Radar Imaging with Compressed Sensing Yang Lu April 2014 Imperial College London Outline • Introduction to Synthetic Aperture Radar (SAR) • Background of Compressed Sensing • Reconstruct Radar Image by CS methods Introduction to SAR Important elements of SAR 1. Range Resolution and Azimuth Resolution 2. Chirp signal and Matched Filter Range Resolution and Azimuth Resolution of SAR http://www.radartutorial.eu/ Range Resolution 1 • Pulse signal (constant frequency signal) Range Resolution 2 • The resolution related with pulse width 𝑟𝑅 = 𝑇𝑝 ×𝑐 2 𝑇𝑝 : pulse width c : speed of pulse 2 : that is a round trip ( slant range resolution) Range Resolution 3 • If the incident angle is 𝜃𝑖 • Then the ground range resolution will be 𝑟𝐺𝑅 = 𝑟𝑅 𝜃𝑖 𝜃𝑖 𝑟𝐺𝑅 𝑟𝑅 𝑠𝑖𝑛𝜃𝑖 = 𝑇𝑝 ×𝑐 2𝑠𝑖𝑛𝜃𝑖 Azimuth Resolution 1 • Assume two points with same range Can’t distinguish A from B if they are in the radar 𝜃𝑎 beam at the same time Azimuth Resolution 2 • The azimuth resolution defined by 𝜃𝑎 : 𝑅×𝜆 𝑟𝐴 = 𝑅 × 𝜃𝑎 = 𝐿 𝜆 𝜃𝑎 = 𝐿 R: the slant range 𝜆: the wavelength L: the length of antenna LFM Signal • Linear Frequency Modulated Signal (Chirp Signal) 𝑡 𝑠 𝑡 = 𝑟𝑒𝑐𝑡 𝑒𝑥𝑝 𝑗𝜋𝐾𝑡 2 𝑇 Where 𝑟𝑒𝑐𝑡 𝑥 = 1 𝑓𝑜𝑟 𝑥 ≤ 1 2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 t: a time variable (fast time) T: duration of the signal K:is the chirp rate So the bandwidth of the signal is: 𝐵𝑊 = 𝐾 𝑇 Matched Filter • The output of the a Matched Filter is: 𝑠𝑜𝑢𝑡 𝑡 = 𝑠𝑟 𝑡 ⨂ℎ 𝑡 +∞ = −∞ 𝑠𝑟 𝑢 ℎ 𝑡 − 𝑢 du 𝑠𝑟 𝑡 : the received signal and ⨂ means convolution ℎ 𝑡 = 𝑔∗ (−𝑡) 𝑔(𝑡) : the duplicated signal of the original signal and ∗ means complex conjugate Matched Filter (example) • If 𝑠 𝑡 = 𝑟𝑒𝑐𝑡 𝑡 𝑇 𝑒𝑥𝑝 𝑗𝜋𝐾𝑡 2 After 𝑡0 delay, we receive the signal 𝑡 − 𝑡0 𝑠𝑟 𝑡 = 𝑟𝑒𝑐𝑡 𝑒𝑥𝑝 𝑗𝜋𝐾(𝑡 − 𝑡0 )2 𝑇 The reference signal will be −𝑡 ∗ ℎ 𝑡 = 𝑔 −𝑡 = 𝑟𝑒𝑐𝑡 𝑒𝑥𝑝 −𝑗𝜋𝐾(−𝑡)2 𝑇 𝑡 = 𝑟𝑒𝑐𝑡 𝑒𝑥𝑝 −𝑗𝜋𝐾𝑡 2 𝑇 Matched Filter (example) • Output signal of the matched filter 𝑠𝑜𝑢𝑡 𝑡 ≈ 𝑇𝑠𝑖𝑛𝑐(𝐾𝑇(𝑡 − 𝑡0 )) 1 −𝑡 𝐾𝑇 0 1 +𝑡 𝐾𝑇 0 𝑡0 So 3dB width of the main lobe= 0.886 𝐾𝑇 ≈ 1 𝐾𝑇 Range resolution improved • The range resolution improved Now we can distinguish B from C