Iterated unscented Kalman filter for phase unwrapping of interferometric fringes

Xianming Xie

doi:10.1364/OE.24.018872

1. Introduction

Phase unwrapping (PU) [1,2] is able to obtain unwrapped or absolute phase from noisy 2π mapped phase through removing the modulus 2π ambiguities of wrapped phase images, and is widely used in various fields including digital holographic microscopy [3], magnetic resonance imaging [4], speckle imaging, optical interferometry [5,6], adaptive optics, synthetic aperture radar interferometry [7,8], etc. Various methods for phase unwrapping have been put forward during the past few years and can be usually classified into following groups: 1) path-following algorithms; 2) minimum-norm algorithms; and 3) Bayesian algorithms; and 4) other miscellaneous methods except above three groups. The path-following algorithms [9–22] generally define suitable paths, and/or mask zones with low-reliability or isolate pixels defined as discontinuous pixels, and then unwrap wrapped phase image alone the paths defined, and/or only process the zones with high-reliability or pixels marked as continuous pixels, which include the branch-cut (BUT) algorithm [9], the quality-guided PU (QGPU) algorithm [14–16], the region growing algorithm [17,18], the network flow (NF) algorithm [19,20], the mask cut algorithm [21], the minimum discontinuity algorithm [22], etc. The path-following algorithms are able to mitigate the accumulative effect of erroneous pixels in the process of phase unwrapping and even obtain perfect solutions with noiseless interferograms, while these methods easily fail with high speckle or high residual noise. The minimum-norm algorithms [23–26] define a cost function under the frame of minimum-norm criterion over whole wrapped images, and then obtain a global solution by minimizing the cost function defined, whose representatives are the FFT-based least square method and the weighted least-square (WLS) method [23] as well as the preconditioned conjugate gradient algorithm [24]. The minimum-norm algorithms are efficient and can usually obtain smooth unwrapped solutions from noisy wrapped phase images, while the large errors will arise if the condition that the phase gradient of the noisy wrapped phase images should be strictly consistent with the true phase gradient, is fully not respected. The Bayesian algorithms [27–36] formulate phase unwrapping problem as an optimum state estimation problem under the frame of Bayesian criterion, which include the extended Kalman filtering PU (EKFPU) algorithm [27,28], particle filtering PU algorithms [29–31], the unscented Kalman filtering PU algorithms [32–36] and so on. Besides these methods mentioned above, there are still some other algorithms [37–42] such as the algorithms based on fringe frequency detection [37,38], and linear dynamic system [39], as well as integration with filters [41] and so on. In my previous works, initial versions of unscented Kalman filtering phase unwrapping (UKFPU) algorithms which have been classified into the above third groups, are proposed in [32,33] by combining an unscented Kalman filter (UKF) with a local phase gradient estimator based on power spectral density (PSD) and traditional paths-following strategies. The enhanced versions of the UKFPU algorithms are further presented in [43,44], where an amended matrix pencil model (AMPM) is deduced to accurately and efficiently estimate phase gradient information from noisy wrapped phase images and is further combined with the UKF to improve the accuracy and the efficiency of phase unwrapping. These methods can usually obtain better results from synthetic data and real data, with respect to some of the most used algorithms, including the BUT, the NF, the QGPU as well as the least-square method, etc.

This paper presents a new phase unwrapping algorithm for wrapped phase images by combining an iterated unscented Kalman filter (IUKF) with a robust phase gradient estimator based on amended matrix pencil model (AMPM), and an efficient quality-guided strategy based on heap sort. Compared with the EKFPU methods [27,28], the unscented Kalman filtering PU algorithms [32–36,43,44], the proposed method incorporates two significant differences which can greatly enhance the performance of this solution. First, the IUKF, which is one of the most robust methods under the frame of Bayesian criterion so far and has been well applied in some nonlinear signal processing fields [45,46], is applied into two-dimensional phase unwrapping of wrapped phase images for the first time, accordingly, the wrapped phase images can be accurately estimated by applying the IUKF to perform simultaneously noise suppression and phase unwrapping, which can simplify the complexity and the difficulty of pre-filtering procedure followed by phase unwrapping procedure, and even can remove the pre-filtering step. Second, the efficient quality-guided strategy based on heap sort presented in [47], which is proposed to accelerate the process of searching the optimum unwrapping paths, is able to ensure that the proposed method fast unwraps wrapped pixels along the path from the high-quality area to the low-quality area of wrapped phase images, which can greatly improve the efficiency of phase unwrapping. In addition, the AMPM-based phase gradient estimator is applied to efficiently and accurately obtain phase gradient information from noisy wrapped phase images, which is needed for the IUKF phase unwrapping model. Results obtained from synthetic data and real data show that the proposed method can obtain robust solutions with an acceptable time consumption, with respect to some of the most used algorithms.

