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 y i = (x ∙ f i ) 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 features Object Conventional Feature-Specific

Noise-free reconstruction: PCA solution : 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

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 = 12.9RMSE = 124 F PCA RMSE = 11.8RMSE = 12 F OPT 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 Optics Noise = 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 y i = f i x + n nNoisy 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 {y i = f i x + n, i=1, …, M} Noise = n Noisy Measurements Object = x Architecture #3: Parallel Compressive Imaging Candidate Architectures

Imaging Optics Polarization Mask Object = x Single Detector y 1 = f 1 x + n Re-Imaging Optics PBS + n y 2 = f 2 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

 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