CN101369019A - Polarization interference synthetic aperture radar three-dimensional imaging method based on polarization data amalgamation - Google Patents

Abstract

Disclosed is a three-dimensional imaging method for a Polarimetric Interferometric Synthetic Aperture Radar (Pol-In-SAR) based on the polarimetric data integration, which belongs to the Polarimetric radar Interferometric measurement data processing technology field. The data is read in and two scattering vectors k1 and k2 are structured to each scattering unit; eigenvalue decomposition is performed on the matrix K1<H> K1-K2<H> K2 and three eigenvalues are obtained: Lambada1>0, Lambada1>0 and Lambada3=0; values of formula(1) can be respectively calculated so as to determine which condition the value belongs to; if in condition 1, w=w<(1)> can be calculated according to formula (6); if in condition 2, w=w<(2)> can be calculated according to formula (14); and then Phi[s]=arg(w<H>k1k2<H>w)can be calculated. An enhanced complex image can be obtained according to formula 3); the matrix Omega12=<K1k2<H>>is calculated; and then Phi[m]=arg(w<H>[Omega12]w) can be calculated according to formula 22). The flat-earth effect is removed on the obtained whole phase map (single view or multi-view); and phase unwrapping is performed by using branch-cut algorithm. The invention has wide application range, and the calculating time is smaller than that of the interference optimization method; and the integrated phase quality is better than that of interference optimization method, the phase quality can be always enhanced despite the original signal level, thereby establishing a fine basis for generating accurate digital elevation model (DEM) in the future.

Description

Polarization interference synthetic aperture radar three-dimensional imaging method based on the polarization data fusion

Technical field

The polarization interference synthetic aperture radar phase place that the present invention relates to radar system strengthens and three-D imaging method.

Background technology

Synthetic-aperture radar (Synthetic Aperture Radar, below represent with SAR) be a kind of round-the-clock, round-the-clock, high-resolution radar, by distance upwards to the linear FM signal of big time-bandwidth product carry out pulse compression and orientation to the echoed signal coherent accumulation obtain two-dimentional high-resolution image.

When flying height is that the Texas tower of H is during along rectilinear flight, towards the positive side launching beam of heading, as shown in Figure 1, the wave beam main lobe covers certain area on ground, motion along with platform, will form a mapping band, the direction of definition Texas tower flight be the orientation to, vertical with it direction be apart to, be with the distance of interior arbitrfary point target C to be called oblique distance to mapping radar antenna, wherein, during radar illumination, the bee-line that antenna is ordered to C is called the shortest oblique distance of this target.Radar is to the observation of point target C, as shown in Figure 2, and at T sometime ₀, radar beam upwards covers the scope of B～C in the orientation, and this moment, point target C just entered wave beam; Through T after a while _sAfter, wave beam upwards covers C～D section in the orientation, and the C point has just broken away from the wave beam irradiation, and Texas tower is at T _sThe distance of passing by in the time is L _sBe called a length of synthetic aperture, with T _sBe called a synthetic aperture time, at this moment between in, point target C is in radar beam irradiation down always, the geometric relationship of point target and radar site, as shown in Figure 3, wherein, definition β is the elevation angle, θ is the angle of squint.

Radar is transponder pulse earthward, after certain hour postpones, receive the echo of different scattering point reflections in the ground scene, to discrete sampling, the accumulation multiecho obtains data array, as shown in Figure 4 through distance, be parallel to the distance to a data represented range line, be parallel to the orientation to a data represented position line, wherein, the parallel orientation that distance constitutes to the sampling interval of sampling rate correspondence to the unit be called range unit.In Texas tower rectilinear motion process, same target constantly changes to the oblique distance of antenna, as shown in Figure 5, has caused the linear frequency modulation characteristic of instantaneous frequency of echo, make SAR can obtain the orientation to high resolving power.Since the orientation to distance to signal be linear FM signal, to SAR signal imaging process be exactly realize the orientation to the process of distance to two-dimentional pulse compression.

The SAR imaging system is compared with other imaging system such as optical imaging system, has the following advantages:

(1) SAR adopts active microwave imaging, has characteristics round-the-clock, the round-the-clock imaging;

(2) select suitable wavelength, utilize the penetrability of microwave, can be to target imaging under the region that covered by vegetation, desert, utmost point ice or shallow water even the face of land;

(3) resolution of SAR and radar operation wavelength, platform flying height, radar horizon have nothing to do, and can both effectively work at space or high-altitude;

(4) synthetic-aperture radar operating distance is far away, the mapping bandwidth;

(5) can realize atural object is carried out multiparameter, multiband, multipolarization and various visual angles mapping.

At present, the SAR system has been widely used in the key areas of national defence such as resource exploration, battle reconnaissance, environmental protection, the condition of a disaster detection, hydrogeology and national economy, and is bringing into play more and more important effect in the national economic development and military field.

Conventional SAR adopts the imaging of single channel to scene large tracts of land static target, two dimensional image can only be obtained, Three-dimension Target information can not be obtained, for this reason, developed interference synthetic aperture radar (Interferometric Synthetic ApertureRadar, below represent) with InSAR.InSAR has added SAR data and has obtained channel on conventional SAR basis, feasible twice observation to the ground Same Scene exists a visual angle difference, is information source by the phase information of extracting complex data, obtains the three-dimensional information and the change information on the face of land.The complex pattern that InSAR observes simultaneously by two slave antennas or twice approximate parallel observation obtains the same view in ground is right.Because the geometric relationship of target and aerial position has produced phase differential on complex pattern, formed interferometric phase line figure.The precise information that has comprised the difference of point that oblique distance makes progress and two aerial positions among the interferometric phase line figure.Therefore, utilize the geometric relationship between sensor height, radar wavelength, sighting distance line of vision and the antenna baseline, can measure the three-dimensional position and the change information of every bit on the image accurately, thereby obtain the 3-D view of sector of observation.At present, InSAR has been widely used in aspects such as topographical surveying, geodynamics application, glacier research, forest survey and drawing, marine charting.