This paper is organized as follows. The basic principle and related algorithms are described in Section 2. The IUKF-based phase unwrapping algorithm is introduced in Section 3. Then, Section 4 illustrates the performances of the proposed method with some examples from synthetic data and real data, and a comparison with the representative methods including the EKFPU algorithm [27,28] and the enhanced UKFPU (EUKFPU) method [43], and some of the most used algorithms including the BUT method [9], the QGPU method [15], the NF method [19], the WLS method [23], is also discussed. Finally, the conclusions are drawn in Section 5.

2. Basic principle and related algorithms

2.1 Basic concepts of phase unwrapping

The relation between the unwrapped phase and its modulo 2π mapped wrapped phase can be expressed as follows:

\tilde{ϕ} (m, n) = {[ϕ (m, n)]}_{2 π} \pm k \cdot 2 π \in (- π, π],

where

ϕ (m, n)

refers to the unwrapped phase, and

\tilde{ϕ} (m, n)

denotes its corresponding modulo 2π mapped wrapped phase. To obtain the unwrapped or absolute phase

ϕ (m, n)

from the noisy 2π mapped phase

\tilde{ϕ} (m, n)

is the final object of phase unwrapping.

2.2 related algorithms

The proposed method in this paper is the result of combining an IUKF with an AMPM-based phase gradient estimator, and an efficient quality-guided strategy based on heap sort. The AMPM-based phase gradient estimator is applied to obtain phase gradient information needed for the IUKF phase unwrapping model, and the efficient quality-guided strategy based on heap sort guides the IUKF to fast unwraps wrapped pixels along the path from the high-quality area to the low-quality area of wrapped phase images, and they will be introduced subsequently.

A. AMPM-based phase gradient estimator

Phase gradient estimation plays an important role in almost all phase unwrapping algorithms, which is directly related to the accuracy and the efficiency of phase unwrapping algorithms. The AMPM-based phase gradient estimator proposed in [43] is effective phase gradient estimation method, and it can efficiently and accurately obtain phase gradient information from noisy wrapped phase images and has been well applied in phase unwrapping algorithms. Phase gradient information at pixel $(a, s)$ can be estimated with the AMPM-based phase gradient estimator as follows:

Step. 1: a complex interferogram is obtained through executing complex trigonometric operations on a wrapped phase image.
Step. 2: build a data matrix $Z_{(a, s)}$ in a local window of the complex interferogram, centered at pixel $(a, s)$ , and remove noise from the data matrix $Z_{(a, s)}$ by executing singular value decomposition on the data matrix to, and then obtain a fresh data matrix ${\overset{⇀}{Z}}_{(a, s)}$ without noise.
Step 3: build sub-matrixes containing phase gradient information by deleting the rows and columns of the data matrix ${\overset{⇀}{Z}}_{(a, s)}$ .
Step. 4: obtain phase gradient estimate at pixel $(a, s)$ from the sub-matrixes constructed in Step. 3. More details about the AMPM-based phase gradient estimator can be found in [43].

B. Efficient quality-guided strategy based on heap sort

The efficient quality-guided strategy based on heap sort [47–49] is proposed to guide unwrapping paths and accelerate the process of searching the pixel with the highest reliance in the queue that is defined to save unwrapped pixels with possible wrapped neighbors. The key of the efficient quality-guided strategy is creating a queue with the data structure based on max-heap (MH) to store and sort the unwrapped pixels with possible wrapped neighbors. The MH-based data structure (MHBDS) is the sorted data structure based on complete binary tree, where the phase quality (PQ) value of unwrapped pixel stored in each node is less than or equal to the value of unwrapped pixel stored in its parent node, consequently, the PQ value of the unwrapped pixel stored in the root node is no less than that of unwrapped pixels stored in all other nodes. The property of max-heap is maintained while deleting unwrapped pixel in the root node and inserting new unwrapped pixels in node, and thus the sorting of unwrapped pixels in the queue is accomplished, which can greatly accelerate the process of searching the pixel with the highest reliance in the queue. More details about the efficient quality-guided strategy can be found in [47]. Main steps of the efficient quality-guided strategy are summarized as follows:

Step. 1: Generate a phase quality map of wrapped phase image to guide the paths of phase unwrapping, and create a queue with the MHBDS to store and sort the unwrapped pixels with possible wrapped neighbors.
Step. 2: The pixel x1 with the highest PQ value is taken as beginning pixel, and the wrapped phase of the beginning pixel is regarded as its unwrapped phase, then the wrapped neighbors of x1 are unwrapped by calculating the modulo 2π mapped phase gradient between x1 and its neighbors. Finally, the unwrapped neighbors of x1 are inserted into the queue with the MHBDS, respectively, and the disorderly heap in the queue with the MHBDS is adjusted to the max-heap according to the PQ value of the above unwrapped pixels.
Step. 3: The pixel x2 in the root node of max-heap is fetched and is deleted from the root node of max-heap in the queue, and then the disorderly heap in the queue is adjusted to the max-heap. The wrapped neighbors of x2 are unwrapped and are inserted into the queue with the MHBDS, respectively, and the disorderly heap in the queue is adjusted to the max-heap.
Step. 4: Are there any pixels in the queue? Yes, go into Step. 3, No, end.

3. Iterated UKF phase unwrapping algorithm

3.1 Two-dimensional iterated UKF phase unwrapping algorithm

Two-dimensional iterated UKF phase unwrapping (IUKFPU) algorithm is carried out by combining the IUKF with the AMPM-based phase gradient estimator, and the efficient quality-guided strategy based on heap sort described above in this paper. The IUKFPU algorithm unwraps wrapped phase images along the path from the high-reliable region to the low-reliable region of the wrapped phase images by applying the phase quality map of the wrapped phase images to guide the unwrapping path of the IUKFPU algorithm. Two-dimensional IUKF system model for phase unwrapping can be written as follows [32,33]:

\begin{array}{l} x (m, n) = x (a, s) + {\overset{⇀}{g}}_{(m, n) | (a, s)} + ε_{(m, n) | (a, s)} = f [x (a, s)] + ε_{(m, n) | (a, s)} \\ y (m, n) = {\begin{cases} \sin [x (m, n)] \\ \cos [x (m, n)] \end{cases}} + {\begin{cases} v_{1} (m, n) \\ v_{2} (m, n) \end{cases}} = h [x (m, n)] + v (m, n), \end{array}

where

x (m, n)

denotes true unambiguous unwrapped phase of the wrapped phase image at pixel

(m, n)

and is taken as state variable behind;

{\overset{⇀}{g}}_{(m, n) | (a, s)}

and

ε_{(m, n) | (a, s)}

refer to phase gradient estimate between pixel

(m, n)

and its neighbor pixel

(a, s)

and its corresponding phase gradient estimation error, respectively, and

{\overset{⇀}{g}}_{(m, n) | (a, s)}

is calculated in Eq. (3) ;

y (m, n)

and

v (m, n)

refer to noisy observation vector at pixel

(m, n)

and additive zero-mean white Gaussian noise vector in real and imaginary part of the complex measurements, respectively. In Eq. (2),

{\overset{⇀}{g}}_{(m, n) | (a, s)}

is obtained as follows:

{\overset{⇀}{g}}_{(m, n) | (a, s)} = {\bar{ϕ}}_{y (a, s)} (m - a) + {\bar{ϕ}}_{x (a, s)} (n - s),

where

{\bar{ϕ}}_{x (a, s)}

and

{\bar{ϕ}}_{y (a, s)}

refer to the unit local phase gradient estimate at pixel

(a, s)

in the row direction and in the column direction, respectively, and can be obtained by applying the AMPM-based phase gradient estimator described in Section 2.2.

