Statistical model for speckle pattern optimization

Yong Su; Qingchuan Zhang; Zeren Gao

doi:10.1364/OE.25.030259

1. Introduction

Image registration is the key technique of optical metrologies such as digital image correlation (DIC) [1], particle image velocimetry (PIV) [2], speckle metrology [3,4], and motion estimation [5]. It registers the same physical points in two or more images of the same scene captured at different times, from different sensors, or from different viewpoints [6]. The metrological performance of image registration depends critically on the quality of image pattern [7]. Two issues then arise: (1) how to assess a pattern; (2) how to generate a good pattern.

The first issue, pattern assessment, has been thoroughly studied [8]. Since a good pattern should correspond to a small measurement error, the overall error is a valid pattern quality measure [9]. The overall error is composed of systematic error and random error [10]. The systematic error is caused by imperfect interpolation [11,12]; it is largely determined by the interpolation bias kernel and the image power spectrum [13,14]. The random error stems from sensor noise [15,16]; it can be accurately characterized by the ratio of the variance of sensor noise and the sum of square of subset intensity gradients [17–19].

The second issue, pattern optimization, has attracted extensive attention [20–23]. However, finding the optimal pattern is difficult, and thus patterns of specific type are generally optimized instead [24,25]. In practical situations, speckle patterns which are composed of random positioned speckles are widely used because of their simplicity, practicability, and effectiveness [26, 27]. The speckles have physical meanings sometimes, for example fluorescent particles of fluorescent stereo microscopy and tracer particles of PIV [23,28]. These patterns can be characterized by several generation parameters such as speckle size and speckle coverage, and hence pattern optimization can be implemented by optimizing these generation parameters [21]. A number of literatures have been devoted to optimizing the generation parameters: Zhou et al. proposed that the optimal size of Gaussian speckle should lie within 2∼5 pixels [20]; Lecompte et al. showed that when a subset 15 is chosen the optimal choice for speckle diameter is 5 pixels and the speckle coverage should lie between 40%∼70% [21]; Sutton et al. suggested that to ensure a reasonable amount of over-sampling the optimal size should be 3∼6 pixels [1]; Hua et al. proposed that the speckle size should be from 2 to 4 pixels [22].

Nowadays, the main problem in the subject of speckle pattern optimization is the lack of a theoretical model. All existing achievements are attained using the numerical methods [20–22]. The numerical methods can only determine the optimal speckle pattern generation parameters in some particular cases, for it is impossible to conduct a numerical experiment for every possible parameter. Furthermore, the numerical methods are incapable of providing a deep physical insight. Thus, there is a strong need to build a theoretical model, and it is the objective of this article.

In this article, a statistical model for speckle pattern optimization is built and validated. The theoretical analyses are divided into three gradually progressive steps: firstly, the statistics (including expectation, variance, and power spectrum) of speckle patterns are derived in Section 2; then, the measurement errors (including systematic errors, random errors, and overall errors) are derived in Section 3; finally, the pattern quality measure is derived and formulas of optimal speckle radius are presented in Section 4. The proposed theoretical model is confirmed by numerical experiments in Section 5. The advantages and limits of this research are discussed in Section 6 and the conclusions are drawn in Section 7. The primary motivation of this article is to establish a speckle pattern generation standard for DIC. However, since the mathematical derivations are general, it is believed that these discussions are beneficial to scholars in other optical measurement communities, such as PIV and speckle metrology.

2. Statistic of speckle patterns

Image patterns play a crucial role in the metrological performance [7]. In practical situations, speckle patterns which are composed of random positioned speckles are widely used [20–23,26–28]. These patterns can be mathematically modeled as

f (x, y) = \sum_{i = 1}^{N} ψ (x - x_{i}, y - y_{i}),

where N is the total number of speckles, ψ(x, y) is the speckle profile, and (x_i, y_i) is the position of the ith speckle. Following the convention in [29], random variables, such as x_i and y_i, are written in boldface letters. As the speckle patterns should be isotropic [1], the speckle profile ψ(x, y) is assumed to be isotropic. The N speckles are randomly scattered over a region [−L, L] × [−L, L], and the positions of every two speckles are independent. Since the speckle positions (x_i, y_i) are random variables, identical speckle pattern generation parameters will generate different speckle patterns, as illustrated in Fig. 1. This randomness is designated as spatial randomness, in contrast with temporal randomness, which is caused by the randomness of sensor noise [30].

Fig. 1 A schematic of speckle patterns which are composed of random positioned speckles. These speckles are randomly scattered over a region, and thus the speckle positions are random variables. Due to this spatial randomness, patterns shown in (a) and (b) are different, though the generation parameters are identical.

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To register the correspondence, a subset is generally chosen around the point of interest. Since the subset is generally much smaller than the speckle region, it is reasonable to assume that, from a performance perspective, patterns described by Eq. (1) are equivalent to patterns filling the whole plane ℝ² with a speckle density λ = N/4L². Namely, we assume L → ∞. These speckle patterns can be rewritten as

f (x, y) = \sum_{i = - \infty}^{\infty} ψ (x - x_{i}, y - y_{i}) .

The question then arises as to character the speckle pattern. One contribution of this work is the find of the relationship between speckle pattern and filtered Poisson process.

If points are randomly scattered over the plane ℝ² with a constant density λ, and each point is stochastically independent of all the other points, then this is a Poisson process [29], which can be described using the Poisson impulses

z (x, y) = \sum_{i = - \infty}^{\infty} δ (x - x_{i}, y - y_{i}),

where δ(x, y) is the Dirac delta function, and (x_i, y_i) is the position of the ith points. This process is named after Poisson because it relates to the Poisson distribution [29].

Convolving the Poisson impulses with a filter ψ(x, y) leads to a filtered Poisson process [29], which can be given by

f (x, y) = ψ \otimes z (x, y) = \sum_{i = - \infty}^{\infty} ψ (x - x_{i}, y - y_{i}) .

