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Cascadic Multilevel Methods for Cascadic Multilevel Methods for Large-Scale Ill-Posed Problems Large-Scale Ill-Posed Problems Fiorella Sgallari University of Bologna , Italy Faculty of Engineering Department of Mathematics - CIRAM Joint work

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Joint work with Serena Morigi, Lothar Reichel, Andriy Shyshkov

Cascadic Multilevel Methods for Cascadic Multilevel Methods for Large-Scale Ill-Posed Problems Large-Scale Ill-Posed Problems

Fiorella Sgallari

University of Bologna, Italy Faculty of Engineering Department of Mathematics - CIRAM

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  • 1. Inverse and ill-posed problems in image

processing

  • 2. Linear and nonlinear regularization methods
  • 3. Cascadic multilevel method for image deblurring
  • 4. Examples

Outline

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Inverse problems arise when one seeks to determine the cause of an observed effect Image restoration: Determine the unavailable exact image from an available contaminated version Ill-posed problems (in the sense of Hadamard)

Inverse problem in image processing

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Ill-posed problem in image processing

U is the representation of the real world into the image plane Let ϕ be the degradation model function

F = ϕ(U)

∀ϕ(U) = U + noise ∀ϕ(U) = K * U + noise

  • ……………………..

U = ϕ-1(F)

F is the observed corrupted image

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Different type of noise

Original salt & pepper speckle Gauss-add Gauss-molt

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Continuous degradation model: Integral equation can be expressed as Discretization yields with matrix A block Toeplitz with Toeplitz blocks

Degradation model

ˆ ( ) ( , ) ( ) ( ) f x h x y u y dy x x

η

Ω

= + ∈ Ω

∫∫

ˆ f h u

η = ∗ +

b = Au

Perturbed observed image Point Spread Function Blur and noise-free image Data noise

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Linear discrete ill-conditioned problems

arise from the discretization of linear ill-posed problems

  • The matrix A is of ill-determined rank, possibly singular.

A may be indefinite.

  • The right-hand side b represents available data that

generally is contaminated by an error.

Au = b

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Available contaminated, possibly inconsistents, linear system (1) Unavailable associated consistent linear system with error-free right-hand side (2) Let denote the desired solution of (2), e.g., the minimal-norm solution. Task: Determine an approximate solution of (1) that is a good approximation of

Au b

=

ˆ Au b

=

ˆ u

ˆ u

Linear discrete ill-conditioned problems

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Solution Au=b: add 0.1% noise to rhs u=A-1b u=A-1b

1 1 1 1

ˆ ˆ ˆ ˆ ( ) b b e u A b e A b A e u A e

− − − −

= + = + = + = +

Shaw.m

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Popular regularization methods

  • 1. Tikhonov regularization
  • 2. Iterative regularization methods:

regularization by truncated iteration

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Tikhonov regularization

Solve the Tikhonov minimization problem where is the regularization operator. Common choices: L = I or a finite difference operator;

μ > 0 is the regularization parameter to be determined. It is important to determine a suitable value; see Engl, Hanke, Neubauer.

{ }

2 2 u mxn pxn

min Au-b Lu , A R ; L R , p n,

µ + ∈ ∈ ≤

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Regularization by truncated iteration

CGLS: CG applied to Define the Krylov subspace

* * A Au A b

=

{

}

− −

=

2 1

( * , * ) * ,( * ) * ,.., ( * ) * ,( * ) *

k k k

K A A A b span A b A A A b A A A b A A A b Minimum-Residual Method

∈

∈ − = − = − ≥ ≥ ≥

( * , * ) 1

Then ( * , * ) and min Therefore discrepancy satisfies ...

k k

k k k u K A A A b j j k

u K A A A b Au b Au b d b Au b d d

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Stopping Criterion

Discrepancy principle

Let α > 1 be fixed, The iterate uk satisfies the discrepancy principle if

(Stopping rule)

Terminate the iterations as soon as iterate uk satisfies Denote the termination index by kδ

δ = − =

b b e ˆ

k

Au b

α δ − ≤

α δ α δ

−

− ≤ − >

1 k k

Au b Au b

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  • ---- exact solution

Hansen, 2005

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An iterative method is a regularization method if

ˆ limsup

k e

u u

δ

δ δ → ≤

− =

[see Nemirovskii,Hanke,Reichel]

Minimum-residual methods and their Krylov subspaces

Matrix Algorithm Krylov subspace Symmetric MINRES MR-II Nonsymmetric and square GMRES RRGMRES Any CGLS, LSQR

( ) ( )

k k

κ κ

A,b A, Ab ( ) ( )

k k

κ κ

A,b A, Ab ( )

k

κ

T T

A A, A b

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Nonlinear denoising-deblurring methods

Return to Tikhonov regularization: Euler-Lagrange equations with gradient descent:

2 2

Tikhonov TV

u

min ( h* u f ) R( u )dx u h* ( h* u f ) D( u ), u f t R( u ) u D( u ) u u R( u ) u D( u ) div( ) u

µ µ

Ω

    − +       ∂ = − − + = ∂ = ∇ ⇒ = ∆ ∇ = ∇ ⇒ = ∇

∫

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TV regularization Tikhonov regularization

Edge-enhanched regularization

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TV regularization Tikhonov regularization

