_{V{S_{\rm out}}}}({x_{VS,1}}) $ and $ {E_{V{S_{\rm out}}}}({x_{VS,2}}) $ are independent for $ {x_{VS,1}} \ne {x_{VS,2}} $.

The field emerging from the virtual source then undergoes free-space propagation on its way to the object, which can be modeled by a Fresnel transformation. Accordingly, the field incident on the hidden object is

$${E_{{O_{\rm in}}}} \propto {p_2}{\cal F}({p_1}{E_{V{S_{\rm out}}}}),$$
where $ {\cal F} $ denotes the Fourier transform operator and $ {p_1} $ and $ {p_2} $ are quadratic phase terms with $ |{p_1}| = |{p_2}| = 1 $.

From the Central Limit Theorem and the independence of the fields at different locations of $ {E_{V{S_{\rm out}}}} $ we have that for all $ {x_O} $, $ {\cal F}({p_1}{E_{V{S_{\rm out}}}})({x_O}) $ follows a circular complex Gaussian distribution. Because the phase of $ {\cal F}({p_1}{E_{V{S_{\rm out}}}})({x_O}) $ is uniformly distributed, multiplying it by a quadratic phase $ {p_2} $ does not change its distribution, thus $ {E_{{O_{\rm in}}}}({x_O}) $ also follows a circular complex Gaussian distribution.

Following the Van Cittert–Zernike theorem, the autocorrelation of $ {E_{{O_{\rm in}}}} $ is determined by the Fourier transform of the intensity of $ {E_{V{S_{\rm in}}}} $ [2]. Thus, assuming we illuminate a sufficiently large spot size on the virtual source, the autocorrelation of $ {E_{{O_{\rm in}}}} $ is $ \sigma _{{O_{\rm in}}}^2\delta (\Delta {x_O}) $ for some constant $ \sigma _{{O_{\rm in}}}^2 $.

At each location $ {x_O} $ on the hidden object, the incident field interacts with multiple scatterers. The resulting scattered field is

$${E_{{O_{\rm out}}}}({x_O}) = {g_O}({x_O}){E_{{O_{\rm in}}}}({x_O}),$$
where for all $ {x_0} $, $ {g_O}({x_O}) $ follows an independent circular complex Gaussian distribution with variance $ r({x_O}) $, where $ r $ denotes the albedo of the hidden object at position $ {x_O} $.

Accordingly, $ {E_{{O_{\rm out}}}} $ follows a circular complex double Gaussian distribution whose autocorrelation is given by

$$r\sigma _{{O_{\rm in}}}^2\delta (\Delta {x_O}).$$

This field propagates to the virtual detector. Thus,

$${E_{V{D_{\rm in}}}} \propto {p_4}{\cal F}({p_3}{E_{{O_{\rm out}}}}),$$
where $ {p_3} $ and $ {p_4} $ are quadratic phase terms. This incident field scatters of multiple scatterers on the virtual detector and we image the intensity of the scattered field. Thus,
$$I({x_D}) = |{g_D}({x_D}){E_{V{D_{\rm in}}}}({x_D}{)|^2} = |{g_D}({x_D}{)|^2}|{E_{V{D_{\rm in}}}}({x_D}{)|^2},$$
where for all $ {x_D} $, $ {g_D}({x_D}) $ follows an independent circular complex Gaussian distribution with unit variance (assuming the virtual detector has albedo one).

To make subsequent analysis simpler, we note that

$$I({x_D}) = |{g_D}({x_D}{)|^2}|{\cal F}({\tilde E_{{O_{\rm out}}}}{)|^2},$$
where $ {\tilde E_{{O_{\rm out}}}} = {p_3}{E_{{O_{\rm out}}}} $. Because $ {E_{{O_{\rm out}}}} $ has uniformly distributed phase, $ {\tilde E_{{O_{\rm out}}}} $ and $ {E_{{O_{\rm out}}}} $ follow the same distribution.