A comparison between Eq. (2) and Eq. (4) implies that the process of speckle pattern generation is essentially a filtered Poisson process. Fortunately, the filtered Poisson process has been thoroughly studied already [31]; the achievements can be exploited directly. The expectation and the variance of a filtered Poisson process can be given by the Campbell’s theorem [31]

𝔼 {f (x, y)} = λ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ (ξ, ζ) d ξ d ζ, Var (f (x, y)) = λ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ^{2} (ξ, ζ) d ξ d ζ .

The power spectrum of a filtered Poisson process is [29]

S_{f f} (ω_{x}, ω_{y}) = 4 π^{2} λ^{2} {\hat{ψ}}^{2} (0, 0) δ (ω_{x}, ω_{y}) + λ {| \hat{ψ} (ω_{x}, ω_{y}) |}^{2},

where ψ̂(ω_x, ω_y) is the Fourier transform of ψ(x, y).

3. Measurement errors

Measurement errors of pure translation are frequently applied to assessing the pattern quality [20]. For a pattern f(x, y), a target pattern f(x − u₀, y) is produced by shifting the original pattern by u₀ units along the x axis. Then, the original pattern and the target pattern are sampled, producing a reference image f[m, n] and a target image g[m, n] respectively. In practice, images are inevitably contaminated by sensor noise. If the noise in the reference image is w_f[m, n] and the noise in the target image is w_g[m, n], the captured images are

f [m, n] = f (m, n) + w_{f} [m, n], g [m, n] = f (m - u_{0}, n) + w_{g} [m, n] .

A area-based registration strategy is commonly employed to estimate the imposed displacement u₀ [6]. A subset with size M is chosen and then specific similarity measure is optimized [32]. Without a loss of generality, the subset center is assumed to be at the origin. If the sum of squared difference (SSD) criterion is utilized, the estimated displacement is given by

u = arg min \sum_{m = - S}^{S} \sum_{n = - S}^{S} {f [m, n] - g (m + u, n)}^{2},

where S = (M −1)/2 denotes the half of subset size, and g(m +u, n) denotes the subpixel intensity of target image, which can be estimated by interpolation [33]. Equation (8) is generally solved by numerical optimization algorithms such as Newton-Raphson method [34], Levenberg-Marquardt method, and Gauss-Newton method. In this article, the inverse compositional Gauss-Newton (IC-GN) algorithm [35], which is the most popular subpixel registration algorithm in the filed of DIC for its high-accuracy and high-efficiency [36, 37], is utilized to estimate the actual displacement u₀.

The estimated displacement u is not equal to the actual displacement u₀ but contains a measurement error Δu = u − u₀. Δu is a random variable due to temporal randomness and can be divided into two parts: a systematic error e_b and a random error e_n, where

e_{b} = 𝔼 {Δ u | f (x, y) |}, e_{n} = \sqrt{Var {Δ u | f (x, y) |}} .

The subscript ‘b’ denotes ‘bias’ and ‘n’ denotes ‘noise’. It is worth noting that the random error e_n is caused by sensor noise, not by speckle generation. If the spatial randomness is considered, both systematic errors and random errors become random variables. The major objective of this section is to derive the spatial expectations of e_b and e_n.

3.1. Systematic errors

In this section, the spatial expectations of systematic errors are derived. If the inverse compositional Gauss-Newton algorithm is utilized, the systematic error e_b of a given pattern f(x, y) can be approximated as [14]

\begin{array}{c} e_{b} \approx C_{b} sin 2 π u_{0}, \\ C_{b} = \frac{\int_{- π}^{π} \int_{- π}^{π} E_{b} (ω_{x}, ω_{y}) {| \hat{f} (ω_{x}, ω_{y}) |}^{2} d ω_{x} d ω_{y}}{\int_{- π}^{π} \int_{- π}^{π} ω_{x} \hat{d} (ω_{x}, ω_{y}) \hat{φ} (ω_{x}, ω_{y}) {| \hat{f} (ω_{x}, ω_{y}) |}^{2} d ω_{x} d ω_{y}}, \\ E_{b} (ω_{x}, ω_{y}) = \hat{d} (ω_{x}, ω_{y}) [\hat{φ} (ω_{x} - 2 π, ω_{y}) - \hat{φ} (ω_{x} + 2 π, ω_{y})] . \end{array}

where C_b is the amplitude of the systematic error, u₀ is the actual displacement, E_b(ω_x, ω_y) is the interpolation bias kernel, f̂(ω_x, ω_y) is the spectrum of the speckle pattern, φ̂(ω_x, ω_y) is the interpolation transfer function of interpolation algorithm for target image, and d̂(ω_x, ω_y) is the magnitude frequency response of gradient estimator for reference image.

In the context of speckle pattern generation, the pattern spectrum f̂(ω_x, ω_y) becomes a stochastic process. Accordingly, the systematic error e_b becomes a random variable. Our aim is to estimate the expectation of e_b. Since the process of speckle pattern generation is essentially a filtered Poisson process, it is rational to substitute |f̂(ω_x, ω_y)|² with the power spectrum of a filtered Poisson process, thus

