Fundamental uncertainty limit for speckle displacement measurements

Andreas Fischer

doi:10.1364/AO.56.007013

1. INTRODUCTION

Speckle phenomena play an important role in optical metrology, e.g., for metrological applications in production engineering [1,2], material science [3], turbomachinery [4,5], and life sciences [6,7]. For instance, speckle photography [8–10], laser triangulation [11], or laser Doppler techniques [12,13] are established techniques to measure position, deformation, shape, velocity, or vibration of optically rough surfaces and, thus, have to cope with or make use of speckles. Note that speckle techniques can be applied on solid nonorganic objects but also on organic materials such as cells or tissues with sufficient roughness.

One basic measurement task is to quantify the displacement of a laterally moved surface section that may be the result of a surface movement or a plastic or elastic deformation. To understand the application potential and the development potential of speckle displacement measurements, this paper focuses on two fundamental questions: what minimal measurement uncertainty of the displacement is achievable and does an ultimate measurement limit exist?

A. State-of-the-Art

A fundamental limit of optical measurement systems follows from Heisenberg’s uncertainty principle [14]. According to the uncertainty principle, the product of the position uncertainty and the momentum uncertainty of a photon are limited. In addition, coherent light is known to minimize the uncertainty product, and the number of photons from a coherent light source fulfills a Poisson distribution [15]. Considering these quantum physical properties of coherent light, a fundamental limit exists for optical measurement systems.

An alternative approach for identifying the fundamental limit of the measurement uncertainty is possible with information theory, i.e., by calculating the Cramér–Rao bound (CRB) [16,17]. The CRB provides the minimal achievable variance of an unbiased or biased estimator or measurement, respectively [18]. The calculation is based on a given signal model, which describes the output signal of the sensor (and its noise) in dependency on the measurand.

Both approaches were successfully applied to identify and to investigate the photon shot noise limits of flow velocity measurements [19], axial distance measurements [20], and angular position measurements of stars [21]. In addition, an error propagation analysis of the latter measurand, which is directly proportional to a lateral displacement of a light spot, was already investigated in [22]. Furthermore, the calculation of the CRB for parameter estimation from images in single-molecule microscopy is described in [23]. However, none of these studies considered speckle images.

The position measurement uncertainty of a speckled intensity distribution was investigated in [11] for characterizing the laser triangulation technique. The measurement uncertainty due to different random speckle patterns was determined using speckle statistics, because different speckle images occur for different axial surface positions and for different surfaces. The random speckle patterns are the domination contribution to the measurement uncertainty of laser triangulation. For the measurement of lateral surface displacements, however, the speckle pattern is shifted but remains the same. For this reason, the uncertainty of the speckle image is mainly due to the photon shot noise. The resulting fundamental measurement limit is unknown.

B. Aim and Outline of the Paper

The aim of the paper is to determine the fundamental uncertainty limit of the speckle displacement measurements resulting from photon shot noise. For this purpose, the measurement arrangement and the general CRB solution for Poisson noise is described in Section 2. As reference, the position measurement of an ideal point light source is considered first in Section 3 by applying the general CRB solution as well as Heisenberg’s uncertainty principle. In Section 4, the same derivations follow for an image with multiple fully developed speckles. The analytical results are verified and validated in Sections 5 and 6, respectively. The paper closes with a summary in Section 7.

2. MEASUREMENT ARRANGEMENT AND CRAMÉR–RAO BOUND

The imaging arrangement of the speckle measurement setup is depicted in Fig. 1. The object surface is located in the object plane, and a point in the object plane is described with the coordinates $(x, y)$ . The object is illuminated with laser light, and the scattered light is imaged to an image plane. The coordinates in the image plane read $(u, v)$ . The image magnification of the optical system amounts to

M = \frac{u}{x} = \frac{v}{y} = \frac{- z_{i}}{z_{o}},

where

z_{o}

is the axial distance from the object plane to the lens, and

z_{i}

is the axial distance from the lens to the image plane. The lens has a certain aperture, and points in the aperture plane are described with the coordinates

(ξ, η)

. The measurement task is to determine a lateral displacement of the object surface by evaluating the acquired image. Subsequently, only a one-component displacement

x_{0}

in the

x

direction is considered, meaning an image displacement in the

- u

direction. This represents the case that the displacement direction is known. However, the considerations can also be adapted for a lateral surface displacement with the

x

and

y

component, i.e., for an unknown displacement direction.

