Terahertz Imaging with Compressed Sensing

1
Terahertz Imaging with Compressed Sensing
Wai Lam Chan
Department of Electrical and Computer
Engineering Rice University, Houston, Texas, USA
December 17, 2007
2
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)
THz Imaging (Chan, Pearce)
3
T-rays and Imaging
4
What Are T-Rays?
T-Rays
X-Rays
Radio Waves
Hz
Visible Light
Microwaves
Gamma Rays
5
Imaging Throughout History
Daguerreotype (1839)
X-rays (1895)
T-rays (1995)
http//inventors.about.com/library/inventors/bldag
uerreotype.htm
http//inventors.about.com/library/inventors/blxra
y.htm
B. B. Hu and M. C. Nuss, Opt. Lett., 20, 1716,
1995
6
Why Can T-Rays Help?
Subpicosecond pulses
Linear Phase
Over 1 THz in Bandwidth
T-Rays Provide
Benefits to Imaging

• Travel-time / Depth Information
• High depth resolution
• High spatial resolution

• Measurement of E(t)
• Subpicosecond pulses
• Submillimeter Wavelengths

7
Material Responses to T-rays
Plastics
Transparent
Metal
Highly Reflective
Water
Strongly Absorbing
8
Promising Applications of T-Rays
Medical Imaging
(Kawase, Optics Photonics News, October 2004)
Diseased Tissue
Concealed Weapon
Wallace, V. P., et. al. Faraday Discuss. 126, 255
- 263 (2004).
Security
Safety
Zandonella, C. Nature 424, 721722 (2003).
(Karpowicz, et al., Appl. Phys. Lett. vol. 86,
054105 (2005))
Space Shuttle Foam
9
THz Time-domain Imaging
THz Transmitter
THz Receiver
Object
10
THz Time-domain Imaging

• Pixel-by-pixel scanning
• Limitations acquisition time vs. resolution
• Faster imaging method

Just take fewer samples!
11
Compressed Sensing (CS)
Candes et al, Donoho
12
Why CS works Sparsity

• Many signals can be compressed in some
  representation/basis (Fourier, wavelets, )

13
High-speed THz Imaging with Compressed Sensing
(CS)

• Take fewer ( ) measurements

• Reconstruct via nonlinear processing
  (optimization)

(Donoho, IEEE Trans. on Information Theory,
52(4), pp. 1289 - 1306, April 2006)
14
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
R
measurements
sparsesignal (image)
Measurement matrix
information rate
15
Compressed Sensing (CS) Theory

• Signal is -sparse
• Few linear projections
• Random measurements will work!

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
R
measurements
sparsesignal (image)
Measurement matrix (e.g., random)
information rate
16
Random can be
Random 0/1 (Bernoulli)

1
2
M
Random 2-D Fourier

1
2
M
and many others
17
CS Signal Recovery

• Reconstruction/decoding given(ill-posed
  inverse problem) find

sparsesignal
measurements
nonzeroentries
18
CS Signal Recovery

• Reconstruction/decoding given(ill-posed
  inverse problem) find
• L2 fast, wrong

19
CS Signal Recovery

• Reconstruction/decoding given(ill-posed
  inverse problem) find
• L2 fast, wrong
• L0 correct, slow only MK1 measurements
  required to perfectly reconstruct
  K-sparse signal Bresler Rice

number ofnonzeroentries
20
CS Signal Recovery

• Reconstruction/decoding given(ill-posed
  inverse problem) find
• L2 fast, wrong
• L0 correct, slow
• L1 correct, mild oversampling Candes et
  al, Donoho

linear program
21
CS in Action Part I CS-THz Fourier Imaging
22
THz Fourier Imaging Setup
object mask
THz transmitter (fiber-coupled PC antenna)
metal aperture
THz receiver
R
6cm
6cm
6cm
6cm
automated translation stage
23
THz Fourier Imaging Setup
Fourier plane
object mask
N Fourier samples
THz transmitter
R
6cm
6cm
6cm
24
Random 2-D Fourier


Measurement matrix
25
THz Fourier Imaging Setup
automated translation stage
THz receiver
object mask R (3.5cm x 3.5cm)
polyethlene lens
26
Fourier Imaging Results
6.4 cm
4.5 cm
R
6.4 cm
4.5 cm
Resolution 1.125 mm
Inverse Fourier Transform Reconstruction
(zoomed-in)
Fourier Transform of object (Magnitude)
27
Imaging Results with CS
4.5 cm
4.5 cm
CS Reconstruction (500 measurements)
CS Reconstruction (1000 measurements)
Inverse FT Reconstruction (4096 measurements)
28
Imaging Using the Fourier Magnitude
object mask
metal aperture
THz receiver
THz transmitter
R
6cm
6cm
variable object position
translation stage
29
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. 2758 - 2769,
August 1982)
(Moravec et al.)
30
Imaging Results with Compressive Phase Retrieval
(CPR)
6 cm
6.4 cm
R
6 cm
6.4 cm
Resolution 1.875 mm
Fourier Transform of object (Magnitude-only)
CPR Reconstruction (4096 measurements)
31
Compressed Sensing Phase Retrieval (CSPR) Results

• Modified CPR algorithm with CS

6.4 cm
6 cm
6 cm
6.4 cm
Fourier Transform of object (Magnitude-only)
CPR Reconstruction (4096 measurements)
CSPR Reconstruction (1000 measurements)
32
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)

33
CS in ActionPart II Single-Pixel THz Camera
34
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))
35
Single-Pixel Camera (Visible Region)
R
DSP
imagereconstruction
DMD
DMD
Random pattern on DMD array
(Baraniuk, Kelly, et al. Proc. of Computational
Imaging IV at SPIE Electronic Imaging, Jan 2006)
36
Random 0/1 Bernoulli

.001010.

Measurement matrix
37
Random patterns for CS-THz imaging

• Random patterns on printed-circuit boards (PCBs)

38
THz Single-Pixel Camera Setup
Random pattern on PCBs
object mask
THz transmitter (fiber-coupled PC antenna)
THz receiver
R
7cm
42cm
6cm
39
THz Single-Pixel Camera Imaging Result
CS resconstruction (200 measurements)
CS resconstruction (400 measurements)
Object mask
40
THz Single-Pixel Camera Imaging Result
CS resconstruction (400 measurements)
CS resconstruction (200 measurements)

• image phase?

41
CS in ActionPart 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

42
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

• Mittleman Group (http//www.ece.rice.edu/daniel)
• Contact info William Chan (wailam_at_rice.edu)
  

Acknowledgement Dr. Daniel Mittleman Dr. Richard
Baraniuk Dr. Kevin Kelly Matthew
Moravec Dharmpal Takhar Kriti Charan
dsp.rice.edu/cs
44
T-Ray System
THz Transmitter
Femtosecond Pulse
Substrate Lens
GaAs Substrate
Picometrix T-Ray Instrumentation System
Picometrix T-Ray Transmitter Module
Femtosecond Pulse
DC Bias
45
T-Ray System
Sample
THz Transmitter
THz Receiver
Optical Fiber
T-Ray Control Box with Scanning Delay Line
Fiber Coupled Femtosecond Laser System
46
Summary of T-Rays

• Broad fractional bandwidth
• Direct measurement of E(t)
• Short wavelengths (good depth resolution)
• Unique material responses

47
Sampling

• Signal is -sparse
• Samples

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
R
sparsesignal
measurements
nonzeroentries