Phase unwrapping of MR phase images using Poisson equation

The authors have developed a technique based on a solution of the Poisson equation to unwrap the phase in magnetic resonance (MR) phase images. The method is based on the assumption that the magnitude of the inter-pixel phase change is less than pi per pixel. Therefore, the authors obtain an estimate of the phase gradient by "wrapping" the gradient of the original phase image. The problem is then to obtain the absolute phase given the estimate of the phase gradient. The least-squares (LS) solution to this problem is shown to be a solution of the Poisson equation allowing the use of fast Poisson solvers. The absolute phase is then obtained by mapping the LS phase to the nearest multiple of 2 K from the measured phase. The proposed technique is evaluated using MR phase images and is proven to be robust in the presence of noise. An application of the proposed method to the 3-point Dixon technique for water and fat separation is demonstrated.     

Phase unwrapping via graph cuts.

Phase unwrapping is the inference of absolute phase from modulo-2pi phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are first-order Markov random fields. We provide an exact energy minimization algorithm, whenever the corresponding clique potentials are convex, namely for the phase unwrapping classical Lp norm, with p > or = 1. Its complexity is KT (n, 3n), where K is the length of the absolute phase domain measured in 2pi units and T (n, m) is the complexity of a max-flow computation in a graph with n nodes and m edges. For nonconvex clique potentials, often used owing to their discontinuity preserving ability, we face an NP-hard problem for which we devise an approximate solution. Both algorithms solve integer optimization problems by computing a sequence of binary optimizations, each one solved by graph cut techniques. Accordingly, we name the two algorithms PUMA, for phase unwrappping max-flow/min-cut. A set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state-of-the-art phase unwrapping algorithms.     

时间位相展开方法研究进展

时间位相展开是位相测量领域内的一种很重要的位相展开方法,被广泛应用于干涉型和结构照明型计量领域中,包括散斑干涉与剪切干涉计量、波长扫描干涉术、先弹测量、位相测量轮廓术、傅立叶变换轮廓术等领域。本文详细分析了近十年所提出的时间位相展开算法的基本原理,讨论和比较了各种算法的设计方法、计算公式和适用范围,对时间位相展开方法的最新进展和应用前景进行了较全面的分析和评述。     

Hierarchical network flow phase unwrapping

The well-studied interferometric synthetic aperture radar (InSAR) problem for digital elevation map generation involves the derivation of topography from the radar phase. The topography is a function of the full phase, whereas the measured phase is known modulo 2/spl pi/, necessitating the process of recovering full phase values via phase unwrapping. This mathematical process becomes difficult through the presence of noise and phase discontinuities. This paper is motivated by research which models phase unwrapping as a network-flow minimization problem. A major limitation is that often a substantial computational effort is required to find solutions. Commonly, these phase images are huge (/spl Gt/10 million pixels), and obviously the sheer size of the problem itself makes phase unwrapping challenging. This paper addresses the development of a computationally efficient hierarchical algorithm, based on a "divide-and-conquer" approach. We have shown that the phase-unwrapping problem can first be partitioned into independent phase-unwrapping subproblems, which can further be recombined to produce the unwrapped phase. Interestingly, the recombination step itself can be interpreted as an unwrapping problem, for which a modified network-flow solution applies! In short, this paper develops a parallelization of the network-flow algorithm, allowing images of virtually unlimited size to be unwrapped and leading to dramatic decreases in the algorithm execution time.     

可靠度导引的相位解包裹算法

设计实现了基于可靠度导引的相位解包裹算法。该算法首先在像素可靠度的基础上计算邻接线可靠度,然后根据邻接线可靠度的高低规划解包裹路径。仿真和实测结果表明,该算法可以有效避免误差扩散引起的拉线现象,提高解包裹相位的可靠性。     

A model-based method for phase unwrapping

Presents a model-based phase unwrapping method which represents the unwrapped phase function by a truncated Taylor series and a residual function. An efficient, noniterative computational algorithm is also proposed for calculating the model parameters from the phase derivatives. Sample experimental results are shown to demonstrate the effectiveness of the algorithm for extracting unwrapped phase images from two-dimensional (2-D) magnetic resonance imaging (MRI) data.     