Fig. 1. Imaging arrangement for the surface displacement measurement.

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The acquired image is evaluated with respect to the desired surface displacement $x_{0}$ . To derive the CRB [16,17] for this estimation task, i.e., the minimal achievable variance of an unbiased estimator for the displacement $x_{0}$ , the image and the noise of the image are described first. The pixel signals of the acquired image in unit number of photons are

N_{i j} = K \cdot f (u_{i}, v_{j}),

with

f

as a continuous function that describes the light intensity distribution in arbitrary units and is evaluated at the discrete pixel positions

(u_{i}, v_{j})

, and

K

as the proportionality factor between the arbitrarily chosen intensity unit and the corresponding number of photons. Note that

K

is later substituted using the definition of the total photon number

N_{total}

of the entire image, obtained by

N_{total} = \sum_{i} \sum_{j} N_{i j} = K \sum_{i} \sum_{j} f (u_{i}, v_{j}) .

Assuming coherent laser light, the signals $N_{i j}$ fulfill uncorrelated Poissonian distributions; i.e., the variance of each photon number equals the respective mean number of photons ${\bar{N}}_{i j}$ . Hence, the resulting joint probability function for the pixel signals is

p = \prod_{i} \prod_{j} \frac{{({\bar{N}}_{i j})}^{N_{i j}}}{N_{i j}!} \exp {- {\bar{N}}_{i j}},

which describes the random behavior of the image. With the known joint probability function, the CRB of the surface displacement

x_{0}

follows according to the relation [24]

CRB (x_{0}) = - {(E (\frac{\partial^{2} \ln p}{\partial x_{0}^{2}}))}^{- 1} .

To calculate the CRB, the natural logarithm of the Poissonian joint probability function

\ln p = \sum_{i} \sum_{j} (N_{i j} \cdot \ln ({\bar{N}}_{i j}) - \ln (N_{i j}!) - {\bar{N}}_{i j})

is derived with respect to

x_{0}

yielding

\frac{\partial \ln p}{\partial x_{0}} = \sum_{i} \sum_{j} (\frac{N_{i j}}{{\bar{N}}_{i j}} - 1) \frac{\partial {\bar{N}}_{i j}}{\partial x_{0}},

\frac{\partial^{2} \ln p}{\partial x_{0}^{2}} = - \sum_{i} \sum_{j} - \frac{N_{i j}}{{({\bar{N}}_{i j})}^{2}} {(\frac{\partial {\bar{N}}_{i j}}{\partial x_{0}})}^{2} = + \sum_{i} \sum_{j} (\frac{N_{i j}}{{\bar{N}}_{i j}} - 1) \frac{\partial^{2} {\bar{N}}_{i j}}{\partial x_{0}^{2}} .

Note that only ${\bar{N}}_{i j}$ depends on $x_{0}$ whereas $N_{i j}$ is a random quantity. Using the identity $E (N_{i j}) = {\bar{N}}_{i j}$ and evaluating the negative expectation value of the second derivative according to Eq. (5), the relation

CRB (x_{0}) = {(\sum_{i} \sum_{j} \frac{1}{{\bar{N}}_{i j}} {(\frac{\partial {\bar{N}}_{i j}}{\partial x_{0}})}^{2})}^{- 1}

is obtained. Inserting Eq. (2), applying the image displacement

u_{0} = M \cdot x_{0}

and the chain rule of differentiation

\frac{\partial \dots}{\partial x_{0}} = \frac{\partial \dots}{\partial u_{0}} \frac{\partial u_{0}}{\partial x_{0}}

with the derivative

\frac{\partial u_{0}}{\partial x_{0}} = M

, introducing the abbreviation

\frac{\partial \bar{f} (u_{i}, v_{j})}{\partial u_{0}} = {\bar{f}}^{'} (u_{i}, v_{j})

