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, Purdue Marc De Graef, CMU 1
Inverse Problems in Imaging Recover information from indirect measurement* x Unknown Quan<ty Physical System Linear/Nonlinear Deterministic/Stochastic y Data Inversion Method ˆx Es<mate φ Other Unknowns (Nuisance Parameters) Regularity Condi<ons (Prior knowledge) Image and system models are cri1cal to accurate inversion 2
Model Based Iterative Reconstruction General framework for solving inverse problems ( ˆx ) argmax p( y x) : Likelihood p(x) : Prior Model x x { p( x y) } = argmin x Physical system { log p( y x) log p( x) } y Prior Model: p(x) Optimization Engine ˆx Forward model : g(.) g(x) Difference 3
Popular models for the Prior Neighborhood based or Local priors: Penalize dissimilarity between voxels: Markov Random Fields Bilateral Filtering Non-local priors: Exploit image information from non-local voxels: Non-Local means K-SVD BM3D x i ρ(x i x j ) : Penalty on the difference Sheep lung image and its learned dictionary with 256 atoms* *Qiong Xu, Low-dose CT reconstruction via Dictionary Learning 4
Physics-based Prior For some inverse problems, Physics can provide more information than local or non-local priors. Example: Microstructure evolution in materials is described by Phase-field model. We explore the idea of using a Physics-based prior in such inverse problems. Microstructure evolution in Cu-Al alloy 5
Cahn-Hilliard Equation as Prior Cahn-Hilliard equation governs the temporal and spatial evolution in binary fluids. We use Cahn-Hilliard equation as the prior for inverse problems. As a first step, we apply the Cahn-Hilliard prior to Image de-noising problem. 6
Cahn Hilliard Equation The Cahn-Hilliard equation for a binary fluid is: where H(x,θ) = x t + a 4 x b 2 f * x = 0 x(r,t) is the concentration of the fluid between (0-1), with 0 representing one phase and 1 representing the other. f *(x) is the dimensionless free energy of the fluid. a and b are parameters of the equation. 7
Image de-noising in presence of Cahn-Hilliard prior Image de-noising problem statement: 2 ˆx = argmin y x D x,θ subject to H(x,θ) = 0 - x, the unknown image - y, the noisy input image D -, a diagonal matrix with ( ) = 0 d i = 1 2σ 2 - H x,θ, Cahn-Hilliard equation 8
De-noising Cost Function For de-noising problem, we form the following cost function: Penalize deviation from H(x,θ) = 0, i.e. deviation from the Physical behavior. L λ (x,θ) = 1 2σ 2 y x 2 + λ H(x,θ) 2 MAP Estimate: ˆx = argmin x,θ L λ (x,θ) 9
Alternate Minimization using ICD Set number of iterations Initialize x y. Low-pass filter to get and initialize For each iteration L λ (x,θ) = 1 2σ 2 y x 2 + λ H(x,θ) 2 x Update to minimize For each pixel y y lp θ argmin θ H(y lp,θ) x s L λ (x,θ) Minimize L λ (x,θ) for between (0-1) Update θ to minimize H(x,θ) Find least-square estimate θ argmin θ H(x,θ). x s 10
Generate phantom images equation. Experiments H(x,θ) = 0 y x that satisfy the Cahn-Hilliard Generate noisy images from : - Add i.i.d. Gaussian noise with σ = 0.05 and 0.1 x Apply ICD to minimize L λ (x,θ) = 1 y x 2 + λ H(x,θ) 2 2σ 2 x and θ jointly over 11
De-noising results for σ = 0.05(5% noise) 12
De-noising Comparison 13
De-noising results for σ = 0.1(10% noise) 14
De-noising Comparison 15
Current and future work Reconstruction in the presence of Cahn-Hilliard Prior: 2 y Ax D min x,θ subject to H(x,θ) = 0 where - A, a matrix implementing the linear forward model Reconstruction with time-interleaving and limited projections. 16
Questions? 17
Supplementary Slides 18
Cahn Hilliard Equation The Cahn-Hilliard equation for a binary fluid is: where u( r, t) u t = γ 2 $ & ε 2 2 (u)+ f % u is the concentration of the alloy between (0-1), with 0 representing one phase and 1 representing the other. f (u) is the free energy of the alloy. Assuming a double-well potential energy functional, we have f (u) = u 2 (u 1) 2 and f u =4u3 6u 2 + 2u ε : controls width of the transition region [µm] γ : controls rate of growth of the phase [µm 2 / s] ' ) ( f *( u) 0 1
Cahn Hilliard Equation The Cahn-Hilliard equation for a binary fluid is: u t = $ a 4 u + b 2 f * & % u where u( r, t) is the concentration of the fluid between (0-1), with 0 representing one phase and 1 representing the other. f *( u) is the dimensionless free energy of the fluid. Assuming a double-well potential energy functional, we have f *(u) = u 2 (u 1) 2 and f * =4u 3 6u 2 + 2u u f *( u) a [µm 4 / s] and b [µm 2 / s] are parameters of the equation ' ) ( 0 1 20
Discrete form of Cahn Hilliard Equation Consider 2D spatial coordinates, and let un, i, j = u(iδx, jδx, nδt s ) be the discrete realization of u at (i, j) spatial coordinates and th n time frame, where Δt s [sec] is time step and Δx [µm] is the spatial step. Finite Difference formulation of CH-equation is: u n+1,i, j u n,i, j = ad 2 (u n, Δx) i, j + bd(4u 3 n 6u 2 n + 2u n, Δx) i, j (1) Δt s where D(u n, Δx) i, j = u n,i+1, j + u n,i 1, j + u n,i, j+1 + u n,i, j 1 4u n,i, j discrete space Laplace operator and (Δx) 2 D 2 (u n, Δx) i, j = D(D(u n, Δx), Δx) i, j is the 21
Parameterization - Discrete form of Cahn Hilliard Equation Re write Cahn Hilliard equation (1) as u n+1,i, j u n,i, j = ad 2 (u n ) i, j + bd(4u n 3 6u n 2 + 2u n ) i, j (2) where a = a (Δx) 4 Δt s, b = b (Δx) 2 Δt s are unitless parameters. H(u n+1,u n,θ) So the Cahn-Hilliard regularization, is: H(u n+1,u n,θ) = u n+1 u n + ad 2 (u n ) bd(4u n 3 6u n 2 + 2u n ) 22
Stability Constraints on discretization Some discretization schemes of the Cahn Hilliard are known to be more stable[1]. Implicit Euler Scheme: Linearly Stabilized Splitting Scheme[1]: u n+1 n ij u ij = γd( ε 2 D(u n+1 ij )+ 2u n+1 ij )+ D((u n ij ) 3 3u n ij ) Δt 4 2 ( u) ( u) - Splits the free energy E( u) = 4 2 into concave and convex parts E ( u) = E1 ( u) + E2( u) - Treats the convex part implicitly and the concave parts explicitly. u ij n+1 u ij n Δt = γd( ε 2 D(u ij n+1 )+ (u ij n+1 ) 3 u ij n+1 ) [1]: D. Eyre, An uncondidonally stable one- step scheme for gradient systems, 1997.
Cost per pixel vs. iterations 24
Regularization per pixel vs. iterations Regularization per pixel after 50 iterations = 8.4469 10 5 25
Comparison with Standard denoising methods RMSE for de-noising with 5% noise: BM3D: 0.012011 BM4D: 0.006212 Cahn-Hilliard prior: 0.02614 26
BM4D De-noising Results 27
De-noising Comparison 28
De-noising Comparison 29