In addition, conventional SAR adopts the emission of single polarization and receiving antenna to obtain data, can only obtain the one-component of scattering wave vector, and scattering properties that can not complete description target has been lost the target polarization information that is comprised in the echo to a considerable extent.Polarimetric synthetic aperture radar (Polarimetric Synthetic Aperture Radar, below represent with PolSAR) has remedied this deficiency of conventional SAR.Target has different scattering propertiess to different polarized electromagnetic waves.PolSAR launches by the antenna that uses different polarization and receives electromagnetic wave, obtain VV, HH, VH, the HV complex pattern of same observation scene, the scattering properties of measurement target under different POLARIZATION CHANNEL, and then obtain the multiple scattering matrix of target, the amplitude and the phase information of target scattering echo can be described fully.Because the electromagnetism echo of different POLARIZATION CHANNEL all there are differences on amplitude and phase place, the quantity of information that is contained in the multiple scattering matrix of target has surpassed the quantity of information that is contained during target radar scattering cross-section is long-pending greatly, therefore, PolSAR has improved the acquisition capability to target information greatly, thereby provides important evidence for furtheing investigate target scattering characteristics more.By Polarization scattering information processing to target, scattering matrix element statistical characteristic analysis for example, target scattering matrix decomposes and the target scattering constituent analysis, find the solution the optimum polarization problem of target etc., not only can analyze the geometrical property of scattering unit inside, and can adjust the antenna polarization state and make interested target obtain to strengthen.At present, polarization SAR is mainly used in the filtering and the sort research of image, landform face of land parameter obtain aspects such as crop analysis, waters observation, land resources utilization, environmental monitoring, lithology breakdown.

In the SAR interferometry, because there is decay in electromagnetic wave in communication process, and the thermonoise of noise on the travel path and noise jamming and transmit receive antenna disturbs, make signal to noise ratio (S/N ratio) very low, severe contamination interferometric phase line figure, cause the generation of a large amount of residual error points, make phase unwrapping comparatively difficult.Polarization interference synthetic aperture radar (PolarimetricInterferometric Synthetic Aperture Radar, below represent with PolInSAR) combine the advantage of PolSAR and InSAR, utilize the complex pattern of a plurality of POLARIZATION CHANNEL, by the complex pattern data fusion, strengthen phase information, improve the signal to noise ratio (S/N ratio) of data, effectively improved interferometric phase line plot quality, the residual error number of spots significantly reduces, and plays important effect for finally obtaining high-quality 3-D view.The gordian technique of PolInSAR three-dimensional imaging processing procedure is that phase place strengthens.In the PolInSAR imaging, the scattering unit packet of each pixel contains two polarization scattering matrix or Scattering of Vector, the antenna that separates corresponding to two radical spaces, therefore, polarization information just can be used for wild phase dryness and improve two phase places between the antenna receiving signal.In phase place enhancing process, polarization scattering matrix or Scattering of Vector with each scattering unit of PolInSAR, be divided into master and slave scattering matrix or Scattering of Vector, be optimized processing, can obtain high-quality interferometric phase line figure, obtain being better than the high-quality three-dimensional images of conventional InSAR then according to the InSAR three-D imaging method.At present, the research contents of PolInSAR mainly concentrates on height of tree inverting, the face of land deformation that the city changes, the mining area subsides, landslide earthquake and volcano etc. cause, aspects such as wave, ocean current motion.

S.R.Cloude in 1998 and K.P.Papathanassiou have delivered paper " Polarimetric SAR interferometry " at periodical IEEE Transactions on Geoscienceand Remote Sensing the 36th volume 1551-1565 page or leaf.Propose the relevant optimization method based on two vectors of interfering related coefficient to be optimized in the literary composition, this algorithm can optimization be interfered the coherence, can effectively improve the phase place quality, but still have more residual error point for the more weak zone of signal.

E.Colin in the POLinSAR 03 Proceedings meeting in 2003, C.Titin-schnaider and W.Tabbara have delivered paper " Investigation on different interferometric coherence optimizationmethods ".Introduced relevant optimization method in the literary composition, and provided analytic solution ideally based on the single vector of interfering related coefficient to be optimized.But, therefore can not well improve the phase place quality because actual conditions usually have bigger deviation with ideal situation.

Tao XIONG in November, the 2007 Progress in Electromagnetic Research Symposium meeting, Jian YANG, people such as Weijie ZHANG have delivered paper " Coherence Enhancement for Polarimetric SARInterferometry ".The phase place enhancement algorithms of the PolInSAR that optimizes based on amplitude has been discussed in the literary composition, the situation that interferometric phase improves under the haplopia situation mainly has been discussed.More original each the POLARIZATION CHANNEL phase place of the phase place that this algorithm obtains has some improvement, but owing to do not consider many visual informations, phase place still exists a large amount of noises, and does not provide the expressed intact form of analytic solution in the literary composition.

By to the USPTO of United States Patent (USP) trademark office, similar patent is not found in the retrieval of EPO of EUROPEAN PATENT OFFICE and the JPO of Jap.P. office.

Summary of the invention

The problem to be solved in the present invention provides a kind of polarization interference synthetic aperture radar three-dimensional imaging method that maximizes the method for interferometric phase enhancing based on signal amplitude.This method is utilized the relation between complex signal amplitude and the phase place, and promptly signal more by a small margin has insecure phase place, and the confidence level of phase place can be improved by the amplitude of maximum signal.PolInSAR provides two scattering matrixes or Scattering of Vector, for each pixel, two Polarization scattering vectors are simultaneously to same optimal direction projection, make two less signal amplitudes of projected length maximize, thereby realize that phase place strengthens, improve interferometric phase line figure, adopt the interference SAR formation method to handle then, obtain high-quality three-dimensional images.

Based on the polarization interference synthetic aperture radar three-dimensional imaging method that polarization data merges, this method step is as follows:

1. for the polarimetric synthetic aperture radar interferometry, for each scattering unit, two polarization scattering matrix [S1] and [S2] are arranged, perhaps two Scattering of Vector

$k_i=\frac{1}{\sqrt{2}}{(s_{\mathrm{HH}}+s_{\mathrm{VV}},s_{\mathrm{HH}}-s_{\mathrm{VV}},s_{\mathrm{HV}}+s_{\mathrm{VH}})}^T,i=\mathrm{1,2}---\left(1\right)$

Wherein ^TThe representing matrix matrix transpose operation, s _Ij(i, j=H or V) is illustrated in the multiple scattering coefficient of the j polarization mode emission down of HV polarization base, the reception of i polarization mode.Here only consider reciprocity situation, i.e. s _HV=s _VH

The coherence matrix [T] of definition 6 * 6:

$\left[T\right]=\&lang;\left[\begin{array}{c}{k}_1\\ k_2\end{array}\right]\left[\begin{array}{cc}{k}_1^H& k_2^H\end{array}\right]\&rang;=\left[\begin{array}{cc}\left[T_{11}\right]& \left[{\mathrm{\&Omega;}}_{12}\right]\\ \left[{\mathrm{\&Omega;}}_{21}\right]& \left[T_{22}\right]\end{array}\right]---\left(2\right)$

Wherein ^HThe expression complex-conjugate transpose.

