Kronecker Products Image Restoration - Emory University

Kronecker Productsin

Image Restoration

James G. Nagy

Emory University

Atlanta, GA

Julie Kamm (Raytheon)

Lisa Perrone (Emory)

Michael Ng (HKU)

Misha Kilmer (Tufts)

Kronecker Product

C ⊗D =

c11D c12D · · · c1nDc21D c22D · · · c2nD

... ... ...cm1D cm2D · · · cmnD

Properties: • (C ⊗D)T = CT ⊗DT

• (C ⊗D)−1 = C−1 ⊗D−1

• (CDE)⊗(FGH) = (C⊗F )(D⊗G)(E⊗H)

• (C ⊗D)x ⇔ DXCT , x = vec(X)

Image Restoration

Given distorted image, reconstruct picture of original scene

b = Ax + n

b = distorted (blurred and noisy) image

x = true image

n = noise

A = matrix modeling distortion process

Computational Problem:

Given b, A, find an approximation of x

Regularization

• Problem: Given b = Ax + n, compute x

• Ill-conditioned A ⇒ x̂ = A−1b is bad!

• Regularization ⇒ math trick to stabilize numerical scheme.

– Regularization by filtering

e.g., using singular value decomposition (SVD)

Regularization

• Problem: Given b = Ax + e, compute x

• Ill-conditioned A ⇒ x̂ = A−1b is bad!

• Regularization ⇒ math trick to stabilize numerical scheme.

– Regularization by filtering

e.g., using singular value decomposition (SVD)

– Regularization by iteration

e.g., using conjugate gradients (CG)

Outline

1. Regularization by filtering.

2. Constructing the matrix A.

(Kronecker products)

3. Regularization by iteration.

4. Summary.

Regularization by Filtering

Suppose A = UΣV T , UTU = I, V TV = I

Σ =diagonal matrix

Then the ”inverse” solution is

x = A−1b

= V Σ−1UTb

=n∑

i=1

uTi b

σivi



Σ =diagonal matrix


x̂ = A−1(b + ε)

= V Σ−1UT (b + ε)

=n∑

i=1

uTi (b + ε)

σivi



Σ =diagonal matrix


x̂ = A−1(b + ε)

= V Σ−1UT (b + ε)

=n∑

i=1

uTi (b + ε)

σivi

=n∑

i=1

uTi b

σivi +

n∑i=1

uTi ε

σivi

= x + error


Basic Idea: Filter out problems caused by small singular values.

That is, compute:

xreg =N∑

i=1

φiuT

i b

σivi

where the “filter factors” satisfy:

φi ≈{

1 if σi is large0 if σi is small


• Truncated SVD (of pseudo inverse filter)

xtsvd =k∑

i=1

uTi b

σivi

• Tikhonov

xtik =N∑

i=1

σ2i

σ2i + α2

uTi b

σivi


One method to choose regularization parameters is:

Generalized Cross Validation

For example, in Tikhonov, choose α to minimize

GCV (α) =

NN∑

i=1

(uT

i b

σ2i + α2

)2

N∑i=1

1

σ2i + α2

2


How to efficiently compute SVD of large matrix?

Exploit structure of A:

1. Circulant ⇒ use Fourier transform:

A = F∗ΛF

2. ”Symmetric” Toeplitz + Hankel ⇒ use cosine transform:

A = CTΛC

3. Kronecker product ⇒ use SVD:

A = C ⊗D = (UcΣcVTc )⊗ (UdΣdV

Td )

= (Uc ⊗ Ud)(Σc ⊗Σd)(Vc ⊗ Vd)T

Constructing matrix A

• Assume distortion operation is ”spatially invariant”

• Choose an appropriate boundary condition

( e.g., zero, periodic, reflexive)

• Build matrix one column at a time:

i-th column = Aei ,

where ei = i-th unit vector


Build matrix one column at a time, using imaging system:

• Suppose ei = ith unit vector

point source


Build matrix one column at a time, using imaging system:

• Suppose ei = ith unit vector

• Then Aei = ith column of A

point source point spread function (PSF)


