designing optimal spectral filters and low-rank matrices ......designing optimal low-rank...

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


What is an inverse problem?

Physical SystemInput Signal Output Signal

Forward Model

Inverse Problem


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


Application: Image Deblurring

b = Aξ + δ

Given: Blurred image, b, andsome information about theblurring, A

Goal: Compute approximation oftrue image, ξ


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


SRimages.aviMedia File (video/avi)

Application: Limited-Angle Tomography

X-ray ImagingDigital TomosynthesisComputed Tomography


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


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.


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


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


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


Designing Optimal Spectral Filters

Outline



3 Conclusions



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



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)



SVD AnalysisThe naïve inverse solution:

ξ = A−1b

= VΣ−1UT b

=n∑

i=1

uTi bσi

vi




ξ̂ = A−1(b + δ)

= VΣ−1UT (b + δ)

=n∑

i=1

uTi (b + δ)σi

vi




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



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



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



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.



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



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



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



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))



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



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



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



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



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



Corresponding Error Images

opt-Tik opt-error Tik-GCV Tik-MSE



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



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



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



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


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


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



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.


Designing Optimal Low-Rank Regularized Inverse Matrices

Outline



3 Conclusions



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)



Rank-constrained Problem

Basic Idea: Enforce ORIM Z to be low-rank

Rank-constrained Problem:

arg minrank(Z)≤r

ρ((ZA− In)ξ + Zδ)



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



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)



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



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.



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)



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))



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))



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)



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)



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



Numerical Results: 1D Example10,000 signal observations, length 150Gaussian blur, white noise level 0.01Compare methods

TSVD-Â: Estimate Â 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



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



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



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 Â from trainingdata, then use TSVDTSVD-A: requires AORIM2: uses training data



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



Reconstructed Images

True

Observed

ORIM2 TSVD-A TSVD-Â

ORIM2 TSVD-A TSVD-Â


Conclusions

Outline



3 Conclusions


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!!


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


Designing Optimal Spectral FiltersDesigning Optimal Low-Rank Regularized Inverse MatricesConclusions

designing optimal spectral filters and low-rank matrices ......designing optimal low-rank...

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