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)