The relationship between observed scene and the data obtained by radar polarimetry is shown in this figure. We assume that the target (or scene) is consisted of scattering points with different polarization properties. And at the each resolution cell, vector sum of the polarization vectors of the scattering points contained in the cell is observed. So the profile of the observed polarization vectors would look like this. If we separately express the Range Profiles of HH, HV and VV channel , it would look like this. I have explained three things here. The polarimetric property of a radar target can be expressed by “Polarization Vector”. Radar Polarimetry is a technique to observe the polarization vector. By Radar Polarimetry, Range profiles of HH, HV, VV channels are obtained and the relationship between this data and polarization vector and the observed scene is shown in this figure. OK? Now let’s move on to the main topic.
Let me start with step 4 first. The Polarimetric Linear Prediction Model assumes that the linear combination of a set of adjacent signal samples of the three polarimetric channels will predict the next sample. As you can see from this figure, a sample outside the transmit bandwidth of HH channel is predicted and extrapolated by the linear combination of these 6 samples of HH channel and VV channel. Here these C’s are the weights or coefficients used to predict the HH channel sample. Please note that I omitted HV channel in this figure to avoid complexity. Then, sample outside the transmit bandwidth of VV channel also is predicted and extrapolated by the linear combination of these 6 samples. And these C’s are the coefficients used to predict the VV channel sample. I guess you probably already noticed that Conventional BWE uses only this upper half for HH channel and this lower half for VV channel. And that’s the difference between BWE and PBWE.
Then step 3 is summarized in this slide. Here we want to determine these C’s, the linear prediction coefficients. Basically the coefficients are set to minimize the mean power of the prediction error “N” shown here. Let this red point a predicted value using these 6 samples and these 6 coefficients. Then the prediction error is defined as the difference between the real or observed value here and this red predicted value. We set these C’s which minimize the mean power of this error. This can be done by simple least squares method shown here. This is actually a vectorized expression of these coefficients. But I’m not going to go into the detail of this, today.
For your intuitive understanding, I’ll give two simple examples. In the first example, a flat plate and a dihedral corner reflector are placed close to each other, and in the second case two flat plates are placed. These targets are observed using a polarimetric radar with bandwidth “B.” And we assume that distance of these two targets are 30% of the range resolution C over 2B. Please note that the flat plate and the dihedral corner reflector have different polarization properties.
This is the range profile obtained for the fist case. Upper Low shows the HH channel profile and Lower shows the VV channel. Two targets are placed here and here. These Blue plots show the Original Range Profile. You can clearly see that the two targets are not resolved in HH channel. And in VV channel peaks are not in the right place. These Purple plots show the Range Profile obtained by the conventional BWE. Again two targets are not resolved in HH channel and in VV channel, two targets are “kind of” resolved. But it looks weird. And the peak level is not right. These red plots show the result obtained by PBWE. The two targets are well resolved in both HH and VV channels.
This is the result for the second case. The two flat plates are not resolved by any of these three. PBWE does not seem to show any advantage over BWE for this case.
Then the question comes “how different the polarization properties of the two targets should be for the PBWE to perform better than BWE” To answer this question a Numerical Simulation has been done. We assume that two point targets are placed close to each other. These yellow and pink arrows express the polarization vectors of the two targets. “PHI” represents the difference of polarization properties of the two targets, and the Second parameter “d” represents the distance between the targets. And the intensities of the two targets are assumed to be the same. The performance was evaluated by the criterion illustrated in this figure. It is the “Notch Depth” between the two peaks in Range Profile corresponding to the two targets. Clearly, when two targets are not resolved, Notch Depth is 0dB, and when the targets are resolved, certain depth of Notch will appear between the peaks. And we will refer to the minimum distance for which Notch Depth is Deeper than –3dB as Minimum Separable Distance.
A Study on The Resolution Limit of Polarimetric Radar and The Performance of Polarimetric Bandwidth Extrapolation Technique Kei Suwa, Toshio Wakayama, Masafumi Iwamoto Mitsubishi Electric Co. Information Technology R&D Center
Outline <ul><li>Background -- SAR resolution, PRF, and Polarimetry -- </li></ul><ul><li>Polarimetric Bandwidth Extrapolation (PBWE) </li></ul><ul><li>The Signal Model </li></ul><ul><li>Statistical Resolution Limit (SRL) </li></ul><ul><li>The Cramér-Rao Bound (CRB) </li></ul><ul><li>SRL and The Performance of PBWE </li></ul><ul><li>Conclusion </li></ul>
Background -- SAR resolution, PRF, and Polarimetry -- <ul><li>Range resolution is determined by the signal bandwidth. </li></ul><ul><li>Azimuth resolution is determined by the synthetic aperture length. </li></ul>Range resolution : signal bandwidth / Azimuth resolution : synthetic aperture length orbit freq. range FT Strip map mode Spotlight mode : Signal Bandwidth : Speed of light : wave length : Synthetic Aperture length : Range Higher resolution with smaller aperture size .