𝔼 {C_{b}} \approx \frac{\int_{- π}^{π} \int_{- π}^{π} E_{b} (ω_{x}, ω_{y}) S_{f f} (ω_{x}, ω_{y}) d ω_{x} d ω_{y}}{\int_{- π}^{π} \int_{- π}^{π} ω_{x} \hat{d} (ω_{x}, ω_{y}) \hat{φ} (ω_{x}, ω_{y}) S_{f f} (ω_{x}, ω_{y}) d ω_{x} d ω_{y}},

where S_ff(ω_x, ω_y), given by Eq. (6), is the power spectrum of a filtered Poisson process. The interpolation bias kernel E_b(ω_x, ω_y) is a high-pass filter and equals zero at ω_x = ω_y = 0 [13]. A further simplification yields

\begin{array}{c} 𝔼 {e_{b}} \approx 𝔼 {C_{b}} sin 2 π u_{0}, \\ 𝔼 {C_{b}} \approx \frac{\int_{- π}^{π} \int_{- π}^{π} E_{b} (ω_{x}, ω_{y}) {| \hat{ψ} (ω_{x}, ω_{y}) |}^{2} d ω_{x} d ω_{y}}{\int_{- π}^{π} \int_{- π}^{π} ω_{x} \hat{d} (ω_{x}, ω_{y}) \hat{φ} (ω_{x}, ω_{y}) {| \hat{ψ} (ω_{x}, ω_{y}) |}^{2} d ω_{x} d ω_{y}}, \\ E_{b} (ω_{x}, ω_{y}) = \hat{d} (ω_{x}, ω_{y}) [\hat{φ} (ω_{x} - 2 π, ω_{y}) - \hat{φ} (ω_{x} + 2 π, ω_{y})] . \end{array}

where ψ̂(ω_x, ω_y) is the spectrum of an individual speckle. Equation (12) represents an estimation of the spatial expectation of the systematic error.

3.2. Random errors

In this section, the spatial expectations of random errors are derived. If the inverse compositional Gauss-Newton algorithm is utilized, the random error e_n of a given speckle pattern f(x, y) can be given by [18]

e_{n} = \sqrt{\frac{σ_{f}^{2} + σ_{g}^{2}}{\sum_{m = - S}^{S} \sum_{n = - S}^{S} f_{x}^{2} [m, n]}},

where σ_f is the standard deviation of sensor noise of reference image, σ_g is the standard deviation of sensor noise of target image, S is the half of subset size, and f_x[m, n] is the image gradient along the x axis.

In the context of speckle pattern generation, the speckle pattern f(x, y) becomes a stochastic process. Accordingly, the random error e_n becomes a random variable. Our aim is to estimate the spatial expectation of e_n. The random error given by Eq. (13) can be rewritten as

e_{n} = \frac{σ}{\sqrt{Q}},

where

σ = {(σ_{f}^{2} + σ_{g}^{2})}^{1 / 2}

, and

Q = \sum_{m = - S}^{S} \sum_{n = - S}^{S} f_{x}^{2} [m, n]

is the sum of square of subset intensity gradients. Deriving 𝔼{e_n} straightforwardly seems difficult, and thus an estimate is presented instead. This estimation is based on a theorem in [29]: the expectation of a random variable y = Φ(x) can be estimated by approximating Φ(x) by a parabola, yielding

𝔼 {Φ (x)} \approx Φ (𝔼 {x}) + \frac{1}{2} Var (x) Φ^{″} (𝔼 {x}) .

The second term, Var(x)Φ″(𝔼{x})/2, acts as a correction [29]. In our case, Φ(x) = σx^−1/2, and thus the expectation of the random error can be approximated by

𝔼 {e_{n}} \approx \frac{σ}{\sqrt{𝔼 {Q}}} [1 + \frac{3}{8} \frac{Var (Q)}{𝔼 {Q}^{2}}] .

Hence, the issue turns to investigate the expectation and the variance of Q.

The expectation of Q can be derived as follows. The pattern gradient is generally evaluated by convolving the pattern with a gradient operator d [5,14], namely,

f_{x} (x, y) = (d \otimes f) (x, y) = (d \otimes ψ \otimes z) (x, y) .

Obviously, the pattern gradient f_x(x, y) is a filtered Poisson process with filter d⊗ψ. Consequently, its expectation and variance can be given by the Campbell’s theorem [see Eq. (5)]. As the speckle patterns should be isotropic and homogeneous in a statistical sense [1], the expectation of the pattern gradient should be zero. A combination of Parseval’s theorem and Campbell’s theorem yields that the expectation of squared pattern gradient is

q = 𝔼 {f_{x}^{2} (x, y)} = \frac{λ}{4 π^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {| \hat{d} (ω_{x}, ω_{y}) |}^{2} {| \hat{ψ} (ω_{x}, ω_{y}) |}^{2} d ω_{x} d ω_{y} .

Since filtered Poisson process is strict-sense stationary [29], all points have the same statistics, and thus the expectation of Q can be expressed as

𝔼 {Q} = \sum_{m = - S}^{S} \sum_{n = - S}^{S} 𝔼 {f_{x}^{2} [m, n]} = M^{2} q,

where M = 2S + 1 is the subset size.

The variance of Q is difficult to derive. However, as the variance acts as a correction, a good estimate seems enough. The sum of the squares of k independent standard normal random variables satisfies a chi-squared distribution $χ_{k}^{2}$ [29]. In practice, there are generally hundreds of points in a subset. Recalling that the gradient of each point is a random variable with zero mean and variance q, it is assumed that q^−1/2Q approximately satisfies a chi-squared distribution $χ_{M^{2}}^{2}$ , then the variance [29]

Var (Q) \approx 2 M^{2} q .

A combination of Eq. (16), Eq. (19), and Eq. (20) yields

𝔼 {e_{n}} \approx \frac{σ}{M \sqrt{q}} (1 + \frac{3}{4 M^{2} q}) .

Equation (21) represents an estimate of the spatial expectation of the random error.

3.3. Overall errors

In this section, the spatial expectations of overall errors are derived. The overall errors are generally characterized by the mean squared errors and the root mean squared errors. The mean squared error e_m is the sum of the squares of systematic error and random error,

e_{m} = e_{b}^{2} + e_{n}^{2},

and the root mean squared error

e_{r} = e_{m}^{1 / 2}

[8,9,24].

In the context of speckle pattern generation, the speckle pattern f(x, y) is a stochastic process.

Accordingly, the mean squared error e_m and the root mean squared error e_r become random variables. The expectation of the mean squared error

𝔼 {e_{m}} = 𝔼 {e_{b}^{2}} + 𝔼 𝔼 {e_{n}^{2}} .

