基于最小二乘法的相位展开
合集下载
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
The most commonly used methods of least squares phase unwrapping to solve the phase problem are iterative methods and direct methods. Simple iterative methods such as those of Jacobi, Gauss–Seidel, or successive over relaxation (SOR) may be feasible, but they can only be applied to reconstruction of the phase on a relatively small grid [64 Â 64 or less] [16]. The direct method could unwrap the phase more quickly. For instance, fast Fourier transform (FFT) and discrete cosine transform (DCT) [20,21], however, the direct method could cause a lot of errors.
& 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Two-dimensional phase unwrapping is an important processing step in many coherent imaging applications, such as synthetic aperture radar interferometry (InSAR) [1,2], magnetic resonance imaging (MRI) [3,4] and electronic speckle pattern interferometry (ESPI) [5,6], etc.
n Corresponding author. E-mail address: chenxiaotian1988@qq.com (X. Chen).
http://dx.doi.org/10.1016/j.optlaseng.2014.06.007 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
2. Description of the phase unwrapping algorithm
In the ESPI, the wrapped phase ψ i;j is obtained by phase shift technology, and ϕi;j is the true phase. The wrapping operator is
ESPI is a non-contact and noninvasive measurement method developed rapidly with the development of CCD and computer technology [7–10]. Displacement fields are obtained by digital correlation of the video images of two speckle interferograms recorded before and after deformation. In ESPI, the computation of phase by a tangential function provides only principal values of the phase that lie
Robust phase unwrapping algorithm based on least squares
Yuan Guo a, Xiaotian Chen a,n, Tao Zhang b
a College of Computer and Control Engineering, Qiqihar University, Qiqihar 161006, Heilongjiang, China b Qiqihar Power Supply Company, State Grid Corporation, Qiqihar 161005, Heilongjiang, China
Therefore, improving the convergence rate is a very important task when using iterative methods. In this paper, we put forward a new phase unwrapping algorithm based on calculating unwrapping coefficient k directly. The new method is based on iterative method and just needs two neighboring variables of k. It can overcome the deficiency of conventional LS phase wrapping methods and provide a new way to speed up the large amount of calculation in phase unwrapping with high precision.
the horizontal and the vertical sizes of the grid.
The phase difference term Δi;j is defined from the original
wrapped
phase
data
ψ i;j
between π and À π rad [11]. In addition, they must be unwrapped to
their true absolute phase values. This is the task of the phase unwrapping (PU), especially for two dimensional problems.
Many algorithms have been proposed for solving the PU problem, which differ in the solution method, application area and robustness. These algorithms can be grouped into three classes: (1) path-following [12–14], (2) minimum-norm methods [15,16], and (3) optimisation estimation [17,18]. Least-Square (LS) PU is one of the most commonly used in the PU algorithms that is a type of minimum norm methods.
article info
Article history: Received 20 March 2014 Received in revised form 5 June 2014 Accepted 5 June 2014
Keywords: Measurement Electronic speckle Phase unwrapping Unwrapping coefficient k Iterative algorithm
The method is divided into unweighted and weighted least squares phase unwrapping [19]. To isolate the phase inconsistencies, a weighted least square method should be used, which suppress the contamination effects by using the weighting arrays. The least squares method is well-defined mathematically and equivalent to the solution of Poisson’s partial differential equation, which can be expressed as a sparse linear equation.
Least squares phase unwrapping is one of the most robust techniques to solve the two-dimensional phase unwrapping problem. This method obtains an unwrapped solution by minimising the differences between the partial derivatives of the wrapped phase data and the unwrapped solution.
26
Y. Guo et al. / Optics and Lasers in Engineering 63 (2014) 25–29
defined as
ϕi;j ¼ ψ i;j þ 2πki;j
ð1Þ
where ki;j is an integer, 0 r i r M À 1 and 0 r j r N À 1, M and N are
Optics and Lasers in Engineering 63 (2014) 25–29
Contents lists available at ScienceDirect
ຫໍສະໝຸດ BaiduOptics and Lasers in Engineering
journal homepage: www.elsevier.com/locate/optlaseng
abstract
We propose an improved phase unwrapping algorithm by calculating the unwrapping coefficient k directly based on a Least-Squares phase unwrapping algorithm. It can unwrap the phase more rapidly and accurately than most previous methods, especially for large grid phase unwrapping and where compulational cost is high. To validate this method, we performed a series studies using a conventional Least-Square phase unwrapping, conventional weighted discrete cosine transform (WDCT) and large grid phase unwrapping. Results show that our method not only improves the accuracy but also reduces the time of phase unwrapping.
