basics of image deblurringweb.ipac.caltech.edu/staff/fmasci/home/astro_refs/image...basics of image...

Basics of Image Deblurring


Math 561

Fall, 2006


Outline

Introduction

Mathematical Model

The Computational Problem

Filtering Algorithms

Fast Computational Methods for FilteringBCCB MatricesSymmetric Toeplitz-plus-Hankel MatricesKronecker Product Matrices


Introduction

Image Restoration: Simple Example

I Given blurred image, and someinformation about the blurring.

I Goal: Compute approximation oftrue image.


Introduction

Applications: Astronomy

Viewing distant star fields using ground based telescopes.

Observed data Restored data


Introduction

Applications: Space Observations

Viewing space vehicles, satellites and other space junk.



Introduction

Applications: Medical Imaging



Introduction

Applications: Microscopy



Introduction

Applications: Iris Recognition



Mathematical Model

Mathematical Model of Image Formation

b(u, v) =

∫ ∫a(u, s, v , t)x(s, t)ds dt + e(u, v)

u

v

s

t


Mathematical Model

”Convolution” implies shift invariance

b(u, v) =

∫ ∫a(u − s, v − t)x(s, t)ds dt + e(u, v)

u

v

s

t


Mathematical Model

Some remarks

I The mathematical model:

b(u, v) =

∫ ∫a(u, s, v , t)x(s, t)ds dt + e(u, v)

is an example of an ill-posed inverse problem.

Small changes in e ⇒ large changes in x .

I Images are usually discrete pixel values, not functions!I Can approximate by matrix-vector equation: b = Ax + e

I A is defined by the ”point spread function” a(u, s, v , t).

I If the PSF is not known, it can be estimated by imaging”point source” objects.


Mathematical Model

Generating Experimental Point Spread Function

Point Source Object PSF: Picture of Point Source Object




From the matrix-vector equation

b = Ax + e

I Given b and A (or the PSF), compute an approximation of x

I Regarding the noise, e:I It is usually not known.I However, some statistical information may be known.I It is usually small, but it cannot be ignored!

That is, solving the linear algebra problem:

Ax = b ⇒ x = A−1b

usually does not work.



SVD Analysis

An important linear algebra tool: Singular Value Decomposition

Let A = UΣVT where

I Σ =diag(σ1, σ2, . . . , σn) , σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0

I UTU = I , VTV = I



SVD Analysis

An important linear algebra tool: Singular Value Decomposition

Let A = UΣVT where

I Σ =diag(σ1, σ2, . . . , σn) , σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0

I UTU = I , VTV = I

For image restoration problems,

I σ1 ≈ 1, small singular values cluster at 0

I small singular values ⇒ oscillating singular vectors



SVD Analysis

The naıve inverse solution can then be represented as:

x = A−1b

= VΣ−1UTb

=n∑

i=1

uTi b

σivi



SVD Analysis


x = A−1(b + e)

= VΣ−1UT (b + e)

=n∑

i=1

uTi (b + e)

σivi



SVD Analysis


x = A−1(b + e)

= VΣ−1UT (b + e)

=n∑

i=1

uTi (b + e)

σivi

=n∑

i=1

uTi b

σivi +

n∑i=1

uTi e

σivi

= x + error




b = Ax + eI Given b and A

I Goal: Compute approximationof true image, x

I Naıve inverse solution

x =n∑

i=1

uTi b

σivi +

n∑i=1

uTi e

σivi

= x + error

is corrupted with noise!



Basic Idea of Filtering

Basic Idea: Filter out effects of small singular values.

xreg =n∑

i=1

φiuT

i b

σivi

where the ”filter factors” satisfy

φi ≈{

1 if σi is large0 if σi is small



Examples of Filtering Methods

1. Truncated SVD

xtsvd =k∑

i=1

uTi b

σivi

2. Tikhonov

xtik =n∑

i=1

σ2i

σ2i + α2

uTi b

σivi

3. Iterative (more in next lecture)



Examples of Filtering Methods

1. Truncated SVD

xtsvd =k∑

i=1

uTi b

σivi

2. Tikhonov

xtik =n∑

i=1

σ2i

σ2i + α2

uTi b

σivi


10−5

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

singular values

filte

r fa

ctor

n = 64, k = 36

10−5

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

singular values

filte

r fa

ctor

α = 0.001



Choosing Regularization Parameters

Lots of choices: Generalized Cross Validation (GCV), L-curve,discrepancy principle, ...