The systematic error can be approximated by Eq. (12), thus

𝔼 {e_{b}^{2}} = 𝔼 {C_{b}^{2}} {sin}^{2} 2 π u_{0} \approx 𝔼 {C_{b}}^{2} {sin}^{2} 2 π u_{0},

where the variance of C_b is ignored for it is generally small. Following the strategy in Section 3.2, the expectation of squared random error can be approximated as

𝔼 {e_{n}^{2}} \approx \frac{σ^{2}}{𝔼 {Q}} [1 + \frac{Var (Q)}{𝔼 {Q}^{2}}] = \frac{σ^{2}}{M^{2} q} (1 + \frac{2}{M^{2} q}) .

A combination of Eq. (23), Eq. (24), and Eq. (25) yields

𝔼 {e_{m}} \approx 𝔼 {C_{b}}^{2} {sin}^{2} 2 π u_{0} + \frac{σ^{2}}{M^{2} q} (1 + \frac{2}{M^{2} q}),

and the corresponding root mean squared error

𝔼 {e_{r}} \approx 𝔼 {e_{m}}^{1 / 2} .

Equation (26) and Eq. (27) represent estimates of the expectations of mean squared error and root mean squared error respectively.

4. Optimal speckle radius

4.1. Pattern quality measure

A good pattern should correspond to a small overall error. Thus, a good pattern should correspond to a small mean squared error. As the mean squared error e_m is a function of the subpixel position u₀ [see Eq. (26)], the integral of e_m over all possible positions is proposed as a pattern quality measure [8], namely,

V = \int_{0}^{1} e_{m} (u_{0}) d u_{0} .

In the context of speckle pattern generation, the pattern quality measure V becomes a random variable. Substituting Eq. (26) into Eq. (28) yields

𝔼 {V} \approx \frac{1}{2} 𝔼 {C_{b}}^{2} + \frac{σ^{2}}{M^{2} q} (1 + \frac{2}{M^{2} q}) .

Equation (29) represents an estimate of the expectation of the pattern quality measure.

4.2. Optimal speckle radius

In practice, a speckle ψ(x, y) is generally characterized by its speckle radius R, and the speckle density λ is generally substituted by another parameter, speckle coverage ρ = λπR² [21]. The speckle coverage is generally specified as 50% [26], then the issue turns to find the optimal R for a given speckle coverage ρ.

The optimal speckle radius should minimize 𝔼{V}, namely

\frac{d 𝔼 {V}}{d R} = 0 .

A combination of Eq. (29) and Eq. (30) yields

𝔼 {C_{b}} \frac{d 𝔼 {C_{b}}}{d R} - \frac{σ^{2}}{M^{2} q^{2}} (1 + \frac{4}{M^{2} q}) \frac{d q}{d R} = 0,

where 𝔼{C_b}, given by Eq. (12), is the expectation of the amplitude of systematic error, and q, given by Eq. (18), is the expectation of squared gradient. The optimal speckle radius R_opt should satisfy Eq. (31).

4.3. Optimal speckle radius for Gaussian speckle patterns

Gaussian speckle patterns are widely used in practice [2, 9, 20, 37]. The profile of a Gaussian speckle is

ψ_{g} (x, y; R) = exp (- \frac{x^{2} + y^{2}}{R^{2}}),

and its spectrum is

{\hat{ψ}}_{g} (ω_{x}, ω_{y}; R) = π R^{2} exp [- R^{2} (ω_{x}^{2} + ω_{y}^{2}) / 4]

. A Gaussian speckle is characterized by its radius R. The optimal radius R_opt can certainly be acquired using Eq. (31). However, the computation may be tedious and cumbersome, and thus a simplification seems necessary.

The gradient operator d[m, n] is an approximation of the ideal gradient operator, whose transfer function is jω_x [38]; the interpolation basis function φ(x, y) is an approximation of the sinc function, whose transfer function is a rectangular function [13,33]. Recalling that d̂(ω_x, ω_y) is the magnitude response of d[m, n] and φ̂(ω_x, ω_y) is the Fourier transform of φ(x, y), we assume that

\hat{d} (ω_{x}, ω_{y}) \approx ω_{x} and \hat{φ} (ω_{x}, ω_{y}) \approx 1, for - π < ω_{x} < π, - π < ω_{y} < π .

In addition, as under-sampling should not occur,

\int_{- π}^{π} \int_{- π}^{π} ω_{x}^{2} {\hat{ψ}}_{g}^{2} (ω_{x}, ω_{y}; R) d ω_{x} d ω_{y} \approx \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ω_{x}^{2} {\hat{ψ}}_{g}^{2} (ω_{x}, ω_{y}; R) d ω_{x} d ω_{y} = 2 π^{3} .

With these approximations [Eq. (33) and Eq. (34)], the systematic errors [Eq. (12)] can be approximated by

𝔼 {C_{b}} \approx \frac{1}{2 π^{3}} \int_{- π}^{π} \int_{- π}^{π} E_{b} (ω_{x}, ω_{y}) ψ_{g}^{2} (ω_{x}, ω_{y}; R) d ω_{x} d ω_{y},

and the squared gradient [Eq. (18)] can be approximated by q ≈ πλ/2 = ρR⁻²/2. If the correction term is ignored, the squared random error [Eq. (25)] can be approximated by

𝔼 {e_{n}^{2}} \approx \frac{σ^{2}}{M^{2} q} (1 + \frac{2}{M^{2} q}) \approx \frac{σ^{2}}{M^{2} q} \approx \frac{2 σ^{2} R^{2}}{ρ M^{2}} .