3. IMPLICATIONS

Correlography is based on relating the ensemble power spectral density (PSD) of the observed images to the autocorrelation function of the hidden object’s albedo. This derivation is based on two steps: (1) showing that the ensemble PSD converges to $ {\mathbb E}[|f \star f{|^2}|](\Delta x) $, where $ f $ is used to denote $ {\tilde E_{{O_{\rm out}}}} $. (2) showing that $ {\mathbb E}[|f \star f{|^2}|](\Delta x) $ is equal to the autocorrelation of the albedo, plus an additional term at $ \Delta x = 0 $.

Changing the distribution of $ {\tilde E_{{O_{\rm out}}}} $ and $ I $ requires us to slightly modify our derivation of each step. It also has minor implications on the predicted variance of our estimate.

A. PSD Estimate

Because $ |{\cal F}(f{)|^2} $ is modulated by speckle $ |{g_D}{|^2} $, as the number of observations goes to infinity our estimate of the autocorrelation converges to the PSD of $ |{g_D}{|^2}|{\cal F}(f{)|^2} $, rather than that of $ |{\cal F}(f{)|^2} $. We now relate the PSD of $ |{g_D}{|^2}|{\cal F}(f{)|^2} $ to the PSD of $ |{\cal F}(f{)|^2} $.

Let $ \beta = |{g_D}{|^2} $ and $ E = |{\cal F}(f{)|^2} $, with $ I = \beta E $. Assuming $ E $ is approximately wide sense stationary, $ I $’s autocorrelation function is

$${R_I}({x_D},\Delta {x_D}) \approx {R_\beta }(\Delta {x_D}){R_E}(\Delta {x_D}).$$

Thus the power spectral density of $ I $ is given by

$${\cal F}({R_I})(\Delta x) \approx {\cal F}({R_\beta })(\Delta x)*{\cal F}({R_E})(\Delta x),$$
where $ * $ denotes convolution.

Because $ {R_\beta }(\Delta {x_D}) = {c_1} + {c_2}\delta (\Delta {x_D}) $ where $ {c_1} $ is a constant due to the non-zero mean of the speckle, the PSD of the speckle is given by $ {\cal F}({R_\beta })(\Delta x) = {c_2} + {c_1}\delta (\Delta x) $. Thus, the PSD of $ I $ is given by

$$\begin{split}{\cal F}({R_I})(\Delta x) &\approx {c_2}*{\cal F}({R_E})(\Delta x) + {c_1}\delta (\Delta x)*{\cal F}({R_E})(\Delta x), \\& \approx {c_3} + {c_1}{\cal F}({R_E})(\Delta x).\end{split}$$

Thus, when one accounts for speckle noise at the virtual detector, the ensemble PSD estimate converges to a scaled version of the unspeckled PSD (which we can equate to $ {\mathbb E}[|f \star f{|^2}|](\Delta x) $ using the original derivation found in the supplement of [1]) plus a constant offset term. This offset is already modeled by the $ b $ term in Eq. (11) of [1], though the paper assigns this term primarily to shot noise.

B. Autocorrelation Estimate

The next step of the proof shows that $ {\mathbb E}[|f \star f{|^2}|](\Delta x) $ is equal to the autocorrelation of the albedo plus a zero lag term. We repeat this proof here with the corrected scattering model. To make notation more concise, we let $ {z_1} = {p_3}{g_O} $ and $ {z_2} = {E_{{O_{\rm in}}}} $ and assume the limits of integration are from $ - \infty $ to $ \infty $ unless otherwise noted.

First, from the independence of $ {z_1} $ and $ {z_2} $ we have that

$$\begin{split}&{\mathbb E}[|(f \star f{)|^2}](\Delta x) \\& = \int_{{x_1}} \int_{{x_2}} {\mathbb E}[{\bar z_1}({x_1} + \Delta x){z_1}({x_1}){z_1}({x_2} + \Delta x){\bar z_1}({x_2})] \\&\quad \cdot {\mathbb E}[{\bar z_2}({x_1} + \Delta x){z_2}({x_1}){z_2}({x_2} + \Delta x){\bar z_2}({x_2})]{\rm d}{x_1}{\rm d}{x_2}.\end{split}$$