2. for the scalar form is expanded to vector form, introduce a complex vector w of unit, so Polarization scattering vector k ₁And k ₂Projection η to w ₁And η ₂Be following two complex signals:

η ₁＝w ^Hk ₁，η ₂＝w ^Hk ₂ (3)

The purpose of this method is to find an optimum vector w to optimize η simultaneously ₁And η ₂Amplitude, promptly maximize η ₁And η ₂In less amplitude.If two amplitudes all become greatly, then interferometric phase will be enhanced.

Above-mentioned optimization problem can be described by following mathematical model:

$\underset{w}{\max }\left(\min (\left|w^H{k}_1\right|,\left|w^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|)\right)---\left(4\right)$

s.t.‖w‖＝1

In order to obtain the analytic solution of the problems referred to above, formula (4) is converted into following equivalence problem.

max(a，b)

$s.t.a=\underset{w}{\max }{\left|w^H{k}_1\right|}^2,\mathrm{if}{\left|w^H{k}_1\right|}^2\&le;{\left|w^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|}^2$

$b=\underset{w}{\max }{\left|{\mathrm{<figure-callout\; id="2"\; label="w\; H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">w</figure-callout>}}^H{k}_{}{}_2\right|}^2,\mathrm{if}{\left|w^H{k}_1\right|}^2>{\left|w^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|}^2$ (5)

‖w‖＝1

According to the quadratic programming theory, the two kinds of situations of existence of separating of formula (5).

3. for first kind of situation, unconfined maximal value | w ^Hk ₁| ²Perhaps | w ^Hk ₂| ²Be positioned at the constraint, then optimum vector w and k ₁And k ₂In to have that direction of smaller length identical:

$w^{\left(1\right)}=\left\{\begin{array}{c}{k}_1/\left|\right|k_1\left|\right|,\mathrm{if}\left|\right|k_1\left|\right|\&le;\left|\right|k_2\left|\right|\\ k_2/\left|\right|k_2\left|\right|,\mathrm{if}\left|\right|k_1\left|\right|>\left|\right|k_2\left|\right|\end{array}\right.---\left(6\right)$

4. for second kind of situation, unconfined maximal value | w ^Hk ₁| ²With | w ^Hk ₂| ²All not in the constraint.This moment, a and b were positioned on the border.Therefore when w was optimal projection direction, two projections had identical amplitude:

|w ^Hk ₁|＝|w ^Hk ₂| (7)

Utilize characteristic value decomposition to obtain analytic solution.Formula (7) can be write as

$w^H(k_1{k}_1^H-k_2{k}_2^H)w=0---\left(8\right)$

If k ₁=ck ₂(c is any non-zero complex) then will be classified as first kind of situation.

Work as k ₁≠ ck ₂The time, make [A] representing matrix

So the order of [A] is 2.[A] is an indefinite matrix in addition.Therefore it has three eigenvalue ₁0, λ ₂＜0 and λ ₃=0, the characteristic of correspondence vector is respectively v ₁, v ₂And v ₃

If w=v ₃, then meet formula (8) fully, because λ ₃=0.But by k ₁And k ₂The space of opening, i.e. Span{k ₁, k ₂And Span{v ₁, v ₂Be of equal value, and v ₃With v ₁And v ₂Quadrature, therefore | w ^Hk ₁|=| w ^Hk ₂|=0.This does not satisfy the target of formula (4).

Therefore w can represent to become v ₁And v ₂Linear combination.

w＝β(αv ₁+v ₂) (9)

Wherein α is a complex coefficient, and β is real normalization coefficient.With formula (9) substitution formula (8),

$w^H(k_1{k}_1^H-k_2{k}_2^H)w={\mathrm{\&beta;}}^2{({\mathrm{\&alpha;v}}_1+v_2)}^H({\mathrm{\&lambda;}}_1{v}_1{v}_1^H+{\mathrm{\&lambda;}}_2{v}_2{v}_2^H)({\mathrm{\&alpha;v}}_1+v_2)$

(10)

$={\mathrm{\&beta;}}^2({\left|\mathrm{\&alpha;}\right|}^2{v}_1^H{v}_1{v}_1^H{v}_1{\mathrm{\&lambda;}}_1+v_2^H{v}_2{v}_2^H{v}_2{\mathrm{\&lambda;}}_2)={\mathrm{\&beta;}}^2({\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}_2^4{\mathrm{\&lambda;}}_1+{\left|\right|v_2\left|\right|}_2^4{\mathrm{\&lambda;}}_2)=0$

Therefore

$\left|\mathrm{\&alpha;}\right|=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}\frac{{\left|\right|v_2\left|\right|}_2^2}{{\left|\right|v_1\left|\right|}_1^2}=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}---\left(11\right)$

With formula (9) substitution $w^H{k}_1{k}_1^{H}w,$

$w^H{k}_1{k}_1^{H}w={\mathrm{\&beta;}}^2{({\mathrm{\&alpha;v}}_1+v_2)}^H{k}_1{k}_1^H({\mathrm{\&alpha;v}}_1+v_2)$

$={\mathrm{\&beta;}}^2({\left|\mathrm{\&alpha;}\right|}^2{\left|v_1^H{k}_1\right|}^2+{\mathrm{\&alpha;v}}_2^H{k}_1{k}_1{k}_1^H{v}_1+{\left({\mathrm{\&alpha;v}}_2^H{k}_1{k}_1^H{v}_1\right)}^{*}+{\left|v_2^H{k}_1\right|}^2)---\left(12\right)$

By Cauchy inequality, the maximal value correspondence of formula (12) $\mathrm{\&alpha;}=d{v}_1^H{k}_1{k}_1^{\mathrm{<figure-callout\; id="2"\; label="H\; v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{v}_{}{}_2,$ Wherein d is a non-zero real.Therefore the argument of α is

$\arg \left(\mathrm{\&alpha;}\right)=\arg \left(v_1^H{k}_1{k}_1^H{v}_2\right)---\left(13\right)$

After with formula (11) and formula (13) substitution (9) and normalization, will obtain at last:

$w^{\left(2\right)}=\frac{{\mathrm{\&alpha;v}}_1+{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}{\left|\right|{\mathrm{\&alpha;v}}_1+{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|\right|},$ $\mathrm{\&alpha;}=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}\exp \{-j\arg \left(v_1^H{k}_1{k}_1^H{v}_2\right)\}---\left(14\right)$

For the w that derives ⁽¹⁾And w ⁽²⁾Rule of judgment, define two function f ₁(| α |) and f ₂(| α |), only contain a variable | α |

