Model-Based Dynamic Sampling (MBDS) in 2D and 3D. Dilshan Godaliyadda Gregery Buzzard Charles Bouman Purdue University

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1 Model-Based Dynamic Sampling (MBDS) in 2D and 3D Dilshan Godaliyadda Gregery Buzzard Charles Bouman Purdue University

2 2D MBDS

3 Raster Sampling: Optimally Bad Each new sample provides the least information.

4 Random Sampling: Better Each new sample provides much more information.

5 Optimal (Greedy) Sampling: Best Each new sample provides the most information.

6 How to Select a New Measurement First model the posterior distribution p ( k x y ) k, with a Gaussian p k ( x y ) k = 1 z exp # 1 2 y 2 % k A k x $ Λk 1 & 2 xt Bx( ' From Forward Model From Prior Model New measurement location The pixel location where the posterior variance is largest

7 Recursion for Optimal Greedy Sampling For each new sample { y k = A k x + w k Step 1: Measure signal µ k = Ε x y k R k = Ε ( x µ k ) x µ k m k argmax m M ( ) T y k ( mt R k m) Step 2: Find posterior covariance of image When everything is Gaussian, R k does not depend on y k! Step 3: Select pixel with largest variance A k = A k m k Step 4: Add new row to A matrix }

8 Non-Gaussian Prior => Dynamic Sampling x 1 best measurement direc:on Likely loca:on of solu:on x 2 prior manifold

9 Non-Gaussian Prior => Dynamic Sampling best measurement direc:on x 1 Likely loca:on of solu:on x 2 prior manifold Optimal sampling depends dynamically on previous samples

k 1 2 {i, j} P x (i) x ( j ) σ c + x(i) x ( j ) σ q q p Same Forward

10 Non-Gaussian Posterior Hence we model the posterior distribution, with a non-gaussian p k ( x y ) k p k ( x y k ) = 1 z exp 1 2 y A x 2 k k Λ k 1 2 {i, j} P x (i) x ( j ) σ c + x(i) x ( j ) σ q q p Same Forward Model Non-Gaussian Prior Model Q-GGMRF All neighboring pairs of pixels

11 2D-MBDS Algorithm For each new sample { y k = A k x + w k Step 1: Measure signal How do you generate samples from the posterior distribution? { x k (1), x k (2 ),!, x k ( L ) } ~ p( x y k ) ˆµ k (n) 1 L i=1 (n) x k (l ) n k = argmax n n (n) ( σˆ k ) 2 1 L ( )( ) T x (n) L 1 ˆµ (n) k (l ) x (n) (n) k k (l ) ˆµ k i=1 (n) ( σˆ k ) 2 Step 2: Generate L samples from posterior distribution Step 3: Estimate posterior variance Step 4: Select pixel with largest variance A k+1 = A k 0,,1,,0 Step 5: Add new row to A matrix n k }

Sampling from the Posterior Distribution The image below is the MAP es:mate aber the k th itera:on The posterior

A x 2 k k 1 Λ k 2 {i, j} P x (i) x ( j ) σ c + x(i) x ( j ) σ q q p sample images Vectors from this distribu:on Use

the MAP es:mate itself) X 1 Draw a sample vector from a proposal distribu:on W ~ q k w X i ( ) Our proposal distribu:on

Because easy to draw samples from a Gaussian Calculate " $ α = min 1, q k ( X i W ) # %$ q k W X i ( ) exp { [ u(w )

12 Sampling from the Posterior Distribution The image below is the MAP es:mate aber the k th itera:on The posterior distribu:on Want sample images from this distribu:on Pick a Monte- Carlo Sampling method p k ( x y k ) = 1 z exp 1 2 y A x 2 k k 1 Λ k 2 {i, j} P x (i) x ( j ) σ c + x(i) x ( j ) σ q q p sample images Vectors from this distribu:on Use Has:ngs- Metropolis (HM) Algorithm Energy - u k (x) HM Algorithm Start with a good guess for a sample vector (can use the MAP es:mate itself) X 1 Draw a sample vector from a proposal distribu:on W ~ q k w X i ( ) Our proposal distribu:on A Gaussian approxima:on to p k (x y k ) Why? Because easy to draw samples from a Gaussian Calculate " $ α = min 1, q k ( X i W ) # %$ q k W X i ( ) exp { [ u(w ) u(x i) ]} & $ ' ($ For i =1:M How? By calcula:ng the second order Taylor series approxima:on to u k (x) at the MAP es:mate " W with probability α X i+1 # $ X i with probability 1-α INTRACTABLE FOR MOST IMAGES!