Using Isserlis’ theorem, substituting in the values of the covariances of $ {z_1} $ and $ {z_2} $, and assuming $ \sigma _{{O_{\rm in}}}^2 = 1 $ we have

$$\begin{split}&{\mathbb E}[|(f \star f{)|^2}](\Delta x) \\& = \int_{{x_1}} \int_{{x_2}} [r({x_1})r({x_2})\delta (\Delta x) + r({x_1})r({x_2})\delta (\Delta x)\delta ({x_2} - {x_1}) \\&\quad + r({x_1} + \Delta x)r({x_1})\delta ({x_2} - {x_1})\delta (\Delta x) \\&\quad + r({x_1} + \Delta x)r({x_1})\delta ({x_2} - {x_1})]{\rm d}{x_1}{\rm d}{x_2} \\& = \delta (\Delta x)\int_{{x_1}} \int_{{x_2}} [r({x_1})r({x_2}) + r({x_1})r({x_2})\delta ({x_2} - {x_1}) \\&\quad + r({x_1} + \Delta x)r({x_1})\delta ({x_2} - {x_1})]{\rm d}{x_1}{\rm d}{x_2} \\&\quad+ \int_{{x_1}} \int_{{x_2}} r({x_1} + \Delta x)r({x_1})\delta ({x_2} - {x_1}){\rm d}{x_1}{\rm d}{x_2} \\& = \delta (\Delta x)W + r \star r(\Delta x),\end{split}$$
where $ W $ is constant.

As desired, $ {\mathbb E}[|(f \star f{)|^2}](\Delta x) $ is still equal to the autocorrelation of the albedo, plus an offset at $ \Delta x = 0 $. While this offset changes with the corrected scattering model, it plays no role in how we generate training data nor reconstruct the hidden object.

C. Variance Estimate

The paper argues that $ I $ follows a Gaussian-like distribution, and thus, following [5], the variance of the autocorrelation estimate is proportional to the true PSD squared divided by the number of observations. However, by incorporating the correct scattering model we find that, ignoring noise, as speckled speckle $ I $ follows a $ K $-distribution (page 60 of [2]), which is not well approximated by a Gaussian. While our empirical results suggest the aforementioned result remains valid, our argument is more heuristic than originally claimed.

Funding

Defense Advanced Research Projects Agency (HR0011-16-C-0028).

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

REFERENCES

1. C. A. Metzler, F. Heide, P. Rangarajan, M. M. Balaji, A. Viswanath, A. Veeraraghavan, and R. G. Baraniuk, “Deep-inverse correlography: towards real-time high-resolution non-line-of-sight imaging,” Optica 7, 63–71 (2020). [CrossRef]  

2. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed. (Roberts and Company Publishers, 2020).

3. N. O’Donoughue and J. M. Moura, “On the product of independent complex Gaussians,” IEEE Trans. Signal Process. 60, 1050–1063 (2011). [CrossRef]  

4. J. R. Fienup, “Emerging systems and technologies,” in The Infrared and Electro-Optical Systems Handbook, Atmospheric propagation of radiation (Infrared Information Analysis Center, 1993), Vol. 8, Chap. 1.5.

5. P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967). [CrossRef]  

References

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  1. C. A. Metzler, F. Heide, P. Rangarajan, M. M. Balaji, A. Viswanath, A. Veeraraghavan, and R. G. Baraniuk, “Deep-inverse correlography: towards real-time high-resolution non-line-of-sight imaging,” Optica 7, 63–71 (2020).
    [Crossref]
  2. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed. (Roberts and Company Publishers, 2020).
  3. N. O’Donoughue and J. M. Moura, “On the product of independent complex Gaussians,” IEEE Trans. Signal Process. 60, 1050–1063 (2011).
    [Crossref]
  4. J. R. Fienup, “Emerging systems and technologies,” in The Infrared and Electro-Optical Systems Handbook, Atmospheric propagation of radiation (Infrared Information Analysis Center, 1993), Vol. 8, Chap. 1.5.
  5. P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
    [Crossref]

2020 (1)

2011 (1)

N. O’Donoughue and J. M. Moura, “On the product of independent complex Gaussians,” IEEE Trans. Signal Process. 60, 1050–1063 (2011).
[Crossref]

1967 (1)

P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
[Crossref]

Balaji, M. M.

Baraniuk, R. G.