$f_i\left(\left|\mathrm{\&alpha;}\right|\right)=\frac{{\left|w^H{k}_i\right|}^2}{{\left|\right|w\left|\right|}^2}=\frac{{\left|{({\mathrm{\&alpha;v}}_1+v_2)}^H{k}_i\right|}^2}{{\left|\right|{\mathrm{\&alpha;v}}_1+{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|\right|}^2}=\frac{{\left|\mathrm{\&alpha;}\right|}^2{\left|v_1^H{k}_i\right|}^2+{\mathrm{\&alpha;}}^{*}{v}_1^H{k}_i{k}_i^{\mathrm{<figure-callout\; id="2"\; label="H\; v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{v}_{}{}_2+{\mathrm{\&alpha;v}}_2^H{k}_i{k}_i^H{v}_1+{\left|v_2^H{k}_i\right|}^2}{{\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}^2+{\mathrm{\&alpha;}}^{*}{v}_1^{\mathrm{<figure-callout\; id="2"\; label="H\; v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{v}_{}{}_2+{\mathrm{\&alpha;v}}_2^H{v}_1+{\left|\right|{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|\right|}^2}$

(15)

$=\frac{{(\left|\mathrm{\&alpha;}\right|\left|v_1^H{k}_i\right|+\left|v_2^H{k}_i\right|)}^2}{{\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}^2+{\left|\right|{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|\right|}^2},i=\mathrm{1,2}$

Order

$\frac{{\mathrm{df}}_i\left(\left|\mathrm{\&alpha;}\right|\right)}{d\left|\mathrm{\&alpha;}\right|}=d\frac{{(\left|\mathrm{\&alpha;}\right|\left|v_1^H{k}_i\right|+\left|v_2^H{k}_i\right|)}^2}{{\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}^2+{\left|\right|{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|\right|}^2}/d\left|\mathrm{\&alpha;}\right|=d\frac{z{\left|\mathrm{\&alpha;}\right|}^2+w\left|\mathrm{\&alpha;}\right|+u}{x{\left|\mathrm{\&alpha;}\right|}^2+y}/d\left|\mathrm{\&alpha;}\right|$

(16)

$=\frac{(2z\left|\mathrm{\&alpha;}\right|+w)(x{\left|\mathrm{\&alpha;}\right|}^2+y)-2\left|\mathrm{\&alpha;}\right|x(z{\left|\mathrm{\&alpha;}\right|}^2+w\left|\mathrm{\&alpha;}\right|+u)}{{(x{\left|\mathrm{\&alpha;}\right|}^2+y)}^2}=0$

So

wx|α| ²+2(xu-zy)|α|-wy＝0 (17)

| α | corresponding separate for

$\left|\mathrm{\&alpha;}\right|=\frac{2\mathrm{zy}}{\mathrm{wx}}=\frac{\left|v_1^H{k}_i\right|}{\left|v_2^H{k}_i\right|}>0---\left(18\right)$

Because f ₁And f ₂Monotonicity, exist following two kinds of situations (synoptic diagram is as shown in Figure 6):

If $\left|\mathrm{\&alpha;}\right|<\left|v_1^H{k}_1\right|/\left|v_2^H{k}_1\right|$ And $\left|\mathrm{\&alpha;}\right|<\left|v_1^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|/\left|v_2^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|,$ Then in the formula (5) a greater than b, and w=w ⁽¹⁾

If $\left|\mathrm{\&alpha;}\right|>\left|v_1^H{k}_1\right|/\left|v_2^H{k}_1\right|$ And $\left|\mathrm{\&alpha;}\right|>\left|v_1^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|/\left|v_2^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|,$ Then in the formula (5) b greater than a, and w=w ⁽¹⁾

Otherwise w=w ⁽²⁾

Therefore, under second kind of situation separate for

$w^{\left(2\right)}=\frac{{\mathrm{\&alpha;v}}_1+{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}{\left|\right|{\mathrm{\&alpha;v}}_1+{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\left|\right|},$ $\mathrm{\&alpha;}=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}\exp \{-j\arg \left(v_1^H{k}_1{k}_1^H{v}_2\right)\}---\left(19\right)$

λ wherein ₁0 and λ ₂The＜0th, matrix

Two nonzero eigenvalues, v ₁And v ₂It is the characteristic of correspondence vector.

5. in sum, modular form (4) is final separate for

$w=\left\{\begin{array}{c}{w}^{\left(2\right)},\mathrm{if}(\sqrt{\frac{-{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}>\max (\frac{\left|v_1^H{k}_1\right|}{\left|v_2^H{k}_1\right|},\frac{\left|v_1^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|}{\left|v_2^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|}))\mathrm{or}(\sqrt{\frac{{-\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}<\min (\frac{\left|v_1^H{k}_1\right|}{\left|v_2^H{k}_1\right|},\frac{\left|v_1^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|}{\left|v_2^{\mathrm{<figure-callout\; id="2"\; label="H\; k"\; filenames="A200810223729E00202.png"\; state="\{\{state\}\}">H</figure-callout>}}{k}_{}{}_2\right|}))\\ w^{\left(1\right)},\mathrm{else}\end{array}\right.---\left(20\right)$

6. after determining w, can calculate interferometric phase.Under the haplopia situation, the phase place of fusion is

${\mathrm{\&phi;}}_s=\arg \left({\mathrm{\&eta;}}_1{\mathrm{\&eta;}}_2^{*}\right)=\arg \left(w^H{k}_1{k}_2^{H}w\right)---\left(21\right)$

How according to circumstances down, the phase place of fusion is

${\mathrm{\&phi;}}_m=\arg ({\&lang;\mathrm{\&eta;}}_1{\mathrm{\&eta;}}_2^{*}\&rang;)=\arg \left(w^H\left[{\mathrm{\&Omega;}}_{12}\right]w\right)---\left(22\right)$

Compared with prior art, advantage of the present invention is as follows:

(1) its key is the amplitude of maximum signal, and this is based on the amplitude of complex signal and the relation between the phase place, and promptly under statistical significance, the amplitude of signal is big more, and then Dui Ying phase place is credible more.Fig. 7 has provided the relevant two-dimentional statistic histogram of signal amplitude and reflection phase place quality.Under the haplopia situation, the phase place of the similarity parameter of the phase place of enhancing and two Scattering of Vector is of equal value.It can think the weighted mean of complete polarization information, and weight is directly proportional with the signal amplitude of each passage.

(2) utilize the polarization interference data that the performance of improving the phase place quality is verified.Lack than the time of relevant optimization method the computing time of this method, and the phase place quality after merging is better than relevant optimization method, especially in the weak signal zone.

