Discrete fringe phase unwrapping algorithm based on Kalman motion estimation for high-speed I/Q-interferometry

Zhangqiang He; Junning Cui; Jiubin Tan; Xingyuan Bian; Wenxue Jiang

doi:10.1364/OE.26.008699

1. Introduction

I/Q-interferometry has been becoming a preferred method in advanced scientific research experiments, ultra-precision instruments and manufacturing equipment for its angstrom-level accuracy, high dynamic-range, and etc [1–4]. The rapid developing high technology fields like semiconductor lithography industry [5], ultra-low frequency vibration calibration [6], etc., are challenging I/Q-interferometers to offer high-speed measurement ability.

Among all technical links of an I/Q-interferometer, successive phase unwrapping (SPUA) which is the process of retrieving unambiguous phase from an array of wrapped phase modulo 2π radians is a main factor that limits the measurement speed. According to Nyquist theorem, a sampling rate of more than double the maximum fringe frequency is required to prevent miscounting the number of whole fringes. For example, during 0.1 Hz ultra-low frequency vibration calibration with an one-meter-long-stroke vibration exciter [7], the maximum velocity of the standard sinusoidal vibration generated is up to 0.8 m/s, and a sampling frequency of more than 4.8 MHz is needed. Consequently, the sampling rate and vast volume of data greatly challenge the hardware performance of the interferometer, and considerable time-consumption is required for data processing, significantly limiting the measurement speed.

In order to break the limitation of Nyquist frequency and time consumption of traditional SPUA, a discrete fringe phase unwrapping algorithm (DFPUA) based on Kalman motion estimation is proposed. Kalman technique which is often used in realms of wavefront distortion testing, surface deformation detection of optics [8–10] is introduced to realize accurate phase unwrapping with deep under-sampled data. Procedures of two typical types of DFPUA, velocity estimation and velocity-acceleration estimation are illustrated. Peak acceleration/jerk instead of peak velocity becomes the factor that determines the minimum sampling rate required, thus the requirements for hardware performance in high speed interferometry applications is effectively reduced.

2. Principle

In an I/Q-interferometer, the displacement $d_{k}$ at the kth sampling cycle can be calculated by demodulating fringe phase signals from the quadrature signal series, $I_{x k}$ and $I_{y k}$ :

\begin{array}{l} d_{k} = \frac{λ}{4 π} (φ_{k} + m_{k} \cdot 2 π), m_{k} = 0, \pm 1, \pm 2... \\ φ_{k} = \arctan \frac{I_{x k}}{I_{y k}} \end{array}

where

λ

is the wavelength of the laser,

m_{k}

is a series of integer numbers that counts the fringes, i.e. phase integer numbers, and

φ_{k}

(

φ_{k} \in [0, 2 π)

) is the series of phase decimal. The task of the phase unwrapping algorithm is to determine the phase integer number

m_{k}

correctly.

The idea of DFPUA is to realize phase unwrapping with deeply under-sampled data based on Kalman motion estimation, thus to break through the limitation of the Nyquist frequency. The basic concept of DFPUA is to estimate the current displacement, i.e. to obtain current estimated phase integer number and current estimated phase decimal, according to the former motion state, then confirm the actual phase integer number by comparing the estimated phase decimal with the actual phase decimal, and finally obtain the actual displacement result and realize the phase unwrapping. The principle is theoretically derived as follows.

Assuming that the displacement d is a continuous function of time, denoted as $d (t)$ , and is differentiable at time t, it can be expressed by Taylor expansion as:

d (t + Δ T) = d (t) + v (t) Δ T + \frac{1}{2} a (t) Δ T^{2} + \frac{1}{3} j (t) Δ T^{3} + ...

where

Δ T

is the sampling interval,

v (t)

,

a (t)

, and

j (t)

are the first, second, and third derivatives of

d (t)

, i.e. velocity, acceleration, and jerk. The state vector of motion is expressed as

x = {[d, v, a, j, ...]}^{T}

, so the state vector at the kth sampling cycle can be expressed as

x_{k} = {[d_{k}, v_{k}, a_{k}, j_{k}, ...]}^{T}

. According to Eq. (2), the state vector

x_{k + 1}

can be evolved from the value of

x_{k}

using a linear stochastic difference model:

x_{k + 1} = A x_{k}

where

A = [\begin{matrix} 1 & Δ T & Δ T^{2} / 2 & ... \\ 0 & 1 & Δ T & ... \\ 0 & 0 & ... & ... \\ 0 & ... & 0 & 1 \end{matrix}] .

