﻿ 不同质量图在卡尔曼滤波相位解缠中应用探讨 The Study on Application of Different Quality Maps in Kalman Filter Phase Unwrapping

Geomatics Science and Technology
Vol. 07  No. 02 ( 2019 ), Article ID: 29542 , 9 pages
10.12677/GST.2019.72011

The Study on Application of Different Quality Maps in Kalman Filter Phase Unwrapping

Man Yan, Chunhua Guo

Media Collage, Weifang University, Weifang Shandong

Received: Mar. 10th, 2019; accepted: Mar. 25th, 2019; published: Apr. 1st, 2019

ABSTRACT

The Kalman Filter transforms the phase unwrapping problem into state estimate issue to deal with phase unwrapping and noise eliminating at the same time. The true phase is obtained through establishment of the dynamic equation and observation equation, noise variance of observation equation is determined by the value of coherence image in the original method. In this paper, noise variance is determined by three kinds of different quality maps, then the Extended Kalman Filter phase unwrapping is implemented, using InSAR data to do the experiment in the steep and flat terrain. Through comparing and analyzing the phase unwrapping results, it is verified that the result using phase derivative variance map is more reliable in the steep terrain, but in the flat terrain, the result using pseudo-coherence map is more precise.

Keywords:Phase Unwrapping, Quality Map, Kalman Filter, InSAR

1. 引言

2. 三种质量图

2.1. 伪相干图

$|{z}_{m,n}|=\frac{\sqrt{{\left(\sum \mathrm{cos}{\psi }_{i,j}\right)}^{2}+{\left(\sum \mathrm{sin}{\psi }_{i,j}\right)}^{2}}}{{k}^{2}}$ (1)

2.2. 相位导数方差图

$|{r}_{m,n}|=\frac{\sqrt{\sum {\left({\Delta }_{i,j}^{x}-{\overline{\Delta }}_{m,n}^{x}\right)}^{2}}+\sqrt{{\sum \left({\Delta }_{i,j}^{y}-{\overline{\Delta }}_{m,n}^{y}\right)}^{2}}}{{k}^{2}}$ (2)

2.3. 最大相位梯度图

$|{q}_{m,n}|=\mathrm{max}\left\{\mathrm{max}\left\{|{\Delta }_{i,j}^{x}|\right\},\mathrm{max}\left\{|{\Delta }_{i,j}^{y}|\right\}\right\}$ (3)

3. 卡尔曼滤波相位解缠基本原理

$y\left(k\right)=\left[\begin{array}{c}\mathrm{Re}\left\{\frac{z\left(k\right)}{a\left(k\right)}\right\}\\ \mathrm{Im}\left\{\frac{z\left(k\right)}{a\left(k\right)}\right\}\end{array}\right]\triangleq \left[\begin{array}{c}\mathrm{cos}\left(\phi \left(k\right)\right)\\ \mathrm{sin}\left(\phi \left(k\right)\right)\end{array}\right]+\left[\begin{array}{c}{v}_{1}\left(k\right)\\ {v}_{2}\left(k\right)\end{array}\right]=h\left(\phi \left(k\right)\right)+v\left(k\right)$ (4)

$E\left\{v\left(k\right)\right\}=0;E\left\{v\left(k\right)v{\left(j\right)}^{T}\right\}=\underset{i=1,2}{diag}\left[\frac{1}{2{|\gamma |}^{2}}-\frac{1}{2}\right]\delta \left(k,j\right)$ (5)

$\left\{\begin{array}{l}x\left(k+1\right)=Ax\left(k\right)+u\left(k\right)\\ u\left(k\right)=\stackrel{^}{u}\left(k\right)+w\left(k\right)\end{array}$ (6)

$\left\{\begin{array}{l}{\stackrel{^}{x}}_{k+1,k}=A{\stackrel{^}{x}}_{k,k}+{\stackrel{^}{u}}_{k,k}\\ {P}_{k+1,k}=A{P}_{k,k}{A}^{T}+{Q}_{k,k}\\ {\stackrel{^}{x}}_{k+1,k+1}={\stackrel{^}{x}}_{k+1,k}+{J}_{k+1}{r}_{k+1,k+1}\\ {P}_{k+1,k+1}=\left(I-{J}_{k+1}{C}_{k+1,k}\right){P}_{k+1,k}\\ {J}_{k+1}={P}_{k+1,k}{C}_{k+1,k}^{T}{\left({C}_{k+1,k}{P}_{k+1,k}{C}_{k+1,k}^{T}+{R}_{k+1,k+1}\right)}^{-1}\\ {r}_{k+1,k+1}={y}_{k+1,k+1}-{C}_{k+1,k}{\stackrel{^}{x}}_{k+1,k}\\ {C}_{k+1,k}={\left[-\mathrm{sin}\left({\stackrel{^}{x}}_{k+1,k}\right),\mathrm{cos}\left({\stackrel{^}{x}}_{k+1,k}\right)\right]}^{T}\end{array}$ (7)

4. 实验结果与分析

4.1. 地形陡峭条件下的实验结果与分析

Figure 1. Interferogram

Figure 2. Phase derivative variance unwrapped result

Figure 3. Pseudo-coherence coefficient unwrapped result

Figure 4. Maximum phase gradient unwrapped result

(a) (b) (c)

Figure 5. (a) Phase derivate variance map; (b) Pseudo coherence map; (c) Maximum phase gradient map

(a) (b) (c)

Figure 6. (a) Phase derivative variance discontinuous points map; (b) Pseudo-coherence coefficient discontinuous points map; (c) Maximum phase gradient discontinuous points map

$\epsilon =\frac{1}{MN}\sum _{i=0}^{M-2}\sum _{j=0}^{N-1}{\omega }_{i,j}^{x}{|{\varphi }_{i+1,j}-{\varphi }_{i,j}-{\Delta }_{i,j}^{x}|}^{p}+\frac{1}{MN}\sum _{i=0}^{M-1}\sum _{j=0}^{N-2}{\omega }_{i,j}^{y}{|{\varphi }_{i,j+1}-{\varphi }_{i,j}-{\Delta }_{i,j}^{x}|}^{p}$ (8)

Table 1. Three quality maps phase unwrapping results’ number of discontinuous points and ε value

4.2. 地形平坦条件下的实验结果与分析

Figure 7. Interferogram

Figure 8. Phase derivative variance unwrapped result

Figure 9. Pseudo-coherence coefficient unwrapped result

Figure 10. Maximum phase gradient unwrapped result

(a) (b) (c)

Figure 11. (a) Phase derivate variance map; (b) Pseudo coherence map; (c) Maximum phase gradient map

(a) (b) (c)

Figure 12. (a) Phase derivative variance discontinuous points map; (b) Pseudo-coherence coefficient discontinuous points map; (c) Maximum phase gradient discontinuous points map

Table 2. Three quality maps phase unwrapping results’ number of discontinuous points and ε value

5. 结论

The Study on Application of Different Quality Maps in Kalman Filter Phase Unwrapping[J]. 测绘科学技术, 2019, 07(02): 65-73. https://doi.org/10.12677/GST.2019.72011

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