(3) for, utilize this method how according to circumstances, the residual error point above 99% in the interferometric phase is removed, and the detailed information that landform changes can be observed more clearly.This greatly reduces the difficulty of phase unwrapping, and will play important effect in the inverting of high accuracy number elevation model.

Description of drawings

Fig. 1 is a synthetic-aperture radar work synoptic diagram;

Fig. 2 is the observation synoptic diagram of synthetic-aperture radar to point target;

Fig. 3 is that the geometry of position of synthetic-aperture radar and point target concerns synoptic diagram;

The data array synoptic diagram that Fig. 4 obtains for the synthetic-aperture radar sampling;

Fig. 5 is the oblique distance variation synoptic diagram of synthetic-aperture radar to target;

Fig. 6 is the synoptic diagram of two kinds of situations of model solution;

Fig. 7 is the two dimension statistics Nogata synoptic diagram of relevant and amplitude;

Fig. 8 (a) is the amplitude synoptic diagram of vegetation and bare area zone HH passage;

(b) be the optical schematic diagram in vegetation and bare area zone;

Fig. 9 (a) is the HH channel amplitude synoptic diagram of mountain region test zone;

(b) be the optical schematic diagram of mountain region test zone;

(c) be the phase place synoptic diagram of the HH passage of mountain region test zone;

(d) the phase place synoptic diagram for obtaining by the amplitude optimization method;

Figure 10 (a) is the amplitude synoptic diagram of the HH passage of square frame A magnification region among Fig. 8 (a);

(b) be the phase place synoptic diagram of HH passage;

(c) the amplitude synoptic diagram for obtaining by the amplitude optimization method;

(d) the phase place synoptic diagram for obtaining by the amplitude optimization method;

(e) the relevant synoptic diagram that relevant optimization method obtains of serving as reasons;

(f) the phase place synoptic diagram that relevant optimization method obtains of serving as reasons;

Figure 11 (a) is the amplitude synoptic diagram of the HH passage of square frame B magnification region among Fig. 8 (a);

(b) be the phase place synoptic diagram of HH passage;

(c) be the residual error point synoptic diagram of HH passage;

(d) the phase place synoptic diagram for obtaining by the amplitude optimization method;

(e) the residual error point synoptic diagram for obtaining by the amplitude optimization method;

(f) the phase place synoptic diagram that relevant optimization method obtains of serving as reasons;

Figure 12 (a) is the phase place synoptic diagram of boxed area in the corresponding diagram 9 (a) that is obtained by the amplitude optimization method;

(b) the phase place synoptic diagram that relevant optimization method obtains of serving as reasons;

(c) the residual error point synoptic diagram for obtaining by the amplitude optimization method;

(d) the residual error point synoptic diagram that relevant optimization method obtains of serving as reasons;

Figure 13 (a) is the phase place synoptic diagram after the effect of level land of going in the vegetation area of Fig. 8 correspondence;

(b) twine the phase place synoptic diagram for separating of landform of reflection;

Figure 14 (a) is the phase place synoptic diagram after the effect of level land of going in the area, mountain region of Fig. 9 correspondence;

(b) twine the phase place synoptic diagram for separating of landform of reflection;

Figure 15 is the schematic flow sheet of invention.

The subordinate list explanation

The comparison in zone among table 1 Figure 10 (a).

The comparison in zone among table 2 Fig. 9 (a).

Embodiment

The invention will be further described below in conjunction with accompanying drawing.

Be the validity of checking this method, the present invention adopts the L-band complete polarization haplopia complex pattern data of SIR-C/X-SAR radar system that algorithm is verified.

These data are right in the L-band complete polarization haplopia complex pattern that is positioned at the Tianshan Mountains test zone of collection 8 to 9 October in 1994 by the SIR-C/X-SAR radar system.Test zone is positioned at the southeast bank in Muscovite Baikal, comprises different atural object, as the woods, cropland, bare area and mountain region.Before any pre-service, interferometric phase is produced a large amount of residual error points by the very noisy severe contamination.

Figure 15 is the schematic flow sheet of invention.In example, the vegetation in the data and bare area, mountain region are handled respectively, by with the comparative illustration of relevant best practice method universality of the present invention and validity.According to the above description, this method step is as follows:

(1) reads in the polarization interference data.

Shown in Fig. 8 (a) is the amplitude of the HH passage of vegetation and bare area test zone (1000 * 1000 pixel) | sHH| has comprised multiple different atural object, as the woods (F), road (R), bare area (BG) and cropland (C).The optical imagery of the equal resolution of the correspondence that is obtained by Google Maps is by shown in Fig. 8 (b).Shown in Fig. 9 (a) is a test zone that comprises the mountain region, and its corresponding optical imagery is by shown in Fig. 9 (b), and the source is Google Map equally.

(2), utilize the polarization interference data by formula (1) structure Scattering of Vector k for each scattering unit ₁And k ₂

(3) to matrix

Carry out characteristic value decomposition, obtain three eigenvalue ₁0, λ ₁0 and λ ₃=0 and the characteristic of correspondence vector v ₁, v ₂And v ₃

(4) calculate respectively And Value.According to formula (20) relatively

With

And

Between relation, judge any situation that belongs to of separating.If belong to first kind of situation, then calculate w=w according to formula (6) ⁽¹⁾If belong to second kind of situation, then calculate w=w according to formula (14) ⁽²⁾

(5) for the haplopia phase place after being enhanced, calculate according to formula (21) ${\mathrm{\&phi;}}_s=\arg \left(w^H{k}_1{k}_2^{H}w\right).$ And the complex pattern after being enhanced according to formula (3).

For vegetation and bare area, the enlarged drawing of square frame A is shown in Figure 10 (a) among Fig. 8 (a).The ground major part is covered by the woods, and three path link mistakes are wherein arranged.Black region is a bare area.Because the influence of noise and decoherence, the noise of HH passage is very serious, causes topographic details to be submerged (Figure 10 (b)).HH, HV and VV passage on average be respectively 0.3604,0.1401 and 0.2578 more by a small margin.Utilize this method, the right less average amplitude of the image after the fusion reaches 0.4768, and the amplitude of the middle η 1 of formula (3) is shown in Figure 10 (c).It obviously than the map of magnitudes of HH passage want " bright " some.

Shown in Figure 11 (a) is the magnification region of square frame B among Fig. 8 (a).Most of ground is covered by short vegetation such as crops and meadow.Wherein parallel straight line looks like ridge.The dihedral angle that forms between ridge and the ground causes strong echoed signal.The phase place of HH passage (shown in Figure 11 (b)) has 6550 residual error points to exist to such an extent as to noise is very big, as Figure 11 (c).

