Radar Imaging with Compressed Sensing

Yang LuApril 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 ( slant range resolution): pulse widthc : speed of pulse2 : that is a round trip

Range Resolution 3

• If the incident angle is • Then the ground range resolution will be

𝑟 𝑅

𝑟 𝐺𝑅

𝜃𝑖𝜃𝑖

Azimuth Resolution 1

• Assume two points with same range Can’t distinguish A from B

if they are in the radar beam at the same time𝜃𝑎

Azimuth Resolution 2

• The azimuth resolution defined by :

R: the slant range : the wavelengthL: the length of antenna

LFM Signal

• Linear Frequency Modulated Signal (Chirp Signal)

Where t: a time variable (fast time)T: duration of the signalK:is the chirp rate So the bandwidth of the signal is:

Matched Filter

• The output of the a Matched Filter is:

: the received signal and means convolution

: the duplicated signal of the original signal and means complex conjugate

Matched Filter (example)

• If After delay, we receive the signal

The reference signal will be

Matched Filter (example)

• Output signal of the matched filter

So 3dB width of the main lobe

𝑡 0

1𝐾𝑇 +𝑡0

1𝐾𝑇 − 𝑡0

Range resolution improved

• The range resolution improved

Now we can distinguish B from C

Range resolution improved

Original ground range resolution:

Now replace with 3dB main lobe width= Finally, the improved ground range resolution will be :

Phase difference: phase difference between the transmitted and the received signal: the distance (round trip): the wavelength of the transmitted signal

http://www.radartutorial.eu/

SAR Azimuth Resolution

• The phase change of the radar signal will be

By Pythagorean theorem

: a time variable (slow time):the speed of plane

Synthetic Aperture Radar Polarimetry (J.V. Zyl and Y. Kim)

SAR Azimuth Resolution

Substitute The instantaneous frequency change of this signal is Which also can be considered as LFM signalAnd the total time

SAR Azimuth Resolution

The The time resolution will be So the azimuth resolution (in distance) will be

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)

• If consider the signal on the two direction simultaneously, that will be a 2-dimensional signal with variable t and .

2D signal of the target

• : a complex constant

: the centre frequency (carrier frequency)

𝜃𝑎

𝜃(𝜂)

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 of , we get the impulse response of SAR sensor:

• 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):

Radar Image

Radar algorithms are trying to obtain the ground reflectivity function based on the received radar signal. • Traditional methodsRange-Doppler AlgorithmChirp scaling algorithmOmega-K algorithms

Background of Compressed Sensing

Assume an N-dimensional signal has a K-sparse representation () in the basis

If we have a measurement matrix () to measure and encode the linear projection of the signal we get measurements

If , there will be enormous possible solutions. And we want the sparsest one.

Compressed Sensing

• CS theory tells us that when the matrix A= has the Restricted Isometry Property (RIP), then it is indeed possible to recover the K-sparse signal from a set of measurement

But RIP condition is hard to check. An alternative way is to measure the mutual coherence

denotes the column of matrix ACompressive Radar Imaging (R. Baraniuk and P.Steeghs)

Compressed Sensing

• We want 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 with high probability

Compressive Radar Imaging (R. Baraniuk and P.Steeghs)

Reconstruct Radar Image by CS methods

• When RIP/incoherency holds, the signal can be recovered exactly from by solving an minimization problem as:

Reconstruct Radar Image by CS methods

• If the measurement matrix is a causal, quasi-Toeplitz matrix , the results also show good performance.

(Right shift distance=

Causal, quasi-Toeplitz Matrix (Example)

If M=4, N=8 then right shift distance D=

is the element of a pseudo-random sequence

Causal, quasi-Toeplitz Matrix

The measurements of the signal will be

CS-based Radar

We already know

For simplicity, just consider 1D range imaging model and ignore the noise

Under this condition, can be considered as the transmitted radar pulse

is the time delay. A is the attenuation.

CS-based Radar

Assume, the target reflectivity function is k-sparse in some basis.

The PN or Chirp signals transmitted as radar waveforms (t) form a dictionary that is incoherent with the time, frequency and time-frequency bases.

CS-based Radar

• Let the transmitted radar signal be the PN signal

• The target reflectivity generated from N Nyquist-rate samples n=1,…,N via where

• The PN signal generated from a N-length Bernoulli vector via

CS-based RadarThe received signal will be

And we sample it every second

CS-based Radar (Results)

The target reflectivity function can be recovered by using an OMP greedy algorithm

y

𝑃 𝑁 𝑠𝑖𝑔𝑛𝑎𝑙

Compressive Radar Imaging (R. Baraniuk and P.Steeghs)

Another Example (2-dimensional)

The 2D received signal of a point target If ignore the antenna pattern =1, be a constant which is the radar cross section of point target

Another Example (2-dimensional)The approximate received signal will be

For a measurement scene () , the recorded echo signal will be

: samples on the azimuth direction (slow time samples): samples on the range direction (fast time samples)i: the point target in the scene

Discrete format of the scene

where =

Discrete format of the scene

Discrete format of the scene

:is additive white noise complete measurement matrix of SAR echo signalAccording to CS theory, we only need a small set of to successfully recover the sparse signal with high probability.

Randomly select rows of matrix A by using random selection matrix

Discrete format of the scene

We assume that have a sparse representation in a certain basis (for example, a set of K point targets corresponds to a sparse sum of delta functions as in , then we have

Where =

, and are complex

So we have

We define signal, and as , ,

Final Format

Sparest solution can be solved by norm minimization

Simulation Results

D

ERS Ship Image Results

d

RD algorithm

CS algorithm

CS algorithm

Noise free SNR=20dB 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.