Adaptive Sampling for Imaging Malena Sabate Landman, Sergey Dolgov, - PowerPoint PPT Presentation

Adaptive Sampling for Imaging Malena Sabate Landman, Sergey Dolgov, Silvia Gazzola, and Tom Davis February 1, 2019

Agenda Problem Recap and Data Multi-resolution Sampling Approach using DEIM Matrix Completion Approach Ideas for Future Work

Problem Recap and Data Scanning a battery to determine presence and distribution of materials. From full scans, we observe A full ∈ R n 1 × n 2 (matrix of absorptions of n 1 energies at n 2 scanned pixels). Aim: to find a reduced scanning pattern which allows us to recover A full .

Problem Recap and Data If the sample contains k components, we can approximate A full by: A full ≈ U spectral C spectral , where U spectral ∈ R n 1 × k are the spectra of the materials and C spectral ∈ R k × n 2 are the coefficients for each pixel

Problem Recap and Data 10 4 2 10 1.8 20 1.6 30 1.4 40 1.2 50 1 60 0.8 70 0.6 80 90 0.4 100 0.2 10 20 30 40 50 60 70 80 90 100

Problem Recap and Data 10 4 10 4 2 1.8 10 10 1.8 20 20 1.6 1.6 30 30 1.4 1.4 40 40 1.2 1.2 50 50 1 1 60 60 0.8 0.8 70 70 0.6 0.6 80 80 90 90 0.4 0.4 100 100 0.2 0.2 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100 100

Problem Recap and Data 10 4 10 4 10 4 2 1.8 1.8 10 10 10 1.8 1.6 20 20 20 1.6 1.6 30 30 30 1.4 1.4 1.4 40 40 40 1.2 1.2 1.2 50 50 50 1 1 1 60 60 60 0.8 0.8 0.8 70 70 70 0.6 0.6 0.6 80 80 80 90 90 90 0.4 0.4 0.4 100 100 100 0.2 0.2 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Problem Recap and Data 10 4 10 4 10 4 10 4 2 1.8 1.8 1.8 10 10 10 10 1.8 1.6 20 20 20 20 1.6 1.6 1.6 30 30 30 30 1.4 1.4 1.4 1.4 40 40 40 40 1.2 1.2 1.2 1.2 50 50 50 50 1 1 1 1 60 60 60 60 0.8 0.8 0.8 0.8 70 70 70 70 0.6 0.6 0.6 0.6 80 80 80 80 90 90 90 90 0.4 0.4 0.4 0.4 100 100 100 100 0.2 0.2 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 60 60 60 60 70 70 70 70 80 80 80 80 90 90 90 90 100 100 100 100

Problem Recap and Data 10 4 10 4 10 4 10 4 2 0.06 1.8 1.8 1.8 10 10 10 10 1.8 0.04 1.6 20 20 20 20 1.6 1.6 1.6 0.02 30 30 30 30 1.4 1.4 1.4 1.4 0 40 40 40 40 1.2 1.2 1.2 1.2 -0.02 50 50 50 50 1 1 1 1 -0.04 60 60 60 60 0.8 0.8 0.8 -0.06 0.8 70 70 70 70 0.6 0.6 -0.08 0.6 0.6 80 80 80 80 -0.1 90 90 90 90 0.4 0.4 0.4 0.4 -0.12 100 100 100 100 0.2 0.2 0 50 100 150 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 60 60 60 60 70 70 70 70 80 80 80 80 90 90 90 90 100 100 100 100