& 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Two-dimensional phase unwrapping is an important processing step in many coherent imaging applications, such as synthetic aperture radar interferometry (InSAR) [1,2], magnetic resonance imaging (MRI) [3,4] and electronic speckle pattern interferometry (ESPI) [5,6], etc.
n Corresponding author. E-mail address: chenxiaotian1988@qq.com (X. Chen).
http://dx.doi.org/10.1016/j.optlaseng.2014.06.007 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
2. Description of the phase unwrapping algorithm
In the ESPI, the wrapped phase ψ i;j is obtained by phase shift technology, and ϕi;j is the true phase. The wrapping operator is
ESPI is a non-contact and noninvasive measurement method developed rapidly with the development of CCD and computer technology [7–10]. Displacement fields are obtained by digital correlation of the video images of two speckle interferograms recorded before and after deformation. In ESPI, the computation of phase by a tangential function provides only principal values of the phase that lie
Robust phase unwrapping algorithm based on least squares
Yuan Guo a, Xiaotian Chen a,n, Tao Zhang b
a College of Computer and Control Engineering, Qiqihar University, Qiqihar 161006, Heilongjiang, China b Qiqihar Power Supply Company, State Grid Corporation, Qiqihar 161005, Heilongjiang, China
Therefore, improving the convergence rate is a very important task when using iterative methods. In this paper, we put forward a new phase unwrapping algorithm based on calculating unwrapping coefficient k directly. The new method is based on iterative method and just needs two neighboring variables of k. It can overcome the deficiency of conventional LS phase wrapping methods and provide a new way to speed up the large amount of calculation in phase unwrapping with high precision.
the horizontal and the vertical sizes of the grid.
The phase difference term Δi;j is defined from the original
wrapped
phase
data
ψ i;j
between π and À π rad [11]. In addition, they must be unwrapped to
their true absolute phase values. This is the task of the phase unwrapping (PU), especially for two dimensional problems.
Many algorithms have been proposed for solving the PU problem, which differ in the solution method, application area and robustness. These algorithms can be grouped into three classes: (1) path-following [12–14], (2) minimum-norm methods [15,16], and (3) optimisation estimation [17,18]. Least-Square (LS) PU is one of the most commonly used in the PU algorithms that is a type of minimum norm methods.
article info
Article history: Received 20 March 2014 Received in revised form 5 June 2014 Accepted 5 June 2014
Keywords: Measurement Electronic speckle Phase unwrapping Unwrapping coefficient k Iterative algorithm
The method is divided into unweighted and weighted least squares phase unwrapping [19]. To isolate the phase inconsistencies, a weighted least square method should be used, which suppress the contamination effects by using the weighting arrays. The least squares method is well-defined mathematically and equivalent to the solution of Poisson’s partial differential equation, which can be expressed as a sparse linear equation.
Least squares phase unwrapping is one of the most robust techniques to solve the two-dimensional phase unwrapping problem. This method obtains an unwrapped solution by minimising the differences between the partial derivatives of the wrapped phase data and the unwrapped solution.
26
Y. Guo et al. / Optics and Lasers in Engineering 63 (2014) 25–29
defined as
ϕi;j ¼ ψ i;j þ 2πki;j
ð1Þ
where ki;j is an integer, 0 r i r M À 1 and 0 r j r N À 1, M and N are
Optics and Lasers in Engineering 63 (2014) 25–29
Contents lists available at ScienceDirect
ຫໍສະໝຸດ BaiduOptics and Lasers in Engineering
journal homepage: www.elsevier.com/locate/optlaseng
abstract
We propose an improved phase unwrapping algorithm by calculating the unwrapping coefficient k directly based on a Least-Squares phase unwrapping algorithm. It can unwrap the phase more rapidly and accurately than most previous methods, especially for large grid phase unwrapping and where compulational cost is high. To validate this method, we performed a series studies using a conventional Least-Square phase unwrapping, conventional weighted discrete cosine transform (WDCT) and large grid phase unwrapping. Results show that our method not only improves the accuracy but also reduces the time of phase unwrapping.