Lecture-by-lecture list of topics EECS 755, F13 topics marked with "X" were not "covered" this year but done in previous years. by "covered" i mean discussed in lecture; those topics are in the book... 1 9/4 do: read ch 1.1-1.8, hw0, download IRT, read doc/doc.pdf Course policies (0) Introduction Overview of model-based image reconstruction object model physics model from C-D to D-D statistical model data-fit term regularization penalized-likelihood estimation X MAP estimation vs MMSE (conditional mean) Convexity 2 9/9 do: read rest of Ch. 1 MAP estimate for gaussian noise / gaussian object closed-form solution (A'A + I) \ A' y Super-resolution application in 1D gradient descent step size in terms of maximum eigenvalue convergence rate in terms of spectral radius rho(I - step H) GD in matlab IRT: fatrix2 for 1D super-resolution, GD 3 9/11 super-resolution in 1D with two sensors (stacking) circulant analysis of \xh and E[\xh] 1D roughness penalty strict convexity of |y-Ax|^2 + |Cx|^2 4 9/16 announce: hw date, a-mat (spectral radius) a-math (conv. rate...) code: 1D restoration with quadratic roughness (C via circshift) code: 2D denoising with quadratic roughness denoise_2d_template.m 2D C via circshift; hint use cat(3, ...) analyze maximum eigenvalue of Hessian / google form (theory vs practice...) X code: 2D denoising with edge-preserving roughness X (1: c-restore) Image restoration X forward model / blur: atmosphere, optics, detector cell X discretization and lexicographic ordering end conditions matrix representation circulant matrix X nonlinearities: saturation and quantization X noise models: poisson and gaussian X ML estimate for gaussian X ML estimate for poisson - richardson/lucy MAP estimate for gaussian noise / gaussian object X circulant PSF analysis of MAP - wiener filter / demo 1D and 2D roughness penalties C matrix X circulant PSF analysis regularized LS / demo 5 9/18 X circulant noise analysis edge-preserving regularization via oracle line-sites leading to broken parabola nonquadratic potential functions weighting function fixed-point condition 6 9/23 max eig for edge-preserving reconstruction hessian A'A + reg C'DC, reviewing matrix norms etc. inpaint1_template -> edge-preserving inpainting example X huber's iteration X edge-preserving restoration example (2: c-reg) Regularization spline interpolation X spline smoothing / nonparametric regression X regression splines variational regularizers thin membrane / thin plate spline bilateral TV total variation 7 9/25 X roughness penalty implementations: Matlab/C regularization parameter selection discrepancy principle vs model order selection RSS, influence matrix, REDF L-curve, CV, GCV, URE 8 9/30 GCV for (nonlinear) edge-preserving image deblurring deblur1_howto.txt deblur1_rms.m deblur1_gcv_template.m 9 10/2 (11: c-opt) Optimization methods: general purpose descent direction convergence rate: root convergence factor preconditioned gradient descent (PGD) Lipschitz condition for convergence Huber example of Lipschitz constant local conv. rate O(1/n) cost function decrease for convex case 10 10/7 convergence conditions preconditioned steepest descent (PSD) preconditioned conjugate gradient (PCG) Barzalai-Borwein Nesterov preconditioners denoise_pgd1.m X Newton-Rhapson X Quasi-Newton 11 10/9 (12: c-ox) Optimization transfer / majorize-minimize general approach convergence rate / spectral radius circulant majorizer for edge-preserving image deblurring maximum curvature == additive half quadratic fall study break 10/14 12 10/16 lab: deblur2_race.m (GD, BBGM, PGD, PCG, etc.) 13 10/21 emission imaging X expectation-maximizaton (EM) algorithm (overview) E-ML-EM derived using De Pierro multiplicative trick (convexity) ir_et_ml_em_demo.m IID exponential prior = l1 regularizer 14 10/23 E-ML-EM vs PGD with iteration-dependent diag precon Constrained optimization Nonnegativity constraints X Box constraints gradient projection method X deblur2_gp.m X Coordinate Descent and CD-NR sparsity-based regularization X time 3D mex / cache with local offsets: demo_reg_times1 l0, l1, wavelets (brief) analysis form synthesis form X Fatrix representation of linear operators iterative soft thresholding (IST) derivation synthesis version (possibly over-complete) dictionary B = [Q I] matrix version of separable quadratic surrogates Lipschitz constant for 1/2 | y - A B \theta |^2 X