Problem Recap and Data 10 4 10 4 10 4 2 0.06 0.06 1.8 1.8 10 10 10 1.8 0.04 0.04 1.6 20 20 20 1.6 1.6 0.02 0.02 30 30 30 1.4 1.4 1.4 0 0 40 40 40 1.2 1.2 1.2 -0.02 -0.02 50 50 50 1 1 -0.04 1 -0.04 60 60 60 0.8 0.8 -0.06 -0.06 0.8 70 70 70 0.6 0.6 -0.08 -0.08 0.6 80 80 80 -0.1 -0.1 90 90 90 0.4 0.4 0.4 -0.12 -0.12 100 100 100 0.2 0.2 0 0 50 50 100 100 150 150 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Problem Recap and Data 10 4 10 4 10 4 2 0.06 0.06 0.06 1.8 1.8 10 10 10 1.8 0.04 0.04 0.04 1.6 20 20 20 1.6 1.6 0.02 0.02 0.02 30 30 30 1.4 1.4 1.4 0 0 0 40 40 40 1.2 1.2 1.2 -0.02 -0.02 -0.02 50 50 50 1 1 -0.04 1 -0.04 -0.04 60 60 60 0.8 0.8 -0.06 -0.06 -0.06 0.8 70 70 70 0.6 0.6 -0.08 -0.08 -0.08 0.6 80 80 80 -0.1 -0.1 -0.1 90 90 90 0.4 0.4 0.4 -0.12 -0.12 -0.12 100 100 100 0.2 0.2 0 0 0 50 50 50 100 100 100 150 150 150 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Problem Recap and Data 10 4 10 4 10 4 2 0.06 0.06 0.06 0.06 1.8 1.8 10 10 10 1.8 0.04 0.04 0.04 0.04 1.6 20 20 20 1.6 1.6 0.02 0.02 0.02 0.02 30 30 30 1.4 1.4 1.4 0 0 0 0 40 40 40 1.2 1.2 1.2 -0.02 -0.02 -0.02 -0.02 50 50 50 1 1 -0.04 1 -0.04 -0.04 -0.04 60 60 60 0.8 0.8 -0.06 -0.06 -0.06 0.8 -0.06 70 70 70 0.6 0.6 -0.08 -0.08 -0.08 -0.08 0.6 80 80 80 -0.1 -0.1 -0.1 -0.1 90 90 90 0.4 0.4 0.4 -0.12 -0.12 -0.12 100 100 100 -0.12 0.2 0.2 0 0 0 50 50 50 100 100 100 150 150 150 0 50 100 150 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Problem Recap and Data 10 4 10 4 10 4 2 0.06 0.06 0.06 0.06 0.06 1.8 1.8 10 10 10 1.8 0.04 0.04 0.04 0.04 0.04 1.6 20 20 20 1.6 1.6 0.02 0.02 0.02 0.02 0.02 30 30 30 1.4 1.4 1.4 0 0 0 0 0 40 40 40 1.2 1.2 1.2 -0.02 -0.02 -0.02 -0.02 -0.02 50 50 50 1 1 -0.04 1 -0.04 -0.04 -0.04 -0.04 60 60 60 0.8 0.8 -0.06 -0.06 -0.06 0.8 -0.06 -0.06 70 70 70 0.6 0.6 -0.08 -0.08 -0.08 -0.08 0.6 -0.08 80 80 80 -0.1 -0.1 -0.1 -0.1 -0.1 90 90 90 0.4 0.4 0.4 -0.12 -0.12 -0.12 100 100 100 -0.12 -0.12 0.2 0.2 0 0 0 50 50 50 100 100 100 150 150 150 0 0 50 50 100 100 150 150 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Problem Recap and Data 10 4 10 4 10 4 2 0.06 0.06 0.06 0.06 0.06 0.2 1.8 1.8 10 10 10 1.8 0.04 0.04 0.04 0.04 0.04 0.15 1.6 20 20 20 1.6 1.6 0.02 0.02 0.02 0.02 0.02 30 30 30 1.4 0.1 1.4 1.4 0 0 0 0 0 40 40 40 1.2 1.2 0.05 1.2 -0.02 -0.02 -0.02 -0.02 -0.02 50 50 50 1 1 -0.04 1 -0.04 -0.04 -0.04 -0.04 0 60 60 60 0.8 0.8 -0.06 -0.06 -0.06 0.8 -0.06 -0.06 70 70 70 -0.05 0.6 0.6 -0.08 -0.08 -0.08 -0.08 0.6 -0.08 80 80 80 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 90 90 90 0.4 0.4 0.4 -0.12 -0.12 -0.12 100 100 100 -0.12 -0.15 -0.12 0.2 0.2 0 0 0 50 50 50 100 100 100 150 150 150 0 0 0 50 50 50 100 100 100 150 150 150 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Problem Recap and Data 10 4 10 4 10 4 2 0.06 0.06 0.06 0.06 0.25 0.06 0.2 1.8 1.8 10 10 10 1.8 0.04 0.04 0.04 0.04 0.04 0.2 0.15 1.6 20 20 20 1.6 1.6 0.02 0.02 0.02 0.02 0.02 0.15 30 30 30 1.4 0.1 1.4 1.4 0 0 0 0 0 40 40 40 0.1 1.2 1.2 0.05 1.2 -0.02 -0.02 -0.02 -0.02 -0.02 50 50 50 0.05 1 1 -0.04 1 -0.04 -0.04 -0.04 -0.04 0 60 60 60 0 0.8 0.8 -0.06 -0.06 -0.06 0.8 -0.06 -0.06 70 70 70 -0.05 -0.05 0.6 0.6 -0.08 -0.08 -0.08 -0.08 0.6 -0.08 80 80 80 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 -0.1 90 90 90 0.4 0.4 0.4 -0.12 -0.12 -0.12 100 100 100 -0.12 -0.15 -0.15 -0.12 0.2 0.2 0 0 0 50 50 50 100 100 100 150 150 150 0 0 0 0 50 50 50 50 100 100 100 100 150 150 150 150 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100

