Terahertz Imaging with Compressed Sensing

Department of Electrical and Computer EngineeringRice University, Houston, Texas, USA

Wai Lam Chan

December 17, 2007

2

Mittleman Group (http://www.ece.rice.edu/~daniel)

THz Near-field microscopy (Zhan, Astley)

THz Imaging (Chan, Pearce)

THz Photonic Crystal structures (Prasad, Jian)

THz waveguides (Mendis, Mbonye, Diebel, Wang)

THz emission spectroscopy (Laib, Zhan)

Terahertz (THz) Research Group at Rice

T-rays and Imaging

What Are T-Rays?

100 103 106 109 101

2

101

5

101

8

102

1

T-Rays

Radio Waves

Microwaves

X-Rays

Gamma Rays

Visible Light

Hz

Imaging Throughout History

Daguerreotype (1839)

http://inventors.about.com/library/inventors/bldaguerreotype.htm

X-rays (1895)

http://inventors.about.com/library/inventors/blxray.htm

T-rays (1995)

B. B. Hu and M. C. Nuss, Opt. Lett., 20, 1716, 1995

Why Can T-Rays Help?

0 20 40 60 80 100

Time (ps)

0.2 0.4 0.6 0.8 1.0

Frequency (THz)

0.2 0.4 0.6 0.8 1.0

Frequency (THz)

E(t) E(f) |E(f)|

•Measurement of E(t)

•Subpicosecond pulses

•Submillimeter Wavelengths

T-Rays Provide

•Travel-time / Depth Information

•High depth resolution

•High spatial resolution

Benefits to Imaging

Subpicosecond pulses Linear Phase Over 1 THz in Bandwidth

Material Responses to T-rays

Water

Metal

Plastics

Strongly Absorbing

Highly Reflective

Transparent

8

Promising Applications of T-Rays

(Karpowicz, et al., Appl. Phys. Lett. vol. 86, 054105 (2005))

Zandonella, C. Nature 424, 721–722 (2003).

Space Shuttle Foam

Wallace, V. P., et. al. Faraday Discuss. 126, 255 - 263 (2004).

Diseased Tissue

Medical Imaging

Safety

SecurityConcealed Weapon

(Kawase, Optics & Photonics News, October 2004)

THz Time-domain Imaging

Object

THz TransmitterTHz Receiver

THz Time-domain Imaging

Object

THz TransmitterTHz Receiver

• 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 largewaveletcoefficients

widebandsignalsamples

largeGaborcoefficients

• Reconstruct via nonlinear processing (optimization)

• Take fewer ( ) measurements

High-speed THz Imaging with Compressed Sensing (CS)

Measurements(projections)

(Donoho, IEEE Trans. on Information Theory, 52(4), pp. 1289 - 1306, April 2006)

“sparse” signal / object(K-sparse)

MeasurementMatrix

M << N

• Signal is -sparse• Few linear projections

Compressed Sensing (CS) Theory

1 2 3 4

5 6 7 8

9 10

11

12

13

14

15

16

sparsesignal (image)

informationrate

measurements

Measurement matrix

• Signal is -sparse• Few linear projections

• Random measurements will work!

Compressed Sensing (CS) Theory

1 2 3 4

5 6 7 8

9 10

11

12

13

14

15

16

sparsesignal (image)

informationrate

measurements

Measurement matrix(e.g., random)

Random can be …

…

1 2 M

…

1 2 M

Random 0/1

(Bernoulli)

Random

2-D Fourier

and many others …

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

CS Signal Recovery

measurementssparsesignal

nonzeroentries

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

• L2 fast, wrong

CS Signal Recovery

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

• L2 fast, wrong

• L0 correct, slowonly M=K+1 measurements required to perfectly reconstruct K-sparse signal[Bresler; Rice]

CS Signal Recovery

number ofnonzeroentries

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

• L2 fast, wrong

• L0 correct, slow

• L1 correct, mild oversampling [Candes et al, Donoho]