In two-dimensional IUKFPU algorithm, any given pixel to be unwrapped will be predicted reliably based on all unwrapped pixels among its eight neighbors as follows [32,33]:

{\vec{x}}^{-} (m, n) = \sum_{(a, s) \in Ψ} d (a, s) \cdot x [(m, n) | (a, s)],

where pixel

(m, n)

refers to the pixel to be unwrapped, and

{\vec{x}}^{-} (m, n)

refers to the prediction estimate of the state variable at pixel

(m, n)

; pixel

(a, s)

refers to the unwrapped pixel among eight neighbors of pixel

(m, n)

, and

Ψ

refers to the collection of unwrapped pixels among eight neighbors of pixel

(m, n)

;

x [(m, n) | (a, s)]

is the evolution model to be applied in the direction from pixel

(a, s)

toward pixel

(m, n)

, and

d (a, s)

refers to the weight of the state estimate of unwrapped phase at pixel

(a, s)

, and are given as follows [43]:

\begin{array}{l} x [(m, n) | (a, s)] = \vec{x} (a, s) + {\overset{⇀}{g}}_{(m, n) | (a, s)} \\ d (a, s) = \frac{{[{\vec{P}}_{x x} (a, s) \cdot \frac{1}{S N R (a, s)}]}^{- 1}}{\sum_{(a, s) \in Ψ} {[{\vec{P}}_{x x} (a, s) \cdot \frac{1}{S N R (a, s)}]}^{- 1}}, \end{array}

where

\vec{x} (a, s)

denotes the state estimate of the unwrapped phase at pixel

(a, s)

, and

{\vec{P}}_{x x} (a, s)

refers to the estimation error covariance of the unwrapped phase at pixel

(a, s)

;

S N R (a, s)

refers to the signal-to-noise ratio (SNR) for pixel

(a, s)

, and is calculated in this paper as follows:

S N R (a, s) = = \frac{γ_{(a, s)}}{1 - γ_{(a, s)}},

where

γ_{(a, s)}

refers to coherence coefficient for pixel

(a, s)

.Note that the coherence coefficient can be replaced with a pseudo-coherence coefficient if the true coherence coefficient is unobtainable, and the (pseudo) coherence coefficient for each pixel usually is estimated within a local window [32,33].

It is worth mentioning that Eqs. (4) and 5 indicate that the unwrapped neighbors with the smaller estimation error and the higher SNR make the larger contribution to the pixel to be unwrapped. Then, the prediction error variance of the state variable at pixel $(m, n)$ can be calculated as follows:

P_{x x}^{-} (m, n) = \sum_{(a, s) \in Ψ} d (a, s) {\vec{P}}_{x x} (a, s) .

The key idea of the IUKF algorithm is to capture the posterior mean value of the state variable and its corresponding variance by Sigma points [32,33]. The Sigma points of the prediction estimate of the state variable at pixel $(m, n)$ can be calculated as follows:

\begin{array}{l} χ_{0} (m, n) = {\vec{x}}^{-} (m, n) \\ χ_{1} (m, n) = {\vec{x}}^{-} (m, n) + \sqrt{(1 + λ) P_{x x}^{-} (m, n)} \\ χ_{2} (m, n) = {\vec{x}}^{-} (m, n) - \sqrt{(1 + λ) P_{x x}^{-} (m, n)}, \end{array}

where

χ_{j} (m, n)

(j = 0, 1, 2)

refers to Sigma points of

{\vec{x}}^{-} (m, n)

,

λ

refers to the scale parameter of the Sigma points, and is calculated as follows:

λ = α^{2} (1 + κ) - 1,

where

α,κ

are preset by the experience to adjust the Sigma points (usually,

α=0 .01,κ=0

). More details of the Sigma points can be found in [50,51].

Then, a new prediction of the state variable at pixel $(m, n)$ can be calculated as follows [32,33]:

\begin{array}{l} \tilde{x} (m, n) = \sum_{j = 0}^{2} w_{j}^{m} χ_{j} (m, n) \\ {\tilde{P}}_{x x} (m, n) = \sum_{j = 0}^{2} w_{j}^{c} [χ_{j} (m, n) - \tilde{x} (m, n)] \cdot, \\ [^{χ_{j} (m, n) - \tilde{x} (m, n)] T} + \sum_{(a, s) \in Ψ} d (a, s) Q_{(m, n) | (a, s)} \end{array}

where

\tilde{x} (m, n)

and

{\tilde{P}}_{x x} (m, n)

refer to the new prediction of the state variable at pixel

(m, n)

and its corresponding prediction error variance, respectively, the superscript

T

stands for vector transpose;

Q_{(m, n) | (a, s)}

refers to the phase gradient estimate error variance matrix between pixel

(m, n)

and pixel

(a, s)