Multi-resolution Sampling Approach using DEIM 1. Use SVD on a coarse resolution scan A coarse of all the energies (aggregated pixels) to identify a matrix U 0 which spans the same space than U spectral . 2. Use DEIM to identify the important energies. High resolution scan (in all pixels) will just be performed for these energies. Compute C 1 ≈ C SVD by imposing A ( p k , :) = U 0 ( p k , :) C 1 ∀ k . 3. Use dictionary of spectra to identify which materials are in the sample.

20 40 60 80 100 120 140 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 10 4 3 20 2.5 40 2 60 1.5 80 100 1 120 0.5 140 20 40 60 80 100 120 140 160 180 200

C 0 20 40 60 U 0 80 100 120 140 20 40 60 80 100 120 140 160 180 200

Multi-resolution Sampling Approach using DEIM 1. Use SVD on a coarse resolution scan A coarse of all the energies (aggregated pixels) to identify a matrix U 0 which spans the same space than U spectral . 2. Use DEIM 1 to identify the important energies. High resolution scan (in all pixels) will just be performed for these energies. Compute C 1 ≈ C SVD by imposing A ( p k , :) = U 0 ( p k , :) C 1 ∀ k . 3. Use dictionary of spectra to identify which materials are in the sample. 1 Saifon Chaturantabut and Danny C. Sorensen. “Nonlinear Model Reduction via Discrete Empirical Interpolation”. In: SIAM Journal on Scientific Computing 32.5 (2010), pp. 2737–2764.

Multi-resolution Sampling Approach using DEIM 1. Use SVD on a coarse resolution scan A coarse of all the energies (aggregated pixels) to identify a matrix U 0 which spans the same space than U spectral . 2. Use DEIM 1 to identify the important energies. High resolution scan (in all pixels) will just be performed for these energies. Compute C 1 ≈ C SVD by imposing A ( p k , :) = U 0 ( p k , :) C 1 ∀ k . 3. Use dictionary of spectra to identify which materials are in the sample. 1 Saifon Chaturantabut and Danny C. Sorensen. “Nonlinear Model Reduction via Discrete Empirical Interpolation”. In: SIAM Journal on Scientific Computing 32.5 (2010), pp. 2737–2764.

Multi-resolution Sampling Approach using DEIM 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0 50 100 150