, and eliminating the factor

K

with Eq. (3) leads finally to the CRB of the displacement for white Poissonian noise in the form

CRB (x_{0}) = \frac{1}{N_{total}} \cdot \frac{\sum_{i} \sum_{j} \bar{f} (u_{i}, v_{j})}{\sum_{i} \sum_{j} \frac{{({\bar{f}}^{'} (u_{i}, v_{j}))}^{2}}{\bar{f} (u_{i}, v_{j})}} \cdot \frac{1}{M^{2}},

\approx \frac{1}{N_{total}} \cdot \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \bar{f} (u, v) d u d v}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{{({\bar{f}}^{'} (u, v))}^{2}}{\bar{f} (u, v)} d u d v} \cdot \frac{1}{M^{2}} .

The integral approximation is valid when the camera pixel dimensions are negligibly small and the light intensities outside the camera chip are negligible. While the formula with the integral expressions is useful for the subsequent analytical considerations, the formula with the sums allows a direct implementation in a computer for a numerical evaluation.

As a consequence, the achievable measurement uncertainty is limited by the intensity distribution $\bar{f} (u, v)$ in the image, the magnification $M$ of the imaging system, and the total number $N_{total}$ of detected photons. Except for the magnification factor $M$ , the result of the CRB in Eq. (11) is identical to the result from Falconi [22], who derived it not from information theory (which is the basis for the CRB) but from elementary statistics. A similar result for a one-dimensional signal also occurs in [20] but without the discrete form. Hence, Eqs. (10) and (11) are consistent with the literature.

3. IDEAL POINT LIGHT SOURCE

The CRB solution for an ideal (infinitesimal) point light source on the surface is considered first, because it is the optimal condition for locating the lateral position of the surface. In contrast to [22], an indirect approach for the solution of Eq. (11) is presented in subsection A, because the same approach is also applicable for the condition of fully developed speckles, which is considered in Section 4. Furthermore, the solution allows a direct comparison with the minimal achievable measurement uncertainty resulting from Heisenberg’s uncertainty principle (see subsection 4.B). The analytical solution for a circular aperture is finally presented in subsection 4.C.

A. CRB Solution with Fourier Optics

The indirect approach means expressing the CRB as a function of the (amplitude) transfer function of the aperture. This is possible by applying Fourier optics, which provides a relation between the image $\bar{f} (u, v)$ and the aperture transfer function $B (ξ, η)$ . All applied theorems of Fourier optics including the subsequently applied definition of the Fourier transform $F {}$ can be extracted from [25].

For an ideal point light source, the image is proportional to the squared amplitude of the Fourier transform of the aperture transfer function, i.e.,

\bar{f} (u, v) = {(F {B (ξ, η)})}^{2} .

Note that only symmetrical aperture transfer functions are considered here so that the Fourier transform is not complex-valued but real-valued, simplifying the subsequent calculations. As a result, the second integrand in Eq. (11) is

\frac{{\bar{f}}^{'} {(u, v)}^{2}}{\bar{f} (u, v)} = 4 {(\frac{\partial}{\partial u} (F {B (ξ, η)}))}^{2} .

Introducing the symbol $k$ as the absolute value of the wave vector, expressing the derivative in Eq. (13) with respect to $u$ as derivative with respect to the frequency variable $ω_{u} = \frac{k}{z_{i}} u$ of the Fourier transform by applying the chain rule $\frac{\partial \dots}{\partial u} = \frac{\partial \dots}{\partial ω_{u}} \frac{\partial ω}{\partial u}$ of the differential calculus, and calculating the derivative in the frequency domain leads to

\frac{{\bar{f}}^{'} {(u, v)}^{2}}{\bar{f} (u, v)} = 4 {(F {(\frac{k}{z_{i}}) ξ \cdot B (ξ, η)})}^{2} .

Using Parseval’s theorem it follows from Eqs. (12) and (14),

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (u, v) d u d v = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} B {(ξ, η)}^{2} d ξ d η,

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{{\bar{f}}^{'} {(u, v)}^{2}}{\bar{f} (u, v)} d u d v = 4 \frac{k^{2}}{z_{i}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ^{2} \cdot B {(ξ, η)}^{2} d ξ d η .

