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– 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

)(log)()( 2 xpxpXH

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

Introduction to CS

sx

sxy

Consider a 1 dimensional signal x of length N and a transform basis s.

Transform matrix of dimension NxN

We require a sensing matrix which is not sparse when transformed (incoherence or restricted isometry property).

Random, length N, sensing matrices satisfy this requirement.

Introduction to CS

Nk

NkM 2log

N

iis

1

min

M random measurements needed to recover signal

Reconstruct image using l1 norm minimization

sy

We use Gradient Projection for Sparse Reconstruction algorithm (noise tolerant BPDN)

Figueiredo et al, IEEE Selected Topics in Signal Processing, 2007

Compressive sensingScene

Reflected light

DMD

Photodiode

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!

Compressive sensing

Why CS?

• Sampling rate – 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 Pros•32x32 APD detectors•Time of Flight Measurements•High Signal to Noise (no amplification noise)•Foliage Penetrating, Aerosol •Low Light Level DetectionCons•Low Fill Factor•Difficult to Scale•Expensive•Resource Heavy•Visible Wavelengths•Large Payload

MIT LL

JIGSAW

Experimental Setup

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

Results: 3D Imaging

U R

Wall

Imaging Through Obscurants

Low Flux 3D Object Tracking

Swinging Ball Trajectory

Frame by Frame

Depth Map of Natural Scene

2nd-Order Correlations

• >7 bits mutual information in X and P• P. Ben Dixon et al PRL 108, 143603 (2012)

Slow Method

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 CameraGregory A. Howland and John C. HowellPhys. 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