Spatially invariant ⇒ distortion independent of position





Each column of A is identical, modulo shift

⇒ one PSF is enough to describe A

⇒ A has Toeplitz structure

Constructing matrix A: Example 1

e5 =

000010000

→

0 0 00 1 00 0 0

→ blur →

p11 p12 p13

p21 p22 p23

p31 p32 p33

→ Ae5 =

p11

p12

p13

p21

p22

p23

p31

p32

p33

A =

p11

p12

p13

p21

p22

p23

p31

p32

p33


e5 =

000010000

→

0 0 00 1 00 0 0

→ blur →

p11 p12 p13

p21 p22 p23

p31 p32 p33

→ Ae5 =

p11

p12

p13

p21

p22

p23

p31

p32

p33

A =

p12 p11

p13 p12 p11

p13 p12

p22 p21

p23 p22 p21

p23 p22

p32 p31

p33 p32 p31

p33 p32


e5 =

000010000

→

0 0 00 1 00 0 0

→ blur →

p11 p12 p13

p21 p22 p23

p31 p32 p33

→ Ae5 =

p11

p12

p13

p21

p22

p23

p31

p32

p33

A =

p22 p21 p12 p11

p23 p22 p21 p13 p12 p11

p23 p22 p13 p12

p32 p31 p22 p21 p12 p11

p33 p32 p31 p23 p22 p21 p13 p12 p11

p33 p32 p23 p22 p13 p12

p32 p31 p22 p21

p33 p32 p31 p23 p22 p21

p33 p32 p23 p22


p11 p12 p13

p21 p22 p23

p31 p32 p33

= cdT =

c1d1 c1d2 c1d3

c2d1 c2d2 c2d3

c3d1 c3d2 c3d3

→ Ae5 =

c1

d1

d2

d3

c2

d1

d2

d3

c3

d1

d2

d3


p11 p12 p13

p21 p22 p23

p31 p32 p33

= cdT =

c1d1 c1d2 c1d3

c2d1 c2d2 c2d3

c3d1 c3d2 c3d3

→ Ae5 =

c1

d1

d2

d3

c2

d1

d2

d3

c3

d1

d2

d3

c1

d1

d2

d3

c2

d1

d2

d3

c3

d1

d2

d3


p11 p12 p13

p21 p22 p23

p31 p32 p33

= cdT =

c1d1 c1d2 c1d3

c2d1 c2d2 c2d3

c3d1 c3d2 c3d3

→ Ae5 =

c1

d1

d2

d3

c2

d1

d2

d3

c3

d1

d2

d3

c1

d2 d1

d3 d2 d1

d3 d2

c2

d2 d1

d3 d2 d1

d3 d2

c3

d2 d1

d3 d2 d1

d3 d2


p11 p12 p13

p21 p22 p23

p31 p32 p33

= cdT =

c1d1 c1d2 c1d3

c2d1 c2d2 c2d3

c3d1 c3d2 c3d3

→ Ae5 =

c1

d1

d2

d3

c2

d1

d2

d3

c3

d1

d2

d3

c2

d2 d1

d3 d2 d1

d3 d2

c1

d2 d1

d3 d2 d1

d3 d2

c3

d2 d1

d3 d2 d1

d3 d2

c2

d2 d1

d3 d2 d1

d3 d2

c1

d2 d1

d3 d2 d1

d3 d2

c3

d2 d1

d3 d2 d1

d3 d2

c2

d2 d1

d3 d2 d1

d3 d2

= C ⊗D

Kronecker Product Summary

If point spread function satisfies: P = cdT , then

• Zero BC ⇒ A = C ⊗D, where C and D are Toeplitz

• Periodic BC ⇒ A = C ⊗D, where C and D are circulant

• Reflexive BC⇒ A = C⊗D, where C and D are Toeplitz+Hankel

General Kronecker Product Decomposition

• If point spread function satisfies: P =r∑

k=1

ckdTk , then

A =r∑

k=1

Ck ⊗Dk

• Optimal approximation using work by Pitsianis and Van Loan

(Kamm, N., 2000; N., Ng, Perrone, 2003)

• For 3-dimensional problems, need tensor decompositions.

Kronecker Product Approximations

Optimal approximations (2-D problems):

• P =r∑

k=1

σkukvTk ⇒ σ1u1v

T1 = best rank-1 approximation

• Setting c1 =√

σ1u1 and d1 =√

σ1v1, then

A ≈ C1 ⊗D1 = best Kronecker product approximation



• P =r∑

k=1

σkukvTk ⇒ σ1u1v

T1 = best rank-1 approximation

• Setting c1 =√

σ1u1 and d1 =√

σ1v1, then

A ≈ C1 ⊗D1 = best Kronecker product approximation

= (UcΣcVTc )⊗ (UdΣdV

Td ) = SVD approximation



• P =m∑

k=1

σkukvTk ⇒

m∑k=1

σkukvTk = rank-m approximation

• Setting ck =√

σkuk and dk =√

σkvk, then

A ≈m∑

k=1

Ck ⊗Dk = sum of Kronecker product approximation



A =r∑

k=1

Ck ⊗Dk ≈ (Uc ⊗ Ud)Σ(Vc ⊗ Vd)T

where C1 = UcΣcVTc , D1 = UdΣdV

Td

minΣ

||A− (Uc ⊗ Ud)Σ(Vc ⊗ Vd)T ||F

Σ = diag((Uc ⊗ Ud)