GCV and Tikhonov: Choose λ to minimize

GCV(λ) =

nn∑

i=1

(uT

i b

σ2i + λ2

)2

(n∑

i=1

1

σ2i + λ2

)2





GCV and Tikhonov: Choose λ to minimize

GCV(λ) =

nn∑

i=1

(uT

i b

σ2i + λ2

)2

(n∑

i=1

1

σ2i + λ2

)2

For L-curve, see G. Rodriguez.


Fast Computational Methods for Filtering

Remarks on Computational Methods

I SVD filtering can be computationally expensive.

I Further simplifying approximations are often used to obtainmore efficient algorithms:

I Spatial invariance and periodic boundary conditions:I A is circulant.I Can replace SVD with fast Fourier transforms (FFT).

I Other fast transforms (e.g., DCT) can sometimes be used.

I Separable blur ⇒ A can be decomposed using Kroneckerproducts:

A = Ar ⊗ Ac



One-Dimensional Problems

Recall:

Each blurred pixel is a weighted sum of the correspondingpixel and its neighbors in the true image.

For example, if

x =

x1

x2

x3

x4

x5

and b =

b1

b2

b3

b4

b5

then

b3 = � x1 + � x2 + � x3 + � x4 + � x5




The weights come from the PSF:

For example, if

x =

x1

x2

x3

x4

x5

, p =

p1

p2

p3

p4

p5

and b =

b1

b2

b3

b4

b5

then





For example, if

x =

x1

x2

x3

x4

x5

,

p5

p4

p3

p2

p1

and b =

b1

b2

b3

b4

b5

then

1. Rotate the PSF, p, by 180 degrees about center.





For example, ifx1

x2

x3

x4

x5

p5

p4

p3

p2

p1

and b =

b1

b2

b3

b4

b5

then

2. Match coefficients of rotated PSF and x





For example, ifx1

x2

x3

x4

x5

p5

p4

p3

p2

p1

and b =

b1

b2

b3

b4

b5

then

3. Multiply corresponding components and sum.





For example, ifx1

x2

x3

x4

x5

p5

p4

p3

p2

p1

and b =

b1

b2

b3

b4

b5

then

b3 = p5x1 + p4x2 + p3x3 + p2x4 + p1x5




If the weights fall outside the true image scene

?x1

x2

x3

x4

x5

p5

p4

p3

p2

p1

b =

b1

b2

b3

b4

b5

then

b2 = p5 ? + p4x1 + p3x2 + p2x3 + p1x4




If the weights fall outside the true image sceneimpose boundary conditions

wx1

x2

x3

x4

x5

p5

p4

p3

p2

p1

b =

b1

b2

b3

b4

b5

then

b2 = p5 w + p4x1 + p3x2 + p2x3 + p1x4




If the weights fall outside the true image sceneimpose boundary conditions, such as zero

0x1

x2

x3

x4

x5

p5

p4

p3

p2

p1

b =

b1

b2

b3

b4

b5

then

b2 = p4x1 + p3x2 + p2x3 + p1x4




If the weights fall outside the true image sceneimpose boundary conditions, such as periodic

x5x1

x2

x3

x4

x5

p5

p4

p3

p2

p1

b =

b1

b2

b3

b4

b5

then

b2 = p5x5 + p4x1 + p3x2 + p2x3 + p1x4




If the weights fall outside the true image sceneimpose boundary conditions, such as reflexive

x1x1

x2

x3

x4

x5

p5

p4

p3

p2

p1

b =

b1

b2

b3

b4

b5

then

b2 = p5x1 + p4x1 + p3x2 + p2x3 + p1x4




In general, we can write

b1

b2

b3

b4

b5

=

p5 p4 p3 p2 p1

p5 p4 p3 p2 p1

p5 p4 p3 p2 p1

p5 p4 p3 p2 p1

p5 p4 p3 p2 p1

w1

w2

x1

x2

x3

x4

x5

y1

y2

where

I zero BC ⇒ wi = yi = 0

I periodic BC ⇒ w1 = x4, w2 = x5, y1 = x1, y2 = x2

I reflexive BC ⇒ w1 = x2, w2 = x1, y1 = x5, y2 = x4




Therefore, for zero boundary conditions we get:

b1

b2

b3

b4

b5

=

p3 p2 p1

p4 p3 p2 p1

p5 p4 p3 p2 p1

p5 p4 p3 p2

p5 p4 p3

x1

x2

x3

x4

x5

Here A is a Toeplitz matrix




For periodic boundary conditions we get:

b1

b2

b3

b4

b5

=

p3 p2 p1 p5 p4

p4 p3 p2 p1 p5

p5 p4 p3 p2 p1

p1 p5 p4 p3 p2

p2 p1 p5 p4 p3

x1

x2

x3

x4

x5

Here A is a circulant matrix




For reflexive boundary conditions we get:

b1

b2

b3

b4

b5

=

p3 p2 p1

p4 p3 p2 p1

p5 p4 p3 p2 p1

p5 p4 p3 p2

p5 p4 p3

+

p4 p5

p5

p1

p1 p2

x1

x2

x3

x4

x5

Here A is a Toeplitz-plus-Hankel



Two-Dimensional Problems

With zero boundary conditions we obtain BTTB matrix:

b11

b21

b31

b12

b22

b32

b13

b23

b33

=

p22 p12 p21 p11

p32 p22 p12 p31 p21 p11

p32 p22 p31 p21

p23 p13 p22 p12 p21 p11

p33 p23 p13 p32 p22 p12 p31 p21 p11

p33 p23 p32 p22 p31 p21

p23 p13 p22 p12

p33 p23 p13 p32 p22 p12

p33 p23 p32 p22

x11

x21

x31

x12

x22

x32

x13

x23

x33

b = vec(B), p = vec(P), x = vec(X)



Two-Dimensional Problems

Matrix structures:

I Zero boundary conditions ⇒ A is BTTB

I Periodic boundary conditions ⇒ A is BCCB

I Reflexive boundary conditions ⇒ A is sum of BTTB, BTHB,BHTB, BHHB

Legend:

BTTB: Block Toeplitz with Toeplitz blocks

BCCB: Block circulant with circulant blocks

BHHB: Block Hankel with Hankel blocks

BTHB: Block Toeplitz with Hankel blocks

BHTB: Block Hankel with Toeplitz blocks



Remark on Boundary Conditions

I Many other choices for boundary conditions.

I For example: Anti-reflective(Arico, Donatelli, Serra-Cappizano)

I Preserve continuity of the image at boundaries.

I Preserve continuity of the normal derivative at the boundary.

I Can help to reduce ringing artifacts.



Separable Two-Dimensional Blurs

Separable Blur ⇒ Horizontal and vertical components separate.

In this case, the PSF array has rank = 1:

P = rcT =

c1

c2

c3

[ r1 r2 r3]

=

c1r1 c1r2 c1r3c2r1 c2r2 c2r3c3r1 c3r2 c3r3





Forming the matrix with this special PSF we obtain (zero BC):

A =

c2r2 c1r2 c2r1 c1r1c3r2 c2r2 c1r2 c3r1 c2r1 c1r1

c3r2 c2r2 c3r1 c2r1c2r3 c1r3 c2r2 c1r2 c2r1 c1r1c3r3 c2r3 c1r3 c3r2 c2r2 c1r2 c3r1 c2r1 c1r1

c3r3 c2r3 c3r2 c2r2 c3r1 c2r1c2r3 c1r3 c2r2 c1r2c3r3 c2r3 c1r3 c3r2 c2r2 c1r2

c3r3 c2r3 c3r2 c2r2






A =

r2

c2 c1

c3 c2 c1

c3 c2

r1

c2 c1

c3 c2 c1

c3 c2

0

r3

c2 c1

c3 c2 c1

c3 c2

r2

c2 c1

c3 c2 c1

c3 c2

r1

c2 c1

c3 c2 c1

c3 c2

0 r3

c2 c1

c3 c2 c1

c3 c2

r2

c2 c1

c3 c2 c1

c3 c2






A = Ar ⊗ Ac

r2 r1r3 r2 r1

r3 r2

⊗ c2 c1

c3 c2 c1

c3 c2

Where ⊗ denotes Kronecker product.




Similar structures occur for other boundary conditions:

A = Ar ⊗ Ac

whereI Zero boundary conditions:

I Ar is Toeplitz, defined by rI Ac is Topelitz, defined by c

I Periodic boundary conditions:I Ar is circulant, defined by rI Ac is circulant, defined by c

I Reflexive boundary conditions:I Ar is Toeplitz-plus-Hankel, defined by rI Ac is Toeplitz-plus-Hankel, defined by c