However, since it is difficult to get values of parameters $v_{k}, a_{k}, j_{k}, ...$ besides displacement $d_{k}$ in the state vector $x_{k}$ , the approximate velocity ${\tilde{v}}_{k}$ , acceleration ${\tilde{a}}_{k}$ , jerk ${\tilde{j}}_{k}$ , etc., can be estimated with the displacement $d_{k - 1}$ :

{\tilde{v}}_{k} = \frac{d_{k} - d_{k - 1}}{Δ T}, {\tilde{a}}_{k} = \frac{{\tilde{v}}_{k} - {\tilde{v}}_{k - 1}}{Δ T}, {\tilde{j}}_{k} = \frac{{\tilde{a}}_{k} - {\tilde{a}}_{k - 1}}{Δ T}, \dots

The state vector is revised as $\tilde{x} = {[d, \tilde{v}, \tilde{a}, \tilde{j}, ...]}^{T}$ . Therefore, Eq. (3) is revised as

x_{k + 1} = A ({\tilde{x}}_{k} + ω_{k})

where

ω_{k}

represents the estimation error vector and is written as

ω_{k} = {[\begin{matrix} 0 & v_{k} - {\tilde{v}}_{k} & a_{k} - {\tilde{a}}_{k} & j_{k} - {\tilde{j}}_{k} & ... \end{matrix}]}^{T}

The first item of $A {\tilde{x}}_{k}$ represents the estimated displacement $d_{k + 1}^{estimated}$ at the (k + 1)th sampling cycle, the first item of $A ω_{k}$ represents the displacement estimation error $e_{k + 1}^{estimated}$ at the (k + 1)th sampling cycle.

The estimated phase integer number $m_{k + 1}^{estimated}$ and the estimated phase decimal $φ_{k + 1}^{estimated}$ can be calculated according to Eq. (1):

m_{k + 1}^{estimated} = r o u n d (\frac{d_{k + 1}^{estimated}}{λ / 2}), φ_{k + 1}^{estimated} = r e m (\frac{4 π d_{k + 1}^{estimated}}{λ}, 2 π)

where

r o u n d (X)

is the integer nearest to X, and

r e m (X, Y)

is the remainder after division, i.e.,

r e m (X, Y) = X - N * Y

, where

N = r o u n d (X / Y)

.

To avoid mistake in the phase integer number, $e_{k + 1}^{estimated}$ should be smaller than $λ / 4$ . Therefore, the difference between the estimated phase decimal $φ_{k + 1}^{estimated}$ and the actual phase decimal $φ_{k + 1}^{}$ should be within $π$ . On the other hand, taking the jumps of phase series into consideration, the actual phase integer number $m_{k + 1}$ is within $[m_{k + 1}^{estimated} - 1, m_{k + 1}^{estimated} + 1]$ , and

m_{k + 1} = {\begin{cases} m_{k + 1}^{estimated}, when | φ_{k + 1}^{} - φ_{k + 1}^{estimated} | \leq π \\ m_{k + 1}^{estimated} - 1, when (φ_{k + 1}^{} - φ_{k + 1}^{estimated}) > π \\ m_{k + 1}^{estimated} + 1, when (φ_{k + 1}^{} - φ_{k + 1}^{estimated}) < - π \end{cases}

In this way, robust phase unwrapping is realized based on Kalman motion estimation. The proposed algorithm relies on the premise condition that the accurate displacement values of former n sampling cycles, i.e. $d_{k - 1}$ , $d_{k - 2}$ , …, $d_{k - n}$ , are available, where n is the length of the state vector $x$ .

3. Procedures

The larger the length of the state vector $x$ is, the more rigorous the premise condition of DFPUA will be. According to the number of initial displacement values used, Two typical types of DFPUA, referred as velocity estimation (n = 2) and velocity-acceleration estimation (n = 3) respectively, are presented.

3.1 Velocity estimation

Assuming initial velocity of motion in a displacement measurement is zero, or approximately zero, to make sure the initial two displacements values $d_{0}$ and $d_{1}$ satisfy $| d_{1} - d_{0} | < λ / 4$ .

