1. Terahertz Imaging with Compressed Sensing and Phase Retrieval Wai Lam Chan Matthew Moravec Daniel Mittleman Richard Baraniuk Department of Electrical and Computer Engineering Rice University, Houston, Texas, USA

2. THz Time-domain Imaging Object THz Transmitter THz Receiver

3. THz Time-domain Imaging Object THz Transmitter THz Receiver Suitcase (weapons) Automobile dashboard (foam layer) Chocolate bar (food) (Mittleman, et al., Appl. Phys. B, vol. 68, 1085-1094 (1999)) (Karpowicz, et al., Appl. Phys. Lett. vol. 86, 054105 (2005))

4. THz Time-domain Imaging Object THz Transmitter THz Receiver Pixel-by-pixel scanning Limitations: acquisition time vs. resolution Faster imaging method

5. Reconstruct via nonlinear processing (optimization) Take fewer ( ) measurements High-speed THz Imaging with Compressed Sensing (CS) Measurements (random projections) (Donoho, IEEE Trans. on Information Theory, 52(4), pp. 1289 - 1306, April 2006) “sparse” signal / object (K-sparse) Measurement Matrix (e.g., random Fourier) M << N

6. Compressed Sensing (CS) Example: Single-Pixel Camera DMD Random pattern on DMD array (Baraniuk, Kelly, et al. Proc. of Computational Imaging IV at SPIE Electronic Imaging, Jan 2006 ) image reconstruction DSP DMD

7. THz Fourier Imaging Setup 12cm6cm12cm object mask THz transmitter (fiber-coupled PC antenna) THz receiver 6cm metal aperture automated translation stage

8. N Fourier samples THz Fourier Imaging Setup 12cm6cm12cm object mask THz transmitter 6cm Fourier plane pick only random measurements for Compressed Sensing

9. THz Fourier Imaging Setup automated translation stage polyethlene lens object mask “R” (3.5cm x 3.5cm) THz receiver

10. Fourier Imaging Results Fourier Transform of object (Magnitude) Inverse Fourier Transform Reconstruction (zoomed-in) 6.4 cm7.2 cm 6.4 cm 7.2 cm Resolution: 1.125 mm

11. Imaging Results with Compressed Sensing (CS) Inverse Fourier Transform Reconstruction (4096 measurements) CS Reconstruction (2000 measurements) 7.2 cm

12. Imaging Using the Fourier Magnitude 12cm object mask THz transmitter THz receiver 6cm metal aperture translation stage variable object position

13. Reconstruction with Phase Retrieval (PR) Reconstruct signal from only the magnitude of its Fourier transform Iterative algorithm based on prior knowledge of signal: –real-valued –positivity –finite support Hybrid Input-Output (HIO) algorithm (Fienup, Appl. Optics., 21(15), pp. 2758 - 2769, August 1982)

14. Imaging Results with Phase Retrieval (PR) 8 cm 6.4 cm Resolution: 3.2mm Fourier Transform of object (Magnitude-only) PR Reconstruction (6400 measurements)

15. Compressed Sensing Phase Retrieval (CSPR) Results Modified PR algorithm with CS Fourier Transform of object (Magnitude-only) PR Reconstruction (6400 measurements) CSPR Reconstruction (1000 measurements) 8 cm 6.4 cm

16. Summary of CSPR Imaging System Novel THz imaging method with compressed sensing (CS) and phase retrieval (PR) Improved acquisition speed Processing time Resolution in reconstructed image

17. Acknowledgements National Science Foundation National Aeronautics and Space Administration Defense Advanced Research Projects Agency

18. 2-D Wavelet Transform (Sparsity)

19. Imaging Results with Phase Retrieval (PR) 6.4 cm 4.8 cm Resolution: 1.5mm Fourier Transform of object (Magnitude-only) PR Reconstruction (4096 measurements)