Inserting Eqs. (15) and (16) into Eq. (11) finally expresses the CRB (for an ideal point source) as a function of the aperture transfer function, as follows:

CRB (x_{0}) = \frac{1}{N_{total}} \cdot \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} B {(ξ, η)}^{2} d ξ d η}{4 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ^{2} \cdot B {(ξ, η)}^{2} d ξ d η} \cdot \frac{z_{o}^{2}}{k^{2}} .

B. Heisenberg’s Uncertainty Principle

It is now shown that the CRB from Eq. (17) is in perfect agreement with Heisenberg’s uncertainty principle. The uncertainty principle reads for $N_{total}$ number of photons:

σ_{x_{0}} \cdot σ_{p} \geq \frac{h}{4 π} \cdot \frac{1}{\sqrt{N_{total}}},

where

h

is the Planck constant,

σ_{x_{0}}

the standard deviation of the photon position

x_{0}

, and

σ_{p_{x}}

the standard deviation of the

x

component of the photon momentum

p = 2 π / λ

. Solving Eq. (18) for

σ_{x_{0}}

and substituting

σ_{p_{x}}

by applying the momentum–wave number relation

p_{x} = \frac{h}{2 π} k_{x}

leads to

σ_{x_{0}}^{2} \geq \frac{1}{N_{total}} \cdot \frac{1}{4 σ_{k_{x}}^{2}} .

Assuming small angles for the light rays, the $x$ component of the wave number at the pupil position $ξ$ fulfills the proportion $\frac{k_{x}}{k} \approx \frac{ξ}{z_{o}}$ , cf. Fig. 1. Hence,

σ_{k_{x}}^{2} = \frac{k^{2}}{z_{o}^{2}} σ_{ξ}^{2} .

The variance of the position $ξ$ is determined by the pupil. Using the amplitude transfer function $B (ξ, η)$ of the pupil, the variance is

σ_{ξ}^{2} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ^{2} \frac{B {(ξ, η)}^{2}}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} B {(\tilde{ξ}, \tilde{η})}^{2} d \tilde{ξ} d \tilde{η}} d ξ d η .

Note that the fraction in the integral represents the relative frequency based on the light intensity. Inserting Eqs. (20) and (21) into Eq. (19) finally yields the expression

σ_{x_{0}}^{2} \geq \frac{1}{N_{total}} \cdot \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} B {(ξ, η)}^{2} d ξ d η}{4 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ^{2} \cdot B {(ξ, η)}^{2} d ξ d η} \cdot \frac{z_{o}^{2}}{k^{2}},

which is identical with Eq. (17). As a conclusion, the approaches from information theory (CRB) and from quantum physics (Heisenberg’s uncertainty principle) result in an identical measurement uncertainty limit for the surface displacement measurement.

C. Circular Aperture

The analytical solution of Eq. (17) for a circular aperture with the radius $w$ is straightforward. Inserting

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} B {(ξ, η)}^{2} d ξ d η = \int_{0}^{2 π} \int_{0}^{w} r d r d θ = π w^{2},

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ^{2} \cdot B {(ξ, η)}^{2} d ξ d η = \int_{0}^{2 π} \int_{0}^{w} r^{3} \cos^{2} θ d r d θ = \frac{π}{4} w^{4}

into Eq. (17) leads to the simple analytical solution:

CRB (x_{0}) = \frac{1}{N_{total}} \cdot \frac{1}{{(w / z_{o})}^{2} k^{2}} .

As a result, the measurement uncertainty limit for a circular aperture is indirectly proportional to the square root of the number of photons, to the absolute value $k$ of the wave vector, and to $(w / z_{o})$ . The last is a measure of the numerical aperture of the receiving optics for small angles. The obtained solution is consistent with the differently derived results in [21,22].

