Terahertz Imaging with Compressed Sensing

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					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)




                                                                                                    2
THz emission spectroscopy (Laib, Zhan)                   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/   http://inventors.about.com/library/   B. B. Hu and M. C. Nuss, Opt.
inventors/bldaguerreotype.htm         inventors/blxray.htm                  Lett., 20, 1716, 1995
                 Why Can T-Rays Help?
                  E(t)                                E(f)                               |E(f)|




0   20     40   60       80   100   0.2   0.4   0.6      0.8     1.0   0.2    0.4   0.6     0.8    1.0
          Time (ps)                   Frequency (THz)                    Frequency (THz)
    Subpicosecond pulses                  Linear Phase                 Over 1 THz in Bandwidth



            T-Rays Provide                                     Benefits to Imaging
         • Measurement of E(t)                        • Travel-time / Depth Information
         • Subpicosecond pulses                       • High depth resolution
         • Submillimeter Wavelengths                  • High spatial resolution
        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)
                      Concealed          Diseased Tissue     Wallace, V. P., et. al. Faraday Discuss.

Security              Weapon                                 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                                large
signal                                  Gabor
samples                                 coefficients
             High-speed THz Imaging with
              Compressed Sensing (CS)


• Take fewer (           ) measurements




    Measurements             Measurement        “sparse” signal / object
     (projections)              Matrix                (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
                                           1    2    3    4
• Few linear projections
                                           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
                                           information
                        (e.g., random)
                                               rate


• Random measurements        will work!
              Random       can be …


Random 0/1
(Bernoulli)                   …
              1        2              M


Random
2-D Fourier                   …

               1       2              M

                            and many others …
               CS Signal Recovery

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




                                        sparse
    measurements                        signal




                                        nonzero
                                        entries
               CS Signal Recovery

• 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, slow
           only M=K+1
           measurements
           required to
           perfectly reconstruct           number of
           K-sparse signal                   nonzero
                                              entries
           [Bresler; Rice]
               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
          CS in Action
Part I: CS-THz Fourier Imaging
        THz Fourier Imaging Setup

THz transmitter         object
 (fiber-coupled         mask            metal     THz receiver
  PC antenna)                          aperture




                  6cm            6cm   6cm
                                                     automated
                                                  translation stage
        THz Fourier Imaging Setup
                                   Fourier plane
                    object                           N Fourier
                    mask
THz transmitter                                       samples




              6cm            6cm         6cm

                                         pick only   random
                                           measurements for
                                          Compressed Sensing
Random 2-D Fourier



   …                        …




       Measurement matrix
           THz Fourier Imaging Setup


                  THz receiver                 automated
                                               translation
                                                  stage




object mask “R”
(3.5cm x 3.5cm)                  polyethlene
                                     lens
              Fourier Imaging Results

                6.4 cm                    4.5 cm




                                                             4.5 cm
6.4 cm




                                   Resolution: 1.125 mm


         Fourier Transform of   Inverse Fourier Transform
          object (Magnitude)    Reconstruction (zoomed-in)
             Imaging Results with CS

          4.5 cm
4.5 cm




         Inverse FT     CS Reconstruction   CS Reconstruction
       Reconstruction  (500 measurements) (1000 measurements)
   (4096 measurements)
     Imaging Using the Fourier Magnitude



                        object
                                                metal
                        mask                              THz receiver
THz transmitter                                aperture




                  6cm        variable object   6cm
                                 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:
  – 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 cm
6.4 cm




                                   Resolution: 1.875 mm

           Fourier Transform of    CPR Reconstruction
         object (Magnitude-only)   (4096 measurements)
            Compressed Sensing Phase Retrieval
                     (CSPR) Results
         • Modified CPR algorithm with CS

               6.4 cm                     6 cm
6.4 cm




                             6 cm




         Fourier Transform           CPR Reconstruction CSPR Reconstruction
             of object              (4096 measurements) (1000 measurements)
         (Magnitude-only)
                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              (Lee A W M, et al., Appl. Phys.
                                     Lett. vol. 89, 141125 (2006))
• Real-time imaging
Single-Pixel Camera (Visible Region)




                                             DSP

                                            image
                                        reconstruction
    DMD                   DMD




     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
                        object
                                        pattern on
                        mask
                                          PCBs
THz transmitter
 (fiber-coupled
  PC antenna)
                                             THz receiver




                  6cm            42cm      7cm
  THz Single-Pixel Camera Imaging Result




               CS resconstruction CS resconstruction
Object mask   (200 measurements) (400 measurements)
THz Single-Pixel Camera Imaging Result




  CS resconstruction    CS resconstruction
 (200 measurements)    (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 (wailam@rice.edu)


                                   Acknowledgement

                                   Dr. Daniel Mittleman
                                   Dr. Richard Baraniuk
                                   Dr. Kevin Kelly

                                   Matthew Moravec
                                   Dharmpal Takhar
                                   Kriti Charan




                                   dsp.rice.edu/cs
                                                43
                   T-Ray System
                                                     THz Transmitter

                                             Femtosecond            Substrate Lens
                                                Pulse




                                                   GaAs Substrate
Picometrix T-Ray Instrumentation System




                        Picometrix T-Ray                                             Femtosecond
                        Transmitter Module                                              Pulse
                                                           +   -



                                               DC Bias                                    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   -sparse      1    2    3    4

                               5    6    7    8

• Samples                      9    10   11   12

                               13   14   15   16




                               sparse
   measurements                signal




                               nonzero
                               entries

                                              47

				
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