Substituting Eq. (35) and Eq. (36) into Eq. (30) yields that the optimal radius satisfies

\begin{array}{l} \int_{- π}^{π} \int_{- π}^{π} E_{b} (ω_{x}, ω_{y}) {\hat{ψ}}_{g}^{2} (ω_{x}, ω_{y}; R) d ω_{x} d ω_{y} \\ \times \int_{- π}^{π} \int_{- π}^{π} E_{b} (ω_{x}, ω_{y}) {\hat{ψ}}_{g} (ω_{x}, ω_{y}; R) \frac{d {\hat{ψ}}_{g} (ω_{x}, ω_{y}; R)}{d R} d ω_{x} d ω_{y} + \frac{8 π^{6} σ^{2} R}{ρ M^{2}} = 0, \end{array}

where E_b(ω_x, ω_y) is the interpolation bias kernel, ψ̂_g(ω_x, ω_y; R) is the spectrum of a Gaussian speckle, R is the speckle radius,

σ = {(σ_{f}^{2} + σ_{g}^{2})}^{1 / 2}

is a characterization of the sensor noise, ρ is the given speckle coverage, and M is the subset size. Equation (37) is a simplification of Eq. (31) and is available for Gaussian speckle patterns.

5. Numerical experiments

5.1. Numerical experiments for measurement errors

In order to verify the correctness of proposed theoretical analyses, the measurement errors of pure translation were evaluated numerically, and then a comparison between the numerical results and the theoretical estimates was made. All source codes are uploaded to provide more implementation details [39].

To evaluate the spatial randomness, K = 1000 Gaussian speckle patterns were generated. The kth pattern is given by

f^{(k)} (x, y) = \sum_{i = 1}^{N} exp {- \frac{{[x - x_{i}^{(k)}]}^{2} + {[y - y_{i}^{(k)}]}^{2}}{R^{2}}},

where N is the total number of the speckles, (

x_{i}^{(k)}

,

y_{i}^{(k)}

) is the position of the ith speckle of the kth pattern, and R is the Gaussian speckle radius. The speckle positions satisfy a uniform distribution over a region [−L, L] × [−L, L], where L = 50. The total number of the speckles is N = 4π⁻¹ρL²R⁻², where ρ denotes the speckle coverage; if N is not an integer, its round is used instead.

Then, the issue was to evaluate the systematic error $e_{b}^{(k)}$ and the random error $e_{n}^{(k)}$ of the kth pattern. A series of translated speckle patterns f^(k)(x − u₀, y) was generated; the displacement u₀ ranged from 0 to 1 pixel, with an increase of 0.05 pixels between successive patterns. The reference pattern and the target patterns were sampled, and exact sampled values were used to remove the quantization error. Images are inevitably contaminated by sensor noise. Sensor noise in the reference image can be virtually removed by averaging many frames, but the averaging approach generally cannot be applied to the target images for they must be captured dynamically [40]. Hence, Gaussian additive white noise w[m, n] was superimposed onto the target images exclusively; this is not fundamental because noise in the reference image does not induce a bias error [19]. Since a typical camera signal-to-noise ratio (SNR) is 40 dB [41], the standard deviation of the noise was specified as σ = 0.01, which is 1% of the speckle intensity. In order to evaluate the random error, the noise addition repeated for I = 1000 times. The reference image f^(k)[m, n] and the target images g^(k,i)[m, n] can be expressed as

f^{(k)} [m, n] = f^{(k)} (m, n), g^{(k, i)} [m, n] = f^{(k)} (m - u_{0}, n) + w^{(k, i)} [m, n],

where the superscript k denotes the kth speckle pattern and i denotes the ith noise addition.

The inverse compositional Gauss-Newton (IC-GN) algorithm [35], which is the most popular subpixel registration algorithm in the filed of DIC for its high-accuracy and high-efficiency [36,37], was utilized to estimate the actual displacement u₀. The inverse compositional Gauss-Newton algorithm requires gradients of reference image and subpixel intensities of target images: the gradients were evaluated using the Barron operator [18, 36]; the intensities were interpolated using the bicubic B-spline, which is proven far more accurate than the cubic spline [11]. The correlation criterion chosen was SSD [see Eq. (8)]. Zero-order shape function was employed due to the underlying deformation is rigid motion [32]. The subset was [−S, S] × [−S, S], so that the subset size was M = 2S + 1 and there were M² points in the subset. The displacement u^(k,i) was evaluated, and then the measurement errors were calculated by

e_{b}^{(k)} = \frac{1}{I} \sum_{i = 1}^{I} [u^{(k, i)} - u_{0}], e_{n}^{(k)} = \sqrt{\frac{1}{I - 1} \sum_{i = 1}^{I} {[u^{(k, i)} - u_{0} - e_{b}^{(k)}]}^{2}},

where

e_{b}^{(k)}

and

e_{n}^{(k)}

respectively denotes the systematic error and the random error of the kth pattern.

Then, the spatial expectation and variance of the systematic error were given by

𝔼 {e_{b}} = \frac{1}{K} \sum_{k = 1}^{K} e_{b}^{(k)}, Var {e_{b}} = \frac{1}{K - 1} \sum_{k = 1}^{K} {(e_{b}^{(k)} - 𝔼 {e_{b}})}^{2} .

Similarly, the spatial expectation and variance of the random error were given by

𝔼 {e_{n}} = \frac{1}{K} \sum_{k = 1}^{K} e_{n}^{(k)}, Var {e_{n}} = \frac{1}{K - 1} \sum_{k = 1}^{K} {(e_{n}^{(k)} - 𝔼 {e_{n}})}^{2} .

The theoretical estimates of the systematic errors, the random errors, and the root mean squared errors were evaluated using Eq. (12), Eq. (21), and Eq. (27) respectively. The double integrals within these equations were rewritten into products of one-dimensional integrals, and then the integrals were calculated using function integral in MATLAB.