4. FULLY DEVELOPED SPECKLE

A. CRB Solution with Fourier Optics

In the case of an illuminated rough surface, which is assumed to result in a fully developed speckle image, the Fourier transform of the pupil transfer function is multiplied by the Fourier transform of the object complex light field $U (x, y)$ [26]. Note that the condition of a fully developed speckle means a uniform probability distribution of the phases of the superposing coherent light portions, corresponding to a random surface topography with sufficiently large surface height variations compared to a fourth of the laser wavelength. Introducing a modified aperture transfer function $\tilde{B} (ξ, η) = F {U (x, y)} \cdot B (ξ, η)$ , Eq. (12) becomes

\bar{f} (u, v) = {| F {\tilde{B} (ξ, η)} |}^{2} .

Since $\tilde{B}$ contains random phases due to the random surface topography, the Fourier transform of $\tilde{B}$ is in general complex-valued for speckles. According to elementary calculations (separating the complex values in real and imaginary parts), the second expression of the CRB formula in Eq. (11) then reads

\frac{{\bar{f}}^{'} {(u, v)}^{2}}{\bar{f} (u, v)} = 2 \cdot {| F^{'} {\tilde{B}} |}^{2} + 2 \cdot Re {{(F^{'} {\tilde{B}})}^{2} \cdot \frac{F^{*} {\tilde{B}}}{F {\tilde{B}}}} .

The symbol $^{'}$ is used again to shorten the notation of the derivative with respect to $u$ . Note that for the special case of $\tilde{B}$ being real-valued (no speckle condition), Eq. (27) is identical to Eq. (13). Owing to the nature of fully developed speckles, the phase of the complex-valued quotient has a mean value of zero over different positions $(ξ, η)$ in the aperture plane. This holds for a sufficiently large aperture in comparison to the speckle size in the aperture plane or a sufficiently large number of speckles in the aperture, respectively. Then, Eq. (27) reduces to

\frac{1}{\bar{f} (u, v)} \cdot {\bar{f}}^{'} {(u, v)}^{2} \approx 2 \cdot {| F^{'} {\tilde{B}} |}^{2} .

Applying Parseval’s theorem, Eqs. (26) and (28) read

\iint f (u, v) d u d v = \iint {| \tilde{B} (ξ, η) |}^{2} d ξ d η,

\iint \frac{{\bar{f}}^{'} {(u, v)}^{2}}{\bar{f} (u, v)} d u d v = 2 \frac{k^{2}}{z_{i}^{2}} \iint ξ^{2} \cdot {| \tilde{B} (ξ, η) |}^{2} d ξ d η .

Inserting Eqs. (29) and (30) into Eq. (11) leads to the following CRB expression for fully developed speckles:

CRB (x_{0}) = \frac{1}{N_{total}} \cdot \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {| \tilde{B} (ξ, η) |}^{2} d ξ d η}{2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ^{2} \cdot {| \tilde{B} (ξ, η) |}^{2} d ξ d η} \cdot \frac{z_{o}^{2}}{k^{2}} .

The result in Eq. (31) is similar to the CRB for an ideal point light source in Eq. (17). However, a factor of 2 instead of a factor of 4 occurs, and absolute values of the modified aperture transfer function, which includes the illumination and surface scattering characteristics, have to be taken into account. Finally, keep in mind that the result was derived for a large number of speckles in the illuminated region. If no speckles occur, Eq. (17) remains valid.

B. Heisenberg’s Uncertainty Principle

The random phases of the scattered light field lead to photon interactions. As a result, the uncertainty relation in Eq. (18) needs to be adapted to this case. A physical derivation is currently missing, but it is assumed that the depolarization of the light scattering leads to an increase of the variance by a factor of 2. Finally, the CRB can be assumed to minimize Heisenberg’s uncertainty principle, because coherent light is considered.

C. Circular Aperture

With a large number of speckles, the integrals over the modified aperture transfer function in Eq. (31) are proportional to the results from Eqs. (23) and (24) due to the spatial averaging. Hence, the CRB for a circular aperture with the radius $w$ and fully developed speckle becomes

CRB (x_{0}) = \frac{1}{N_{total}} \cdot \frac{2}{{(w / z_{o})}^{2} k^{2}} .

As a result, the CRB for multiple speckles is 2 times the CRB for an ideal point light source, cf. Eq. (25). Note further that both CRBs coincide when the number of speckles is negligibly small ( $≪ 1$ ).

