Optical Architectures for Compressive Imaging

1. 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

2. Photons Object N - Detector Array Direct Image features M - Detector Array Object Compressive Imaging Conventional Feature-Specific • 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) random projections PCA, ICA, Wavelets, Fisher, … DCT, Hadamard, …

3. Feature-Specific Imaging for Reconstruction • PCA features provide optimal measurements in the absence of noise • Limit attention to linear reconstruction operators Noise-free reconstruction: PCA solution : General solution : photon count constraint depends on optics Result using PCA features:

4. FPCA RMSE = 12 RMSE = 11.8 FOPT RMSE = 12.9 RMSE = 124 What Happens in the Presence of Noise ? Problem statement with Noise: Stochastic tunneling to find optimal • 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

5. 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

6. Feature-Specific Imaging Results Summary • 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

7. N - Detector Array Noisy Measurements S y = x + n Object= x Imaging Optics Noise = n Candidate Architectures • Architecture Comparison Assumes • - Equal total photon number • - Equal total measurement time Architecture #1: Conventional Imaging Characteristics • All N measurements made in a single time step (noise BW ~ 1/T) • Photons divided among many detectors (measured signal ~ 1/N)

8. Noisy Measurement Single Photo-Detector Programmable Mask S n yi = fi x + n Object= x Light Collection Optics Imaging Optics Candidate Architectures 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

9. Noisy Measurements Noise = n Fixed Mask S M – Detector Array Object= x {yi = fi x + n, i=1, …, M} Imaging m-Optics Candidate Architectures Architecture #3: Parallel Compressive Imaging 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

10. y1 = f1 x + n y2 = f2 x + n Polarization Mask Re-Imaging Optics PBS PBS Imaging Optics Object= x n + n + Single Detector Candidate Architectures Architecture #4: Pipeline Compressive Imaging 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

11. Architecture Comparison – Full Image FSI • 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. Reconstruction RMSE versus SNR

12. Architecture Comparison – Full Image FSI • RMSE Trend 1: Pipeline < Parallel < Sequential • RMSE Trend 2: Compressive < Conventional for low SNR

13. Architecture Comparison – Full Image FSI

14. Architecture Comparison – Blockwise FSI • Evaluate compressive imaging architectures for 16x16 block-wise feature extraction • All other conditions remain unchanged Reconstruction RMSE versus SNR • Block-wise operation shifts crossover to larger SNR • RMSE Trend 1: Pipeline < Parallel < Sequential • RMSE Trend 2: Compressive < Conventional for low SNR

15. Architecture Comparison – Full Image FSI

16. Conclusions • 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.