Designing Optimal Spectral Filters and Low-RankMatrices for Inverse Problems
Julianne Chung
Department of MathematicsVirginia Tech
Joint work with:
Matthias Chung, Virginia Tech
Dianne O’Leary, University of Maryland
Julianne Chung, Virginia Tech
What is an inverse problem?
Physical SystemInput Signal Output Signal
Forward Model
Julianne Chung, Virginia Tech
What is an inverse problem?
Physical SystemInput Signal Output Signal
Forward Model
Inverse Problem
Julianne Chung, Virginia Tech
Discrete Linear Inverse Problem
b = Aξ + δ
whereξ ∈ Rn - unknown parametersA ∈ Rm×n - large, ill-conditioned matrixδ ∈ Rm - additive noiseb ∈ Rm - observation
Goal: Given b and A, compute approximation of ξ
Julianne Chung, Virginia Tech
Application: Image Deblurring
b = Aξ + δ
Given: Blurred image, b, andsome information about theblurring, A
Goal: Compute approximation oftrue image, ξ
Julianne Chung, Virginia Tech
Application: Image Deblurring
b = Aξ + δ
Given: Blurred image, b, andsome information about theblurring, A
Goal: Compute approximation oftrue image, ξ
Julianne Chung, Virginia Tech
Application: Super-Resolution Imaging
bi = A(yi) ξ + δi
Given: LR images
b1...bm
︸ ︷︷ ︸
=
A(y1)...A(ym)
︸ ︷︷ ︸
ξ+
δ1...δm
︸ ︷︷ ︸
b = A(y) ξ + δ
Goal: Improve parameters andapproximate HR image
Julianne Chung, Virginia Tech
SRimages.aviMedia File (video/avi)
Application: Limited-Angle Tomography
X-ray ImagingDigital TomosynthesisComputed Tomography
Julianne Chung, Virginia Tech
tomomovie.movMedia File (video/quicktime)
Application: Tomosynthesis Reconstruction
Given: 2D projection images
Goal: Reconstruct a 3D volume
bi = Υ[Aiξ] + δi
where Υ[·] represents nonlinearenergy dependent transmissiontomography
True Images
Julianne Chung, Virginia Tech
What is an Ill-posed Inverse Problem?
Hadamard (1923): A problem is ill-posed if the solutiondoes not exist,is not unique, ordoes not depend continuously on the data.
Julianne Chung, Virginia Tech
What is an Ill-posed Inverse Problem?Hadamard (1923): A problem is ill-posed if the solution
does not exist,is not unique, ordoes not depend continuously on the data.
Forward Problem
Inverse Problem
True image: x Blurred & noisy image: b
Inverse solu:on: A-‐1b
Julianne Chung, Virginia Tech
Regularization
Incorporate prior knowledge:1 Knowledge about the noise in the data2 Knowledge about the unknown solution
Goals of this work:Incorporate probabilistic information in the form of training dataCompute optimal regularization:
Optimal Spectral FiltersOptimal Low-Rank Inverse Matrices
Julianne Chung, Virginia Tech
Outline
1 Designing Optimal Spectral FiltersBackground on Spectral FiltersComputing Optimal FiltersNumerical Results
2 Designing Optimal Low-Rank Regularized Inverse MatricesRank-Constrained ProblemBayes Problem: Theoretical ResultsEmpirical Bayes Problem: Numerical Results
3 Conclusions
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Outline
1 Designing Optimal Spectral FiltersBackground on Spectral FiltersComputing Optimal FiltersNumerical Results
2 Designing Optimal Low-Rank Regularized Inverse MatricesRank-Constrained ProblemBayes Problem: Theoretical ResultsEmpirical Bayes Problem: Numerical Results
3 Conclusions
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Regularization and Filtering
Singular Value Decomposition:Let A = UΣVT where
Σ =diag(σ1, σ2, . . . , σn) , σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0UT U = I , VT V = I
For ill-posed inverse problems,Singular values σi decrease to and cluster at 0There is no gap separating large and small singular valuesSmall singular values⇒ highly oscillatory singular vectors
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Discrete Picard Condition
Inverse Solution, A−1b
Investigate behavior of:Singular values, σiSingular vectors, viSVD coefficients, uTi bSolution coefficients, u
Ti bσi
Two toy problems:1D deconvolutionGravity surveying
Hansen, Discrete Inverse Problems (2010)
