Optical Architectures for Compressive Imaging Mark A. Neifeld and Jun Ke Electrical and Computer...
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Transcript of Optical Architectures for Compressive Imaging Mark A. Neifeld and Jun Ke Electrical and Computer...
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