Terahertz Imaging with Compressed Sensing and Phase Retrieval

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

THz Time-domain Imaging THz Transmitter THz Receiver Object

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

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

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

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

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

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

THz Fourier Imaging Setup THz receiver object mask “R” (3. 5 cm x 3. 5 cm) automated translation stage polyethlene lens

Fourier Imaging Results 8 cm 6 cm Resolution: 3 mm Fourier Transform of object (Magnitude) Inverse Fourier Transform Reconstruction (zoomed-in)

Imaging Results with Compressed Sensing (CS) 6 cm Inverse Fourier Transform Reconstruction (6400 measurements) CS Reconstruction (1000 measurements)

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

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

Imaging Results with PR 8 cm 6. 4 cm Resolution: 3. 2 mm Fourier Transform of object (Magnitude) PR Reconstruction (6400 measurements)

Compressed Sensing Phase Retrieval (CSPR) Results • Modified PR algorithm with CS 6. 4 cm 8 cm Fourier Transform of object (Magnitude) PR Reconstruction CSPR Reconstruction (6400 measurements) (1000 measurements)

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

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

Compressed Sensing (CS) Theory 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 …. 001010…. sparse signal (image) Measurement matrix (e. g. , random) information rate

Compressed Sensing (CS) Theory 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 measurements sparse signal (image) Measurement matrix (e. g. , random) information rate

THz Tomography • Other imaging methods: – Pulsed THz Tomography (S. Wang & X. C. Zhang) – WART (J. Pearce & D. Mittleman) – Interferometric and synthetic aperture imaging (A. Bandyopadhyay & J. Federici) • Limitations in speed and resolution

Future Improvements • • • Higher imaging resolution Higher SNR Using Broad spectral information Reconstruction of “complex” objects CS and CSPR detection

2 -D Wavelet Transform (Sparsity)