1、 中文 3300 字 附录 A：英文原文 Least squares phase unwrapping in wavelet domain Abstract: 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 method
2、s have been applied to improve this condition. In this paper, 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 wavel
3、et 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. 1 introduction Two-dimensional phase unwrapping is an important processing step in some coherent imaging a
4、pplications, such as synthetic aperture radar interferometry(InSAR) and magnetic resonance imaging(MRI).In these processes, three-dimensional information of the measured objects can be extracted from the phase of the sensed signals ,However, the obseryed phase data are wrapped principal values, whic
5、h are restricted in a 2 modulus ,and they must be unwrapped to their true absolute phase values .This is the task of the phase unwrapping, especially for two-dimensional problems. The basic assumption of the general phase unwrapping methods is that the discrete derivatives of the unwrapped phase at
6、all grid points are less than in absolute value .With this assumption satisfied ,the absolute phase can be reconstructed perfectly by integrating the partial derivatives of the wrapped phase data. In the general case, however, it is not possible to recover unambiguously the absolute phase from the m
7、easured wrapped phase which is usually corrupted by noise or aliasing effects such as shadow, layover, etc. In such cases, the basic assumption is violated and the simple integration procedure cannot be applied owing to the phase inconsistencies caused by the contaminations. After Goldstein-et al in
8、troduced the concept of residues in the two-dimensional phase unwrapping problem of InSAR, many phase unwrapping approaches to cope with this problem have been investigated. Path-following (or integration-based) methods and least squares methods are the most representative two basic classes in this
9、field. There have also been some other approaches such as Green methods, Bayesian regularization methods ,image processing-based methods, and model-based methods. Least squares phase unwrapping ,established by Ghiglia and Romero, is one of the most robust techniques to solve the two-dimensional phas
10、e unwrapping problem. This method obtains an unwrapped solution by minimizing the differences between the partial derivatives of the wrapped phase data and the unwrapped solution .Least squares method is divided into unweighted and weighted least squares phase unwrapping. To isolate the phase incons
11、istencies, a weighted least squares method should be used, which depresses the contamination effects by using the weighting arrays. Green methods and Bayesian methods are also based on the least squares scheme .But these methods are different from those of ,in the concept of phase inconsistency trea
12、tment. Thus, this paper concerns only the least squares phase unwrapping problem of Ghiglias category. The least squares method is well-defined mathematically and equivalent to the solution of Poissons partial differential equation, which can be expressed as a sparse linear equation. anterior method
13、 is usually used to solve this large linear equation. However, a large computation time is required and therefore improving the convergence rate is a very important task when using this method. Some numerical algorithms have been applied to this problem to improve convergence conditions. An approach
14、 for fast convergence of a sparse linear equation is to transfer the original equation system into a new system with larger supports .Multiresolution or hierarchical representation concepts have often been used for this purpose. Recently, wavelet transform has been investigated deeply in science and
15、 engineering fields as a sophisticated tool for the multiresolution analysis of signals and systems. It decomposes a signal space into its low-resolution subspace and the complementary detail subspaces. In our method, the discrete wavelet transform is applied to the linear system of least squares ph
16、ase unwrapping problem to represent the original system in separate multiresolution spaces .In this new transferred system, a better convergence condition can be achieved. This method was briefly introduced in out previous work ,where the proposed method was applied only to the unweighted problem, I
17、n this paper, this new method is extended to the weighted least squares problem. Also, a full description of the proposed method is given here. 2 Weighted least squares phase unwrapping: a review Least squares phase obtains an unwrapped solution by minimizing the 2L -norm between the discrete partia
18、l derivatives of the wrapped phase data and those of the unwrapped solution function. Given the wrapped phase ,ij on an MN rectangular grid( 01iM , 01jN ),the partial derivatives of the wrapped phase are defined as , 1 , ,xi j i j i jW , , , 1 ,yi j i j i jW (1) Where W is the wrapping operator that
19、 wraps the phase into the interval , .The differences between the partial derivatives of the solution ,ij and those in (1) can be minimized in the weighted least squares sense, by differentiating the sum 22, 1 , , , , , 1 , ,x x y yi j i j i j i j i j i j i j i ji j i jww (2) With respect to ,ij and
20、 setting the result to zero. In (2),the gradient weights ,xijw and ,yijw,are used to prevent some phase values corrupted by noise or aliasing from degrading the unwrapping , and are defined by 22, 1 , ,m i n ,x i j i j i jw w w , 22, , 1 ,m i n ,y i j i j i jw w w , ,01ijw (3) The weighted least squ
21、ares phase unwrapping problem is to find the solution ,ij that minimizes the sum of (2).The initial weight array ,ijw is user-defined and some methods for defining these weights are presented in 1,11. When all the weights , 1ijw , the above equation is the unweighted phase unwrapping problem. Since
22、weight array is related to the exactitude of the resultant unwrapped solution , it must be defined properly. In this paper, however, it is assumed that the weight array is defined already for the given phase data and how to define it is not covered here. Only the convergence rates issue of the weigh
23、ted least squares phase unwrapping problem is considered here. The least squares solution to this problem yields the following equation: , 1 , , 1 , , 1 , , , 1 , , 1 , , 1 ,x x y yi j i j i j i j i j i j i j i j i j i j i j i j i jw w w w (4) Where ,ij is the weighted phase Laplacian defined by , ,
24、 , 1 , 1 , , , , 1 , 1x x x x x x x xi j i j i j i j i j i j i j i j i jw w w w (5) The unwrapped solution ,ij is obtained by iteratively solving the following equation , , 1 , 1 , 1 , , , 1 , 1 , 1 , , 1 , , , 1/x x y y x x y yi j i j i j i j i j i j i j i j i j i j i j i j i j i jw w w w w w w w (6) Equation (4) is the weighted and discrete version of the Poissons partial differential equation (PDE), 2.By concatenating all the nodal variables ,ij into MN1 one column vector , the above equation is expressed as a
