Terahertz Imaging with Compressed Sensing and Phase Retrieval Wai Lam Chan Matthew Moravec Daniel Mittleman Richard Baraniuk Good afternoon. My name is William Chan. In today’s talk, I would like to introduce to you our latest research at Rice University on THz imaging which incorporates a new signal processing theory called compressed sensing. Department of Electrical and Computer Engineering Rice University, Houston, Texas, USA

THz Time-domain Imaging THz Transmitter THz Receiver Object THz imaging has been studied extensively in time-domain THz beam focused on position on an object System scans object and forms an image based on THz beam transmitted and detected at the receiver at each scan position (Briefly explain traditional setup (transmission/reflection))

THz Time-domain Imaging Object THz Transmitter THz Receiver Examples (no need to distinguish pulsed vs. cw imaging at this point since these examples contain both) Chocolate bar (food) Automobile dashboard (foam layer) Suitcase (weapons) (Karpowicz, et al., Appl. Phys. Lett. vol. 86, 054105 (2005)) (Mittleman, et al., Appl. Phys. B, vol. 68, 1085-1094 (1999))

THz Time-domain Imaging Object THz Transmitter THz Receiver Pixel-by-pixel scanning Limitations: acquisition time vs. resolution Faster imaging method Limitations in acquisition speed vs. resolution Therefore, our THz group, joint effort with Rice signal processing group, designs a new THz imaging system which aims to address this issue Other groups not time-domain : such as transmission/reflection tomography, interferometric THz imaging

High-speed THz Imaging with Compressed Sensing (CS) Take fewer ( ) measurements “sparse” signal / image (K-sparse) information rate R Reconstruct via nonlinear processing (optimization) Measurements (random projections) Measurement Matrix (e.g., random Fourier) So, how can we reduce the image acquisition time? A straightforward, but seemingly naïve solution to take fewer measurements. In time-domain imaging, that’s equivalent to acquiring lower resolution image because one measurement directly corresponds to one pixel. By CS theory, it’s possible to take fewer measurements than we need in a time-domain scan, and still achieve the same image resolution. But we need to modify the design of the system to fit in a CS mathematical model, summarized in one equation. - How to improve speed? By taking fewer measurements … -> compressed sensing - In the traditional regime, we are always over-sampling … - Reconstruction of target from random measurements based on its spatial structure - Impossible with previous THz time-domain transmission/reflection setup (each sample does not contain information about the whole image) (Donoho, IEEE Trans. on Information Theory, 52(4), pp. 1289 - 1306, April 2006)

Compressed Sensing (CS) Example: Single-Pixel Camera DSP image reconstruction DMD DMD Briefly explain concepts in single-pixel camera (explain y, phi, x) Say what “DMD” and “RNG” are: “Digital Micromirror Device” and “Random Number Generator” This single-pixel camera is an example to apply CS to imaging in the optical regime. Advantage: simple hardware requirement (i.e., works well with single THz detector) We are now going to see CS in action for THz imaging, which is the main focus of research in this talk. 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 THz receiver aperture R mention again that this is a pulsed THz system (large fractional bandwidth) and that you are using fiber-couple antennas Explain why we need aperture: silicon dome is too large 6cm 6cm 12cm 12cm 12cm automated translation stage

THz Fourier Imaging Setup pick only random measurements for Compressed Sensing Fourier plane object mask N Fourier samples THz transmitter R Explain scanning each “pixel” is a waveform, pick intensity at one frequency For our simulation, picking only M random measurements, equivalently, scanning a random path for only M positions (save in acquisition time) 6cm 6cm 12cm 12cm 12cm

THz Fourier Imaging Setup THz receiver automated translation stage Beam about 5cm diameter, object about 4cm x 4cm Explain TOPAS Scan speed limited by pixel averaging, stage motion, communication with motion controller object mask “R” (3.5cm x 3.5cm) polyethlene lens

Fourier Imaging Results 8 cm 6 cm R 6 cm 8 cm 150 GHz Describe experimental results (resolution, blurring effect of aperture and limited scan range) Resolution: 3mm Fourier Transform of object (Magnitude) Inverse Fourier Transform Reconstruction (zoomed-in)

Imaging Results with Compressed Sensing (CS) 6 cm R 6 cm Compare results, mention convergence issue Size of object -> # measurements 40x40 -> we need only a subset of 40x40 to reconstruct Inverse Fourier Transform Reconstruction (6400 measurements) CS Reconstruction (1000 measurements)

Imaging Using the Fourier Magnitude object mask THz receiver THz transmitter aperture R Phase only accurate when object is one focal length away from focusing lens 6cm 12cm variable object position 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 R 6.4 cm 8 cm Position of object varies in reconstruction (explain) Resolution: 3.2mm Fourier Transform of object (Magnitude) PR Reconstruction (6400 measurements)

Compressed Sensing Phase Retrieval (CSPR) Results Modified PR algorithm with CS 8 cm 6.4 cm R 6.4 cm 8 cm Fourier Transform of object (Magnitude) PR Reconstruction (6400 measurements) CSPR Reconstruction (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 Ending: do a quick one sentence statement on your summary slide hinting at future directions (improving resolution and other types of objects). We see that this Thz imaging method has great potentials and research is still in progress to …

Acknowledgements National Science Foundation National Aeronautics and Space Administration Finally, I would like to thank these funding agencies for their contribution to our research. Thank you. 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 R ….001010…. Measurement matrix (e.g., random) sparse signal (image) - How to improve speed? By taking fewer measurements … -> compressed sensing - In the traditional regime, we are always over-sampling … - Reconstruction of target from random measurements based on its spatial structure - Impossible with previous THz time-domain transmission/reflection setup (each sample does not contain information about the whole image) information rate

Compressed Sensing (CS) Theory 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 R measurements sparse signal (image) - How to improve speed? By taking fewer measurements … -> compressed sensing - In the traditional regime, we are always over-sampling … - Reconstruction of target from random measurements based on its spatial structure - Impossible with previous THz time-domain transmission/reflection setup (each sample does not contain information about the whole 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)