1. 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

2. 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))

3. 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, (2005))
(Mittleman, et al., Appl. Phys. B,
vol. 68, (1999))

4. 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

5. 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 , April 2006)

6. 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)

7. 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

8. 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

9. 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

10. 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)

11. 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)

12. 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

13. 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 , August 1982)

14. 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)

15. 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)

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
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 …

17. 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

18. Compressed Sensing (CS) Theory1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 R
… ….
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

19. Compressed Sensing (CS) Theory1 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

20. 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

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

22. 2-D Wavelet Transform (Sparsity)