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.