Point spread function reconstruction from the image of a sharp edge

Transcription

1 DOE/NV/ Point spread function reconstruction from the image of a sharp edge John Bardsley, Kevin Joyce, Aaron Luttman The University of Montana National Security Technologies LLC Montana Uncertainty Quantification June 4, 5

2 Stochastic Imaging Model Inverse problem Ideal Image Measured image Image system response k(x s) ds Measurement error +ε N(, I)

3 Point Spread Function Estimation Point Source Impulse Response PSF estimate Image System Response k(x s)(s) ds = k(x) Measurement error +ε N(, I)

4 Point Spread Function Estimation Heaviside function Edge Spread Noisey Data Image System Response f (x) = k(x s)e(s) ds Measurement error +ε N(, I)

5 Underdetermined Inverse problem f (x i ) = k(s, t)e(x i s)dtds + ε i y System Response t s??

6 x Radially Symmetric PSF If we assume radial symmetry, k(x, y) = ρ( x + y ), and that the edge is indicated at x = by E, then f (x i ) = = = k(s, t)e(x i s)dtds + ε i ( π ρ(r) ρ(r) g(x i, r) rdr + ε i. ) E(x i r cos θ)dθ rdr + ε i g(x,r).8 6 θ x < r x r x > r r Observe that g is symmetric about the x =.

7 Bayesian Stochastic Inverse Problem For the inverse problem f = Gρ + ɛ, regularization and uncertainty quantification can be achieved by modeling the PSF as a random quantity with an appropriate prior. Measurement error is given by Gaussian likelihood with E f = Gρ and Var f = I. We use a Gaussian prior defined implicitly on k(s, t). Denote as the squared Laplacian defined on a certain subspace of L (R ), then the prior density satisfies E k = and Var k = I [Stu3]. For the polar coordinate map T (r, θ), the radial symmetry assumption ( gives (Lρ)(r) := ( d k T )(r) = r dr r d ) ρ(r). Here, we impose a dr left zero derivative and right zero boundary condition on ρ.

8 Monte Carlo Sampling of the Posterior Since both densities are Gaussian, Bayes Theorem leads to a posterior density satisfying p(ρ f,, ) exp ( f Gρ ) Lρ. We impose uninformative Gamma hyper-priors for and with hyper-parameters α = and β = 6 [CS8]. It can be shown theoretically that for medium to large scale discretizations of the unknown ρ, the Hierarchical Gibbs sampler gives unsatisfactory samples of as the discretization for ρ becomes more refined [ABPS4]. To illustrate this, we implement the hierarchical Gibbs sampler for the discretization ρ at various levels of refinement on a synthetic example.

9 A Synthetic Example A radially symmetric two-dimensional Gaussian PSF has the form ρ(r j ) = (πσ ) e r j σ and it can be shown analytically that the analytic forward blur is an error function [Gρ](x i ) = πσ xi e s σ ds 8 6 ρ f G Gaussian radial PSF profile Line out from blurred edge r.5.5 x

10 Hierarchical Gibbs Sampler For ρ discretized with n points and f with n points, the joint posterior density of the unknown and hyperpriors satisfies ( p(ρ,, f ) ) R(ρ µ), (ρ µ) β( + ), where R = (G G + L L)ρ and µ = R G f. In order to draw efficiently from the conditional density for ρ, we use the Cholesky decomposition of R = S S which dominates the computation at O(n 3 ). The Hierarchical Gibbs sample is: : Let and be given and set k =. : Compute R k = S( T k S k. ) 3: Draw ρ k from N k R k G T f, R k. ( ) 4: Draw k+ from Γ n + α, Gρ k f β ( ) 5: Draw k+ from Γ n/ + α, Lρ k β 6: Set k = k + and return to.

11 Correlated chains.5 Disc. n = 5, for samples with ESS M/τ autocorrelation Disc. n = 5, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n=5 x Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x

12 Correlated chains.5 Disc. n = 5, for samples with ESS M/τ autocorrelation Disc. n = 5, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n=5 x Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x

13 Marginalized Posterior Density In order to marginalize out ρ, we complete the square of the quadratic form in ρ and integrate out the multivariate Gaussian, ( p(ρ,, f )dρ ( ) n/+α exp Gρ f ) Lρ β( ) dρ ( = ( ) n/+α exp R(ρ µ), (ρ µ) where U = ) U(,, b) β( ) dρ = ( ) n/+α c(,, f ) exp ( ) U(,, b) β( ), (I GR G )f, f and c = det R /. To draw from this density, we embed a Metropolis-Hastings algorithm within the Gibbs sampler. Both U and c can be carried out using a Cholesky factorization (O(n 3 )), and will be required for each Metropolis step.

14 Metropolis-Hastings within Gibbs Sampling Since is the problematic parameter, we implment the Metropolis-Hastings step on its conditional marginalized density p( f, ). We use a Gaussian proposal with adaptive variance. We use an initial variance estimate and support by running a Hierarchical Gibbs sampler. Due to numerical overflow issues, all computations are carried out on the log scale. : Let, and γ be given. : Compute R k = S( T k S k. ) 3: Draw x k from N R k k A T b, R k. ( ) 4: Draw k+ from Γ n/ + α, Ax k b β. 5: Set j =, = k k i= i. Compute c = log c( k, ) and then U = log U( k,, b), and p = log p( f k ). 6: while j < M do 7: Draw j from N( j, γ). 8: Compute U j = U( k, j ), c j = c( k, j ), and then p j = log p( j f k ). 9: Draw u j from Unif[, ] and if p j p j < log u j set p j = p j and k = j. Otherwise, continue. : Set j = j +. : end while : Set k = k + and return to.

15 Marginalized chains.5 Disc. n = 5, for samples with ESS M/τ autocorrelation Disc. n = 5, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n=5 x Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x

16 Marginalized chains.5 Disc. n = 5, for samples with ESS M/τ autocorrelation Disc. n = 5, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n=5 x Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x

17 Results Marginalized joint density of (,) with MCMC samples Disc. n =, for samples with ESS M/τ autocorrelation Disc. n =, for samples with ESS M/τ autocorrelation MCMC chains for, n= MCMC chains for, n= x x 4

18 Future Work The current implementation takes approximately 5-6 hours on a Intel(R) Core(TM) i5-43u core for a reconstruction with n =. Other factorizations based on spectral methods might allow for faster than O(n 3 ) computation for calucating U and c in each Metropolis step. Analyze how uncertainty in the blurring kernel affects deconvolution uncertainty. Theoretical details related to the infinite dimensional prior Lρ are not fully developed. Explore other priors that might enforce higher correlation near the peak of the PSFs.

19 References Sergios Agapiou, Johnathan M Bardsley, Omiros Papaspiliopoulos, and Andrew M Stuart. Analysis of the Gibbs sampler for hierarchical inverse problems. SIAM/ASA Journal on Uncertainty Quantification, ():5 544, 4. Johnathan M Bardsley. MCMC-based image reconstruction with uncertainty quantification. SIAM Journal on Scientific Computing, 34(3):A36 A33,. Daniela Calveti and Erkki Somersalo. Hypermodels in the Bayesian imaging framework. Inverse Problems, 4(3):343, 8. Andrew M Stuart. The Bayesian approach to inverse problems. arxiv preprint arxiv:3.6989v, 3.