example X De Pierro's additive trick X general form of image reconstruction cost functions X optimization transfer / IST for phase retrieval ("fail") X CG with monotonic line search FISTA 15 10/28 (dan weller) constrained optimization motivated by splitting methods for analysis-form regularization penalty method Lagrange multipliers Augmented Lagrangian ADMM Split Bregman for deblurring 16 10/30 lab on split Bregman for deblurring, using PCG inner solver deblur3_sb_template.m 17 11/4 Constrained optimization Coordinate Descent (nonnegativity easy) Diagonal majorizer theorem: B'B <= diag(|B|' |B| 1) using convexity inequality Separable quadratic surrogate for 1/2 ||y - Ax||_W^2 Huber quadratic surrogates for regularizer separable surrogates for convex image reconstruction problems acceleration via Nesterov (cf PGD and SQS) Cfd2_abs.m deblur2_nesterov.m 18 11/6 tn-1 / t_{n+1} -> 1 better diagonal majorizer? (via ADMM) ir_diag_majorize_admm1.m stopping rules (KKT) for nonnegativity constraint acceleration by better pi_ij values: NU-SQS kim:13:aos-tmi 19 11/11 incremental gradients / ordered subsets erdogan:99:osa example: c19_super_ig.m (22: c-srp) Spatial resolution properties LIR for QPWLS definition(s) of local impulse response 20 11/13 implicit function to derive LIR X eml_osem_example.m with many impulses LIR for PWLS estimators origins of space-varying resolution MR-SENSE (parallel MRI) example (Q)PL for Poisson example modified regularization design for uniform resolution 21 11/18 (c-mav) Noise properties (covariance analysis) QPWLS (linear estimator) fully-sampled parallel MRI example implicit estimator (delta method) for cov(nonlinear estimator) PL for Poisson example local variance approximation X calculating LIR: matrix approach, FFT approach X local shift invariance, local frequency response X penalized-likelihood emission image reconstruction X demo_empl.m showing beta/delta and trade-offs (c-basis) Signal models / basis functions subspace model for dimension reduction train using SVD union-of-subspaces model 22 11/20 demo_subspace1.m learning over-complete dictionary K-SVD, matching pursuit, basis pursuit atom update by block coordinate descent 23 11/25 using dictionaries for image reconstruction mathematical/trained dictionary min L(x) st l0 <= r min l0 st L(x) = eps min l1 st L(x) = eps min L(x) + reg l1(x) etc. equivalent prior (IID Laplacian) union of subspaces vs gaussian mixture importance of scale? (illumination...) relaxing the strict synthesis formulation alt. min over image x and sparse coef. 24 11/27 (cancelled, day before thanksgiving) 25 12/2 patch-based use of dictionary for recon adapted / jointly learned dictionary recent examples from literature: ravishankar:11:mir xu:12:ldx smith:13:ldl (improved K-SVD) gramfort:14:daf X sadeghi:13:dlf (ADMM for DL?) X semerci:12:air, tensors? (c-dyn) Dynamic imaging motivations: bulk motion, flow, diffusion, ... typical series expansion with spatio-temporal basis functions sampling patterns: video, CT, SPECT, MRI, list-mode PET 26 12/4 Simple dynamic reconstruction methods data sharing, sliding window, gating Motion-compensated image reconstruction deformation parameterization interpolation reconstruct-then-average motion-compensated temporal regularization parametric motion model (PMM) Joint estimation of images and motion (brief) 27 12/9 Space-time separable object models / linear measurement model y = A(X) + noise Analysis example (running 1D+time example fig_dyn_heart1.m) quadratic in space and time edge-preserving in space and time (and nearly l1) "TV" in space-time Synthesis models temporal subspace: X = Z B known subspace PCA oracle subspace oracle DFT subspace oracle Fourier series subspace 28 12/11 last class / (1 student presentation) Sparsity of temporal dictionary coefficients iterative hard thresholding Matrix completion singular value thresholding fixed-point (majorize-minimize) method failed illustration on 1D+time example ** 12/17 student presentations ---------------------------------------------- below here from W10 todo: AMP methods? s,opt,amp X (8) Emission imaging applications list-mode / binned-mode / current integration emission rate density function poisson point process list-mode likelihood derivation ML-EM derived for list-mode data demo_list_mode_em penalized-likelihood and surrogates (brief) EM via complete data (brief)