5. VERIFICATION

The derived analytical CRB solutions Eqs. (25) and (32) for an ideal point light source and for multiple speckles, respectively, are verified by simulating the speckle image and directly calculating the CRB with the general solution in Eq. (11). Furthermore, the CRB expressions in Eqs. (17) and (31) are verified, based on the light field in the aperture plane.

The calculated CRB results (for a circular aperture) are normalized by the CRB for an ideal point light source. According to Eq. (25), the reference CRB reads

{CRB}_{0} = \frac{1}{N_{total}} \cdot \frac{s_{speckle}^{2} / M^{2}}{4 π},

where the speckle size

s_{speckle} = \frac{λ \cdot z_{i}}{\sqrt{π w^{2}}}

in the image plane is introduced following the definition from Goodman [26]. As a result, the only remaining parameters are the speckle size projected on the original object and the total number of detected photons.

The simulation parameters are the wavelength $λ = 532 nm$ , the circular aperture with the radius $w = 10 mm$ , the distances $z_{i} = z_{o} = 200 mm$ , and the image magnification $M = - 1$ , so that the speckle size in the object plane amounts to $s_{speckle} / | M | = 6 μm$ . The number of occurring speckles is varied by varying the spot diameter $D$ of the circular illumination assuming a constant intensity of the incident light. The number of speckles is defined as

N_{speckle} = \frac{π {(D / 2)}^{2}}{s_{speckle}^{2}} .

Examples of the different speckle images for 0.01, 1, 100, and 10,000 speckles are shown in Fig. 2.

Fig. 2. Simulated speckle images for different illumination spot diameters $D$ , meaning different number of speckles $N_{speckle}$ , cf. Eq. (35): (a) $N_{speckle} = 0.01$ , (b) $N_{speckle} = 1$ , (c) $N_{speckle} = 100$ , and (d) $N_{speckle} = 10000$ .

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The roughness of the object surface is modeled with random, equally distributed phase shifts between 0 and $2 π$ . For each value of $N_{speckle}$ , the images for 400 different surface samples are simulated. The resulting mean value (solid blue line) and the standard deviation (dashed blue lines) of the general CRB solution are shown Fig. 3 over the number of speckles. As a result, the general CRB solution asymptotically reaches the analytical solutions ${CRB}_{0}$ for $N_{speckle} ≪ 1$ and $2 \cdot {CRB}_{0}$ for $N_{speckle} ≫ 1$ . Hence, the analytical CRB solutions in Eqs. (25) and (32) are verified.

Fig. 3. Mean value (solid lines) and standard deviation (dashed lines) of the CRB solution from Eq. (11) (blue) and Eq. (31) (red) considering 400 simulated speckle images at each number of speckles, cf. Fig. 2.

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The mean value (solid red line) and the standard deviation (dashed red lines) of the CRB expression from Eq. (31) for multiple speckles are also shown in Fig. 3. The mean value equals $2 \cdot {CRB}_{0}$ , verifying Eq. (31). Since the expression in Eq. (17) for an ideal point light source differs only by a factor of 1/2, it is verified as well.

6. VALIDATION

To validate the theoretical results, a speckle image and a shifted copy of the speckle image superposed by Poissonian noise are correlated. Using a copy of the original speckle image allows a known speckle displacement to be applied. For speckle displacement measurements with subpixel resolution, different correlation algorithms exist [27,28]. Here, the algorithm from Nobach and Honkanen [28] is used.

The validation is performed first with a simulated speckle image and then with a measured speckle image. Both speckle images are shown in Fig. 4, illustrating the same speckle size of 4.2 pixels. The Monte Carlo study has a sample size of 400. Note that the reference speckle image is the original speckle image without added noise (calibrated image), while the copied speckle image is shifted in the $x$ direction by 1 pixel and is superposed with Poissonian noise (measured image). The total number of detected photons in the evaluation window is set to $N_{total} = 100, 000$ . Note further that the image processing considers only a displacement in the $x$ direction, meaning that the displacement direction is known. When the displacement direction is not known a priori, a two-component displacement has to be determined. The size of the evaluation window amounts to 200 times 200 pixels and, thus, the evaluation window contains about 2268 speckles.

