Terahertz Imaging with Compressed Sensing Wai Lam Chan

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

Terahertz (THz) Research Group at Rice Mittleman Group (http: //www. ece. rice. edu/~daniel) THz Near-field microscopy (Zhan, Astley) THz waveguides (Mendis, Mbonye, Diebel, Wang) THz Photonic Crystal structures (Prasad, Jian) THz emission spectroscopy (Laib, Zhan) 2 THz Imaging (Chan, Pearce)

T-rays and Imaging

What Are T-Rays? T-Rays X-Rays Radio Waves 100 103 106 109 1012 1015 1018 1021 Hz Visible Light Microwaves Gamma Rays

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

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

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

Promising Applications of T-Rays Medical Imaging (Kawase, Optics & Photonics News, October 2004) Security Concealed Weapon Diseased Tissue Wallace, V. P. , et. al. Faraday Discuss. 126, 255 - 263 (2004). Safety Zandonella, C. Nature 424, 721– 722 (2003). (Karpowicz, et al. , Appl. Phys. Lett. vol. 86, 054105 (2005)) Space Shuttle Foam 8

THz Time-domain Imaging THz Transmitter THz Receiver Object

THz Time-domain Imaging THz Transmitter THz Receiver Object • 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

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

Compressed Sensing (CS) Theory • Signal is -sparse • Few linear projections 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 measurements sparse signal (image) Measurement matrix information rate

Compressed Sensing (CS) Theory • Signal is -sparse • Few linear projections 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) • Random measurements will work! information rate

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

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

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

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

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

CS in Action Part I: CS-THz Fourier Imaging

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

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

Random 2 -D Fourier … … Measurement matrix

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

Fourier Imaging Results 6. 4 cm 4. 5 cm Resolution: 1. 125 mm Fourier Transform of object (Magnitude) Inverse Fourier Transform Reconstruction (zoomed-in)

Imaging Results with CS 4. 5 cm CS Reconstruction Inverse FT (500 measurements) (1000 measurements) Reconstruction (4096 measurements)

Imaging Using the Fourier Magnitude object mask THz transmitter 6 cm variable object position metal aperture THz receiver 6 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: – real-valued – positivity – finite support • Hybrid Input-Output (HIO) algorithm (Fienup, Appl. Optics. , 21(15), pp. 2758 - 2769, August 1982) • Compressive Phase Retrieval (CPR) (Moravec et al. )

Imaging Results with Compressive Phase Retrieval (CPR) 6. 4 cm 6 cm Resolution: 1. 875 mm Fourier Transform of object (Magnitude-only) CPR Reconstruction (4096 measurements)

Compressed Sensing Phase Retrieval (CSPR) Results • Modified CPR algorithm with CS 6 cm 6. 4 cm Fourier Transform of object (Magnitude-only) CPR Reconstruction CSPR Reconstruction (4096 measurements) (1000 measurements)

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? • Continuous-Wave (CW) THz imaging with a detector array • Real-time imaging (Lee A W M, et al. , Appl. Phys. Lett. vol. 89, 141125 (2006))

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

Random 0/1 Bernoulli … …. 001010…. … Measurement matrix

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

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

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

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

Mittleman Group (http: //www. ece. rice. edu/~daniel) Contact info: William Chan (wailam@rice. edu) Acknowledgement Dr. Daniel Mittleman Dr. Richard Baraniuk Dr. Kevin Kelly Matthew Moravec Dharmpal Takhar Kriti Charan 43 dsp. rice. edu/cs

T-Ray System THz Transmitter Femtosecond Pulse Substrate Lens Ga. As Substrate Picometrix T-Ray Instrumentation System Picometrix T-Ray Transmitter Module + DC Bias - Femtosecond Pulse 44

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

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

Sampling • Signal is • Samples measurements -sparse 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 sparse signal nonzero entries 47