Systematic errors e_b, random errors e_n, and root mean squared errors e_r of Gaussian speckle patterns with speckle radii R = 1.3, 1.5, 2.0, and 3.0 pixels are shown in Fig. 2. The subset size M is 31 pixels, and the speckle coverage ρ is 50%. The theoretical estimates show excellent agreements with the numerical results. Furthermore, it can be found that, (1) the spatial expectations of the systematic errors are sinusoidal-shaped and decrease as the speckle radius increases; (2) the random errors nearly remain constant and increase as the speckle radius increases; (3) the root mean squared errors are governed by the systematic errors when the speckle radius is small and are governed by the random errors when the speckle radius is large. These observations are consistent with literatures and can be explained by the proposed model [11,18,19]. (1) Equation (12) clearly indicates the sinusoidal-shape of systematic errors. Since the interpolation bias kernel E_b(ω_x, ω_y) is a high-pass filter [13,14], a negative correlation exists between systematic error and speckle radius. (2) According to Eq. (21), the random errors are independent of the actual displacement u₀, and thus they appear as horizontal lines in Fig. 2. Equation (36) indicates that 𝔼{e_n} ∼ R, and thus there is a positive correlation between random error and speckle radius. (3) As mentioned before, an increase in speckle radius will give rise to a decrease in systematic error and an increase in random error, leading to a bias-variance trade-off.

Fig. 2 Systematic errors e_b, random errors e_n, and root mean squared errors e_r of Gaussian speckle patterns with speckle radii (a) R = 1.3 pixels, (b) R = 1.5 pixels, (c) R = 2.0 pixels, and (d) R = 3.0 pixels. The theoretical estimates are evaluated using Eq. (12), Eq. (21), and Eq. (27). The variability of the numerical results is due to the spatial randomness illustrated in Fig. 1.

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In following sections, the influences of subset size, speckle coverage, and speckle radius are investigated. To characterize the systematic errors and the random errors, the actual displacement u₀ is set as 0.25 pixels.

5.1.1. Influence of subset size

Firstly, the influence of subset size on measurement errors are investigated. The subset size M ranged from 21 to 91 pixels, in increments of 10 pixels. The speckle radius R was respectively specified as 2 and 5 pixels; the speckle coverage ρ was respectively set as 50% and 80%.

Numerical results and theoretical estimates of the measurement errors corresponding to displacement u₀ = 0.25 pixels are depicted in Fig. 3. The theoretical estimates show good agreements with the numerical results. Figure 3 exhibits that, as the subset size increases, (a) the expectations of the systematic errors roughly remain invariant, and (b) the expectations of the random errors will decrease. These two observations can be explained by the proposed model. (a) Equation (12) implies that the expectation of the systematic error is specified by the gradient estimator d[m, n], the interpolation algorithm φ(x, y), and the speckle profile ψ(x, y) exclusively. As a consequence, 𝔼{C_b} is independent of subset size M and speckle coverage ρ. (b) Equation (21) clearly indicates that increasing subset size M induces an decrease in 𝔼{e_n}, and the leading order is 𝔼{e_n} ∼ M⁻¹.

Fig. 3 Influence of subset size on measurement errors: systematic errors for speckle radius (a1) R = 2.0 pixels and (a2) R = 5.0 pixels; random errors for speckle radius (b1) R = 2.0 pixels and (b2) R = 5.0 pixels. The actual displacement is u₀ = 0.25 pixels. The theoretical estimates are evaluated using Eq. (12) and Eq. (21).

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The numerical results also indicate a negative correlation between the subset size and the variability of the measurement errors. An intuitive explanation is that, with the increase of subset size, more texture information will be exploited, thus reducing the measure uncertainty.

5.1.2. Influence of speckle coverage

Then, the influence of speckle coverage on measurement errors are investigated. The speckle coverage ranged from 20% to 80%, in increments of 10%. The speckle radius R was respectively specified as 2 and 5 pixels; the subset size was respectively specified as 31 and 91 pixels.

Numerical results and theoretical estimates of the measurement errors corresponding to displacement u₀ = 0.25 pixels are depicted in Fig. 4. The data corresponding to ρ = 20% and M = 31 pixels is missing because it happens that there is no speckle in a subset and therefore the iteration cannot converge. The theoretical estimates show good agreements with the numerical results. Figure 4 exhibits that, as the speckle coverage increases, (a) the expectations of the systematic errors nearly remain constant, and (b) the expectations of the random errors will decrease. These two observations can be explained by the proposed model. (a) As stated in Section 5.1.1, the systematic error is independent of the speckle coverage. (b) The gradient variance q increases with increasing speckle coverage ρ [see Eq. (18)], thus decreasing the random error [Eq. (21)]; the leading order is 𝔼{e_n} ∼ ρ^−1/2.

Fig. 4 Influence of speckle coverage on measurement errors: systematic errors for speckle radius (a1) R = 2.0 pixels and (a2) R = 5.0 pixels; random errors for speckle radius (b1) R = 2.0 pixels and (b2) R = 5.0 pixels. The actual displacement is u₀ = 0.25 pixels. The theoretical estimates are evaluated using Eq. (12) and Eq. (21).

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5.1.3. Influence of speckle radius

Finally, the influence of speckle radius on measurement errors are investigated. As significant aliasing effect occurs when R < 1.3 pixels [14], the first speckle radius was set as R = 1.3 pixels, then R ranged from 1.5 to 5 pixels, in increments of 0.5 pixels. The subset size M was respectively specified as 31 and 91 pixels; the speckle coverage was respectively specified as 50% and 80%.

Numerical results and theoretical estimates of the measurement errors corresponding to displacement u₀ = 0.25 pixels are depicted in Fig. 5. The theoretical estimates show good agreements with the numerical results. Figure 5 indicates that, an increase in speckle radius will give rise to (a) a decrease in systematic error and (b) an increase in random error. Theses two observations can be explained by the proposed model. (a) It follows from Eq. (35) that the systematic error is determined by the integral of the product of the interpolation bias kernel E_b(ω_x, ω_y) and the power spectrum |ψ̂_g(ω_x, ω_y; R)|². With the increase of speckle radius, more energy will concentrate on the low-frequency domain. Since the interpolation bias kernel is a high-pass filter [13,14], the systematic error will decrease. (b) It follows from Eq. (36) that the random error

𝔼 {e_{n}} \approx \frac{\sqrt{2} σ}{\sqrt{ρ} M} R,

thus the random error is approximately proportional to the speckle radius R with a proportionality constant

\sqrt{2} σ M^{- 1} ρ^{- 1 / 2}

. The random errors in Fig. 5(b1)–(b2) are normalized by a factor Mσ⁻¹(ρ/2)^1/2 and then are illustrated in Fig. 6. Figure 6 implies that the approximation [Eq. (43)] is more accurate when the subset size is large. This is because the linearity is based on an elimination of the correction term, which is related to the variance, and meanwhile the variability of a small subset size is more significant.