Fig. 4. Comparison of (a) simulated with (b) measured speckle image. The speckle size amounts to 4.2 pixel.

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The resulting systematic errors of the displacement measurement are below 0.002 pixel, demonstrating the achievable subpixel precision of the evaluation algorithm. According to the CRB, the minimum achievable standard deviation for the displacement measurement reads

\sqrt{2 \cdot {CRB}_{0}} = \frac{1}{\sqrt{N_{total}}} \cdot \frac{4.2}{\sqrt{2 π}} pixel = 0.005 pixel .

For the simulated as well as for the measured speckle image, the determined standard deviation is about only 3.5% larger. Hence, the CRB is almost attained with the evaluation algorithm, and the theoretically derived CRB is validated.

7. CONCLUSIONS

The fundamental uncertainty limit of lateral displacement measurements due to photon shot noise was studied using information and quantum theoretic approaches. Note that speckle noise was of minor interest here, because the speckle pattern is shifted according to the surface displacement but remains the same. Starting with a general solution of the CRB for an image with an arbitrary intensity distribution, formulas for calculating the CRB in dependency on the aperture were derived for an ideal point light source and a speckle pattern, respectively. Assuming a circular aperture, analytic CRB expressions were finally derived for both cases.

For an ideal point light source, the CRB result was shown to be identical with the fundamental uncertainty limit that follows from Heisenberg’s uncertainty principle. The uncertainty limit of displacement measurements is indirectly proportional to the square root of the number of detected photons and is directly proportional to the speckle size. Note that the speckle size describes the diffraction limit of the imaging system divided by the image magnification.

For a fully developed speckle pattern, the CRB differs from the CRB for the ideal case of a point light source only by a factor of 2 when the speckle pattern contains a large number of speckles. Since coherent light is considered, the CRB is assumed to minimize Heisenberg’s uncertainty principle. Hence, speckle displacement measurements are ultimately limited by the quantum characteristics of light.

The analytical CRB results are verified with simulations and validated for a simulated and a measured speckle image. For 100,000 detected photons and a speckle size of 4.2 pixels, for instance, the resulting lower limit of the displacement measurement uncertainty amounts to 0.005 pixel. With a pixel size of 5.5 μm and an image magnification of 4, the displacement standard deviation is then limited to 7 nm. As a result, nanometer resolution (and below) is achievable with speckle displacement measurements especially with additional spatial and/or temporal averaging.

The measurement uncertainty can be reduced further, e.g., by superposing coherent light sources with different colors and separately evaluating the different speckle images, by using multiple cameras with different observation directions, or by using robust image processing algorithms coping with speckle image disturbances in in-process applications.

REFERENCES

1. A. Tausendfreund, D. Stöbener, G. Dumstorff, M. Sarma, C. Heinzel, W. Lang, and G. Goch, “Systems for locally resolved measurements of physical loads in manufacturing processes,” CIRP Ann. 64, 495–498 (2015). [CrossRef]

2. R. Kuschmierz, A. Davids, S. Metschke, F. Löffler, H. Bosse, J. Czarske, and A. Fischer, “Optical, in-situ, three dimensional, absolute shape measurements in CNC metal working lathes,” Int. J. Adv. Manuf. Technol. 84, 2739–2749 (2016). [CrossRef]

3. K. Philipp, A. Filippatos, R. Kuschmierz, A. Langkamp, M. Gude, A. Fischer, and J. Czarske, “Multi-sensor system for in-situ shape monitoring and damage identification of high-speed composite rotors,” Mech. Syst. Signal Process. 76-77–77, 187–200 (2016). [CrossRef]

4. T. Pfister, L. Büttner, J. Czarske, H. Krain, and R. Schodl, “Turbo machine tip clearance and vibration measurements using a fibre optic laser Doppler position sensor,” Meas. Sci. Technol. 17, 1693–1705 (2006). [CrossRef]

5. M. Neumann, F. Dreier, P. Günther, U. Wilke, A. Fischer, L. Büttner, F. Holzinger, H.-P. Schiffer, and J. Czarske, “A laser-optical sensor system for vibration detection of high-speed compressors,” Mech. Syst. Signal Process. 64-65–65, 337–346 (2015). [CrossRef]

