Phase difference-based multichannel phase unwrapping.

This work addresses the derivation of the phase difference-based maximum likelihood (ML) phase unwrapping algorithm. To this end, we derive the joint statistics of the phase differences on a two-dimensional grid for the multichannel case, where several scaled wrapped phase values are available. Subsequently, we determine and study the structure of the phase difference-based ML estimator and compare it to known phase unwrapping techniques. This work allows us to frame single and multichannel algorithms in a common formulation. Moreover, among the known single-channel phase difference-based procedures, we identify those attaining an ML solution. We also show that multichannel phase difference-based and, recently proposed, phase-based ML algorithms achieve equivalent solutions.     

Phase unwrapping for 2-D blind deconvolution of ultrasound images

In most approaches to the problem of two-dimensional homomorphic deconvolution of ultrasound images, the estimation of a corresponding point-spread function (PSF) is necessarily the first stage in the process of image restoration. This estimation is usually performed in the Fourier domain by either successive or simultaneous estimation of the amplitude and phase of the Fourier transform (FT) of the PSE This paper addresses the problem of recovering the FT-phase of the PSF, which is an important reconstruction problem by itself. The purpose of this paper is twofold. First, it provides a theoretical framework, establishing that the FT-phase of the PSF can be effectively estimated by a proper smoothing of the FT-phase of the appropriate radio-frequency (RF) image. Second, it presents a novel approach to the estimation of the FT-phase of the PSF, by solving a continuous Poisson equation over a predefined smooth subspace, in contrast to the discrete Poisson equation solver used for the classical least mean squares phase unwrapping algorithms, followed by a smoothing procedure. The proposed approach is possible due to the distinct properties of the FT-phases, among which the most important property is the availability of precise values of their partial derivatives. This property overcomes the main disadvantage of the discrete schemes, which routinely use wrapped (principal) values of the phase in order to approximate its partial derivatives. Since such an approximation is feasible subject to the restriction that the partial phase differences do not exceed pi in absolute value, the discrete schemes perform satisfactory only for few practical situations. The proposed approach is shown to be independent of this restriction and, thus, it performs for a wider class of the phases with significantly lower errors. The main advantages of the novel method over the algorithms based on discrete schemes are demonstrated in a series of computer simulations and for in vivo measurements.     

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     

Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting

Displacement encoding with stimulated echoes (DENSE) encodes myocardial tissue displacement into the phase of the MR image. Cine DENSE allows for rapid quantification of myocardial displacement at multiple cardiac phases through the majority of the cardiac cycle. For practical sensitivities to motion, relatively high displacement encoding frequencies are used and phase wrapping typically occurs. In order to obtain absolute measures of displacement, a two-dimensional (2-D) quality-guided phase unwrapping algorithm was adapted to unwrap both spatially and temporally. Both a fully automated algorithm and a faster semi-automated algorithm are proposed. A method for computing the 2-D trajectories of discrete points in the myocardium as they move through the cardiac cycle is introduced. The error in individual displacement measurements is reduced by fitting a time series to sequential displacement measurements along each trajectory. This improvement is in turn reflected in strain maps, which are derived directly from the trajectories. These methods were validated both in vivo and on a rotating phantom. Further measurements were made to optimize the displacement encoding frequency and to estimate the baseline strain noise both on the phantom and in vivo. The fully automated phase unwrapping algorithm was successful for 767 out of 800 images (95.9%), and the semi-automated algorithm was successful for 786 out of 800 images (98.3%). The accuracy of the tracking algorithm for typical cardiac displacements on a rotating phantom is 0.24 +/- 0.15 mm. The optimal displacement encoding frequency is in the region of 0.1 cycles/mm, and, for 2 scans of 17-s duration, the strain noise after temporal fitting was estimated to be 2.5 +/- 3.0% at end-diastole, 3.1 +/- 3.1% at end-systole, and 5.3 +/- 5.0% in mid-diastole. The improvement in intra-myocardial strain measurements due to temporal fitting is apparent in strain histograms, and also in identifying regions of dysfunctional myocardium in studies of patients with infarcts.     

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     

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     

Unwrapping of MR phase images using a Markov random field model

Phase unwrapping is an important problem in many magnetic resonance imaging applications, such as field mapping and flow imaging. The challenge in two-dimensional phase unwrapping lies in distinguishing jumps due to phase wrapping from those due to noise and/or abrupt variations in the actual function. This paper addresses this problem using a Markov random field to model the true phase function, whose parameters are determined by maximizing the a posteriori probability. To reduce the computational complexity of the optimization procedure, an efficient algorithm is also proposed for parameter estimation using a series of dynamic programming connected by the iterated conditional modes. The proposed method has been tested with both simulated and experimental data, yielding better results than some of the state-of-the-art method (e.g., the popular least-squares method) in handling noisy phase images with rapid phase variations.     

Regenerating MR tagged images using harmonic phase (HARP) methods

Magnetic resonance tagging has proven useful in the visualization and quantification of cardiac motion. Traditionally, tags are designed to have crisp geometric profiles in order to enhance both visualization and detection of tags. Recent image acquisition and analysis methods, however, have been designed to exploit sinusoidal tag profiles. This paper presents a method based on harmonic phase (HARP) concepts to synthesize tag lines that have both crisp profiles and alternative orientations from the original sinusoidal patterns. Results are demonstrated on images acquired with SPAMM, CSPAMM, and fast-HARP pulse sequences.     