Range resolution improved Original ground range resolution: 𝑟𝐺𝑅 = 𝑟𝑅 𝑠𝑖𝑛𝜃𝑖 = 𝑇𝑝 ×𝑐 2𝑠𝑖𝑛𝜃𝑖 Now replace 𝑇𝑝 with 3dB main lobe width= 1 𝐾𝑇 Finally, the improved ground range resolution will be : 𝑟𝑅 𝑐 𝑐 𝑟𝐺𝑅 = = = 𝑠𝑖𝑛𝜃𝑖 2𝐵𝑊𝑠𝑖𝑛𝜃𝑖 2 𝐾 𝑇𝑠𝑖𝑛𝜃𝑖 Phase difference 2𝑟 × 2𝜋 𝜑= 𝜆 𝜑: phase difference between the transmitted and the received signal 2𝑟: the distance (round trip) 𝜆: the wavelength of the transmitted signal http://www.radartutorial.eu/ SAR Azimuth Resolution • The phase change of the radar signal will be 4𝜋𝑅(𝜂) 𝜑 𝜂 = 𝜆 By Pythagorean theorem 𝑅 𝜂 = 𝑅02 + (𝑣𝜂)2 𝑣 2 𝜂2 ≈ 𝑅0 + 2𝑅0 𝜂: a time variable (slow time) 𝑣:the speed of plane Synthetic Aperture Radar Polarimetry (J.V. Zyl and Y. Kim) SAR Azimuth Resolution Substitute 𝑅 𝜂 ≈ 𝑅0 + 𝑣 2 𝜂2 2𝑅0 4𝜋𝑅(𝜂) 4𝜋𝑅0 2𝜋𝑣 2 2 𝜑 𝜂 = ≈ + 𝜂 𝜆 𝜆 𝑅0 𝜆 The instantaneous frequency change of this 1 𝜕𝜑 𝜂 2𝑣 2 signal is 𝑓 𝜂 = = 𝜂 2𝜋 𝜕𝜂 𝑅0 𝜆 Which also can be considered as LFM signal And the total time 𝜂𝑡𝑜𝑡 ≈ 𝜃𝑎 𝑅0 𝑣 = 𝜆𝑅0 𝐿𝑣 SAR Azimuth Resolution The 𝐵𝑊𝑎 = 𝑓 𝜂𝑡𝑜𝑡𝑎𝑙 = 2𝑣 𝐿 The time resolution will be 1 𝐵𝑊𝑎 = 𝐿 2𝑣 So the azimuth resolution (in distance) will be 1 𝐿 𝑟𝐴 = 𝑣 × = 𝐵𝑊𝑎 2 2D signal of the target • One target have two equations-one is in the range direction (variable: fast time t) and another is in the azimuth direction (variable :slow time 𝜂) • If consider the signal on the two direction simultaneously, that will be a 2-dimensional signal with variable t and 𝜂. 2D signal of the target • 𝑠 𝑡, η = 𝐴0 𝜔𝑟 (𝑡 −𝑗4𝜋𝑓0 2𝑅(𝜂) − ) 𝑐 𝑅(𝜂) 𝑐 𝑒 ×𝑒 𝐴0 : a complex constant 𝑡 𝜔𝑟 t = rect( ) 𝑇 𝜃(𝜂) 2 𝜔𝑎 (𝜂) ≈ 𝑠𝑖𝑛𝑐 𝜃𝑎 𝜔𝑎 𝜂 × 𝑗𝜋𝐾(𝑡− 2𝑅(𝜂) 2 ) 𝑐 𝜃𝑎 𝜃(𝜂) 𝑓0 : the centre frequency (carrier frequency) Signal Energy 2D signal space • The received signals are stored in the signal space Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation ( G.Cumming and H.Wong) SAR impulse response • If we ignore the constant 𝐴0 of 𝑠 𝑡, η , we get the impulse response of SAR sensor: • ℎ𝑖𝑚𝑝 𝑡, η = 𝜔𝑟 (𝑡 − −𝑗4𝜋𝑓0 𝑅(𝜂) 𝑐 2𝑅(𝜂) ) 𝑐 𝜔𝑎 𝜂 × 𝑗𝜋𝐾(𝑡− 2𝑅(𝜂) 2 ) 𝑐 𝑒 ×𝑒 The received signal of the ground model can be built as the convolution of the ground