, and is obtained in Eq. (12);

w_{j}^{m}

and

w_{j}^{c}

refer to the weight factors, given as follows [32,33]:

\begin{array}{l} w_{0}^{m} = λ / (1 + λ), w_{0}^{c} = λ / (1 + λ) + (1 - α^{2} + β), \\ w_{j}^{m} = w_{j}^{c} = 1 / [2 (1 + λ)] \end{array}

where

β

is preset to adjust the Sigma points, similar to the parameter

α,κ

, usually,

α=0 .01,κ=0, β =2

. In Eq. (10),

Q_{(m, n) | (a, s)}

is calculated as follows:

Q_{(m, n) | (a, s)} = G ({\bar{ϕ}}_{y (a, s)}) {(m - a)}^{2} + G ({\bar{ϕ}}_{x (a, s)}) {(n - s)}^{2},

where

G ({\bar{ϕ}}_{y (a, s)})

and

G ({\bar{ϕ}}_{x (a, s)})

refer to the unit local phase gradient estimate error variance matrix in the row direction and in the column direction of the AMPM local window centered at pixel

(a, s)

, respectively, and are calculated as follows [43]:

\begin{array}{l} G ({\bar{ϕ}}_{y (a, s)}) = 4 π^{2} \frac{1 - γ_{(a, s)}^{2}}{γ_{(a, s)}^{2} Κ_{n} (Κ_{m}^{2} - 1)}, \\ G ({\bar{ϕ}}_{x (a, s)}) = 4 π^{2} \frac{1 - γ_{(a, s)}^{2}}{γ_{(a, s)}^{2} Κ_{m} (Κ_{n}^{2} - 1)} \end{array}

where

Κ_{n}

and

Κ_{m}

refer to the row length and the column length of the AMPM local window, respectively [43]. Then, the state update at pixel

(m, n)

can be performed as follows [32,33]:

\begin{array}{l} ξ_{j}^{-} (m, n) = h [χ_{j} (m, n)] \\ {\vec{y}}^{-} (m, n) = \sum_{j = 0}^{2} w_{j}^{m} ξ_{j}^{-} (m, n) \\ P_{y y} (m, n) = \sum_{j = 0}^{2} w_{j}^{c} [ξ_{j}^{-} (m, n) - {\vec{y}}^{-} (m, n)] \cdot \\ [^{ξ_{j}^{-} (m, n) - {\vec{y}}^{-} (m, n)] T} + R (m, n), \\ P_{x y} (m, n) = \sum_{j = 0}^{2} w_{j}^{c} [χ_{j} (m, n) - \tilde{x} (m, n)] \cdot \\ [^{ξ_{j}^{-} (m, n) - {\vec{y}}^{-} (m, n)] T} \\ ρ (m, n) = P_{x y} (m, n) / P_{y y} (m, n) \\ {\vec{x}}^{0} (m, n) = \tilde{x} (m, n) + ρ (m, n) [y (m, n) - {\vec{y}}^{-} (m, n)] \\ {\vec{P}}_{x x}^{0} (m, n) = {\tilde{P}}_{x x} (m, n) - ρ (m, n) P_{y y} (m, n) ρ (m, n) \end{array}

where

{\vec{x}}^{0} (m, n)

and

{\vec{P}}_{x x}^{0} (m, n)

denote state estimate with zero iteration and its corresponding estimation error covariance at pixel

(m, n)

, respectively, and will continue to be applied into the following iterated operations; and

R (m, n)

denotes the observed error variance matrix at pixel

(m, n)

, and is calculated in this paper as follows:

R (m, n) = \frac{1}{S N R (m, n)} = \frac{1 - γ_{(m, n)}}{γ_{(m, n)}},

where

S N R (m, n)

denotes the SNR for pixel

(m, n)

, and

γ_{(m, n)}

refers to coherence coefficient for pixel

(m, n)

. Suppose that the number of iterations is

N_{iter}

in the IUKFPU algorithm, then the iterated operations in two-dimensional IUKF phase unwrapping are described in following section.

