1. Entropy, Information and Compressive Sensing in the Quantum Domain.
John Howell
Greg Howland
James Schneeloch
Daniel Lum
Sam Knarr
Clemente Cuevas(REU)
Matt Ware (REU)
Robert Boyd
Cliff Chan
Petros Zerom

2. Outline Introduction to compressive sensing Shannon entropy
Nyquist sampling
Lossy compression
K-sparse sensing
L1 norm reconstruction
Advantages
Applications
Ghost imaging with entangled photons
Photon counting Lidar
Depth Maps
Object tracking
High dimensional entanglement characterization

3. Shannon Entropy Entropy Example (alphanumeric)
Measure uncertainty of random variable X with distribution p(x).
Find number of symbols and bits per symbol (e.g., 0 or 1 binary)
Compression removes intersymbol correlations
Example (alphanumeric)
_ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ R _ _ E _ _ _ _ _
Q _ _ _ R _ _ E _ T _ _ _
Q S T A R T M E E T _ N G
Shannon showed there are approximately 1 bit per symbol in English language

4. Information Theorist  Marriage Therapist
“I see that your channel capacity is quite low. You need to spend more time maximizing your mutual information. Please increase the entropy of your communication while minimizing the noise in your classical channels.”

5. Shannon Entropy for Images: Compression After Sensing
Compression removes interpixel correlations
Decompose in decorrelated transform basis
k-sparse representation
DCT, DFT, wavelets etc.
Significant reduction in memory requirements, reduced uncertainty

6. Standard Sensing Paradigm (e.g., jpeg)
Sample (at least two times above Nyquist)
Transform to sparse basis
Preferentially attenuate high frequency components
Round coefficients
Inverse transform

7. Sensing Paradigms Typical Sensing: Compress after you sense
Compressed sensing: Compress while you sense

8. CS Literature of Interest
Tutorials on CS
R.G. Baraniuk, IEEE SIGNAL PROCESSING MAGAZINE [118] JULY 2007
E.J. Candes and M.J. Wakin, IEEE SIGNAL PROCESSING MAGAZINE [21] MARCH 2008
Single Pixel Camera
Duarte et al, IEEE SIGNAL PROCESSING MAGAZINE [83] MARCH 2008

9. Transform matrix of dimension NxN
Introduction to CS
Consider a 1 dimensional signal x of length N and a transform basis s.
Transform matrix of dimension NxN
We require a sensing matrix F which is not sparse when transformed (incoherence or restricted isometry property).
Random, length N, sensing matrices satisfy this requirement.

10. Introduction to CS M random measurements needed to recover signal
Reconstruct image using l1 norm minimization
We use Gradient Projection for Sparse Reconstruction algorithm (noise tolerant BPDN)
Figueiredo et al, IEEE Selected Topics in Signal Processing, 2007

11. Compressive sensing
Reflected light
Scene
Photodiode
DMD

12. Compressive sensing
We take M << N measurements with different random DMD patterns and then reconstruct x
Outside information is needed to solve our underdetermined linear system to reconstruct our image!

13. Why CS? Sampling rate Resource efficient Above information rate
NOT above Nyquist rate.
Resource efficient
Single pixel measurements
Fewer measurements
Automatically finds large k-sparse coefficients

14. Cool stuff recently done
“Compressive Sensing Hyperspectral Image” T. Sun and K. Kelly (COSI) (2009)
“Sparsity-based single-shot sub-wavelength coherent diffractive imagin” A. Szameit et al (M. Segev), Nature Materials 11, 455 (2012)
“Compressive Depth Map Acquisition Using a Single Photon-Counting Detector” A. Colaco et al Proc. IEEE Conf. Computer Vision and Pattern Recognition (2012)
“Compressive Sensing for Background Subtraction” Volkan Cevher, (Baraniuk)
3D COMPUTATIONAL IMAGING WITH SINGLE-PIXEL DETECTORS (Padgett) Science 340, 844 (2013)

15. Some of our applications
Ghost imaging with entangled photons
Photon counting Lidar
Depth Maps
Object tracking
High dimensional entanglement characterization

16. Entangled Photon Compressive Imaging

17. Comparison to Raster Scanning
Compressive Sensing
4500 measurements
N=128x128 pixels
9 seconds/measurement
SNR 8
Acquisition time
4500 x 9s~ ½ day
Raster Scanning
For same SNR and resolution it would take almost 3 years to acquire image with same flux

18. LIDAR with JIGSAW MIT LL Pros 32x32 APD detectors
Time of Flight Measurements
High Signal to Noise (no amplification noise)
Foliage Penetrating, Aerosol
Low Light Level Detection
Cons
Low Fill Factor
Difficult to Scale
Expensive
Resource Heavy
Visible Wavelengths
Large Payload
MIT LL

19. JIGSAW

20. G. Howland, P.B. Dixon and J.C. Howell, Appl. Optics 50, 5917 (2011)
Experimental Setup
G. Howland, P.B. Dixon and J.C. Howell, Appl. Optics 50, 5917 (2011) 20

21. Results: 3D Imaging
Wall U R 21

22. Imaging Through Obscurants

23. Low Flux 3D Object Tracking

24. Swinging Ball Trajectory

25. Frame by Frame

26. Depth Map of Natural Scene

27. 2nd-Order Correlations
Slow Method
>7 bits mutual information in X and P
P. Ben Dixon et al PRL 108, (2012)

28. With CS Replace Raster with CS N log N scaling rather than N3 to N4.
8 hours instead of a year
Efficient High-Dimensional Entanglement Imaging with a Compressive-Sensing Double-Pixel Camera
Gregory A. Howland and John C. Howell
Phys. Rev. X 3, (2013).

29. 32x32 Position Position Correlations (3 raster)

30. Mutual Information in X and P
Violation of Continuous-Variable Einstein-Podolsky-Rosen Steering with Discrete Measurements J. Schneeloch, P. Ben Dixon, G. A. Howland, C. J. Broadbent, and J. C. Howell Phys. Rev. Lett. 110, (2013).

31. Background Subtraction Object Tracking
Compressive object tracking using entangled photons
Omar S. Magana-Loaiza, Gregory A. Howland, Mehul Malik, John C. Howell, and Robert W. Boyd Appl. Phys. Lett (2013).

32. Ghost Object Tracking

33. Novel Acquisition Paradigm
Quantum imaging
Entanglement mutual information
Low flux LIDAR
Precision measurements
Real-time video