reflectivity with the SAR impulse response (with additive white noise): 𝑠 𝑡, 𝜂 = 𝑔 𝑡, 𝜂 ⊗ ℎ𝑖𝑚𝑝 𝑡, 𝜂 + 𝑛(𝑡, 𝜂) Radar Image • 𝑠 𝑡, 𝜂 = 𝑔 𝑡, 𝜂 ⊗ ℎ𝑖𝑚𝑝 𝑡, 𝜂 + 𝑛(𝑡, 𝜂) Radar algorithms are trying to obtain the ground reflectivity function 𝑔 𝑡, 𝜂 based on the received radar signal. • Traditional methods Range-Doppler Algorithm Chirp scaling algorithm Omega-K algorithms Background of Compressed Sensing Assume an N-dimensional signal 𝑥 has a Ksparse representation (𝜃) in the basis Ψ 𝑥 = Ψ𝜃 If we have a measurement matrix Φ (𝑀 × 𝑁) to measure and encode the linear projection of the signal we get measurements 𝑦 = Φ𝑥 = ΦΨ𝜃 If 𝑀 < 𝑁, there will be enormous possible solutions. And we want the sparsest one. Compressed Sensing • CS theory tells us that when the matrix A=ΦΨ has the Restricted Isometry Property (RIP), then it is indeed possible to recover the K-sparse signal from a set of measurement 𝑀 = 𝑂(𝐾𝑙𝑜𝑔(𝑁/𝐾)) But RIP condition is hard to check. An alternative way is to measure the mutual coherence 𝑎𝑖 , 𝑎𝑗 𝜇 = 𝜇 𝐴 = max 𝑖≠𝑗 𝑎𝑖 2 𝑎𝑗 2 𝑎𝑖 denotes the 𝑖 𝑡ℎ column of matrix A Compressive Radar Imaging (R. Baraniuk and P.Steeghs) Compressed Sensing • We want 𝜇 to be small (incoherence) CS theory has shown that many random measurement matrices are universal in the sense that they are incoherent with any fixed basis Ψ with high probability Compressive Radar Imaging (R. Baraniuk and P.Steeghs) Reconstruct Radar Image by CS methods • When RIP/incoherency holds, the signal 𝑥 can be recovered exactly from 𝑦 by solving an 𝑙1 minimization problem as: 𝜃 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝜃 1 𝑠. 𝑡. 𝑦 = ΦΨ𝜃 Reconstruct Radar Image by CS methods • If the measurement matrix Φ is a causal, quasi-Toeplitz matrix , the results also show good performance. (Right shift distance= 𝑁 𝑀 ) Causal, quasi-Toeplitz Matrix (Example) 𝑁 𝑀 If M=4, N=8 then right shift distance D= =2 𝑝1 0 0 0 0 0 0 0 𝑝3 𝑝2 𝑝1 0 0 0 0 0 𝑝5 𝑝4 𝑝3 𝑝2 𝑝1 0 0 0 𝑝7 𝑝6 𝑝5 𝑝4 𝑝3 𝑝2 𝑝1 0 𝑝𝑖 is the 𝑖 𝑡ℎ element of a pseudo-random sequence 𝑃 Causal, quasi-Toeplitz Matrix The measurements of