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

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

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

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.”

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

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

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

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

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.

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

Compressive sensing Reflected light Scene Photodiode DMD

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!

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

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)

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

Entangled Photon Compressive Imaging

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

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

JIGSAW

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

Results: 3D Imaging Wall U R 21

Imaging Through Obscurants

Low Flux 3D Object Tracking

Swinging Ball Trajectory

Frame by Frame

Depth Map of Natural Scene

2nd-Order Correlations Slow Method >7 bits mutual information in X and P P. Ben Dixon et al PRL 108, 143603 (2012)

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, 011013 (2013).

32x32 Position Position Correlations (3 raster)

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, 130407 (2013).

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. 102 231104 (2013).

Ghost Object Tracking

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