The following procedures for the DFPUA based on velocity estimation are implemented as:

Step (1): Apply gain and offset correction method to the raw interference quadrature signals $I_{x}$ and $I_{y}$ .

Step (2): Calculate the phase series $φ_{}$ as follows taking into account the sign of $I_{x}$ and $I_{y}$ :

φ = {\begin{cases} \arctan (I_{x} / I_{y}) if I_{x} and I_{y} \geq 0 \\ \arctan (I_{x} / I_{y}) + π if I_{x} and I_{y} < 0 \\ \arctan (I_{x} / I_{y}) + 2 π if I_{x} * I_{y} < 0 \end{cases}

Step (3): Provided that the first two phases at the first two sampling cycles, φ₀ and φ₁, are within half fringe, $d_{0}$ and $d_{1}$ are calculated as

\begin{matrix} d_{0} = \frac{λ}{4 π} φ_{0} \\ d_{1} = {\begin{cases} \frac{λ}{4 π} φ_{1}, if | φ_{0} - φ_{1} | \leq π \\ \frac{λ}{4 π} (φ_{1} + 2 π), if (φ_{0} - φ_{1}) > π \\ \frac{λ}{4 π} (φ_{1} - 2 π), if (φ_{0} - φ_{1}) < - π \end{cases} \end{matrix}

Step (4): As shown in Fig. 1, for k = 1, 2, …, the velocity estimation is implemented as

Fig. 1 Principle of DFPUA based on velocity estimation (sampling interval: $Δ T$ ).

Download Full Size | PDF

\begin{array}{l} {\tilde{v}}_{k - 1} = \frac{d_{k} - d_{k - 1}}{Δ T}, d_{k + 1}^{estimated} = d_{k} + {\tilde{v}}_{k - 1} * Δ T ​ \\ m_{k + 1}^{estimated} = r o u n d (\frac{d_{k + 1}^{estimated}}{λ / 2}), φ_{k + 1}^{estimated} = r e m (\frac{4 π d_{k + 1}^{estimated}}{λ}, 2 π) \end{array}

Step (5): According to Eq. (9), the actual phase integer number $m_{k + 1}$ and the actual displacement $d_{k + 1}$ can be confirmed.

According to Eq. (2), there exists:

e_{}^{estimated} \leq a_{p e a k} Δ T^{2}

where

a_{p e a k}

is the peak acceleration of motion.

As mentioned in Section 2, to avoid mistake in the phase integer number, $e_{k + 1}^{estimated}$ should be smaller than $λ / 4$ , i.e. the maximum value of $e_{k + 1}^{estimated}$ is $λ / 4$ . Therefore, the sampling rate $f_{s} = 1 / Δ T$ should satisfy

f_{s} \geq 2 \sqrt[]{\frac{a_{p e a k}}{λ}}

When the nonlinearity error of the I/Q-interferometer is taken into consideration, the requirement on sampling rate $f_{s}$ is revised:

f_{s} \geq \sqrt{\frac{a_{p e a k}}{λ / 4 - 2 e r r o r_{PV}}}

where

e r r o r_{PV}

is the peak-valley value of nonlinearity error.

3.2 Velocity-acceleration estimation

Assuming initial velocity and acceleration of motion in a displacement measurement are zero, or approximately zero, to make sure the initial three displacement values $d_{0}$ , $d_{1}$ , and $d_{2}$ satisfy $| d_{1} - d_{0} | < λ / 4$ and $| d_{2} - d_{1} | < λ / 4$ .

The following procedures for the DFPUA based on velocity- acceleration estimation are implemented as:

Steps (1) and (2) are the same as the DFPUA based on velocity estimation.

Step (3): Provided that the first three phases at the first two sampling cycles, φ₀, φ₁, and φ₂, are all within half fringe, the initial three displacement values $d_{0}$ , $d_{1}$ , and $d_{2}$ are calculated as:

\begin{matrix} d_{0} = \frac{λ}{4 π} φ_{0} \\ d_{1} = {\begin{cases} \frac{λ}{4 π} φ_{1}, if | φ_{0} - φ_{1} | \leq π \\ \frac{λ}{4 π} (φ_{1} + 2 π), if (φ_{0} - φ_{1}) > π \\ \frac{λ}{4 π} (φ_{1} - 2 π), if (φ_{0} - φ_{1}) < - π \end{cases}, d_{2} = {\begin{cases} \frac{λ}{4 π} φ_{2}, if | φ_{1} - φ_{2} | \leq π \\ \frac{λ}{4 π} (φ_{2} + 2 π), if (φ_{1} - φ_{2}) > π \\ \frac{λ}{4 π} (φ_{2} - 2 π), if (φ_{1} - φ_{2}) < - π \end{cases} \end{matrix}