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
SVD AnalysisThe naïve inverse solution:
ξ = A−1b
= VΣ−1UT b
=n∑
i=1
uTi bσi
vi
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
SVD AnalysisThe naïve inverse solution:
ξ̂ = A−1(b + δ)
= VΣ−1UT (b + δ)
=n∑
i=1
uTi (b + δ)σi
vi
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
SVD AnalysisThe naïve inverse solution:
ξ̂ = A−1(b + δ)
= VΣ−1UT (b + δ)
=n∑
i=1
uTi (b + δ)σi
vi
=n∑
i=1
uTi bσi
vi +n∑
i=1
uTi δσi
vi
= ξ + error
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Regularization via Spectral Filtering
Filtered Solution
ξfilter =n∑
i=1
φiuTi bσi
vi
= VCΣ−1φ
φi - filter factorsφ ∈ Rn contains φiC = diag(UT b) 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Singular values
Filt
er fa
ctor
s
TSVDTikhonov
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Some Filter RepresentationsTruncated SVD (TSVD)
φtsvdi (α) =
{1, if i ≤ α,0, else,
α ∈ {1, . . . ,n}
Tikhonov filter
φtiki (α) =σ2i
σ2i + α2, α ∈ R
Error filterφerri (α) = αi , α ∈ R
n
Spline filterφ
spli (α) = s(τ ,α;σi), α ∈ R
`, ` < n
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
How to choose α?Previous approaches (1 parameter)
Discrepancy PrincipleGeneralized Cross-Validation (GCV)L-Curve
Our approach to compute optimal parameters:Stochastic programming formulation to incorporate probabilisticinformationUse training data and numerical optimization to minimize errors
Shapiro, Dentcheva, Ruszczynski. SIAM, 2009.
Vapnik. Wiley & Sons, 1998.
Tenorio. SIAM Review, 2006.
Horesh, Haber, Tenorio. Inverse Problems, 2008.
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Some Assumptions
Suppose we havea set of possible signals Ξ ⊆ Rn
a set of possible noise samples ∆ ∈ Rn
Selectsignal ξ ∈ Ξ according to probability distribution Pξnoise sample δ ∈ ∆ according to probability distribution Pδ
Inverse Problem: determine ξ given A and b, where
b = Aξ + δ
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
How to define error?
Error vector e(α, ξ, δ) = xfilter(α, ξ, δ)− ξ
Error function: ρ : Rn → R+0
Errorerr(α, ξ, δ) = ρ(e(α, ξ, δ))
For example, ρ(x) = ‖x‖pp , err(α, ξ, δ) = ‖e(α, ξ, δ)‖pp
Goal: Find optimal parameters α that minimize average error over setof signals and noise
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Bayes Risk Minimization
Ideally...
Optimal filter φ(α̌) where
α̌ = arg minα
Eξ,δ{
err(α, ξ, δ)}
Remarks:Minimizing expected value is difficultUse Monte Carlo sampling approach
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Empirical Bayes Risk MinimizationIf Pξ and Pδ known→ use Monte Carlo samples as training dataIf Pξ and Pδ unknown but samples are given→ use samples as training data
Training data:
ξ(1), . . . , ξ(K ) realizations of random variable ξ
δ(1), . . . , δ(K ) realizations of random variable δ
b(k) = Aξ(k) + δ(k) , k = 1, ...,K
Empirical Bayes risk:
Eξ,δ{
err(α, ξ, δ)}≈ 1
K
K∑k=1
ρ(e(k)(α))
Optimal filter φ(α̂) where
α̂ = arg minα
1K
K∑k=1
ρ(e(k)(α))
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Suggested Optimization Methods
error function\ filter tsvd tik spl errHuber GSS GN GN GN
1 < p < 2 (smoothing) GSS GN GN GNp = 2 GSS GN GN LSp > 2 GSS GN GN GN
p =∞,p = 1 GSS IPM-N IPM-N IMP-L
GSS - discrete golden section search algorithmGN - Gauss-Newton methodLS - linear least squares systemIPM-N / IPM-L -interior point methods for nonlinear / linearproblems
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Gauss-Newton Optimization
x(k)filter = VC(k)Σ−1φ
Jacobian
J =1K
J(1)
...J(K )
where J(k) = VC(k)Σ−1φαφα - partial derivatives of φ w.r.t. α
Gradient and GN Hessian
g =1K
K∑k=1
J(k)>
g(k) , H =1K
K∑k=1
J(k)>
D(k)J(k)
g(k) and D(k) contain information regarding derivatives of ρ
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Special Case: 2-norm
α̂ = arg minα
12K
K∑k=1
‖e(k)‖22
1 If δ ∼ N(0, βδI) and φerri (α) = αi , approximate Wiener filter
2 If, in addition, ξ ∼ N(0, βξI) , get Tikhonov filter with α = βδβξ