3.2 Iterated operations

Suppose that ${\vec{x}}^{i} (m, n)$ and ${\vec{P}}_{x x}^{i} (m, n)$ denote the state estimate at pixel $(m, n)$ and its corresponding estimation error covariance obtained by performing the $i th$ iterated operation, respectively. If $(i +1) \leq N_{iter}$ , then the Sigma points of the state estimate ${\vec{x}}^{i} (m, n)$ can be calculated as follows:

\begin{array}{l} χ_{0}^{i + 1} (m, n) = {\vec{x}}^{i} (m, n) \\ χ_{1}^{i + 1} (m, n) = {\vec{x}}^{i} (m, n) + \sqrt{(1 + λ) {\vec{P}}_{x x}^{i} (m, n)} \\ χ_{2}^{i + 1} (m, n) = {\vec{x}}^{i} (m, n) - \sqrt{(1 + λ) {\vec{P}}_{x x}^{i} (m, n)}, \\ {\overset{⌢}{x}}^{i + 1} (m, n) = \sum_{j = 0}^{2} w_{j}^{m} χ_{j}^{i + 1} (m, n) \end{array}

Then, the state estimate with the

(i +1)th

iterated operation, at pixel

(m, n)

, can be obtained as follows:

\begin{array}{l} ξ_{j}^{i + 1} (m, n) = h [χ_{j}^{i + 1} (m, n)] \\ {\vec{y}}^{i + 1} (m, n) = \sum_{j = 0}^{2} w_{j}^{m} ξ_{j}^{i + 1} (m, n) \\ P_{y y}^{i + 1} (m, n) = \sum_{j = 0}^{2} w_{j}^{c} [ξ_{j}^{i + 1} (m, n) - {\vec{y}}^{i + 1} (m, n)] \cdot, \\ [^{ξ_{j}^{i + 1} (m, n) - {\vec{y}}^{i + 1} (m, n)] T} + R (m, n) \\ P_{x y}^{i + 1} (m, n) = \sum_{j = 0}^{2} w_{j}^{c} [χ_{j}^{i + 1} (m, n) - {\overset{⌢}{x}}^{i + 1} (m, n)] \cdot [^{ξ_{j}^{i + 1} (m, n) - {\vec{y}}^{i + 1} (m, n)] T} \\ ρ^{i + 1} (m, n) = P_{x y}^{i + 1} (m, n) / P_{y y}^{i + 1} (m, n) \\ {\vec{x}}^{i + 1} (m, n) = \tilde{x} (m, n) + ρ^{i + 1} (m, n) [y (m, n) - {\vec{y}}^{i + 1} (m, n)] \\ {\vec{P}}_{x x}^{i + 1} (m, n) = {\vec{P}}_{x x}^{i} (m, n) - ρ^{i + 1} (m, n) P_{y y}^{i + 1} (m, n) ρ^{i + 1} (m, n) \end{array}

where

{\vec{x}}^{i + 1} (m, n)

and

{\vec{P}}_{x x}^{i + 1} (m, n)

denote the state estimate obtained by performing the

(i +1)th

iterated operation and its corresponding estimation error covariance, respectively, and will continue to be applied into the recursive iterated operations if the iterated operations continue to be performed. Suppose that the state estimate obtained by performing

N_{iter}

iterated operations and its corresponding estimation error covariance are

{\vec{x}}^{N_{iter}} (m, n)

and

{\vec{P}}_{x x}^{N_{iter}} (m, n)

, respectively. To be coherent with the notation of state variable above,

{\vec{x}}^{N_{iter}} (m, n)

and

{\vec{P}}_{x x}^{N_{iter}} (m, n)

are denoted as

\vec{x} (m, n)

and

{\vec{P}}_{x x} (m, n)

, respectively. More details about the iterated operations of the IUKF can be found in [45,46].

3.3 Main steps of the proposed method

Preprocessing stage

Step. 1: Generate a complex interferogram through executing complex trigonometric operations on a wrapped phase image, and then obtain local phase gradient at each pixel from the complex interferogram through the AMPM-based phase gradient estimator, required by the IUKF phase unwrapping model.
Step. 2: Generate a phase quality map according to interferogram or complex interferogram. Various kinds of phase quality map including coherent coefficient map, pseudo coherence map, phase derivative variance map, maximum phase gradient map, the combination of these phase quality and so on, can be used to guide phase unwrapping paths. Phase derivative variance map of wrapped phase image is applied to guide phase unwrapping paths in the experiments of the proposed method in this paper.
Step. 3: Create a queue with the MHBDS (QMHBDS) to store and sort the waiting pixels that are waiting to be unwrapped and are further defined in unwrapping stage.