Two-dimensional phase unwrapping using wavelet transform

A method for solving the least squares two-dimensional phase unwrapping problem is presented. This technique is based on the multiresolution representation of a linear system using the discrete wavelet transform. By applying the wavelet transform to the original system, a better convergence condition of an equivalent new system can be achieved     

On frequency-domain and time-domain phase unwrapping

Time-domain phase unwrapping of analytic or complex functions of time is an unavoidable operation when processing such signals within an FM-modeling framework or when attempting to establish the instantaneous frequency of such signals. With the help of an alternate derivation of his equations, Tribolet's frequency-domain phase unwrapping algorithm is easily rewritten for time-domain application. The time derivative of the argument of the signal, a quantity needed in these modified equations, is developed and furthermore recognized to be proportional to the instantaneous frequency of the signal.     

Least-squares two-dimensional phase unwrapping using FFT's

A solution of the least-squares two-dimensional phase-unwrapping problem is presented that is simpler to understand and implement than previously published solutions. It extends the phase function to a periodic function using a mirror reflection, and the resulting equation is solved using the Fourier transform     

基于标记理论和相位滤波的相位解缠绕算法

相位解缠绕是磁共振成像技术应用中准确获取相位信息的关键所在,如温度监控,油水分离,测定血流速度等.然而,由于磁共振成像噪声比较严重,而且组织交界区域附近相位本身就存在突变,造成解缠绕的精度下降.本文将相位解缠绕转化为计算机视觉中的标记问题,并构造出相应的能量函数,为利用网络优化对其求解提供方便.同时,针对磁共振相位解缠绕中的噪声问题,本文提出一个新的滤波方法,可以高效地消除噪声导致的解缠绕的不一致性.通过仿真实验和磁共振图像的相位解缠绕实验,表明本文方法相位解缠绕精度更高,且计算速度快.     

区域标记法位相展开

在干涉型和结构照明型计量领域中 ,对具有病态特性的截断位相进行有效位相展开是这一领域中关键步骤之一。本文基于位相展开过程中出现的一些问题提出了一种新的算法 ,该算法利用区域标记的方法并结合虚拟极点的概念 ,针对展开过程中经常出现的由于极点问题处理不当造成的错误展开情况进行合理展开 ,以期获得更加合理的结果。本文给出的实验结果证实了这一算法的有效性     

合成孔径雷达干涉测量中的相位展开

合成孔径雷达干涉测量技术 (InSAR)是以合成孔径雷达复数据提取的相位信息作为信息源来获取地表的三维信息和变化信息的一项技术。相位展开是InSAR数据处理中的关键步骤之一。本文介绍了合成孔径雷达干涉测量的原理 ,重点讨论了InSAR中的相位展开算法 ,特别对与路径有关的算法和与路径无关的算法进行了分析 ,比较了各类方法的优缺点。指出一些在结构照明型计量中有效的相位展开算法也可以用于合成孔径雷达干涉测量之中的相位展开     

Interferometric SAR phase unwrapping using Green's formulation

Any method that permits retrieving full range (unwrapped) phase values starting from their (-π,π) determination (wrapped phase) can be defined as a phase unwrapping technique. This paper addresses a new procedure for phase unwrapping especially designed for interferometric synthetic aperture radar applications. The proposed algorithm is based on use of Green's first identity. Results on simulated as well as on real data are presented. They both confirm the excellent performance of the procedure     

小波变换在位相展开中的应用

在结构照明型光学三维传感和干涉型计量技术中,都需要将计算出的截断位相恢复为物理参数对应的连续位相,位相展开的过程中,对于十分复杂的截断位相,如何有效地标记出极点和截断线是至关重要的。本文提出利用小波变换寻找截断线的方法,利用小波变换系数的模极大值与局部奇异性的关系,通过检测小波变换系数模极大值的位置来标记截断线的基本走向,进而采用图像处理中的二值数学形态学处理方法形成完整的截断线,构成指导位相展开的控制模板,新算法有效地标记了极点和截断线,将位相展开的错误限制在局部范围。本文给出了这种算法的详细分析和计算机模拟实验的结果。     

Least squares phase unwrapping in wavelet domain

Two-dimensional phase unwrapping is an important processing step in some coherent imaging applications. Least squares phase unwrapping is one of the robust techniques used to solve two-dimensional phase unwrapping problems. However, owing to its sparse structure, the convergence rate is very slow, and some practical methods have been applied to improve this condition. A new method for solving the least squares two-dimensional phase unwrapping problem is presented. This technique is based on the multiresolution representation of a linear system using the discrete wavelet transform. By applying the wavelet transform, the original system is decomposed into its coarse and fine resolution levels. Fast convergence in separate coarse resolution levels makes the overall system convergence very fast.     