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
1D Deconvolution
Convolution kernel Columns used for training
0 50 100 150 200 2500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Blurred image
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Compare to Standard Approaches
600 training signals2800 validation signalsNoise level: 0.001− 0.01
For each validation signal,reconstruct:
1 opt-Tik2 opt-error3 Tik-GCV4 Tik-MSE
Box and whisker plot (err)
opt−Tik opt−error Tik−GCV Tik−MSE
0
0.5
1
1.5
2
x 10−3
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Corresponding Error Images
opt-Tik opt-error Tik-GCV Tik-MSE
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Pareto Curve
100
101
102
103
10−2
10−1
number of training signals
aver
age
RRE
opt−Tik−SVDopt−error−SVDopt−Tik−GSVD
minx‖Aξ − b‖22 + λ
2 ‖Lξ‖2 , L =
1
−1 . . .. . . 1
−1
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
2D DeconvolutionTraining Validation
800 training images800 validation imagesGaussian point spread functionNoise level: 0.1− 0.15
Filter, φ(α) :opt-TSVDopt-Tikopt-splineopt-errorsmooth
Error Function, ρ(z) :Huber function2-norm4-norm
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Numerical Results
Huber function 2-norm 4-norm
opt−TSVD opt−Tik opt−spline opt−error smooth0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
opt−TSVD opt−Tik opt−spline opt−error smooth0
1
2
3
4
5
6
7
8
x 10−3
opt−TSVD opt−Tik opt−spline opt−error smooth
0
1
2
x 10−4
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Filter Factors and Error Images
Huber function 2-norm 4-norm
0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Singular values
Filt
er fa
ctor
s
opt−erroropt−TSVDopt−Tikopt−spline
0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Singular values
Filt
er fa
ctor
s
opt−erroropt−TSVDopt−Tikopt−spline
0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
Singular values
Filt
er fa
ctor
s
opt−erroropt−TSVDopt−Tikopt−spline
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
Julianne Chung, Virginia Tech
Designing Optimal Spectral Filters
Summary for Optimal Filters
Computing good regularization parameters can be difficult
Use training data to get optimal parameters/filters
Different error measures and filter representations can be used
Optimal filters can be computed off-line
Chung, Chung, O’Leary. SISC, 2011.Chung, Chung, O’Leary. JMIV, 2012.Chung, Español, Nguyen. ArXiv, 2014.
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Outline
1 Designing Optimal Spectral FiltersBackground on Spectral FiltersComputing Optimal FiltersNumerical Results
2 Designing Optimal Low-Rank Regularized Inverse MatricesRank-Constrained ProblemBayes Problem: Theoretical ResultsEmpirical Bayes Problem: Numerical Results
3 Conclusions
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
What is an optimal regularized inverse matrix(ORIM)?
Let Z ∈ Rn×m be a reconstruction matrixError vector: Zb− ξ
Error function: ρ : Rn → R+0
Error: err(Z) = ρ(Zb− ξ)= ρ(Z(Aξ + δ)− ξ)= ρ((ZA− In)ξ + Zδ)
For example, ρ(y) = ‖y‖pp , err(Z) = ‖Zb− ξ‖pp
Goal: Find a regularized inverse matrix Z ∈ Rn×m that minimizes
minZρ((ZA− In)ξ + Zδ)
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Rank-constrained Problem
Basic Idea: Enforce ORIM Z to be low-rank
Rank-constrained Problem:
arg minrank(Z)≤r
ρ((ZA− In)ξ + Zδ)
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Why does a low-rank inverse approximation makesense?
Rank-r truncated SVD (TSVD) solution can be written as
ξTSVD =r∑
i=1
u>i bσi
vi = VrΣ−1r U>r b ,
whereVr and Ur contain the first r vectors of V and U respectivelyΣr is the r × r principal submatrix of Σ
Matlab demo
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Bayes Risk Minimization
Suppose wetreat ξ as a random variable with given probability distributiontreat δ as a random variable with given probability distribution
Define the Bayes risk:
f (Z) = Eξ,δ(ρ((ZA− In)ξ + Zδ))
where E is the expected value.
