Terahertz Imaging with Compressed Sensing Department of Electrical and Computer Engineering Rice University, Houston, Texas, USA Wai Lam Chan December 17, 2007

2 Mittleman Group ( THz Near-field microscopy (Zhan, Astley) THz Imaging (Chan, Pearce) THz Photonic Crystal structures (Prasad, Jian) THz waveguides (Mendis, Mbonye, Diebel, Wang) THz emission spectroscopy (Laib, Zhan) Terahertz (THz) Research Group at Rice

T-rays and Imaging

What Are T-Rays? T-Rays Radio Waves Microwaves X-Rays Gamma Rays Visible Light Hz

Imaging Throughout History Daguerreotype (1839) inventors/bldaguerreotype.htm X-rays (1895) inventors/blxray.htm T-rays (1995) B. B. Hu and M. C. Nuss, Opt. Lett., 20, 1716, 1995

Why Can T-Rays Help? E(t)  E(f)|E(f)| Measurement of E(t) Subpicosecond pulses Submillimeter Wavelengths T-Rays Provide Travel-time / Depth Information High depth resolution High spatial resolution Benefits to Imaging Subpicosecond pulsesLinear PhaseOver 1 THz in Bandwidth

Material Responses to T-rays Water Metal Plastics Strongly Absorbing Highly Reflective Transparent

8 Promising Applications of T-Rays (Karpowicz, et al., Appl. Phys. Lett. vol. 86, (2005)) Zandonella, C. Nature 424, 721– 722 (2003). Space Shuttle Foam Wallace, V. P., et. al. Faraday Discuss. 126, (2004). Diseased Tissue Medical Imaging Safety Security Concealed Weapon (Kawase, Optics & Photonics News, October 2004)

THz Time-domain Imaging Object THz Transmitter THz Receiver

THz Time-domain Imaging Object THz Transmitter THz Receiver Pixel-by-pixel scanning Limitations: acquisition time vs. resolution Faster imaging method Just take fewer samples!

Compressed Sensing (CS) [Candes et al, Donoho]

Why CS works: Sparsity Many signals can be compressed in some representation/basis (Fourier, wavelets, …) pixels large wavelet coefficients wideband signal samples large Gabor coefficients

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

Signal is -sparse Few linear projections Compressed Sensing (CS) Theory sparse signal (image) information rate measurements Measurement matrix

Signal is -sparse Few linear projections Random measurements will work! Compressed Sensing (CS) Theory sparse signal (image) information rate measurements Measurement matrix (e.g., random)

Random can be … … 12 M … 12 M Random 0/1 (Bernoulli) Random 2-D Fourier and many others …

Reconstruction/decoding:given (ill-posed inverse problem)find CS Signal Recovery measurements sparse signal nonzero entries

Reconstruction/decoding:given (ill-posed inverse problem)find L 2 fast, wrong CS Signal Recovery

Reconstruction/decoding:given (ill-posed inverse problem)find L 2 fast, wrong L 0 correct, slow only M=K+1 measurements required to perfectly reconstruct K-sparse signal [Bresler; Rice] CS Signal Recovery number of nonzero entries

Reconstruction/decoding:given (ill-posed inverse problem)find L 2 fast, wrong L 0 correct, slow L 1 correct, mild oversampling [Candes et al, Donoho] CS Signal Recovery linear program

CS in Action Part I: CS-THz Fourier Imaging

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

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

Random 2-D Fourier … Measurement matrix …

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

Fourier Imaging Results Fourier Transform of object (Magnitude) Inverse Fourier Transform Reconstruction (zoomed-in) 6.4 cm4.5 cm 6.4 cm 4.5 cm Resolution: mm

Imaging Results with CS Inverse FT Reconstruction (4096 measurements) CS Reconstruction (500 measurements) 4.5 cm CS Reconstruction (1000 measurements)

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

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 Compressive Phase Retrieval (CPR) (Fienup, Appl. Optics., 21(15), pp , August 1982) (Moravec et al.)

Imaging Results with Compressive Phase Retrieval (CPR) 6 cm Resolution: mm Fourier Transform of object (Magnitude-only) CPR Reconstruction (4096 measurements) 6.4 cm

Compressed Sensing Phase Retrieval (CSPR) Results Modified CPR algorithm with CS Fourier Transform of object (Magnitude-only) CPR Reconstruction (4096 measurements) CSPR Reconstruction (1000 measurements) 6.4 cm 6 cm

CS in Action Part I: CSPR Imaging System THz Fourier imaging with compressed sensing (CS) and phase retrieval (PR) Improved acquisition speed Processing time Potential for: –Flaw or impurity detection –Imaging with CW source (e.g., QCL)

CS in Action Part II: Single-Pixel THz Camera

Imaging with a Single-Pixel detector? (Lee A W M, et al., Appl. Phys. Lett. vol. 89, (2006)) Continuous-Wave (CW) THz imaging with a detector array Real-time imaging

Single-Pixel Camera (Visible Region) 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

Random 0/1 Bernoulli … Measurement matrix … … ….

Random patterns for CS-THz imaging Random patterns on printed-circuit boards (PCBs)

THz Single-Pixel Camera Setup THz receiver Random pattern on PCBs THz transmitter (fiber-coupled PC antenna) object mask 7cm 6cm 42cm

THz Single-Pixel Camera Imaging Result Object mask CS resconstruction (200 measurements) CS resconstruction (400 measurements)

THz Single-Pixel Camera Imaging Result CS resconstruction (400 measurements) CS resconstruction (200 measurements) image phase?

CS in Action Part II: Single-Pixel THz camera First single-pixel THz imaging system with no raster scanning Potential for: –Low cost (simple hardware) –near video-rate acquisition Faster acquisition: –film negatives (wheels/sprockets) –more advanced THz modulation techniques

Conclusions Terahertz imaging with Compressed Sensing –Acquire fewer samples high-speed image acquisition –THz Fourier imaging with CSPR –Single-pixel THz camera Ongoing research –THz camera with higher speed and resolution –Imaging phase with CS –CS-THz tomography –Imaging with multiple THz sensors

43 dsp.rice.edu/cs Mittleman Group ( Contact info: William Chan Acknowledgement Dr. Daniel Mittleman Dr. Richard Baraniuk Dr. Kevin Kelly Matthew Moravec Dharmpal Takhar Kriti Charan

T-Ray System THz Transmitter Substrate Lens Femtosecond Pulse GaAs Substrate DC Bias Picometrix T-Ray Instrumentation System Picometrix T-Ray Transmitter Module Femtosecond Pulse

45 T-Ray System T-Ray Control Box with Scanning Delay Line Fiber Coupled Femtosecond Laser System Sample THz TransmitterTHz Receiver Optical Fiber

46 Summary of T-Rays Broad fractional bandwidth Direct measurement of E(t) Short wavelengths (good depth resolution) Unique material responses

47 Signal is -sparse Samples sparse signal nonzero entries measurements Sampling