TA(Vc ⊗ Vd))

= diag

(Uc ⊗ Ud)T

r∑k=1

Ck ⊗Dk

(Vc ⊗ Vd)


For 3-D problems:

• Need orthogonal tensor decompositions

SIMAX: de Lathauwer, de Moor, Vandewalle, ’00;

Kolda, ’01;

Zhang, Golub, ’01

• We use HOSVD (de Lathauwer, de Moor, Vandewalle, ’00):

P =r1∑

i=1

r2∑j=1

r3∑k=1

σijkui ◦ vj ◦wk .

• These vectors define matrices Ci, Dj and Fk, with

A =∑ ∑

σijk 6=0

∑Ci ⊗Dj ⊗ Fk ,

SVD Summary

The PSF and boundary condition define A:

• Periodic BC ⇒ FFT decomposition O(n2 logn)

• Reflexive BC, symmetric PSF⇒ DCT decomposition O(n2 logn)

• Otherwise Kronecker product approximation⇒ SVD O(n3)

SVD Summary

The PSF and boundary condition define A:

• Periodic BC ⇒ FFT decomposition O(n2 logn)

• Reflexive BC, symmetric PSF⇒ DCT decomposition O(n2 logn)

• Otherwise Kronecker product approximation⇒ SVD O(n3)

In MATLAB:

>> A = psfMatrix(PSF, ’boundary condition’);

>> [U, S, V] = svd(A);

>> x = TSVD(U, S, V, b);

Iterative Regularization

We consider algorithms (e.g., conjugate gradients) of the form

x0 = initial estimate of xfor j = 0,1,2, . . .

• xj+1 = computations involving xj, A,preconditioner matrix, M

• determine if stopping criteria are satisfiedend

Need to efficiently:

multiply with A, AT

solve linear systems with M , MT .

Multiply with A

• Use FFTs to form convolutions on padded arrays

• Padding depends on:

– boundary condition

– support (size) of PSF

• Can also do spatially variant blurs.

Preconditioning

Typical approach for Ax = b

Find matrix M satisfying the following properties:

• Inexpensive to “construct” M .

• Inexpensive to solve Mz = w.

• M−1A ≈ I (singular values of M−1A cluster around 1).

Preconditioning

“Ideal” preconditioner: M = A = UΣV T

• Solving Mz = w ⇒ z = V Σ−1UTw.

• For ill-posed problems, noise/errors in w are amplified.

Preconditioning

“Ideal” preconditioner: M = A = UΣV T

• Solving Mz = w ⇒ z = V Σ−1UTw.

• For ill-posed problems, noise/errors in w are amplified.

• Remedy: Regularize using truncated SVD

z = V Σ†UTw

where Σ† =diag(1/σ1, . . . ,1/σk,0, . . . ,0)

• Problem: The matrix M is singular.

Preconditioning Ill-Posed Problems

Modify TSVD idea (Hanke, N., Plemmons, ’93)

z = V Σ−1τ UTw

where

• Σ−1τ =diag(1/σ1, . . . ,1/σk,1, . . . ,1)

• σk ≥ τ > σk

That is, the preconditioner is:

Mτ = UΣτV T

Preconditioning Ill-Posed Problems

Notice that the preconditioned system is:

M−1τ A = (UΣτV T )−1(UΣV T )

= V Σ−1τ ΣV T

= V ∆V T

where ∆ =diag(1, . . . ,1, σk+1, . . . , σn)

That is,

• Large (good) singular values clustered at 1.

• Small (bad) singular values not clustered.

Preconditioning Summary

• Construct preconditioner by approximating A with either

FFT, DCT or SVD

• Use modified TSVD to regularize preconditioner.

In MATLAB:

>> A = psfMatrix(PSF, ’boundary condition’);

>> M = svdPrec(A, b);

The End!

• Exploiting Kronecker product structure in image restoration

– Can be effective in filtering methods that use SVD

– Can be effective as preconditioners

• Matlab software:

http://www.mathcs.emory.edu/∼nagy/RestoreTools/

• Related software for ill-posed problems (Hansen, Jacobsen)

http://www.imm.dtu.dk/∼pch/Regutools/

Kronecker Products Image Restoration - Emory University

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