Rank-constrained Bayes risk minimization problem:
minrank(Z)≤r
f (Z) = Eξ,δ (ρ((ZA− In)ξ + Zδ)) (1)
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Bayes Risk Minimization for ρ = ‖·‖22
minrank(Z)≤r
f (Z) = Eξ,δ(‖(ZA− In)ξ + Zδ)‖22
)Assume
ξ and δ are statistically independentµξ = 0 (mean) and C
−1ξ = MξM
>ξ for Pξ
µδ = 0 (mean) and C−1δ = η
2Im for Pδ
Then Eξ,δ(‖(ZA− In)ξ + Zδ‖22
)= ‖(ZA− In)Mξ‖2F + η
2 ‖Z‖2F
In addition, if C−1ξ = β2In,
Eξ,δ(‖(ZA− In)ξ + Zδ‖22
)= β2 ‖ZA− In‖2F + η
2 ‖Z‖2F
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
A Theoretical ResultTheoremGiven a matrix A ∈ Rm×n of rank r ≤ n ≤ m and an invertible matrixM ∈ Rn×n, let their generalized singular value decomposition beA = UΣG−1, M = GS−1V>. Let η be a given parameter, nonzero ifr < m. Let J ≤ r be a given positive integer. DefineD = ΣS−2Σ> + η2Im. Let the symmetric matrix H = GS−4Σ>D−1ΣG>
have eigenvalue decomposition H = V̂ΛV̂> with eigenvalues orderedso that λj ≥ λi for j < i ≤ n. Then a global minimizer Ẑ ∈ Rn×m of theproblem
Ẑ = arg minrank(Z)≤J
‖(ZA− In)M‖2F + η2 ‖Z‖2F
isẐ = V̂J V̂>J GS
−2Σ>D−1U>,
where V̂J contains the first J columns of V̂. Moreover this Ẑ is theunique global minimizer if and only if λJ > λJ+1.
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
A Special Case
Theorem
A global minimizer Ẑ ∈ Rn×m of the problem
Ẑ = arg minrank(Z)≤r
‖ZA− In‖2F + α2 ‖Z‖2F
is Ẑ = VrΨr U>r , where Vr contains the first r columns of V, Ur containsthe first r columns of U, and Ψr = diag
(σ1
σ21+α2 , . . . ,
σrσ2r +α
2
). Moreover, Ẑ
is unique if and only if σr > σr+1.
Remarks on Bayes problem:Expected value difficult to evaluateIn real applications, Pξ and Pδ are unknownNo theory for cases with ρ 6= ‖·‖22
Chung, Chung, and O’Leary (LAA 2014), Spantini et. al. (ArXiv 2014)
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Empirical Bayes Risk MinimizationIf Pξ and Pδ known→ use Monte Carlo samples as training dataIf Pξ and Pδ unknown but samples are given→ use samples as training data
Training data:
ξ(1), . . . , ξ(K ) realizations of random variable ξ
δ(1), . . . , δ(K ) realizations of random variable δ
b(k) = Aξ(k) + δ(k) , k = 1, ...,K
Empirical Bayes risk:
Eξ,δ (ρ((ZA− In)ξ + Zδ)) ≈1K
K∑k=1
ρ(Zb(k) − ξ(k))
Rank-constrained Empirical Bayes Minimization Problem:
Ẑ = arg minrank(Z)≤r
1K
K∑k=1
ρ(Zb(k) − ξ(k))
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Empirical Bayes Minimization Problem
Let Z be of rank r < min(m,n), then
Z = XY> =r∑
j=1
xjy>j
where X = [x1, ...xr ] ∈ Rn×r and Y = [y1, ...yr ] ∈ Rm×r
Rank-constrained problem:
(X̂, Ŷ) = arg minX,Y
1K
K∑k=1
ρ(XY>b(k) − ξ(k))
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Special Case: 2-norm
Let B = [b(1),b(2), ...,b(K )] and C = [ξ(1), ξ(2), ..., ξ(K )], then
minrank(Z)≤r
1K
K∑k=1
‖Zb(k) − ξ(k)‖22 =1K‖ZB− C‖2F (2)
Theorem
Ẑ = Pr B†
is a solution to the minimization problem (2), where Ṽ is the matrix ofright singular vectors of B and P = CṼs(Ṽs)> where s = rank(B). Thissolution is unique if and only if either r ≥ rank(P) or 1 ≤ r ≤ rank(P)and σ̄r > σ̄r+1, where σ̄r and σ̄r+1 denote the r and (r + 1)st singularvalues of P.