CS Signal Recovery

linear program

CS in Action Part I: CS-THz Fourier

Imaging

THz Fourier Imaging Setup

6cm 6cm 6cm

objectmask

THz transmitter (fiber-coupled PC antenna)

THz receiver

6cm

metal aperture

automated translation stage

N Fourier samples

THz Fourier Imaging Setup

6cm6cm

objectmask

THz transmitter

6cm

Fourier plane

pick only random measurements for

Compressed Sensing

Random 2-D Fourier

…

Measurement matrix…

THz Fourier Imaging Setup

automated translation

stage

polyethlene lens

object mask “R”(3.5cm x 3.5cm)

THz receiver

Fourier Imaging Results

Fourier Transform of object (Magnitude)

Inverse Fourier Transform Reconstruction (zoomed-in)

6.4 cm 4.5 cm

6.4

cm

4.5

cm

Resolution: 1.125 mm

Imaging Results with CS

Inverse FT Reconstruction

(4096 measurements)

CS Reconstruction (500 measurements)

4.5 cm

4.5

cm

CS Reconstruction (1000 measurements)

Imaging Using the Fourier Magnitude

6cm

objectmask

THz transmitterTHz receiver

6cm

metalaperture

translationstage

variable objectposition

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.)

Imaging Results with Compressive Phase Retrieval (CPR)

6 cm

6 cm

Resolution: 1.875 mm

Fourier Transform of object (Magnitude-only)

CPR Reconstruction(4096 measurements)

6.4 cm

6.4

cm

Compressed Sensing Phase Retrieval (CSPR) Results

• Modified CPR algorithm with CS

Fourier Transform of object

(Magnitude-only)

CPR Reconstruction (4096 measurements)

CSPR Reconstruction (1000 measurements)

6.4 cm

6.4

cm

6 cm

6 c

m

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 ActionPart II: Single-Pixel THz

Camera

Imaging with a Single-Pixel detector?

(Lee A W M, et al., Appl. Phys. Lett. vol. 89, 141125 (2006))

• Continuous-Wave (CW) THz imaging with a detector array

• Real-time imaging

Single-Pixel Camera (Visible Region)

DMD

Random pattern onDMD array

(Baraniuk, Kelly, et al. Proc. of Computational Imaging IV at SPIE Electronic Imaging, Jan 2006)

imagereconstruction

DSP

DMD

Random 0/1 Bernoulli

…

Measurement matrix

…

….001010….

Random patterns for CS-THz imaging

• Random patterns on printed-circuit boards (PCBs)

THz Single-Pixel Camera Setup

THz receiver

Random pattern on

PCBsTHz transmitter (fiber-coupled PC antenna)

object mask

7cm6cm 42cm

THz Single-Pixel Camera Imaging Result

Object maskCS resconstruction

(200 measurements)CS resconstruction

(400 measurements)

THz Single-Pixel Camera Imaging Result

CS resconstruction (400 measurements)

CS resconstruction (200 measurements)

• image phase?

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

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

43dsp.rice.edu/cs

Mittleman Group (http://www.ece.rice.edu/~daniel)

Contact info: William Chan

(wailam@rice.edu) Acknowledgement

Dr. Daniel MittlemanDr. Richard BaraniukDr. Kevin Kelly

Matthew MoravecDharmpal TakharKriti Charan

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T-Ray System

THz Transmitter

Substrate LensFemtosecond Pulse

GaAs Substrate

DC Bias

Picometrix T-Ray Instrumentation System

Picometrix T-Ray Transmitter Module

Femtosecond Pulse

45

T-Ray System

T-Ray Control Box with Scanning Delay Line

Fiber Coupled Femtosecond Laser System

Sample

THz Transmitter THz Receiver

Optical Fiber

46

Summary of T-Rays

• Broad fractional bandwidth

• Direct measurement of E(t)

• Short wavelengths (good depth resolution)

• Unique material responses

47

• Signal is -sparse

• Samples

sparsesignal

nonzeroentries

measurements

Sampling1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16