Reconstruction with Adaptive Feature-Specific Imaging

1. Reconstruction with Adaptive Feature-Specific Imaging Jun Ke1 and Mark A. Neifeld1,2 1Department of Electrical and Computer Engineering, 2College of Optical Sciences University of Arizona Frontiers in Optics 2007

2. Outline • Motivation for FSI and adaptation. • Adaptive FSI using PCA/Hadamard features. • Adaptive FSI in noise. • Conclusion. Frontiers in Optics 2007

3. object reconstruction object Imaging optics light collection Reconstruction matrix G (NxM) feature single detector DMD projection vector FSI benefits: • Lower hardware complexity • Smaller equipment size/weight • Higher measurement SNR • High data acquisition rate • Lower operation bandwidth • Less power consumption LCD LCD G (NxM) LCD Motivation - FSI Reconstruction with Feature-specific Imaging (FSI) : Sequential architecture: Parallel architecture: Frontiers in Optics 2007

4. Adaptive PCA Static PCA Projection value Training samples for 2nd projection vector Projection axis 2 Testing sample Projection axis 1 Training samples Projection axis 1 Reconstruction Projection axis 2 Reconstruction Motivation - Adaptation • The design of projection vector effects reconstruction quality. • Using Principal Component Analysis (PCA) projection as example • Acquire feature measurements sequentially • Use acquired feature measurements and training data to adapt the next projection vector Frontiers in Optics 2007

5. Object estimate Object x Calculate R1from A1 Reconstruction Computational Optics Calculate f1 yi = fiTx Update Ai to Ai+1 according to yi Calculatefi+1 Ri+1 PCA-Based AFSI Adaptive FSI (AFSI) – PCA: K(1) nearest samples Projection axis 1 Selected samples K(1) nearest samples Projection axis According to 1st feature K(2) nearest samples Testing sample Projection axis 2 According to 2nd feature Testing sample i: adaptive step index Ai: ith training set K(i): # of samples for Ai+1 Ri: autocorrelation matrix of Ai fi: dominate eigenvector of Ai yi: feature value measured by fi • High diversity of training data helps adaptation Frontiers in Optics 2007

6. is the total # of features • Feature measurements: where, • Reconstructed object: • RMSE: PCA-Based AFSI Object examples (32x32): • Number of training objects: 100,000 • Number of testing objects: 60 Frontiers in Optics 2007

7. K(i) decreases iteration index i iteration index i Reconstruction from static FSI (i = 100) Reconstruction from AFSI (i = 100) PCA-Based AFSI AFSI – PCA: • RMSE reduces using more features • RMSE reduces using AFSI compare to static FSI • Improvement is larger for high diversity data • RMSE improvement is 33% and 16% for high and low diversity training data, when M = 250. Frontiers in Optics 2007

8. Hadamard-Based AFSI AFSI – Hadamard: • Projection vector’s implementation order is adapted. First 5 Hadamard basis ←Static FSI AFSI→ projection axis 1 projection axis 2 projection axis 1 sample mean <A1> Selected samples according to 1st feature K(2) nearest samples testing sample testing sample K(1) nearest samples K(1) nearest samples according to 2nd feature sample mean <A2> • Sample mean for training set Ai is <Ai> • yj= fiT<Ai>j = 1,…,M • max{yj} corresponds to the dominant Hadamard projection vector Frontiers in Optics 2007

9. Selected samples according to 1st 2 features <Ai> Object estimate K(1) nearest samples Sort Hadamard basis vectors Reconstruction Object x sample mean Choose f1~fL Computational Optics yiL+j = f iL+jTx (j=1,…,L) projection axis 2 testing sample Choose fiL+1 ~ f(i+1)L Update Ai to Ai+1 according to yiL+j projection axis 1 Sort <Ai+1> Hadamard-Based AFSI AFSI – Hadamard: • L : # of features in each adaptive step <Ai>: sample mean of Ai fi: ith Hadamard vector for Ai Frontiers in Optics 2007

10. L decreases L increases K(i) decreases number of features M = Li number of features M = Li Reconstruction from static FSI Hadamard-Based AFSI AFSI – Hadamard: • RMSE reduces in AFSI compared with static FSI • RMSE improvement is 32% and 18% for high and low diversity training data, when M = 500 and L = 10. • AFSI has smaller RMSE using small L when M is also small • AFSI has smaller RMSE using large L when M is also large Reconstruction from adaptive FSI Frontiers in Optics 2007

11. Object estimate <A1> Object x Sort Hadamard bases Reconstruction Computational Optics Calculate Ri for Ai yiL+j = fiL+jTx+niL+j (j = 1,2,…L) Choose f1~fL ChoosefiL+1~f(i+1)L from de-noising yiL+j Sort Update Ai to Ai+1 according to <Ai+1> Hadamard-Based AFSI – Noise AFSI – Hadamard: • T : integration time • σ02 = 1 • detector noise variance: σ22 = σ02/T • Hadmard projection is used because of its good reconstruction performance • Feature measurements are de-noised before used in adaptation • Wiener operator is used for object reconstruction • Auto-correlation matrix is updated in each adaptation step Frontiers in Optics 2007

12. High diversity training data; σ02 = 1 High diversity training data; σ02 = 1 AFSI – adapted Rx AFSI – fixed Rx L decreases L increases K(i) decreases Static FSI Hadamard-Based AFSI – Noise • RMSE in AFSI is smaller than in static FSI • RMSE is reduced further by modifyingRx in each adaptation step • RMSE improvement is larger using small L when M is also small • RMSE is small using large L when M is also large Frontiers in Optics 2007

13. High diversity training data; σ02 = 1 High diversity training data; σ02 = 1 Hadamard-Based AFSI – Noise T : integration time/per feature; M0: the number of features Total feature collection time = T × M0 Increasing T • Reducing Measurement error • Losing adaptation advantage Trade-off Minimum total feature collection time Frontiers in Optics 2007

14. Conclusion • Noise free measurements: • PCA-based and Hadmard-based AFSI system are presented • AFSI system presents lower RMSE than static FSI system • Noisy measurements: • Hadamard-based AFSI system in noise is presented • AFSI system presents smaller RMSE than static FSI system • There is a minimum total feature collection time to achieve a reconstruction quality requirement Frontiers in Optics 2007