Fig. 5 Influence of speckle radius on measurement errors. (a1) Systematic errors; (a2) the difference between systematic errors by theoretical estimations and by numerical experiments. Random errors for subset size (b1) M = 31 pixels and (b2) M = 91 pixels. The actual displacement u₀ = 0.25 pixels. The theoretical estimates are evaluated using Eq. (12) and Eq. (21).

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Fig. 6 The normalized random errors are approximately proportional to the speckle radius.

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5.2. Numerical experiments for optimal speckle radius

In order to validate the formulas of pattern quality measure V [Eq. (29)] and optimal speckle radius R_opt [Eq. (31) and Eq. (37)], numerical experiments were conducted and then a comparison between the numerical results and the theoretical estimates was made.

The numerical method utilized is similar to that in Section 5.1; the difference is that, in order to enhance the evaluation accuracy of the pattern quality measure V, 100 translated patterns were generated, with a displacement of 0.01 pixels between successive patterns. Recall that in Section 5.1, there are 20 translated patterns and the relative displacement is 0.05 pixels. The mean square errors were evaluated and then the pattern quality measure V, an average of the mean squared errors, was numerically evaluated by the trapezoidal rule [42]. Theoretical estimations of the pattern quality measure are based on Eq. (29). The optimal speckle radii R_opt are numerically evaluated using Eq. (31) and Eq. (37) respectively by the bisection method [42].

The pattern quality measure by numerical experiments and by theoretical estimations are illustrated in Fig. 7. The subset size M was specified as 31 and 91 pixels respectively; the speckle coverage ρ was specified as 50% and 80% respectively; the speckle radius R was firstly selected as 1.3 pixels, and then R ranged from 1.5 to 5 pixels, with an increase of 0.25 pixels. The numerical results show good agreements with the theoretical estimations. Evidently, there is a trade-off between systematic errors and random errors: when the speckle radius is excessively small, the systematic error is significant; when the speckle radius is too large, the random error is significant. Therefore, an optimal speckle radius exists, which can be estimated using Eq. (31), the full form, or Eq. (37), the approximation form for Gaussian speckle patterns. For the cases in Fig. 7, the optimal radii given by Eq. (31) are 1.977, 2.075, 2.453, and 2.570 pixels respectively, and the optimal radii given by Eq. (37) are 1.974, 2.071, 2.448, and 2.564 pixels respectively, which are almost the same as Eq. (31). The corresponding speckle sizes are 4 ∼ 5 pixels, which are consistent with the results in other literatures [1,20,22]. It can also be found that, as the increase of subset size or speckle coverage, the optimal speckle radius will increase. This is because the systematic errors are independent of subset size and speckle coverage, while the random error will decrease if the subset size or the speckle coverage increase. Besides, it seems that the variance of V will increase in response to the increase of 𝔼{V}.

Fig. 7 Relationship between pattern quality measure and speckle radius: (a) subset size M = 31 pixels, speckle coverage 50%; (b) subset size M = 31 pixels, speckle coverage 80%; (c) subset size M = 91 pixels, speckle coverage 50%; (d) subset size M = 91 pixels, speckle coverage 80%. The theoretical estimations of the pattern quality measure are evaluated using Eq. (29) and the optimal speckle radii R_opt are evaluated using Eq. (31).

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As a whole, the numerical results demonstrate the correctness of proposed theoretical model.

6. Discussion

6.1. Advantages

The theoretical model is superior to the traditional numerical methods in three aspects. (1) Generality: the theoretical model is applicable to all parameters; by contrast, conducting a simulation for every possible parameter is impossible. (2) Efficiency: it takes days to evaluate the numerical results in Fig. 7; by contrast, the theoretical estimates can be attained in about 1 second. (3) Deep understanding: the model explains why the systematic error is independent of the speckle density and why the random error is approximately proportional to the speckle radius.

6.2. Limits and outlook

Although the proposed theoretical model has the advantages as described above, there are still four limits exist. (1) The proposed model fails to characterize the spatial variances of measurement error and pattern quality measure because of the difficulties faced in mathematics. (2) The optimal speckle radius provided by the model is optimal in a sense of statistics. How to optimize the position of every speckle is still a problem. (3) The proposed model cannot suggest an optimal speckle coverage due to the assumption that the noise σ is a constant. In practice, the image noise depends on the intensity [16, 41]. Thus, a more appropriate assumption may be that the signal-to-noise ratio is a constant. However, it will make the theoretical analyses and the numerical simulations far more complicated. (4) The proposed model can suggest an optimal speckle radius for pure translation. Nevertheless, whether this value can be instructive in cases of deformations or rotations is still unclear. These problems need further research.

The key to our model is to derive the spatial expectations of systematic errors and random errors. For the systematic errors, the key is to derive the pattern power spectrum, which is the Fourier transform of the auto-correlation function due to Wiener-Khinchin theorem. For the random errors, the key is to derive the expectation of squared gradient. Speckle patterns which are composed of random positioned speckles are studied because of their wide application. However, there is a great hope that this method can be applied to other types of pattern, provided that the power spectrum and the expectation of squared gradient are known.

