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

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)
0
20
40
60
|E(f)|
E(f)
80
100
0.2
0.4
0.6
0.8
Time (ps)
Frequency (THz)
Subpicosecond pulses
Linear Phase
T-Rays Provide
1.0
0.2
0.4
0.6
0.8
Frequency (THz)
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
1.0
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)
• L2
fast, wrong
given
find
CS Signal Recovery
• Reconstruction/decoding:
(ill-posed inverse problem)
• L2
fast, wrong
• L0
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)
• L2
fast, wrong
• L0
correct, slow
• L1
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
6cm
metal
aperture
6cm
THz receiver
6cm
automated
translation stage
THz Fourier Imaging Setup
Fourier plane
THz transmitter
6cm
object
mask
N Fourier
samples
6cm
6cm
pick only
random
measurements for
Compressed Sensing
Random 2-D Fourier
…
…
Measurement matrix
THz Fourier Imaging Setup
THz receiver
object mask “R”
(3.5cm x 3.5cm)
automated
translation
stage
polyethlene
lens
Fourier Imaging Results
6.4 cm
4.5 cm
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
4.5 cm
CS Reconstruction
CS Reconstruction
Inverse FT
(500 measurements) (1000 measurements)
Reconstruction
(4096 measurements)
Imaging Using the Fourier Magnitude
object
mask
THz transmitter
6cm
variable object
position
metal
aperture
THz receiver
6cm
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
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 cm
6.4 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
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
6cm
42cm
7cm
THz Single-Pixel Camera Imaging Result
Object mask
CS resconstruction
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 ([email protected])
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
GaAs 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