6. J. Qiu, Y. Li, Q. Huang, Y. Wang, and P. Li, “Correcting speckle contrast at small speckle size to enhance signal to noise ratio for laser speckle contrast imaging,” Opt. Express 21, 28902–28913 (2013). [CrossRef]

7. K. Philipp, A. Smolarski, N. Koukourakis, A. Fischer, M. Stürmer, U. Wallrabe, and J. Czarske, “Volumetric HiLo microscopy employing an electrically tunable lens,” Opt. Express 24, 15029–15041 (2016). [CrossRef]

8. K. A. Stetson, “A review of speckle photography and interferometry,” Opt. Eng. 14, 145482 (1975). [CrossRef]

9. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981). [CrossRef]

10. M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32, 395–403 (2000). [CrossRef]

11. R. G. Dorsch, G. Häußler, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994). [CrossRef]

12. T. Pfister, A. Fischer, and J. Czarske, “Cramér–Rao lower bound of laser Doppler measurements at moving rough surfaces,” Meas. Sci. Technol. 22, 055301 (2011). [CrossRef]

13. R. Kuschmierz, J. Czarske, and A. Fischer, “Multiple wavelength interferometry for distance measurements of moving objects with nanometer uncertainty,” Meas. Sci. Technol. 25, 085202 (2014). [CrossRef]

14. W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927). [CrossRef]

15. M. C. Teich and B. Q. A. Saleh, “Squeezed states of light,” Quantum Opt. 1, 153–191 (1989). [CrossRef]

16. C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

17. H. Cramér, Mathematical Methods of Statistics (Princeton University, 1946).

18. A. Fischer and J. Czarske, “Measurement uncertainty limit analysis with the Cramér–Rao bound in case of biased estimators,” Measurement 54, 77–82 (2014). [CrossRef]

19. A. Fischer, “Fundamental uncertainty limit of optical flow velocimetry according to Heisenberg’s uncertainty principle,” Appl. Opt. 55, 8787–8795 (2016). [CrossRef]

20. P. Pavliček and G. Häusler, “Methods for optical shape measurements and their measurement uncertainty,” Int. J. Optomechatron. 8, 292–303 (2014). [CrossRef]

21. L. Lindegren, “High-accuracy positioning: astrometry,” in Observing Photons in Space, Vol. 9 of ISSI Scientific Report Series (Springer, 2013), pp. 299–311.

22. O. Falconi, “Maximum sensitivities of optical direction and twist measuring instruments,” J. Opt. Soc. Am. 54, 1315–1320 (1964). [CrossRef]

23. J. Chao, E. A. Ward, and R. J. Ober, “Fisher information theory for parameter estimation in single molecule microscopy: tutorial,” J. Opt. Soc. Am. A 33, B36–B57 (2016). [CrossRef]

24. G. Casella and R. L. Berger, Statistical Inference (Duxbury, 1990).

25. J. W. Goodman, Fourier Optics (Roberts & Company, 2005).

26. J. W. Goodman, Speckle Phenomena in Optics (W. H. Freeman, 2010).

27. P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40, 1613–1620 (2001). [CrossRef]

28. H. Nobach and M. Honkanen, “Two-dimensional Gaussian regression for sub-pixel displacement estimation in particle image velocimetry or particle position estimation in particle tracking velocimetry,” Exp. Fluids 38, 511–515 (2005). [CrossRef]

Fundamental uncertainty limit for speckle displacement measurements

Abstract

1. INTRODUCTION

A. State-of-the-Art

B. Aim and Outline of the Paper

2. MEASUREMENT ARRANGEMENT AND CRAMÉR–RAO BOUND

3. IDEAL POINT LIGHT SOURCE

A. CRB Solution with Fourier Optics

B. Heisenberg’s Uncertainty Principle

C. Circular Aperture

4. FULLY DEVELOPED SPECKLE

A. CRB Solution with Fourier Optics

B. Heisenberg’s Uncertainty Principle

C. Circular Aperture

5. VERIFICATION

6. VALIDATION

7. CONCLUSIONS

REFERENCES

Cited By

Figures (4)

Equations (36)

Applied Optics