7. Conclusion

This paper concentrates on speckle patterns which are composed of random positioned speckles. A statistical model for speckle pattern optimization is built and validated. All source codes are uploaded and ready to be downloaded. The conclusions of this work are drawn as follows:

The process of speckle pattern generation is essentially a filtered Poisson process.
Spatial expectations of systematic errors [Eq. (12)], random errors [Eq. (21)], and overall errors [Eq. (26) and Eq. (27)] are characterized mathematically. The expectations of systematic errors are independent of speckle coverage and subset size; the expectations of random errors are approximately proportional to the speckle radius. The formulas proposed will facilitate the estimation of metrological performance.
The relationship between spatial expectation of pattern quality measure and speckle pattern generation parameters is deduced [Eq. (29)]. Assuming the noise variance remains constant, a formula for the optimal speckle radius is derived [Eq. (31)] and a simplified version is presented for Gaussian speckle patterns [Eq. (37)].

The original contribution of this work is the presentation of a theoretical model for speckle pattern optimization. As the mathematical derivations merely needs pattern power spectrum and expectation of squared gradient, there is a great hope that this method can be applied to patterns of other type. The primary motivation of this research is to establish a speckle pattern generation standard for DIC. However, as the model is general, the authors believe that scholars in other optical measurement communities, such as PIV and speckle metrology, will benefit from these discussions.

Funding

National Natural Science Foundation of China (11702287, 11332010, 11627803, 11472266); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB22040502).

References and links

1. M. A. Sutton, J. J. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications (Springer Science & Business Media, 2009).

2. M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: a Practical Guide (Springer, 2013).

3. M. Sjödahl and L. R. Benckert, “Systematic and random errors in electronic speckle photography,” Appl. Opt. 33(31), 7461–7471 (1994). [CrossRef] [PubMed]

4. M. Chakrabarti, M. L. Jakobsen, and S. G. Hanson, “Speckle-based spectrometer,” Opt. Lett. 40(14), 3264–3267 (2015). [CrossRef] [PubMed]

5. R. Szeliski, Computer Vision: Algorithms and Applications (Springer Science & Business Media, 2010).

6. B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003). [CrossRef]

7. D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006). [CrossRef]

8. Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016). [CrossRef]

9. L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011). [CrossRef] [PubMed]

10. Y. Su, Q. Zhang, Z. Gao, and X. Xu, “Noise-induced bias for convolution-based interpolation in digital image correlation,” Opt. Express 24(2), 1175–1195 (2016). [CrossRef] [PubMed]

11. H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000). [CrossRef]

12. T. Roesgen, “Optimal subpixel interpolation in particle image velocimetry,” Exp. Fluids 35(3), 252–256 (2003). [CrossRef]

13. Y. Su, Q. Zhang, Z. Gao, X. Xu, and X. Wu, “Fourier-based interpolation bias prediction in digital image correlation,” Opt. Express 23(15), 19242–19260 (2015). [CrossRef] [PubMed]

14. Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018). [CrossRef]

15. Y. Wang, M. Sutton, H. Bruck, and H. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009). [CrossRef]

16. B. Blaysat, M. Grédiac, and F. Sur, “On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study,” Exp. Mech. 56(6), 919–944 (2016). [CrossRef]

17. B. Pan, H. Xie, Z. Wang, K. Qian, and Z. Wang, “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037–7048 (2008). [CrossRef] [PubMed]

18. X. Shao, X. Dai, and X. He, “Noise robustness and parallel computation of the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 71, 9–19 (2015). [CrossRef]

19. W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017). [CrossRef]

20. P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40(8), 1613–1620 (2001). [CrossRef]

21. D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006). [CrossRef]

22. T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011). [CrossRef]

23. Z. Hu, T. Xu, H. Luo, R. Z. Gan, and H. Lu, “Measurement of thickness and profile of a transparent material using fluorescent stereo microscopy,” Opt. Express 24(26), 29822–29829 (2016). [CrossRef]

24. P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015). [CrossRef]

25. G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017). [CrossRef]

26. Z. Chen, C. Quan, F. Zhu, and X. He, “A method to transfer speckle patterns for digital image correlation,” Meas. Sci. Technol. 26(9), 095201 (2015). [CrossRef]

27. A. Lavatelli and E. Zappa, “A displacement uncertainty model for 2-D DIC measurement under motion blur conditions,” IEEE Trans. Instrum. Meas. 66(3), 451–459 (2017). [CrossRef]

28. J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8(12), 1379 (1997). [CrossRef]

29. A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (Tata McGraw-Hill Education, 2002).

30. Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

31. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23(3), 282–332 (1944). [CrossRef]

32. B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009). [CrossRef]

33. P. Thevenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000). [CrossRef] [PubMed]

34. H. A. Bruck, S. R. McNeill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989). [CrossRef]

35. S. Baker and I. Matthews, “Lucas-Kanade 20 years on: a unifying framework,” Int. J. Comput. Vision 56(3), 221–255 (2004). [CrossRef]

36. B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013). [CrossRef]

37. Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015). [CrossRef]

38. B. Kumar and S. C. D. Roy, “Design of digital differentiators for low frequencies,” Proc. IEEE 76(3), 287–289 (1988). [CrossRef]

39. Y. Su, “Optimal speckle radius for image registration,” figshare (2017) [retrieved 01 Nov 2017], http://dx.doi.org/10.6084/m9.figshare.5558542.v1.

40. W. Tong, “Subpixel image registration with reduced bias,” Opt. Lett. 36(5), 763–765 (2011). [CrossRef] [PubMed]

41. Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016). [CrossRef]

42. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).

Statistical model for speckle pattern optimization

Abstract

1. Introduction

2. Statistic of speckle patterns

3. Measurement errors

3.1. Systematic errors

3.2. Random errors

3.3. Overall errors

4. Optimal speckle radius

4.1. Pattern quality measure

4.2. Optimal speckle radius

4.3. Optimal speckle radius for Gaussian speckle patterns

5. Numerical experiments

5.1. Numerical experiments for measurement errors

5.1.1. Influence of subset size

5.1.2. Influence of speckle coverage

5.1.3. Influence of speckle radius

5.2. Numerical experiments for optimal speckle radius

6. Discussion

6.1. Advantages

6.2. Limits and outlook

7. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (43)

Optics Express