Fienup, J. R.

J. R. Fienup, “Emerging systems and technologies,” in The Infrared and Electro-Optical Systems Handbook, Atmospheric propagation of radiation (Infrared Information Analysis Center, 1993), Vol. 8, Chap. 1.5.

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed. (Roberts and Company Publishers, 2020).

Heide, F.

Metzler, C. A.

Moura, J. M.

N. O’Donoughue and J. M. Moura, “On the product of independent complex Gaussians,” IEEE Trans. Signal Process. 60, 1050–1063 (2011).
[Crossref]

O’Donoughue, N.

N. O’Donoughue and J. M. Moura, “On the product of independent complex Gaussians,” IEEE Trans. Signal Process. 60, 1050–1063 (2011).
[Crossref]

Rangarajan, P.

Veeraraghavan, A.

Viswanath, A.

Welch, P.

P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
[Crossref]

IEEE Trans. Audio Electroacoust. (1)

P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
[Crossref]

IEEE Trans. Signal Process. (1)

N. O’Donoughue and J. M. Moura, “On the product of independent complex Gaussians,” IEEE Trans. Signal Process. 60, 1050–1063 (2011).
[Crossref]

Optica (1)

Other (2)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed. (Roberts and Company Publishers, 2020).

J. R. Fienup, “Emerging systems and technologies,” in The Infrared and Electro-Optical Systems Handbook, Atmospheric propagation of radiation (Infrared Information Analysis Center, 1993), Vol. 8, Chap. 1.5.

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Equations (13)

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E V S i n ( x V S ) = 1.
E V S o u t ( x V S ) C N ( 0 , σ E V S o u t 2 ) ,
E O i n p 2 F ( p 1 E V S o u t ) ,
E O o u t ( x O ) = g O ( x O ) E O i n ( x O ) ,
r σ O i n 2 δ ( Δ x O ) .
E V D i n p 4 F ( p 3 E O o u t ) ,
I ( x D ) = | g D ( x D ) E V D i n ( x D ) | 2 = | g D ( x D ) | 2 | E V D i n ( x D ) | 2 ,
I ( x D ) = | g D ( x D ) | 2 | F ( E ~ O o u t ) | 2 ,
R I ( x D , Δ x D ) R β ( Δ x D ) R E ( Δ x D ) .
F ( R I ) ( Δ x ) F ( R β ) ( Δ x ) F ( R E ) ( Δ x ) ,
F ( R I ) ( Δ x ) c 2 F ( R E ) ( Δ x ) + c 1 δ ( Δ x ) F ( R E ) ( Δ x ) , c 3 + c 1 F ( R E ) ( Δ x ) .
E [ | ( f f ) | 2 ] ( Δ x ) = x 1 x 2 E [ z ¯ 1 ( x 1 + Δ x ) z 1 ( x 1 ) z 1 ( x 2 + Δ x ) z ¯ 1 ( x 2 ) ] E [ z ¯ 2 ( x 1 + Δ x ) z 2 ( x 1 ) z 2 ( x 2 + Δ x ) z ¯ 2 ( x 2 ) ] d x 1 d x 2 .
E [ | ( f f ) | 2 ] ( Δ x ) = x 1 x 2 [ r ( x 1 ) r ( x 2 ) δ ( Δ x ) + r ( x 1 ) r ( x 2 ) δ ( Δ x ) δ ( x 2 x 1 ) + r ( x 1 + Δ x ) r ( x 1 ) δ ( x 2 x 1 ) δ ( Δ x ) + r ( x 1 + Δ x ) r ( x 1 ) δ ( x 2 x 1 ) ] d x 1 d x 2 = δ ( Δ x ) x 1 x 2 [ r ( x 1 ) r ( x 2 ) + r ( x 1 ) r ( x 2 ) δ ( x 2 x 1 ) + r ( x 1 + Δ x ) r ( x 1 ) δ ( x 2 x 1 ) ] d x 1 d x 2 + x 1 x 2 r ( x 1 + Δ x ) r ( x 1 ) δ ( x 2 x 1 ) d x 1 d x 2 = δ ( Δ x ) W + r r ( Δ x ) ,

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