Summary of Matrix Structures

BC Non-separable PSF Separable PSF

zero BTTB Kronecker of Toeplitz matrices

periodic BCCB Kronecker of circulant matrices

reflexive BTTB+BTHB Kronecker of+BHTB+BHHB Toeplitz-plus-Hankel matrices



BCCB Matrices

BCCB Matrices

With periodic boundary conditions A is a BCCB matrix:

b11

b21

b31

b12

b22

b32

b13

b23

b33

=

p22 p12 p32 p21 p11 p31 p23 p13 p33

p32 p22 p12 p31 p21 p11 p33 p23 p13

p12 p32 p22 p11 p31 p21 p13 p33 p23

p23 p13 p33 p22 p12 p32 p21 p11 p31

p33 p23 p13 p32 p22 p12 p31 p21 p11

p13 p33 p23 p12 p32 p22 p11 p31 p21

p21 p11 p31 p23 p13 p33 p22 p12 p32

p31 p21 p11 p33 p23 p13 p32 p22 p12

p11 p31 p21 p13 p33 p23 p12 p32 p22

x11

x21

x31

x12

x22

x32

x13

x23

x33

b = vec(B), p = vec(P), x = vec(X)



BCCB Matrices

Important BCCB Matrix Property

I Every BCCB matrix has the same set of eigenvectors:

A = F∗ΛF

whereI F is the two-dimensional discrete Fourier transform matrixI F∗F = FF∗ = II Λ = diagonal containing eigenvalues of A

I Computations with F can be done very efficiently:

O(N log N)

using Fast Fourier Transforms (FFT)s.



BCCB Matrices

BCCB and FFT Relations

A = F∗ΛF ⇒ FA = ΛF ⇒ Fa1 = Λf1

where

I a1 = first column of AI f1 = first column of F,

f1 =1√N

11...1

I Thus,

Fa1 = Λf1 =1√N

λ

where λ is a vector containing the eigenvalues of A.



BCCB Matrices

Some BCCB Computations

If A is BCCB defined by PSF P, and

b = Ax = F∗ΛFx

then to compute b use

S = fft2( circshift(P) );B = ifft2(S .* fft2(X));

whereb = vec(B) and x = vec(X)



BCCB Matrices


If A is BCCB defined by PSF P, and

xnaive = A−1b = F∗Λ−1Fb

then to compute xnaive use

S = fft2( circshift(P) );X = ifft2(fft2(B) ./ S);




BCCB Matrices


If A is BCCB defined by PSF P, and Φ contains filter factors,

xfilt = F∗ΦΛ−1Fb

then to compute xfilt use

S = fft2( circshift(P) );Sfilt = Phi ./ S;X = ifft2(fft2(B) .* Sfilt);




Symmetric Toeplitz-plus-Hankel Matrices



zero BTTB Kronecker of Toeplitz matrices

periodic BCCB Kronecker of circulant matrices

reflexive BTTB+BTHB Kronecker of+BHTB+BHHB Toeplitz-plus-Hankel matrices




Toeplitz-plus-Hankel Matrices

With reflexive boundary conditions A is a

BTTB + BTHB + BHTB + BHHB

matrix defined by the PSF.

”Strong” symmetry condition: If

P =

0 0 0

0 P 00 0 0

where

I P is (2k − 1)× (2k − 1) with center located at the (k, k)

I P = fliplr(P) = flipud(P) = fliplr(flipud(P))




BTTB+BTHB+BHTB+BHHB Matrix Properties

If the PSF satisfies strong symmetry condition, then:

I A is symmetric

I A is block symmetric

I Each block in A is symmetric

I A has the spectral decomposition

A = CTΛC

where C is the two-dimensional discrete cosine transform(DCT) matrix.

I As with FFTs, computations with C cost O(N log N).




Toeplitz-plus-Hankel and DCT Relations

A = CTΛC ⇒ CA = ΛC ⇒ Ca1 = Λc1

where

I a1 = first column of A

I c1 = first column of C,

I Thus, the eigenvalues of C are given by

Ca1 = Λc1 ⇒ λi =[Ca1]i[c1]i




Additional DCT Computations

If A is defined by strongly symmetric PSF with reflexive boundaryconditions, and

b = Ax = CTΛCx

then to compute b use

e1 = zeros(size(P));, e1(1,1) = 1;S = dct2( dctshift(P) ) ./ dct2(e1);B = idct2(S .* dct2(X));






If A is defined by strongly symmetric PSF with reflexive boundaryconditions, and

xnaive = A−1b = CTΛ−1Cb

then to compute xnaive use

e1 = zeros(size(P));, e1(1,1) = 1;S = dct2( dctshift(P) ) ./ dct2(e1);X = idct2(dct2(B) ./ S);