使用区域重构技术的路径无关相位解包方法

相位解包在光学测量领域有着广泛的应用,其中路径相关类解包方法通常计算速度较快,但抗噪声能力较弱;而路径无关类方法的鲁棒性好,但大多存在迭代收敛慢、计算量大的问题。因此,将区域重构技术引入解包运算中,得到一种新的路径无关类解包方法。该方法对于矩形区域数据能够直接求解解包相位,无需迭代过程;设计了采样-重组加速方法使运算量大幅减小,计算速度优于同类解包方法;使用Gerchberg式迭代过程能够解决不规则孔径内的数据解包问题,可在少量迭代后收敛。通过仿真验证了该方法在矩形区域内的直接求解精度和加速法的效能,通过对实验数据的解包验证了算法处理不规则孔径数据的能力。     

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.     

Two-dimensional phase unwrapping using a minimum spanning treealgorithm

Phase unwrapping refers to the determination of phase from modulo 2pi data, some of which may not be reliable. In 2D, this is equivalent to confining the support of the phase function to one or more arbitrarily shaped regions. A phase unwrapping algorithm is presented which works for 2D data known only within a set of nonconnected regions with possibly nonconvex boundaries. The algorithm includes the following steps: segmentation to identify connectivity, phase unwrapping within each segment using a Taylor series expansion, phase unwrapping between disconnected segments along an optimum path, and filling of phase information voids. The optimum path for intersegment unwrapping is determined by a minimum spanning tree algorithm. Although the algorithm is applicable to any 2D data, the main application addressed is magnetic resonance imaging (MRI) where phase maps are useful.     

双频外差结合相位编码的相位解包裹方法

为了使用高频条纹进行一次外差完成相位解包裹,实现高精度测量,提出了双频外差结合相位编码的相位解包裹方法。首先,用两组正弦条纹获得两个包裹相位,进行外差处理得到外差相位;其次,相位编码条纹得到条纹级次后对外差相位展开;最后,由连续的外差相位对两个包裹相位进行展开,用最高频率的连续相位求得物体的相位信息。实验结果表明:RMS误差为0.038 mm。双频外差合成的周期不需要覆盖整个视场,打破了传统双频外差方法对频率选择的限制,可以用更高频率的条纹进行高精度测量。克服了相位主值误差对使用更高频率条纹进行高精度测量的局限性。     

Two-dimensional phase unwrapping using robust derivative estimationand adaptive integration

The adaptive integration (ADI) method for two-dimensional (2-D) phase unwrapping is presented. The method uses an algorithm for noise robust estimation of partial derivatives, followed by a noise robust adaptive integration process. The ADI method can easily unwrap phase images with moderate noise levels, and the resulting images are congruent modulo 2pi with the observed, wrapped, input images. In a quantitative evaluation, both the ADI and the BLS methods of Strand et al. (see ibid., vol.8, p.375-86, Mar. 1999) were better than the least-squares methods of Ghiglia and Romero (1994) (GR), and of Marroquin and Rivera (1995) (MRM). In a qualitative evaluation, the ADI, the BLS, and a conjugate gradient version of the MRM method (MRMCG), were all compared using a synthetic image with shear, using 115 magnetic resonance images, and using 22 fiber-optic interferometry images. For the synthetic image and the interferometry images, the ADI method gave consistently visually better results than the other methods. For the MR images, the MRMCG method was best, and the ADI method second best. The ADI method was less sensitive to the mask definition and the block size than the BLS method, and successfully unwrapped images with shears that were not marked in the masks. The computational requirements of the ADI method for images of nonrectangular objects were comparable to only two iterations of many least-squares-based methods (e.g., GR). We believe the ADI method provides a powerful addition to the ensemble of tools available for 2-D phase unwrapping.     

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。     

三维形貌测量中基于错位条纹的解相位方法

针对光栅投影法三维测量中的相位展开问题,提出一种用四步相移法求取折叠相位的解决方案.在相移图中,提取带有90°相移的4个单周期正弦条纹,对4个单周期正弦条纹进行编码组合,生成错位条纹.测量中,摄像机获取的错位条纹图像和相移图像按灰度值相减,得到4幅条纹差图,提取各差图中灰度值为零的条纹,依此确定错位条纹图像中各个单周期...     

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.     

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.     

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。     

相位展开的6种算法比较

相位展开算法主要可以分为路径跟踪算法和全局展开算法。为了比较这些算法的性能,采用计算机模拟斜面和高斯面,并加入不同程度噪声产生包裹相位;从两大类中各选取3种比较典型的算法分别对其进行相位展开,计算它们的均方根误差;最后对实验数据进行比较和理论分析。结果表明,基于网络规划的最小费用流算法和基于贝叶斯推断的ZπM算法是较为有效的算法,在实际应用中值得选择和改进。     

相位解包裹中欠采样问题的实验研究

为了解决相位解包裹时由于相位变化快而出现的欠采样问题,采用剪切干涉的原理建立光场,利用加权离散余弦变换算法求解泊松方程的方法,实现了对横向剪切最小二乘相位解包裹算法的改进。在理论分析的基础上给出了具体的相位解包裹算法,并通过实验研究验证了改进后算法的可行性。结果表明,改进后的算法能够有效地避免由相位变化过快带来的欠采样问题。