Background -- SAR resolution, PRF, and Polarimetry orbit Beam Illumination area Platform trajectory Due to PRF limitation, Polarimetric SAR often give up the resolution. <ul><li>Conditions of the PRF </li></ul><ul><ul><li>PRF must be higher than the signal Doppler bandwidth, which is determined by the azimuth beam width. </li></ul></ul><ul><ul><li>PRF must be low enough so that the range ambiguity is sufficiently low. </li></ul></ul><ul><li>The problem with Polarimetry </li></ul><ul><ul><li>Polarimetry requires sending H-polarization pulse and V-polarization pulse. </li></ul></ul><ul><ul><li>PRF gets lower for each polarization channel, so polarimetric SAR often need to give up the resolution. </li></ul></ul>It would be nice if polarization information helps enhancing the resolution Doppler Freq. Time PRF
Background -- SAR resolution, PRF, and Polarimetry Dr. Mihai Datcu “Semantic Content Extraction from High Resolution Earth Observation Images”
Background -- SAR resolution, PRF, and Polarimetry … resolution refers to separate two or more nearby targets, i.e., to determine that there are two instead of one. ~Radar Handbook (1970) 4-2~ What is “resolution”?
Background -- SAR resolution, PRF, and Polarimetry range 1. Flat plate and dihedral corner reflector case 2. Two flat plates case Polarization vector range HH VV HH VV Flat plate Dihedral corner reflector 1 0 1 1 0 -1 polarization vectors S hh S hv S vv = Two vectors are orthogonal to each other. Polarization information would not help It seems to be possible that polarization information helps improving the “resolution”.
Polarimetric Bandwidth Extrapolation (PBWE) FFT IFFT Polarimetric Linear Prediction Model Estimation range range i ii iii iv v freq. HH HV VV B freq. B’ HH HV VV HH HV VV HH HV VV <ul><li>A polarimetric linear prediction model is fitted to the spectral data. </li></ul><ul><li>Then the HH,HV,VV spectral data is extrapolated up to B’ . </li></ul>2B c 2B’ c
Polarimetric Bandwidth Extrapolation (PBWE) http://www.emi.dtu.dk/research/DCRS/Emisar/emisar.html EMISAR : dual frequency (L- and C-band) polarimetric Synthetic Aperture Radar (SAR) system developed at Technical University of Denmark. Storebaelt : the great belt range azimuth We have empirically shown that Polarization information does contribute to the resolution enhancement
Resolution “ Nominal” Resolution does not reflect polarization properties <ul><li>Fourier analysis would give the nominal resolution. </li></ul><ul><li>Parametric spectrum analysis would achieve higher resolution. </li></ul><ul><li> --- e.g. PBWE, MUSIC, ESPRIT, MVM, MEM, etc </li></ul>Quantitative analysis on the influence of polarization information on the resolution is provided in this presentation. range azimuth We are interested in the resolution achievable by “polarimetric” parametric spectrum analysis method such as PBWE.
Statistical Resolution Limit (SRL) <ul><li>“ Statistical Resolution” : Δ d </li></ul><ul><li>“ Statistical Resolution Limit” : min( Δ d) </li></ul> S.T.Smith, “Statistical Resolution Limits and the Complexified Cramér-Rao Bound,” IEEE Trans. Signal Process., vol. 53, no. 5, pp1597—1609, May 2005 The estimate of target separation contains error due to noise. If the expected error is sufficiently small compared to the real separation, we can claim that these targets are resolvable. true target separation = d SRL is a simple metric that provides the highest resolution achievable by any unbiased parametric spectral estimator.
The Signal Model clutter range target HH HV VV power scene Polarization property Polarimetric channels The signal at each resolution cell is represented by a polarization vector.
The Signal Model denotes the long column vector formed by concatenating the columns of X. n: Gaussian noise V: matrix of steering vectors a: complex amplitudes of the targets N: the number of samples The probability density function of the data z is: freq. B range d VV HH HH VV target #2 target #1 Two closely located point targets signal model is considered.