It is worth mentioning that only wrapped phase map is required in the experiments of the proposed method in this paper since both phase gradient information and phase derivative variance map can obtained directly from the wrapped phase map.

Unwrapping stage

Flowchart of unwrapping stage in the proposed method is shown in Fig. 1, and main steps of the unwrapping stage are as follows:

Fig. 1 Flowchart of unwrapping stage in the proposed method.

Mean Square Root Errors (radian)
SNR /dB	BUT	NF	WLS	QGPU	EKFPU	EUKFPU
6.11	0.1226	0.8551	0.4736	0.1255	0.2123	0.0936
2.18	0.2364	1.0431	1.2623	0.2363	0.2291	0.1684
0.89	0.3079	1.4064	1.5532	0.3107	0.5216	0.2166
−0.51	0.4508	2.5756	3.0916	0.4258	0.8266	0.2776

Mean Square Root Errors (radian)
SNR /dB	IUKFP (Niter = 0)	IUKFPU (Niter = 1)	IUKFPU (Niter = 2)	IUKFPU (Niter = 3)	IUKFPU (Niter = 4)	IUKFPU (Niter = 5)
6.11	0.0936	0.0782	0.0716	0.0708	0.0699	0.0690
2.18	0.1684	0.1361	0.1192	0.1176	0.1166	0.1158
0.89	0.2166	0.1703	0.1524	0.1502	0.1489	0.1481
−0.51	0.2776	0.2180	0.1977	0.1948	0.1937	0.1933

Run-time (second)
IUKFPU (Niter = 0)	IUKFPU (Niter = 1)	IUKFPU (Niter = 2)	IUKFPU (Niter = 3)	IUKFPU (Niter = 4)	IUKFPU (Niter = 5)
17.37	20.74	23.81	27.59	31.18	36.66

Mean Square Root Errors (radian)
SNR /dB 6.11 2.18 0.89	IUKFPU (Niter = 0) 0.1694 0.2429 0.2749	IUKFPU (Niter = 1) 0.1239 0.1968 0.2311	IUKFPU (Niter = 2) 0.0907 0.1480 0.1774	IUKFPU (Niter = 3) 0.0887 0.1429 0.1708
−0.51	0.3036	0.2643	0.2108	0.2034

Mean Square Root Errors (radian)
SNR /dB	QGPU	EKFPU	IUKFPU (Niter = 0)	IUKFPU (Niter = 1)	IUKFPU (Niter = 2)	IUKFPU (Niter = 3)
6.11	0.2943	0.3737	0.1661	0.1205	0.0939	0.0915
2.18	2.3612	0.5917	0.2611	0.2013	0.1649	0.1602
0.89	5.2249	1.3495	0.3012	0.2397	0.1986	0.1934
−0.51	8.5322	1.5043	0.3401	0.2810	0.2417	0.2371

Mean Square Root Errors (radian)
SNR /dB	QGPU	EKFPU	IUKFPU (Niter = 0)	IUKFPU (Niter = 1)	IUKFPU (Niter = 2)	IUKFPU (Niter = 3)
6.11	0.2943	0.3737	0.1661	0.1205	0.0939	0.0915
2.18	2.3612	0.5917	0.2611	0.2013	0.1649	0.1602
0.89	5.2249	1.3495	0.3012	0.2397	0.1986	0.1934
−0.51	8.5322	1.5043	0.3401	0.2810	0.2417	0.2371

Abstract

1. Introduction

2. Basic principle and related algorithms

2.1 Basic concepts of phase unwrapping

2.2 related algorithms

A. AMPM-based phase gradient estimator

B. Efficient quality-guided strategy based on heap sort

3. Iterated UKF phase unwrapping algorithm

3.1 Two-dimensional iterated UKF phase unwrapping algorithm

3.2 Iterated operations

3.3 Main steps of the proposed method

Preprocessing stage

Unwrapping stage

4. Experimental results and analysis

4. 1 Synthetic data experiments

A1. Unwrapping with a pre-filtering procedure

A2. Unwrapping without a pre-filtering procedure

A3. Solutions with interferogram over peaks without the pre-filtering procedure

A4. Solutions with interferogram over cone without the pre-filtering procedure

4.2. Real data experiments

5. Conclusions

Funding

Acknowledgments

References and Links

Cited By

Figures (34)

Tables (7)

Equations (17)

Optics Express