13 Posterior Sampling for Images (2D) Compute posterior variance for each voxel loca:on Find pixel loca:on with largest posterior variance (index n k ) { x k (1), x k (2 ),!, x k ( L ) } MAP es:ma:on to reconstruct block n k Formulate Proposal distribu:on for Has:ngs- Metropolis Sampler ˆx k Proposal Distribution for HM Sampler Generate block sample vector using Has:ngs- Metropolis Sampler Extract blocks surrounding measured voxel loca:on from the l th sample image x Generate L Block Sample Images x k (l ) Do 5ll enough measurements are taken x k (1), x k (2 ),...x k ( L )

14 Dynamic Sampling for BSE Image x 200 Segment (Image from Georgia Tech Team) 45% of image sampled Original Image Selected Samples Measured Image Reconstructed Image

15 2D MBDS With a Distance Penalty

16 Incorporating a Distance Penalty For each new sample { y k = A k x + w k { x k (1), x k (2 ),!, x k ( L ) } ~ p( x y k ) ˆµ k (n) 1 L n i=1 (n) ( σˆ k ) 2 1 (n) x k (l ) L ( )( ) T x (n) L 1 ˆµ (n) k (l ) x (n) (n) k k (l ) ˆµ k i=1 Step 1: Measure signal Step 2: Generate L samples from posterior Step 3: Estimate posterior variance n k = max n (n) ( σ k ) 2 Δt (i,s) where Δt (i,s) = α + βd (i,s) Step 4: Select pixel with largest variance according to the distance penalty d (i,s) α β - distance between location s and i. - time taken to acquire a measurement after beam is moved - time taken to move beam a unit distance A k+1 = A k 0,,1,,0 n (k) Step 5: Add new row to A matrix }

17 Incorporating a Distance Penalty Reconstructed Images 5% 10% 15% 20% d 0 =50 (Penalized rela:vely more for distance) d 0 =10,000 (Penalized rela:vely less for distance)

18 3D MBDS

19 3D - MBDS Algorithm For each slice (for t = 1:T){ For each new measurement (for k=1:k){ y k,t = A k,t x t + w k,t { ( )} ˆx t argmax p x t y t,k, y t,k, x t i for i = 1,2,3... x t Step 1: Measure Image Step 2: Estimate the image using MAP Estimation { x, x,!, x t (1),k t (2 ),k t ( L ),k} ~ p( x t y t,k, y t i,k, x t i for i = 1,2,3... ) ˆµ (n) t,k 1 L L i=1 x (n) t (i ),k ( σ ˆ (n) t,k ) 2 1 L 1 L ( x (n) ˆµ (n) t (i ),k t,k ) x (n) i=1 ( ˆµ (n) t (i ),k t,k ) T Step 3: Generate L samples from posterior y t i,k x t i - Measurements from previous slices - Previous slices Step 4: Estimate posterior variance n t,k = max n (n) ( σ t,k ) 2 Δt (i,s) where Δt (i,s) = α + βd (i,s) Step 5: Select voxel with largest variance } A t,k+1 = } A t,k 0,,1,,0 n t,k Step 6: Add new row to A t matrix

20 Posterior Sampling for Images (3D) Compute posterior variance for each voxel loca:on Find pixel loca:on with largest posterior variance (index n t,k ) ˆx t 1,K, ˆx t 2,K, ˆx t 3,K { x t,k 1 (1), x t,k 1 (2 ),!, x t,k 1 ( L ) } n t,k Formulate Proposal distribu:on for Has:ngs- Metropolis Sampler Proposal Distribution for HM Sampler Generate block sample vector using Has:ngs- Metropolis Sampler x t,k (l ) Extract blocks surrounding measured voxel loca:on from the l x th sample image x t 1,K (l ), x t 2,K (l ), x t 3,K (l ) Do 5ll enough measurements are taken MAP es:ma:on to reconstruct block ˆx t,k Generate L Block Sample Images x t,k (1), x t,k (2 ),...x t,k ( L )

The Setup for the experiment Temporal neighborhood (z-direction) = 1 Spacial neighborhood (x-y plane) = 1 z x Need a condition to stop measuring a slice and move on to the next one y We use a

21 The Setup for the experiment Temporal neighborhood (z-direction) = 1 Spacial neighborhood (x-y plane) = 1 z x Need a condition to stop measuring a slice and move on to the next one y We use a threshold on Expected Variance For the experiment the threshold used is 0.3 This means for any voxel in the 1 st time slice we consider 8 neighbors and for all the others we consider 17 neighbors.

22 Results Reconstructions when Sampling Terminates Slice 0 (x 0 ) Slice 1 (x 1 ) Slice 2 (x 2 ) Slice 3 (x 3 ) Reconstructed Image Sampled locations Measured Image Original Images Percentage when sampling terminated for each slice ~64% ~34% ~40% ~33%

Physics-based Prior modeling in Inverse Problems

Physics-based Prior modeling in Inverse Problems MURI Meeting 2013 M Usman Sadiq, Purdue University Charles A. Bouman, Purdue University In collaboration with: Jeff Simmons, AFRL Venkat Venkatakrishnan,