Optical Architectures for Compressive Imaging

Mark A. Neifeld and Jun Ke Electrical and Computer

Engineering Department College of Optical Sciences University of

Arizona

OUTLINE

1. Compressive (a.k.a. Feature-Specific) Imaging

2. Candidate Optical Architectures

3. Results and Conclusions

Compressive Imaging

Conventional imagers measure a large number (N) of pixels Compressive imagers measure a small number (M<<N) of features

Features are simply projections yi = (x ∙ fi) for i = 1, …, M

Benefits of projective/compressive measurements include - increased measurement SNR improved image fidelity

- more informative measurements reduced sensor power and bandwidth

- enable task-specific imager deployment information optimal Previous feature-specific imaging: Neifeld (2003), Brady (2005), Donoho (2004), Baraniuk (2005)

PCA, ICA, Wavelets, Fisher, …DCT, Hadamard, …

random projections

Object N - Detector Array Direct Image

Photons

M - Detector Array featuresObject

Conventional Feature-Specific

2

11

ˆ

ˆmin {|| || }

|| || max{ | |} 1m

iji

j

E

subject to f

y Fx x My

x x

F

Noise-free reconstruction:

= pca pca pcaTF M F

PCA solution :

1( )general T

x xM R F FR F

is any invertable matrix

pcaF AF

A

General solution :

Result using PCA features:

PCA features provide optimal measurements in the absence of noise Limit attention to linear reconstruction operators

photon count constraint depends on optics

Feature-Specific Imaging for Reconstruction

2 -1( I) Wiener - operatoropt T Tx xM R F FR F

2 2 -1{ ( I) } { }Tr Tr T Tx x xFR F FR F R

MyxnFxy ˆ

)}||{||

log(102

2

xE

SNR

Problem statement with Noise:

What Happens in the Presence of Noise ?

PCA features are sub-optimal in the presence of noise. PCA tradeoff between truncation error and measurement fidelity Optimal features improve performance by

- using optimal basis functions- using optimal energy allocation

RMSE = 12.9 RMSE = 124

FPCA

RMSE = 11.8RMSE = 12

FOPT

Stochastic tunneling to find optimal

Optimal Features in Noise

Optimal solution always improves with number of features FSI results can be superior to conventional imaging Optical implementation require non-negative projections

Optimal FSI is always superior to conventional imaging Non-negative solution is a good experimental system candidate All these results rely on optimal photon utilization

Feature-Specific Imaging Results Summary

Architecture #1: Conventional Imaging

y = x + n

Object = x

N - Detector Array

Imaging OpticsNoise = n

Noisy Measurements

Characteristics

All N measurements made in a single time step (noise BW ~ 1/T) Photons divided among many detectors (measured signal ~ 1/N)

Candidate Architectures

Architecture Comparison Assumes

- Equal total photon number

- Equal total measurement time

Architecture #2: Sequential Compressive Imaging

Characteristics

A single feature is measured in each time step (noise BW ~ M/T) Photons collected on a single detector (measured signal ~ 1/M) Unnecessary photons discarded in each time step (1/2) Reconstruction computed via post-processing

Imaging Optics Light Collection Optics

Programmable Mask

Single Photo-

Detector

yi = fi x + n

n

Noisy Measurement

Object = x

Candidate Architectures

Characteristics

All M features are measured in a single time step (noise BW ~ 1/T) Photons collected on M << N detectors (measured signal ~ 1/M) Unnecessary photons discarded in each channel (1/2) Reconstruction computed via post-processing

Imaging -Optics

Fixed Mask

M – Detector Array

{yi = fi x + n, i=1, …, M}

Noise = n Noisy Measurements

Object = x

Architecture #3: Parallel Compressive Imaging

Candidate Architectures

Imaging Optics

Polarization Mask

Object = xSingle

Detector

y1 = f1 x + n

Re-Imaging OpticsPBS

+ n

y2 = f2 x + n

PBS

+ n

Characteristics All M features are measured in a single time step (noise BW ~ 1/T) Photons collected on M << N detectors (measured signal ~ 1/M) No discarded photons most photon efficient Most complex hardware Reconstruction computed via post-processing

Architecture #4: Pipeline Compressive Imaging

Candidate Architectures

All results use 80x80 pixel object. All results use PCA features with optimal energy allocation among time/space channels. All results use optimal linear post-processor (LMMSE) for reconstruction. Measurement noise is assumed AWGN iid.

Architecture Comparison – Full Image FSI

Reconstruction RMSE versus SNR

RMSE Trend 1: Pipeline < Parallel < Sequential RMSE Trend 2: Compressive < Conventional for low SNR

Architecture Comparison – Full Image FSI

Architecture Comparison – Full Image FSI

Evaluate compressive imaging architectures for 16x16 block-wise feature extraction All other conditions remain unchanged

Block-wise operation shifts crossover to larger SNR RMSE Trend 1: Pipeline < Parallel < Sequential RMSE Trend 2: Compressive < Conventional for low SNR

Architecture Comparison – Blockwise FSI

Reconstruction RMSE versus SNR

Architecture Comparison – Full Image FSI

Compressive imaging (FSI) exploits projective measurements.

There are many potentially useful measurement bases.

FSI can yield high-quality images with relatively few detectors.

FSI can provide performance superior to conventional imaging for low SNR.

Various optical architectures for FSI are possible.

FSI Reconstruction fidelity is ordered as Sequential, Parallel, Pipeline.

Pipeline offer best performance owing to optimal utilization of photons.

Conclusions