For the interferometric phase of mountain region HH passage shown in Fig. 9 (c), in 1000 * 1000 pixels, corresponding 95329 residual error points, so its density approaches 10%.With the information fusion of complete polarization passage, what image was right on average can increase to 0.4574 from 0.3250,0.1310 and 0.2831 of HH, HV and VV passage more by a small margin.

Because the haplopia phase place after strengthening still exists bigger noise, therefore do not consider here.

(6) for looking phase place, at first compute matrix after being enhanced more ${\mathrm{\&Omega;}}_{12}=\&lang;k_1{k}_2^H\&rang;,$ Calculate φ according to formula (22) then _m=arg (w ^H[Ω ₁₂] w).

For the regional A of vegetation and bare area, to merge back phase place (Figure 10 (d)) noise and obviously removed, the saltus step of phase place can distinguish clearly.99.76% residual error point is removed.After it should be noted that enhancing, the phase place in the woods and bare area zone does not have tangible border.The phase place that this means woods area can be considered to the ground phase place, because the phase place of bare area is affirmed corresponding landing ground phase place.

As a comparison, the phase place that is obtained by relevant optimization method is shown in Figure 10 (f).Though corresponding optimised 1 (shown in Figure 10 (e)) that approach of coherent value, and the most of noise in woods zone is removed, but the phase place of bare area still is subjected to The noise, and this does not conform to flatness observed by optical imagery and the bare area that radar image kind ground amplitude (primary scattering) is reflected.

Table 1 has been listed the comparison between more data that strengthen about raw data, by amplitude optimization method and relevant optimization method, comprises on average more by a small margin, and average coherence value and haplopia and many residual errors under are according to circumstances counted.

For the area B of vegetation and bare area, the phase place of utilizing the enhancing of amplitude optimization and relevant optimization method is respectively shown in Figure 11 (d) and 11 (f).Significantly, the phase place that is obtained by the amplitude optimization method has best quality and minimum noise.99.63% residual error point as Figure 11 (e), has been proved the advantage of this method by successful removal.Though the coherent value that relevant optimization method obtains is the highest, and approaches 1, corresponding phase place still is subjected to the influence of noise and decoherence.

For the mountain region, the phase place of fusion is shown in Fig. 9 (d).99.96% residual error point is removed, only remaining 34.Though for the mountain region with medium and strong signal, relevant optimization method can effectively be improved the phase place quality, for the level land with feeble signal, the phase place of fusion still exists a large amount of residual error points.Note the zone in the white edge among Fig. 9 (a), the right half part amplitude is less.The enhancing phase place that is obtained by amplitude optimization and relevant optimization is shown in Figure 12 (a) and 12 (b) respectively.Corresponding to the zone of low amplitude, there is a large amount of residual error points in the right half part of Figure 12 (d), and phase place corresponding among Figure 12 (b) is subjected to interference of noise.

Based on the amplitude of maximum signal, the phase place that is obtained by this method still is that the weak signal zone all exists seldom residual error point in strong signal area, shown in Figure 12 (c).This shows the robustness of this method.

More relatively list in the table 2.

(7) go the level land effect for resulting view picture phase diagram (haplopia or look) more.The interference fringe that highly identical level land forms in interferometric phase image with distance to the orientation to variation be periodically variable phenomenon and be called " level land effect " phenomenon." level land effect " makes interferometric phase image can not reflect intuitively that landform changes, and brought difficulty for simultaneously noise reduction and phase unwrapping, therefore must carry out " level land effect " and eliminate processing." level land effect " can be by removing interfering line to multiply by the complex phase bit function.

If baseline is apart from being B, wavelength is λ, and downwards angle of visibility is θ ₀, the angle of baseline and horizontal direction is α, and oblique distance is R, and 2 range differences to antenna of distance direction are Δ R, and then the phase differential between these 2 is

${\mathrm{\&Delta;\&phi;}}_R=-\frac{4\mathrm{\&pi;}}{\mathrm{\&lambda;}}\frac{B\cos ({\mathrm{\&theta;}}_0-\mathrm{\&alpha;})\mathrm{\&Delta;R}}{R\tan {\mathrm{\&theta;}}_0}$

The i.e. phase place that causes by the level land.Is reference phase with distance to certain point, calculates the level land phase in whole zone according to following formula _Flat, then go the phase place after the effect of level land to be

φ _{M, f}=W (φ _m-φ _Flat) or φ _{S, f}=W (φ _s-φ _Flat)

Figure 13 (a) and Figure 14 (a) have provided vegetation respectively and level land effect phase diagram is afterwards removed in bare area zone, zone, mountain region.

(8) to going the phase place after the effect of level land to carry out phase unwrapping, because the phase noise after strengthening reduced greatly, therefore corresponding irregularly count considerably lessly, therefore the branch cutting method of utilizing Goldstein to propose is carried out phase unwrapping.Roughly be divided into four steps: 1. the irregular point that calculates phase place; 2. the position according to irregular point is connected the branch tangent line with electric charge; 3. carry out the phase gradient integration; 4. carry out aftertreatment, the pixel on the branch tangent line is separated extort final result.

Figure 13 (b) and Figure 14 (b) have provided vegetation respectively and bare area zone, mountain region final the separating in zone twine phase diagram.As we can see from the figure, landform changes and can be reflected more clearly, and detailed information keeps finely.

The comparison in zone among table 1 Figure 10 (a)

	On average more by a small margin	The average coherence value	Residual error is counted
				The HH passage	0.3604	0.7843	8331
The HV passage	0.1401	0.7077	12033
				The VV passage	0.2578	0.7450	10279
The amplitude optimization, haplopia	0.4768	0.8523	1699
				Relevant optimization, haplopia	0.2795	0.8858	1889
The amplitude optimization is looked more	--------	0.8228	20
				Relevant optimization is looked more	--------	0.9634	173

The comparison in zone among table 2 Fig. 9 (a)

	On average more by a small margin	The average coherence value	Residual error is counted
				The HH passage	0.3250	0.7632	95329
The HV passage	0.1310	0.6974	125954
				The VV passage	0.2831	0.7536	100533
The amplitude optimization, haplopia	0.4574	0.8309	21851
				Relevant optimization, haplopia	0.2535	0.8623	24788
The amplitude optimization is looked more	--------	0.7998	34
				Relevant optimization is looked more	--------	0.9551	2078

Claims (1)

1. based on the polarization interference synthetic aperture radar three-dimensional imaging method of polarization data fusion, it is characterized in that this method step is as follows:

(1) for the polarimetric synthetic aperture radar interferometry, each scattering unit has two polarization scattering matrix [S ₁] and [S ₂], perhaps two Scattering of Vector

$k_i=\frac{1}{\sqrt{2}}{[s_{\mathrm{HH}}+s_{\mathrm{VV}},s_{\mathrm{HH}}-s_{\mathrm{VV}},s_{\mathrm{HV}}+s_{\mathrm{VH}}]}^T,i=\mathrm{1,2}---1)$

Wherein, ^TThe representing matrix matrix transpose operation, s _Ij(i, j=H or V) is illustrated in the multiple scattering coefficient of the j polarization mode emission down of HV polarization base, the reception of i polarization mode; Here we only consider reciprocity situation, i.e. s _HV=s _VH,

The coherence matrix [T] of definition 6 * 6:

$\left[T\right]=\&lang;\left[\begin{array}{c}{k}_1\\ k_2\end{array}\right]\left[\begin{array}{cc}{k}_1^H& k_2^H\end{array}\right]\&rang;=\left[\begin{array}{cc}\left[T_{11}\right]& \left[{\mathrm{\&Omega;}}_{12}\right]\\ \left[{\mathrm{\&Omega;}}_{21}\right]& \left[T_{22}\right]\end{array}\right]---2)$

Wherein ^HThe expression complex-conjugate transpose;

(2) introduce the complex vector w of unit, Polarization scattering vector k ₁And k ₂Projection η to w ₁And η ₂Be following two complex signals:

η ₁＝w ^Hk ₁，η ₂w ^Hk ₂ 3)

Set up mathematical model:

$\underset{w}{\max }\left(\min (\left|w^H{k}_1\right|,\left|w^H{k}_2\right|)\right)$

$\begin{array}{cc}s.t.& \left|\right|w\left|\right|=1\end{array}$ 4)

In order to obtain the analytic solution of the problems referred to above, with formula 4) transform as follows:

$\max (a,b)$

$s.t.a=\underset{w}{\max }{\left|w^H{k}_1\right|}^2,\mathrm{if}{\left|w^H{k}_1\right|}^2\&le;{\left|w^H{k}_2\right|}^2$

5)

$b=\underset{w}{\max }{\left|w^H{k}_2\right|}^2,\mathrm{if}{\left|w^H{k}_1\right|}^2>{\left|w^H{k}_2\right|}^2$

$\left|\right|w\left|\right|=1$

According to the quadratic programming theory, formula 5) separate the two kinds of situations that exist:

(2.1) for first kind of situation, unconfined maximal value | w ^Hk ₁| ²Perhaps | w ^Hk ₂| ²Be positioned at the constraint, then optimum vector w and k ₁And k ₂In to have that direction of smaller length identical:

$w^{\left(1\right)}=\left\{\begin{array}{c}{k}_1/\left|\right|k_1\left|\right|,\mathrm{if}\left|\right|k_1\left|\right|\&le;\left|\right|k_2\left|\right|\\ k_2/\left|\right|k_2\left|\right|,\mathrm{if}\left|\right|k_1\left|\right|>\left|\right|k_2\left|\right|\end{array}\right.---6)$

(2.2) for second kind of situation, unconfined maximal value | w ^Hk ₁| ²With | w ^Hk ₂| ²All not in the constraint; This moment, a and b were positioned on the border; Therefore when w was optimal projection direction, two projections had identical amplitude:

|w ^Hk ₁|＝|w ^Hk ₂| 7)

Utilize characteristic value decomposition to obtain analytic solution, formula 7) write as

$w^H(k_1{k}_1^H-k_2{k}_2^H)w=0---8)$

If k ₁=ck ₂, c is any non-zero complex, then will be classified as first kind of situation;

Work as k ₁≠ ck ₂The time, make [A] representing matrix So the order of [A] is 2; [A] is an indefinite matrix in addition; Therefore it has three eigenvalue ₁0, λ ₂＜0 and λ ₃=0, the characteristic of correspondence vector is respectively v ₁, v ₂And v ₃

If w=v ₃, then meet formula 8 fully) because λ ₃=0; But by k ₁And k ₂The space of opening, i.e. Span{k ₁, k ₂And Span{v ₁, v ₂Be of equal value, and v ₃With v ₁And v ₂Quadrature, therefore | w ^Hk ₁|=| w ^Hk ₂|=0; This does not satisfy formula 4) target;

Therefore w represents to become v ₁And v ₂Linear combination be

w＝β(αv ₁+v ₂) 9)

Wherein, a is a complex coefficient, and β is real normalization coefficient; With formula 9) substitution formula 8),

$w^H(k_1{k}_1^H-k_2{k}_2^H)w={\mathrm{\&beta;}}^2{(\mathrm{\&alpha;}{v}_1+v_2)}^H({\mathrm{\&lambda;}}_1{v}_1{v}_1^H+{\mathrm{\&lambda;}}_2{v}_2{v}_2^H)(\mathrm{\&alpha;}{v}_1+v_2)$

10)

$={\mathrm{\&beta;}}^2({\left|\mathrm{\&alpha;}\right|}^2{v}_1^H{v}_1{v}_1^H{v}_1{\mathrm{\&lambda;}}_1+v_2^H{v}_2{v}_2^H{v}_2{\mathrm{\&lambda;}}_2)={\mathrm{\&beta;}}^2({\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}_2^4{\mathrm{\&lambda;}}_1+{\left|\right|v_2\left|\right|}_2^4{\mathrm{\&lambda;}}_2)=0$

Therefore

$\left|\mathrm{\&alpha;}\right|=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}\frac{{\left|\right|v_2\left|\right|}_2^2}{{\left|\right|v_1\left|\right|}_1^2}=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}---11)$

With formula 9) substitution

$w^H{k}_1{k}_1^{H}w={\mathrm{\&beta;}}^2{(\mathrm{\&alpha;}{v}_1+v_2)}^H{k}_1{k}_1^H(\mathrm{\&alpha;}{v}_1+v_2)$

$={\mathrm{\&beta;}}^2({\left|\mathrm{\&alpha;}\right|}^2{\left|v_1^H{k}_1\right|}^2+\mathrm{\&alpha;}{v}_2^H{k}_1{k}_1^H{v}_1+{\left(\mathrm{\&alpha;}{v}_2^H{k}_1{k}_1^H{v}_1\right)}^{*}+{\left|v_2^H{k}_1\right|}^2)---12)$

By Cauchy inequality, formula 12) maximal value correspondence $\mathrm{\&alpha;}=d{v}_1^H{k}_1{k}_1^H{v}_2,$ Wherein d is a non-zero real; Therefore the argument of a is