Friedland and Torokhti (2007), Sondermann (1986), Chung and Chung (2013)
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
For general error measures and large-scale problems
Rank-` update approach for computing a solution
Initialize Ẑ = 0n×m, X̂ = [ ], Ŷ = [ ]
while rank Ẑ < r
• Compute matrices X̂` ∈ Rn×` and Ŷ` ∈ Rm×` such that
(X̂`, Ŷ`) = arg minX∈Rn×`,Y∈Rm×`
1K
K∑k=1
ρ((Ẑ + XY>)b(k) − ξ(k))
• update matrix inverse approximation: Ẑ←− Ẑ + X̂`Ŷ>`• update solutions: X̂←− [X̂, X̂`], Ŷ←− [Ŷ, Ŷ`]
end
Chung and Chung (Inverse Problems, 2014)
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Empirical Bayes Problem
Remarks:Knowledge about forward model can be incorporated, not required
Need adequate number of training data
Solve inverse problems with only a matrix-vector multiplication: Ẑb
Framework allows more general error measures and morerealistic probability distributions
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Numerical Results: 1D Example10,000 signal observations, length 150Gaussian blur, white noise level 0.01Compare methods
TSVD-Â: Estimate  from training data, then use TSVDTSVD-A: requires AORIM2: uses training data
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
signal1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
signal2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
signal3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
signal4
time
TrueBlurred
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Reconstruction Errors for Validation Data
10 20 30 40 50 60 70 80 90 100
100
rank r
fK(Z
)
TSVD-ÂTSVD-AORIM2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
ORIM2
TSVD-A
TSVD-Â
sample error
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Pareto Curves
100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
sample
errorfor
trainingset
100 200 300 400 500 600 700 800 900 10000
1
2
3
# of training signals
sample
errorfor
validationset
25 to 75 percentilemedian error
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Numerical Results: 2D Example
6,000 observations, 128× 128spatially invariant Gaussian blur,reflexive BCGaussian white noise, levels rangefrom 0.1 to 0.15Compare methods
TSVD-Â: Estimate  from trainingdata, then use TSVDTSVD-A: requires AORIM2: uses training data
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Reconstruction Errors for Validation Data
100 200 300 400 500 600 700 800 900 1000101
102
103
104
rank r
fK(Z
)
TSVD-ÂTSVD-AORIM2
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
sample error
den
sity
ORIM2TSVD-Â
Julianne Chung, Virginia Tech
Designing Optimal Low-Rank Regularized Inverse Matrices
Reconstructed Images
True
Observed
ORIM2 TSVD-A TSVD-Â
ORIM2 TSVD-A TSVD-Â
Julianne Chung, Virginia Tech
Conclusions
Outline
1 Designing Optimal Spectral FiltersBackground on Spectral FiltersComputing Optimal FiltersNumerical Results
2 Designing Optimal Low-Rank Regularized Inverse MatricesRank-Constrained ProblemBayes Problem: Theoretical ResultsEmpirical Bayes Problem: Numerical Results
3 Conclusions
Julianne Chung, Virginia Tech
Conclusions
Concluding Remarks
New framework for solving inverse problems.Bayes problem:
Theoretical results can be derived for 2-normBayes problem provides insight
Empirical Bayes problem:Use training data to get
optimal spectral filters - for cases where A and its SVD are availableoptimal low-rank regularized inverse matrix - for cases where theforward model is unknown
Incorporate probabilistic informationOptimal filters and matrices can be computed off-lineReconstruction can be done efficiently and quality is good
Thank you!!
Julianne Chung, Virginia Tech
Conclusions
Some References on Inverse Problems
Discrete Inverse Problems: Insights and Algorithms - Per ChristianHansen
Deblurring Images: Matrices, Spectra, and Filtering - Hansen, Nagy andO’Leary
Introduction to Bayesian Scientific Computing: Ten Lectures onSubjective Computing - Calvetti and Somersalo
Computational Methods for Inverse Problems - Vogel
Introduction to Inverse Problems in Imaging - Bertero and Boccacci
Linear and Nonlinear Inverse Problems with Practical Applications -Mueller and Siltanen
Julianne Chung, Virginia Tech
Designing Optimal Spectral FiltersDesigning Optimal Low-Rank Regularized Inverse MatricesConclusions