Step (4): As shown in Fig. 2, for k = 2, 3, …, the velocity-acceleration estimation is implemented as

\begin{array}{l} {\tilde{v}}_{k - 2} = \frac{d_{k - 1} - d_{k - 2}}{Δ T}, {\tilde{v}}_{k - 1} = \frac{d_{k} - d_{k - 1}}{Δ T}, {\tilde{a}}_{k - 2} = \frac{{\tilde{v}}_{k - 1} - {\tilde{v}}_{k - 2}}{Δ T} \\ {\tilde{v}}_{k}^{estimated} = {\tilde{v}}_{k - 1} + {\tilde{a}}_{k - 2} * Δ T, d_{k + 1}^{estimated} = d_{k} + {\tilde{v}}_{k}^{estimated} * Δ T \\ m_{k + 1}^{estimated} = r o u n d (\frac{d_{k + 1}^{estimated}}{λ / 2}), φ_{k + 1}^{estimated} = r e m (\frac{4 π d_{k + 1}^{estimated}}{λ}, 2 π) \end{array}

where

{\tilde{v}}_{k}^{estimated}

is the estimated velocity at the kth sampling cycle.

Fig. 2 Principle of DFPUA based on velocity-acceleration estimation (sampling interval: $Δ T$ ).

Download Full Size | PDF

Step (5): According to Eq. (9), the actual phase integer $m_{k + 1}$ and the actual displacement $d_{k + 1}$ can be confirmed.

According to Eq. (2), there exists:

e_{}^{estimated} \leq j_{p e a k} Δ T^{3}

where

j_{p e a k}

is the peak jerk of motion.

Since the maximum value of $e_{k + 1}^{estimated}$ is $λ / 4$ , the sampling rate $f_{s} = 1 / Δ T$ should satisfy

f_{s} \geq \sqrt[3]{\frac{4 j_{p e a k}}{λ}}

When nonlinearity error the I/Q-interferometer is considered, the minimum requirement on sampling rate $f_{s}$ is revised:

f_{s} \geq \sqrt[3]{\frac{j_{p e a k}}{λ / 4 - 2 e r r o r_{PV}}}

where

e r r o r_{PV}

is the peak-valley value of nonlinearity error.

4. Simulation experiments

Simulation experiments have been carried out to investigate the feasibility and capacity of the DFPUA. An acceleration signal $a (t)$ from the field of ultra-low frequency vibration calibration is used to simulate a vibration movement:

a (t) = {\begin{cases} 0.25 a_{p e a k} \sin (2 π f_{v} t), 0 \leq t \leq T_{v} \\ - 0.5 a_{p e a k} \sin (2 π f_{v} (t - T_{v})), T_{v} < t \leq 1.5 T_{v} \\ a_{p e a k} \sin (2 π f_{v} (t - 1.5 T_{v})), t > 1.5 T_{v} \end{cases}

where

f_{v}

is the vibration frequency,

T_{v}

is the vibration period,

a_{p e a k}

is the peak acceleration of motion. Therefore, the acceleration, velocity, and displacement are continuous functions of time and time differentiable, the initial acceleration, velocity, and displacement are zero, and the displacement offset of the vibration is zero. The peak velocity of motion can be calculated as

v_{p e a k} = \frac{a_{p e a k}}{2 π f_{v}}

An I/Q-interferometer whose laser wavelength

λ

is 632.8 nm modulates the vibration movement and generate quadrature signals

I_{x}

and

I_{y}

:

I_{x} = \cos φ_{M o d}, I_{y} = \sin (φ_{M o d} + α)

where

φ_{M o d}

is the modulated phase and

φ_{M o d} = 4 π \iint a (t) d^{2} t / λ

,

α

is the residual quadrature phase error after correction, and

α

is supposed to be 0.1°. The residual quadrature phase error results in a nonlinearity error whose peak-valley value

e r r o r_{PV}

= 0.09 nm. In a practical I/Q-interferometer, various noise components (detector noise, shot noise, laser noise, etc.) contributes a nonlinearity error of 10 pm level, thus the condition of the simulation appropriately reflects the practical situation.