干涉合成孔径雷达边缘检测解相位模糊的方法

给出了一种干涉合成孔径雷达相位解模糊方法,该方法利用边缘检测算子确定干涉因子中干涉纹的位置。为了提高干涉纹的检测精度,将纹线检测和纹线跟踪相结合,并采用了中值滤波和相对平均灰度梯度法,该方法具有无积累误差且稳定性好的优点。利用三峡地区的L波段SIR-C并行轨道数据,运用上述方法得到了解模糊的相位图和局部区域的相对地形图,获得了比较满意的结果。     

Two-dimensional phase unwrapping using a block least-squares method

We present a block least-squares (BLS) method for two-dimensional (2-D) phase unwrapping. The method works by tessellating the input image into small square blocks with only one phase wrap. These blocks are unwrapped using a simple procedure, and the unwrapped blocks are merged together using one of two proposed block merging algorithms. By specifying a suitable mask, the method can easily handle objects of any shape. This approach is compared with the Ghiglia-Romero (1994) method and the Marroquin-Rivera (1995) method. On synthetic images with different noise levels, the BLS method is shown to be superior, both with respect to the resulting gray values in the unwrapped image as well as visual inspection. The method is also shown to successfully unwrap synthetic and real images with shears, fiber-optic interferometry images, and medical magnetic resonance images. We believe the new method has the potential to improve the present quality of phase unwrapped images of several different image modalities.     

基于模拟退火的全局相位展开算法

与常用的最小二乘和最小费用流相位展开算法求满足待求相位梯度与包裹相位梯度最小范数解不同,提出一种以待求相位的二阶相位差在整个相位场上的总和为目标函数,并利用模拟退火算法实现其最小化的相位展算法。计算表明：该算法可以对最小二乘和最小费用流算法不能展开的欠采样包裹相位和含有噪声的欠采样包裹相位,进行有效的相位展开。     

2-D phase unwrapping and instantaneous frequency estimation

The phase of complex signals is wrapped since it can only be measured modulo-2π; unwrapping searches for the 2π-combinations that minimize the discontinuity of the unwrapped phase, as only the unwrapped phase can be analyzed and interpreted by further processing. Given an estimate of the phase gradient (i.e., of the instantaneous frequency), the 2-D unwrapped phase can be obtained as a solution of a variational problem. The analysis of unwrapping is done quite separately from instantaneous frequency estimation so that the reliability of both steps can be assessed independently. Various methods for evaluating 2-D instantaneous frequency are presented and compared in the presence of noise and amplitude variations. A study has also been made on aliasing arising in areas where, with respect to instantaneous frequency, spatial sampling is insufficient. The presence of noise in the data further complicates phase aliasing analysis since there is no way to distinguish between the aliasing due to noise or that due to steep phase slopes     

A complementary binary code based phase unwrapping method

Phase unwrapping is used to establish the mapping relationship between camera and projector,which is one of the key technologies in fringe projection profilometry(FPP)based three-dimensional(3D)measurement.Although comple-mentary Gray code assisted phase unwrapping technology can get a good result on the periodic boundary,it needs more coded images to obtain a high frequency fringe.Aiming at this problem,a complementary binary code assisted phase unwrapping method is proposed in this paper.According to the periodic consistency between the wrapping phase and binary codes,the coded patterns are generated.Then the connected domain strategy is performed to calculate the fringe orders using the positive and negative image binaryzation.To avoid the mistake near the periodic boundary,comple-mentary binary code inspired by the complementary Gray code is proposed.The fringe order correction is also dis-cussed for different situations in the first measured period.Only two binary images are needed in the proposed method,and the fringe frequency is not limited.Both the simulation and experiment have verified the feasibility of proposed method.