Compressive Sensing Applications and Demonstrations: Synthetic Aperture Radar

Transcription

1 Compressive Sensing Applications and Demonstrations: Synthetic Aperture Radar Shaun I. Kelly The University of Edinburgh 1

2 Outline 1 SAR Basics 2 Compressed Sensing SAR 3 Other Applications of Sparsity in SAR 4 Compressed Sensing SAR: Example Code 2

3 Outline 1 SAR Basics 2 Compressed Sensing SAR 3 Other Applications of Sparsity in SAR 4 Compressed Sensing SAR: Example Code 3

4 SAR Data Acquisition 4

5 SAR Data Acquisition 5

6 SAR Data Acquisition 6

7 SAR Data Acquisition 7

8 SAR Data Acquisition 7

9 SAR Data Acquisition 7

10 SAR Data Acquisition 7

11 SAR Data Acquisition 8

12 SAR Data Acquisition 8

13 SAR Data Acquisition 8

14 SAR Data Acquisition 8

15 SAR Data Acquisition 9

16 SAR Data Acquisition 1

17 SAR Data Acquisition 11

18 SAR Data Acquisition 12

19 SAR Data Acquisition 12

20 SAR Data Acquisition 12

21 SAR Data Acquisition 12

22 SAR Data Acquisition 13

23 SAR Data Acquisition 13

24 SAR Data Acquisition 13

25 SAR Data Acquisition 13

26 SAR Data Acquisition 14

27 SAR Data Acquisition 15

28 SAR Data Acquisition 16

29 SAR Data Acquisition 17

30 SAR Data Acquisition 18

31 SAR System Model 19

32 SAR System Model 19

33 SAR System Model Spatial Fourier Domain ky kx 2

34 SAR System Model Spatial Fourier Domain ky kx 21

35 SAR System Model Spatial Fourier Domain ky kx 21

36 SAR System Model Spatial Fourier Domain ky kx 21

37 SAR System Model Spatial Fourier Domain ky kx 21

38 SAR System Model Spatial Fourier Domain ky kx 21

39 SAR system model Spatial Fourier Domain ky kx 22

40 Standard Image Formation Approximate 2-D Matched Filter 1 Back-projection Algorithm 3 Range Migration Algorithm 2 Polar Format Algorithm 4 Range/Doppler Algorithm Example SAR Image y (m) 3 4 ky (rad m 1) x (m) kx (rad m 1)

41 Outline 1 SAR Basics 2 Compressed Sensing SAR 3 Other Applications of Sparsity in SAR 4 Compressed Sensing SAR: Example Code 24

42 Why is Compressed Sensing Interesting for SAR? Data Reduction If the phase history could be sampled at less than the Nyquist rate it would lead to improved storage and transmission capabilities. Sensor Constraints Missing radar echoes: interrupted pulses allow other tasks to be performed (multi-function radar) Missing frequency bands: the radar band may contain frequencies where there is interference or transmission is not allowed 25

43 Aperture Undersampling Undersampling of K-space 26

44 Aperture Undersampling Standard Image Formation y (m) 3 4 ky (rad m 1) x (m) kx (rad m 1)

45 Compressed Sensing of a SAR Phase History First Ingredient: Sparsity The signal/image x CN must be sparse or well approximated by a sparse signal/image (compressible): kxk = K N Will be considered later. Second Ingredient: Good Measurements Measurement equation y = SFx(+e), with F the DFT matrix and S {, 1}mq n a subset selection SH S is the projector on the selected subset, PSF = (SF)H (SF) is the point spread function (more precisely PSF = F 1 diag(sh S) and PSFx = PSF x) An approximate sub-sampling of the k-space! Third Ingredient: Reconstruction Algorithm Optimisation algorithm. ex: constrained `1 minimisation Greedy algorithm. ex: Orthogonal Matching Pursuit (OMP), CoSaMP, Iterative Hard Thresholding (IHT) Many fast algorithms available if there are fast operators available! 28

46 Sparsity of SAR Image Sparsity in Wavelets? Image Domain Wavelet Domain SAR images are not significantly compressible in any basis! 29

47 SAR Image Statistics Interaction of Reflectors in a Range Cell Random interference: Speckle dominates images due to many random reflectors in a range cell inducing multiplicative noise in the reconstructed image - not compressible. Coherent interference: Coherent reflectors (often targets of interest) whose intensity tend to be much larger than incoherent reflections - compressible in spatial domain. 3

48 Aperture Undersampling Compressed Sensing Image Formation ` arg. min. y SF xbg + xs 2 arg. min. kxs k1 s.t. ky SFxs k2 xbg xs y (m) y (m) x (m) Bright Targets x (m) Background Speckle 31

49 Aperture Undersampling Compressed Sensing Image Formation y (m) y (m) x (m) Standard Image Formation x (m) Compressed Sensing Image Formation Significant improvement in imaging of bright targets! 32

50 Aperture Undersampling Compressed Sensing Image Formation y (m) y (m) x (m) Reference Standard Image Formation x (m) Compressed Sensing Image Formation Degradation in background speckle! 33

51 Issues in Compressed Sensing SAR Topics to Consider: Fast operators Sparsity-parameter selection Off-the-grid targets Deterministic undersampling 34

52 Outline 1 SAR Basics 2 Compressed Sensing SAR 3 Other Applications of Sparsity in SAR 4 Compressed Sensing SAR: Example Code 35

53 Phase Calibration (Autofocus) Motion and propagation errors produce unknown phase errors. Each aperture position has a phase error. Without a sparsity assumption the problem ill-posed. eθ N ky... kx eθ eθ 1 36

54 Ground Moving Target Indication Targets may have a significant velocity. Target velocities can cause target displacement and/or blurring. Without a sparsity assumption on the number of moving targets, many antennas and a large amount of aperture oversampling is required. 37

55 3-D Imaging Multi-pass/2-D aperture is required. Meeting the Nyquist sampling rate requirement for a 2-D aperture is typically not possible. Scattering occurs at propagation medium boundaries so 3-D images are sparse. 38

56 Outline 1 SAR Basics 2 Compressed Sensing SAR 3 Other Applications of Sparsity in SAR 4 Compressed Sensing SAR: Example Code 39

57 Compressed Sensing SAR: Example Code simulation cs sar.m simple sar simulator.m Main function of example code. Mono-static strip-map SAR. Range/Doppler algorithm for operators. Returns are delayed and scaled versions of a reference chirp signal. Random aperture undersampling. Dechirp-on-receive. fista.m Implementation of the Fast Iterative ShrinkageThresholding Algorithm (Beck et al. 29). Optimal first order method. 4

58 Compressed Sensing SAR: Example Code Results Reference Standard Image Formation Standard Image Formation Compressed Sensing Image Formation Things to Experiment With Use a different solver. Use different operators. Change the λ parameter. Use a more complicated SAR simulator. Change the amount of undersampling. Add in clutter. 41

59 Thank you for your attention! Questions? 42