When $f_{v}$ = 500Hz, $a_{p e a k}$ = 10 m/s², an experiment to illustrate the unwrapping of phase and phase integer number is carried out. The quadrature signals of the vibration movement are shown in Fig. 3(a). The peak velocity $v_{p e a k}$ is calculated as 3.184 mm/s, thus the sampling rate should be higher than 20.12 kHz according to Nyquist theorem when the traditional SPUA algorithm is used. However, according to Eq. (15), the sampling rate is 7955 Hz when the DFPUA based on velocity estimation is applied. In Fig. 3(a), the red circles and blue squares are corresponding deeply under-sampled data. The acquired signals I_x and I_y are processed to obtain phases and phase integer numbers, as shown in Fig. 3(b) and Fig. 3(c). The sampling cycles when the difference between the actual phase $φ$ and the estimated phases $φ^{estimated}$ are greater than π are marked with black dashed lines, the actual phase integer numbers m are revised according to Eq. (9) by plus one or minus one to the estimated phase integer number $m^{estimated}$ . It also can be seen that when the fringe signals vary quickly, the fringes are discretely sampled, i.e. some fringes have not been sampled, and thus the corresponding two neighboring phase integer numbers, which are marked with ellipses in Fig. 3(c), have a difference of two.

Fig. 3 Data processing of quadrature signals in the measurement of vibration (f_v = 500 Hz, a_peak = 10 m/s²) using DFPUA based on velocity estimation (f_s = 7955 Hz). (a) Quadrature signals. (b) Phases. (c) Phase integer numbers unwrapped.

Download Full Size | PDF

When $a_{p e a k}$ = 10 m/s², $f_{v}$ = 100 Hz, and $f_{v}$ = 1 kHz, an experiment is carried out to show the unwrapped displacement signals. The unwrapped displacement signals using the DFPUA based on velocity estimation and velocity-acceleration estimation are shown in Fig. 4(a) and Fig. 4(b) respectively. In case of $f_{v}$ = 100 Hz, the peak velocity $v_{p e a k}$ is calculated as 15.92 mm/s, thus the minimum sampling rate is 100.6 kHz using traditional SPUA, and 7955 Hz, 3414 Hz using DFPUA based on velocity estimation and velocity-acceleration estimation respectively. In case of $f_{v}$ = 1 kHz, the peak velocity $v_{p e a k}$ is calculated as 1.592 mm/s, thus the minimum sampling rate is 10.06 kHz using traditional SPUA, and 7955 Hz, 7354 Hz using the two DFPUAs based on Kalman motion estimation. It is can be seen that that the DFPUAs accurately demodulate deeply under-sampled quadrature signals. In case of $f_{v}$ = 100 Hz, since the sampling rate using the two DFPUAs are much higher than the frequency of the displacement signal, the displacement signal of low frequency vibration is deeply over-sampled. Meanwhile, in case of $f_{v}$ = 1 kHz, since the sampling rate using the two DFPUAs are at the same level of the frequency of the displacement signal, the displacement signal of vibration is not over-sampled.

Fig. 4 Unwrapped displacement signals (a_peak = 10 m/s²) using DFPUA based on velocity estimation and DFPUA velocity-acceleration estimation. (a) f_v = 100 Hz. (b) f_v = 1 kHz.

Download Full Size | PDF

When $a_{p e a k}$ = 10 m/s², $f_{v}$ = 0.1 Hz ~10 kHz, an experiment on minimum sampling rate and volume of data is carried out. The volume of data is evaluated by the number of sampling per channel for 10 cycles of vibration signal, and can be calculated as $10 f_{s} / f_{v}$ . Minimum sampling rate and the volume of data for traditional SPUA and proposed DFPUAs are shown in Fig. 5(a) and Fig. 5(b). As concluded before, the minimum sampling rate required for traditional SPUA depends on the peak velocity, while the minimum sampling rate required for the two types of DFPUAs proposed depends on the peak acceleration or peak jerk. Therefore, for low frequency vibration measurement, since the peak velocity $v_{p e a k}$ is relatively very high, high performance hardware whose sampling rate is up to dozens of mega-hertz and memory capacity is up to several giga sampling per channel is required by traditional SPUA, while the proposed DFPUAs enable a significant reduction in the hardware requirement. For example, for 0.1 Hz low frequency vibration measurement, the peak velocity $v_{p e a k}$ is calculated as 15.92 m/s, the minimal sampling rate is 100.6 MHz for SPUA, 7955 Hz for velocity estimation, 342 Hz for velocity-acceleration estimation, respectively. For high frequency vibration measurement above 1.265 kHz, DFPUAs have no advantage over SPUA on sampling rate and memory requirement. The main reason is that the displacement amplitude is so small and less than λ/2 that it can be handled by any algorithm easily. However, in the case of vibration displacement signal reconstruction, the sampling rate is required to be higher than double vibration frequency.