The Cramér-Rao Bound (CRB) The CRB for the parameters are derived. <ul><li>Likelihood function of the parameters </li></ul><ul><li>Fisher Information Matrix </li></ul><ul><li>Cramér-Rao Bound on the covariance of the estimator of the parameters </li></ul><ul><li>Convert the CRB to the lower bound of the target separation d </li></ul>
The Cramér-Rao Bound (CRB) The CRB for 2 polarimetric channels case is derived. <ul><li>The lower bound for the target separation </li></ul><ul><ul><li>(Polarimetric / 2 channels / White Gaussian noise) </li></ul></ul><ul><li>cf) when ∆ φ ->large (two targets are far away) </li></ul>
SRL and the Performance of PBWE The PBWE achieves CRL and SRL fairly well. (a) ∆ Ψ = 0 (b) ∆ Ψ = π /6 Δ d=0.4 Δ d=0.15 <ul><li>Monte Carlo iteration 1,000 times. </li></ul><ul><li>N=50 / SNR=40dB / order of the linear prediction filter = 6 </li></ul><ul><li>If the polarization properties of the two targets are the same polarization information does not help (a). </li></ul>range d a a HH VV
SRL and the Performance of PBWE The SRL is minimum when the polarization properties of the two targets are orthogonal (c) ∆ Ψ = π /2 (d) ∆ Ψ = π <ul><li>The SRL is improved to ¼ by using polarization information, when the polarization properties of the two targets are orthogonal (d). </li></ul><ul><li>(d) corresponds to a case where two targets are trihedral and dihedral corner reflectors. </li></ul>range d a a HH VV
Conclusion <ul><li>The SRL of polarimetric radar is derived and compared with that of single polarization radar </li></ul><ul><li>It has been shown analytically that the polarization information helps improving the resolution when the polarization properties of the two closely located targets are different, and thus on the average, the polarimetric radar outperforms single polarization radar. </li></ul><ul><ul><li>e.g. If the polarization properties of the two closely located targets are orthogonal to each other, the SRL is improved to ¼ by using polarization information. </li></ul></ul><ul><li>It has been shown that the resolution of the PBWE previously proposed by the authors almost achieves the SRL. </li></ul>
Radar polarimetry H trans. rec. Horizontal antenna (H) time time observation S hh S vh S hv S vv transmit receive incident wave scattered wave target ” polarization vector” direction : polarization property length : intensity Vertical antenna (V) rec. V trans. . . . represents
FRC is equivalent to “ Rayleigh resolution limit” incoherent average 1 If there are two targets with same amplitude and the separation between them is FRC , then the two targets are not necessarily resolvable with FT. It depends on the phase difference. On average, they are resolved.
1 CRB of the target separation for Different (SNR=42dB) Average CRB of the target separation average
2.2 レーダポラリメトリ クラッタ range 観測シーンとポラリメトリックレーダの観測値の関係 ターゲット HH HV VV power 観測シーン 偏波特性 偏波チャネル
3.2 スペクトル外挿方法 freq freq freq freq VV HH ポラリメトリック線形予測モデル L : モデル次数 ( m = L+1, L+2, …, M ) C l X ( m - l ) X ( m ) = c 1 11 c 2 11 c 3 11 c 1 12 c 2 12 c 3 12 c 1 21 c 2 21 c 3 21 c 1 22 c 2 22 c 3 22 l = 1 L
3.3 ポラリメトリック線形予測係数の推定 正規方程式の解 予測誤差 where, 予測係数は，予測誤差の二乗平均値を最小化するように決定 Γ = R -1 U f ( m ) = X ( m ) – Γ T Y m ( m = L+1, L+2, …, M ) freq freq VV HH c 1 12 c 2 12 c 3 12 f p ( m ) c 1 11 c 2 11 c 3 11 E f ( m ) Y m = 0 * E Y m * Y m T R = E U = Y m －１ * X ( m ) T
3.4 計算機シミュレーション 平板 二面コーナリフレクタ 点目標の偏波特性 = 1. 平板と二面コーナリフレクタが近接して存在する場合 2. 二つの平板が近接して存在する場合 帯域 B のパルス 点目標が二つ近接して存在する場合の時間波形の観察 2 B 0.3 c 1 0 1 1 0 -1 S hh S hv S vv
3.5 計算機シミュレーション Original number of samples M=128 Order L =10 SNR = 20 dB Expanded bandwidth B’ = 8 B 1. 平板と二面コーナリフレクタが近接して存在する場合 HH VV Original BWE PBWE
3.6 計算機シミュレーション ２． 二つの平板が近接して存在する場合 HH VV Original number of samples M=128 Order L =10 SNR = 20 dB Expanded bandwidth B’ = 8 B Original BWE PBWE
4.1 PBWE 法の性能評価 ND p = P p,n – P p,s P p,s ND p P p,n 最小分離可能距離 : ND p –3dB を満たす最小の距離 d 2. “ 分解能”の評価指標 1. パラメータ <ul><li> : 二つの目標の偏波特性の相違を示すパラメータ </li></ul><ul><li>d : 二つの目標間の距離を決めるパラメータ </li></ul>range d 偏波特性 2 二つのピーク間の極小値 = 二つのピークのうちの小さいほうの値