the signal will be 𝑦 = Φ𝑥 𝑁 𝑦 𝑚 = 𝑃 𝐷𝑚 − 𝑛 𝑥(𝑛) 𝑛=1 CS-based Radar We already know 𝑠 𝑡, 𝜂 = 𝑔 𝑡, 𝜂 ⊗ ℎ𝑖𝑚𝑝 𝑡, 𝜂 + 𝑛(𝑡, 𝜂) For simplicity, just consider 1D range imaging model and ignore the noise 𝑠 𝑡 = 𝑔 𝑡 ⊗ ℎ𝑖𝑚𝑝 𝑡 Under this condition, ℎ𝑖𝑚𝑝 𝑡 can be considered as the transmitted radar pulse 𝑠 𝑡 =𝐴 𝜏= 2𝑟 𝑐 𝑔(𝜏)𝑠𝑇 (𝑡 − 𝜏)𝑑𝜏 is the time delay. A is the attenuation. CS-based Radar Assume, the target reflectivity function 𝑔(𝑡) is ksparse in some basis. The PN or Chirp signals transmitted as radar waveforms 𝑠𝑇 (t) form a dictionary that is incoherent with the time, frequency and timefrequency bases. CS-based Radar • Let the transmitted radar signal be the PN signal 𝑠𝑇 • The target reflectivity generated from N Nyquist-rate samples 𝑥(𝑛) n=1,…,N via 𝑔 𝑡 = 𝑥 𝑡 ∆ where 0 ≤ 𝑡 ≤ 𝑁∆ • The PN signal generated from a N-length Bernoulli ∓1 vector 𝑃 𝑛 via 𝑠𝑇 𝑡 = 𝑃 𝑡 ∆ CS-based Radar The received signal will be 𝑠 𝑡 = 𝐴 𝑔(𝜏)𝑠𝑇 (𝑡 − 𝜏)𝑑𝜏 And we sample it every 𝐷∆ second 𝑦 𝑚 = 𝑠 𝑡 |𝑡=𝑚×𝐷∆ 𝑁∆ =𝐴 =𝐴 𝑔 𝜏 𝑠𝑇 (𝑚𝐷∆ − 𝜏)𝑑𝜏 0 𝑁∆ 𝑔 0 𝑁 =𝐴 𝜏 ∆ 𝑛∆ 𝜏 𝑃 (𝑚𝐷 − )𝑑𝜏 𝑃(𝑚𝐷 − 𝑛) 𝑛=1 𝑁 =𝐴 (𝑛−1)∆ 𝑃 𝑚𝐷 − 𝑛 𝑥(𝑛) 𝑛=1 𝑔(𝜏)𝑑𝜏 CS-based Radar (Results) The target reflectivity function can be recovered by using an OMP greedy algorithm 𝑃𝑁 𝑠𝑖𝑔𝑛𝑎𝑙 y(𝑛) Compressive Radar Imaging (R. Baraniuk and P.Steeghs) Another Example (2-dimensional) The 2D received signal of a point target 𝑠 𝑡, η = 𝐴0 𝜔𝑟 (𝑡 2𝑅(𝜂) − ) 𝑐 −𝑗4𝜋𝑓0 𝑅(𝜂) 𝑐 𝜔𝑎 𝜂 × 𝑗𝜋𝐾(𝑡− 2𝑅(𝜂) 2 ) 𝑐 𝑒 ×𝑒 If ignore the antenna pattern 𝜔𝑎 𝜂 =1, 𝐴0 𝜔𝑟 (𝑡 − 2𝑅(𝜂) ) 𝑐 be a constant 𝜎(𝑃𝜔 ) which is the radar cross section of point target 𝑃𝜔 Another Example (2-dimensional) The approximate received signal will be −𝑗4𝜋𝑓0 𝑅(𝜂,𝑃𝜔 ) 𝑐 2𝑅(𝜂,𝑃𝜔 ) 2 𝑗𝜋𝐾(𝑡− ) 𝑐 𝑒 𝑠 𝑡, η, 𝑃𝜔 = 𝜎(𝑃𝜔 )𝑒 × For a measurement scene Ω (𝑁 = 𝑁𝑎 × 𝑁𝑟 ) , the recorded echo signal will be 𝑁 𝑆𝑐 𝑡, η = 𝑠 𝑡, η, 𝑖 𝑖=1 𝑁𝑎 : samples on the azimuth direction (slow time samples) 𝑁𝑟 : samples on the range direction (fast time samples) i: the 𝑖𝑡ℎ point target in the scene Discrete format of the scene 𝑁 𝑆𝑐 𝑡, η = 𝜎𝑖 𝑒 𝑖=1 𝑁 = 𝑖=1 −𝑗4𝜋𝑓0 𝑅(𝜂,𝑖) 𝑐 × 2𝑅(𝜂,𝑖) 𝑗𝜋𝐾(𝑡− 𝑐 )2 𝑒 2𝑅(𝜂,𝑖) 𝑅(𝜂,𝑖) 𝑗𝜋𝐾(𝑡− 𝑐 )2 −𝑗4𝜋𝑓0 𝑐 𝜎𝑖 𝑒 𝑁 𝜎𝑖 𝑒 −𝑗∅𝑖 (𝑡,𝜂) = 𝑖=1 where 𝑒 −𝑗∅𝑖(𝑡,𝜂) =𝑒 𝑗𝜋𝐾(𝑡− 2𝑅(𝜂,𝑖) 2 𝑅(𝜂,𝑖) ) −𝑗4𝜋𝑓0 𝑐 𝑐 Discrete format of the scene 𝑁 𝜎𝑖 𝑒 −𝑗∅𝑖 (𝑡,𝜂) = 𝐴(𝑡,𝜂) 𝑇 𝜎 𝑆𝑐 𝑡, η = 𝑖=1 𝐴(𝑡,𝜂) 𝑒 −𝑗∅1 (𝑡,𝜂) −𝑗∅2 (𝑡,𝜂) 𝑒 = ⋮ 𝑒 −𝑗∅𝑁 (𝑡,𝜂) 𝐴(1,1) 𝑇 𝑆𝑐(1,1) 𝐴(2,1) 𝑇 ⋮ 𝐴 = 𝐴(𝑁𝑟,1) 𝑇 𝑆𝑐(2,1) ⋮ 𝑆 = 𝑆𝑐(𝑁𝑟,1) 𝐴(1,2) 𝑇 ⋮ 𝐴(𝑁𝑟,𝑁𝑎) 𝑇 𝑆𝑐(1,2) ⋮ 𝑆𝑐(𝑁𝑟,𝑁𝑎) Discrete format of the scene 𝑆 = 𝐴𝜎 + 𝑛 𝑛 :is additive white noise 𝐴: complete measurement matrix of SAR echo signal According to CS theory, we only need a small set of 𝑆 to successfully recover the sparse signal 𝝈 with high probability. Randomly select 𝑀 = 𝑂(𝐾𝑙𝑜𝑔(𝑁/𝐾)) rows of matrix A by using random selection matrix Φ Discrete format of the scene We assume that 𝜎 have a sparse representation𝛼 in a certain basisΨ (for example, a set of K point targets corresponds to a sparse sum of delta functions as in 𝜎𝑚 = 𝐾 𝑛=1 𝛼𝑛 𝛿(𝑚 − 𝑛)), then we have 𝑆𝑝 = 𝛷𝐴𝜎 + 𝑛 = 𝛷𝐴𝛹𝛼 + 𝑛 = 𝛩𝛼 + 𝑛 Where Θ= 𝛷𝐴𝛹 𝑆𝑝 , Θ and 𝛼 are complex 𝑆𝑝 = 𝛩𝛼 + 𝑛 𝑅𝑒 𝑆𝑝 + 𝑗𝐼𝑚 𝑆𝑝 = 𝑅𝑒 𝛩 + 𝑗𝐼𝑚 𝛩 𝑅𝑒 𝛼 + 𝑗𝐼𝑚 𝛼 = Re 𝛩 Re 𝛼 − Im 𝛩 Im(𝛼) +𝑗 𝑅𝑒 𝛩 𝐼𝑚 𝛼 + 𝐼𝑚 𝛩 𝑅𝑒 𝛼 So we have 𝑅𝑒 𝑆𝑝 = 𝑅𝑒 𝛩 𝑅𝑒 𝛼 − 𝐼𝑚 𝛩 𝐼𝑚(𝛼) 𝐼𝑚 𝑆𝑝 = 𝑅𝑒 𝛩 𝐼𝑚 𝛼 + 𝐼𝑚 𝛩 𝑅𝑒(𝛼) We define signal𝑆𝑝 , 𝛩 and 𝛼 as 𝑅𝑒 𝑆𝑝 𝑅𝑒 𝛩 𝑆𝑝 = ,𝛩 = 𝐼𝑚 𝛩 𝐼𝑚(𝑆𝑝 ) −𝐼𝑚 𝛩 𝑅𝑒 𝛩 ,𝛼 = 𝑅𝑒 𝛼 𝐼𝑚 𝛼 Final Format 𝑆𝑝 = 𝛩𝛼 + 𝑛 Sparest solution can be solved by 𝑙1 norm minimization min 𝜆 𝛼 1 𝑠. 𝑡. 𝑆 − 𝛩𝛼 2 < 𝜀 Simulation Results D ERS Ship Image Results SNR=20dB Noise free RD algorithm d CS algorithm CS algorithm SNR=10dB Reference • • • • • • R. Baraniuk and P. Steeghs. Compressive Radar Imaging. IEEE Radar Conference, April 2007. S.J. Wei, X.L. Zhang, J.Shi and G.Xiang. Sparse Reconstruction For SAR Imaging Based On Compressed Sensing. Progress In Electromagnetics Research, P63-81, 2010. J.V. Zyl and Y. Kim. Synthetic Aperture Radar Polarimetry. Dec 2010. G. Cumming and H. Wong. Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation. Dec 2007. Radar Basics available at http://www.radartutorial.eu/index.en.html#this Y.K. Chan and V.C. Koo. An Introduction to Synthetic Aperture Radar (SAR). Progress In Electromagnetics Research. P27-60, 2008.