Fig. 5 The minimum requirement for SPUA and two proposed DFPUAs vs. vibration frequency. (a) Sampling rate requirement. (b) Number of samples per channel.

Download Full Size | PDF

5. Conclusion

A DFPUA based on Kalman motion estimation is proposed to breaks through the limitations of the Nyquist frequency for high-speed I/Q-interferometry measurement. Procedures of two types of practical DFPUAs including velocity estimation and velocity-acceleration estimation are detailedly illustrated. The effectiveness and capacity of proposed DFPUA has been verified by simulation experiments with a time-continuous acceleration vibration signal whose initial parameters of motion are all zero. The peak acceleration/jerk instead of peak velocity becomes the factor that determines the minimum sampling rate required, resulting in a significant reduction in the sampling rate and volume of data, thus the requirements for hardware performance requirements in high speed interferometry applications can be significantly reduced. Simulation experiment results show that for 0.1 Hz low frequency vibration with peak acceleration of 1g (10 m/s²), the minimal required sampling rates are 7955 Hz and 342 Hz for velocity estimation and velocity-acceleration estimation respectively, while 100.6 MHz for SPUA. The proposed algorithm meets the urgent demand for high-speed interferometry measurement in fields like ultra-low frequency vibration calibration, and can be applied in other phase quadrature detection technology.

Funding

National Natural Science Foundation of China (NSFC) (51675139); Fundamental Research Funds for the Central Universities.

References and links

1. P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17(18), 16322–16331 (2009). [CrossRef] [PubMed]

2. S. H. Eang and K. Cho, “Balanced-path homodyne I/Q-interferometer scheme with very simple optical arrangement using a polarizing beam displacer,” Opt. Express 25(7), 8237–8244 (2017). [CrossRef] [PubMed]

3. S. H. Eang, S. Yoon, J. G. Park, and K. Cho, “Scanning balanced-path homodyne I/Q-interferometer scheme and its applications,” Opt. Lett. 40(11), 2457–2460 (2015). [CrossRef] [PubMed]

4. J. G. Park and K. Cho, “High-precision tilt sensor using a folded Mach-Zehnder geometry in-phase and quadrature interferometer,” Appl. Opt. 55(9), 2155–2159 (2016). [CrossRef] [PubMed]

5. D. Musinski, “Displacement-measuring interferometers provide precise metrology,” Laser Focus World 39(12), 80 (2003).

6. T. Bruns and S. Gazioch, “Correction of shaker flatness deviations in very low frequency primary accelerometer calibration,” Metrologia 53(3), 986–990 (2016). [CrossRef]

7. W. He, X. Zhang, C. Wang, R. Shen, and M. Yu, “A long-stroke horizontal electromagnetic vibrator for ultralow-frequency vibration calibration,” Meas. Sci. Technol. 25(8), 085901 (2014). [CrossRef]

8. L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014). [CrossRef] [PubMed]

9. X. M. Xie and Q. N. Zeng, “Efficient and robust phase unwrapping algorithm based on unscented Kalman filter, the strategy of quantizing paths-guided map, and pixel classification strategy,” Appl. Opt. 54(31), 9294–9307 (2015). [CrossRef] [PubMed]

10. R. Kulkarni and P. Rastogi, “Phase derivative estimation from a single interferogram using a Kalman smoothing algorithm,” Opt. Lett. 40(16), 3794–3797 (2015). [CrossRef] [PubMed]

Discrete fringe phase unwrapping algorithm based on Kalman motion estimation for high-speed I/Q-interferometry

Abstract

1. Introduction

2. Principle

3. Procedures

3.1 Velocity estimation

3.2 Velocity-acceleration estimation

4. Simulation experiments

5. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (23)

Optics Express