$\arg \left(\mathrm{\&alpha;}\right)=\arg \left(v_1^H{k}_1{k}_1^H{v}_2\right)---13)$

With formula 11) and formula 13) substitution 9) and normalization after, will obtain at last:

$w^{\left(2\right)}=\frac{\mathrm{\&alpha;}{v}_1+v_2}{||\mathrm{\&alpha;}{v}_1+v_2||},\mathrm{\&alpha;}=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}\exp \{-j\arg \left({v_1}^H{k}_1{k_1}^H{v}_2\right)\}---14)$

For the w that derives ⁽¹⁾And w ⁽²⁾Rule of judgment, define two function f ₁(| α |) and f ₂(| α |), only contain a variable | α |

$f_i\left(\left|\mathrm{\&alpha;}\right|\right)=\frac{{\left|w^H{k}_i\right|}^2}{{\left|\right|w\left|\right|}^2}=\frac{{\left|{({\mathrm{\&alpha;v}}_1+v_2)}^H{k}_i\right|}^2}{{\left|\right|{\mathrm{\&alpha;v}}_1+v_2\left|\right|}^2}=\frac{{\left|\mathrm{\&alpha;}\right|}^2{\left|v_1^H{k}_i\right|}^2+{\mathrm{\&alpha;}}^{*}{v}_1^H{k}_i{k}_i^H{v}_2+{\mathrm{\&alpha;v}}_2^H{k}_i{k}_i^H{v}_1+{\left|v_2^H{k}_i\right|}^2}{{\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}^2+{\mathrm{\&alpha;}}^{*}{v}_1^H{v}_2+{\mathrm{\&alpha;v}}_2^H{v}_1+{\left|\right|v_2\left|\right|}^2}---15)$

$=\frac{{(\left|\mathrm{\&alpha;}\right|\left|v_1^H{k}_i\right|+\left|v_2^H{k}_i\right|)}^2}{{\left|\mathrm{\&alpha;}\right|}^2{\left|\right|v_1\left|\right|}^2+{\left|\right|v_2\left|\right|}^2},i=\mathrm{1,2}$

Order

$\frac{{\mathrm{df}}_i\left(\left|\mathrm{\&alpha;}\right|\right)}{d\left|\mathrm{\&alpha;}\right|}=d\frac{{(\left|\mathrm{\&alpha;}\right|\left|{v_1}^H{k}_i\right|+\left|{v_2}^H{k}_i\right|)}^2}{{\left|\mathrm{\&alpha;}\right|}^2{||v_1||}^2+{||v_2||}^2}/d\left|\mathrm{\&alpha;}\right|=d\frac{z{\left|\mathrm{\&alpha;}\right|}^2+w\left|\mathrm{\&alpha;}\right|+u}{x{\left|\mathrm{\&alpha;}\right|}^2+y}/d\left|\mathrm{\&alpha;}\right|$

$=\frac{(2z\left|\mathrm{\&alpha;}\right|+w)(x{\left|\mathrm{\&alpha;}\right|}^2+y)-2\left|\mathrm{\&alpha;}\right|x(z{\left|\mathrm{\&alpha;}\right|}^2+w\left|\mathrm{\&alpha;}\right|+u)}{{(x{\left|\mathrm{\&alpha;}\right|}^2+y)}^2}=0---16)$

So

wx|α| ²+2(xu-zy)|α|-wy＝0 17)

| α | corresponding separate for

$\left|\mathrm{\&alpha;}\right|=\frac{2\mathrm{zy}}{\mathrm{wx}}=\frac{\left|v_1^H{k}_i\right|}{\left|v_2^H{k}_i\right|}>0---18)$

Because f ₁And f ₂Monotonicity, exist following two kinds of situations:

If $\left|\mathrm{\&alpha;}\right|<\left|v_1^H{k}_1\right|/\left|v_2^H{k}_1\right|$ And $\left|\mathrm{\&alpha;}\right|<\left|v_1^H{k}_2\right|/\left|v_2^H{k}_2\right|,$ Then in the formula (5) a greater than b, and w=w ⁽¹⁾,

If $\left|\mathrm{\&alpha;}\right|>\left|v_1^H{k}_1\right|/\left|v_2^H{k}_1\right|$ And $\left|\mathrm{\&alpha;}\right|>\left|v_1^H{k}_2\right|/\left|v_2^H{k}_2\right|,$ Then in the formula (5) b greater than a, and w=w ⁽¹⁾

Otherwise w=w ⁽²⁾

Therefore, under second kind of situation separate for

$w^{\left(2\right)}=\frac{\mathrm{\&alpha;}{v}_1+v_2}{\left|\right|\mathrm{\&alpha;}{v}_1+v_2\left|\right|},\mathrm{\&alpha;}=\sqrt{-\frac{{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}\exp \{-j\arg \left(v_1^H{k}_1{k}_1^H{v}_2\right)\}---19)$

λ wherein ₁0 and λ ₂The＜0th, matrix

Two nonzero eigenvalues, v ₁And v ₂It is the characteristic of correspondence vector;

(3) therefore, modular form 4) final separate for

$w=\left\{\begin{array}{c}{w}^{\left(2\right)},\mathrm{if}(\sqrt{\frac{-{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}>\max (\frac{\left|v_1^H{k}_1\right|}{\left|v_2^H{k}_1\right|},\frac{\left|v_1^H{k}_2\right|}{\left|v_2^H{k}_2\right|}))\mathrm{or}(\sqrt{\frac{-{\mathrm{\&lambda;}}_2}{{\mathrm{\&lambda;}}_1}}<\min (\frac{\left|v_1^H{k}_1\right|}{\left|v_2^H{k}_1\right|},\frac{\left|v_1^H{k}_2\right|}{\left|v_2^H{k}_2\right|}))\\ w^{\left(1\right)},\mathrm{else}\end{array}\right.---20)$

(4) determine w after, can calculate interferometric phase;

Under the haplopia situation, the phase place of fusion is

${\mathrm{\&phi;}}_s=\arg \left({\mathrm{\&eta;}}_1{\mathrm{\&eta;}}_2^{*}\right)=\arg \left(w^H{k}_1{k}_2^{H}w\right)---21)$

How according to circumstances down, the phase place of fusion is

${\mathrm{\&phi;}}_m=\arg (\&lang;{\mathrm{\&eta;}}_1{\mathrm{\&eta;}}_2^{*}\&rang;)=\arg \left(w^H\left[{\mathrm{\&Omega;}}_{12}\right]w\right)---22)$

Images