n Corresponding author. E-mail address: chenxiaotian1988@ (X. Chen).
/10.1016/j.optlaseng.2014.06.007 0143-8166/& 2014 Elsevier Ltd. All rights reserved.
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.
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
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.
Optics and Lasers in Engineering 63 (2014) 25–29
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Optics and Lasers in Engineering
journal homepage: /locate/optlaseng
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.
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.
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
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
& 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.
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
26
Y. Guo et al. / Optics and Lasers in Engineering 63 (2014) 25–29
defined asϕi;j来自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
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.