If A is defined by strongly symmetric PSF with reflexive boundaryconditions, and Φ contains filter factors,

xfilt = CTΦΛ−1Cb

then to compute xfilt use

e1 = zeros(size(P));, e1(1,1) = 1;S = dct2( dctshift(P) ) ./ dct2(e1);Sfilt = Phi ./ S;X = idct2(dct2(B) .* Sfilt);




Kronecker Product Matrices



zero BTTB Kronecker of Toeplitz matricesperiodic BCCB Kronecker of circulant matricesreflexive BTTB+BTHB Kronecker of

+BHTB+BHHB Toeplitz-plus-Hankel matricesreflexive BTTB+BTHB Kronecker of symmetric

strongly symmetric +BHTB+BHHB Toeplitz-plus-Hankel matrices




Separable PSFs

Recall: If the PSF has rank = 1,

P = crT =

c1

c2...

cm

[ r1 r2 · · · rn]

then the blurring matrix has the form

A = Ar ⊗ Ac

where Ar is defined by r and Ac is defined by c.

Assume for now Ar and Ac are known.




Useful Kronecker Product Properties

I b = (Ar ⊗ Ac)x ⇔ B = AcXATr

where b = vec(B) and x = vec(X)

I (Ar ⊗ Ac)T = AT

r ⊗ ATc

I (Ar ⊗ Ac)−1 = A−1

r ⊗ A−1c

I (A(1)r ⊗ A(1)

c )(A(2)r ⊗ A(2)

c ) = (A(1)r A(2)

r )⊗ (A(1)c A(2)

c )




Exploiting Kronecker Product Properties in Matlab

Using the property:

b = (Ar ⊗ Ac)x ⇔ B = AcXATr

in Matlab we can compute

B = Ac*X*Ar’;





Using the property:

b = (Ar ⊗ Ac)x ⇔ B = AcXATr

and if Ar and Ac are nonsingular,

(Ar ⊗ Ac)−1 = A−1

r ⊗ A−1c

we obtainX = A−1

c BA−Tr

In Matlab we can compute

X = Ac \ B / Ar’;





We can compute SVD of small matrices:

Ar = UrΣrVTr and Ac = UcΣcV

Tc

Then

A = Ar ⊗ Ac

= (UrΣrVTr )⊗ (UcΣcV

Tc )

= (Ur ⊗Uc)(Σr ⊗Σc)(Vr ⊗ Vc)T

= SVD of big matrix A

Note: Do not need to explicitly form big matrices

Ur ⊗Uc, Σr ⊗Σc, Vr ⊗ Vc





To compute inverse solution from SVD of small matrices:

xnaive = A−1b = VΣ−1UTb

is equivalent to

Xnaive = A−1c BA−T

r = VcΣ−1c UT

c BUrΣ−1r VT

r

A Matlab implementation could be:

[Ur, Sr, Vr] = svd(Ar);

[Uc, Sc, Vc] = svd(Ac);

S = diag(Sc) * diag(Sr)’;

X = Vc * ( (Uc’ * B * Ur)./S ) * Vr’;





If Φ contains filter factors, the filtered solution

xfilt = A−1b = VΦΣ−1UTb

can be computed as:

[Ur, Sr, Vr] = svd(Ar);

[Uc, Sc, Vc] = svd(Ac);

S = diag(Sc) * diag(Sr)’;

Sfilt = Phi ./ S;

X = Vc * ( (Uc’ * B * Ur) .* Sfilt ) * Vr’;




Summary of Fast Algorithms

For spatially invariant PSFs, we have the following fast algorithms.

Boundary Matrix FastPSF condition structure algorithm

arbitrary periodic BCCB 2-d FFT

strongly symmetric reflexive sum of BXXB 2-d DCT

rank one arbitrary Kronecker product 2 SVDs




Matlab Examples

Recall some examples of filtering methods:

1. Truncated SVD

xtsvd =k∑

i=1

uTi b

σivi

2. Tikhonov

xtik =n∑

i=1

σ2i

σ2i + α2

uTi b

σivi


10−5

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

singular values

filte

r fa

ctor

n = 64, k = 36

10−5

10−4

10−3

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

singular values

filte

r fa

ctor

α = 0.001




The End

I Image deblurring examples arise in many applications.

I Fast algorithms can produce good reconstructions for manyproblems.

I Further details and software can be found in:Deblurring Images: Matrices, Spectra and FilteringP. C. Hansen, J. G. Nagy and D. P. O’LearySIAM, 2006

http://www2.imm.dtu.dk/∼pch/HNO/

I Next time: Iterative methods for harder problems.

basics of image deblurringweb.ipac.caltech.edu/staff/fmasci/home/astro_refs/image...basics of image...

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