Edge-enhanched regularization

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Perona-Malik diffusivity

2

1 1 D( u ) div( g( u ) u ) g( s ) /( s / ),

κ ρ = ∇ ∇ = + >

g(s) s

u ( g( u ) u ) t

ρ

∂ − ∇ • ∇ ∇ = ∂

2

[ Perona Malik ‘87, Cattè Lions Morel ‘92]

Perona-Malik denoising model

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Computational effort

1 1 g( s ) s / κ

= +

Space discretization Explicit time schemes:

  • Easy implementation
  • CFL stability conditions (small step, many iterations)

⇒ method expensive Semi-implicit time schemes:

  • Solve a linear system at each time step
  • Stability conditions due to the blur term

⇒ method expensive

2

u h* ( h* u f ) ( g( u ) u ), u f t u

µ ∂ = − − + ∇ ⋅ ∇ ∇ = ∂ =

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Combine: Computational efficiency of truncated iteration for linear systems Edge-preserving property

  • f nonlinear models

Au b

=

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Multilevel methods

W1 Wk

nested linear subspaces coarsest level finest level

n k n k

W W ... W W W

⊂ ⊂ ⊂ ⊂ ℜ = ℜ

1 2 1

1

4

j j j j

dimW n n n

+

= =

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Multilevel methods

W1 Wk

1 1 1 1 2

1 L

i i+1 i

: Res triction Operator

n i i i i i i i i k * i i i

R W M :W W b M b R M M M A R AR i k

+ + + + +

ℜ → → = = = ≤ <

  • averaging (replaces groups of four adjacent

by one pixel, whose value is the average of the values of the four pixels it replaces)

  • local weighted least-squares

approximation

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Multilevel methods

W1 Wk

i-1 i

Prolongation Oper ato 1 r <i k

i i

P P :W W

→ ≤

  • Piecewise Linear Interpolation
  • Nonlinear prolongation
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Edge-preserving Prolongation operator

q q

u D ( u ) u 1 q 2 u ε

ε

  ∇   = ∇ ∇ • ≤ ≤   ∇  

1

i i-1

du ˆ D( u ) u(0) = P u dt

=

( )

D ( u ) g( u ) u

= ∇ • ∇ ∇

2 2

May integrate for a few, say ≤ 10, time steps by explicit method

uij uij+1 uij-1 ui+1j ui-1j N W E S NE NW SE SW

Use the Perona-Malik operator D(u) as a nonlinear prolongation

  • perator for the multilevel scheme

P ˆ is a linear interpolation operator

q=1.5

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Cascadic multilevel methods: Practice

  • Solve linear system A1u=b1 in W1, map solution to W2 using P2,
  • Solve linear system in W2 for correction, map to W3 using P3,
  • Solve linear system in W3 for correction, map to W4

...

  • An approximate solution in Wk is found

Prolongation

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Algorithm: Cascadic multiresolution

k

Input ( =number of levels Output approximate solution u

k

: , ,k ) : W

≥ ∈

A b 1

i i

i i i,0 i i i ,m i i i i,0 i i,0 i ,m

Compute A ,b ,i : , ,..,k by restriction u : for i : , ,..,k do u : P u u : linear _solver( A u b A u ) Correction step: u : u u endfor

−

= = = = ∆ = = − = + ∆

1

1 2 1 2

Prolongation

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Cascadic Multiresolution method

LSQR, GMRES, RRGMRES for unsymmetric case Theorem:

Let Assume that computed by MR II ( or LSQR) Then

i

i i i i i * i i

ˆ b b c i k c c R(P ) R( A ) i k u

δ

+

− ≤ ≤ ≤ = ⊂ < ≤

1

1 1 2 1

Pi Linear prolongation

i i i i

i i ˆ b b c

ˆ ˆ

ˆ lim sup u u i k

δ δ → − ≤

− = ≤ ≤

1

where

+ u = A b i i i

Thus, cascadic multilevel methods are regularization methods.

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Image denoising and deblurring

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256x256 A 2562x2562 sym. matrix with Toeplitz blocks, band specifies the half-

bandwidth Toeplitz blocks; sigma is the variance of the Gaussian PSF. Band=3 sigma=1

Gaussian PSF, (band ,sigma) Matlab -Regularization Tool – P.C. Hansen (blur.m)

Band=3 sigma=5 Band=3 sigma=3

Blur and noisy contamined image:

e has normally distributed entries with mean zero, scaled to obtain a desired

noise level: from which we determine the value of δ.

ˆ b Au e

= +

Blur and noise-free image ˆ

u

ˆ e b

ν =

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Example 1: Multilevel with linear and nonlinear P

Band=3 Sigma=3

Linear P

Band=5 Sigma=3

Blurred and noisy image: Gaussian blur, ν=0.05 noise

Nonlinear P (use D2)

4 levels

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Multilevel vs nonlinear method

Example: Original 256 × 256 image

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Multilevel vs nonlinear method

Image contaminated by Gaussian blur and ν=0.05 noise

dB u u u u PSNR

k k

ˆ 255 log 20 ) ˆ , (

10

− =

Peak Signal to Noise Ratio

Sigma=3 Band=5

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Multilevel vs nonlinear method

Restoration by nonlinear PDE model, D2-regularization, 500 time-steps, 1000 mvp, PSNR=28.84 dB u u u u PSNR

k k

ˆ 255 log 20 ) ˆ , (

10

− =

Peak Signal to Noise Ratio

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Multilevel vs nonlinear method

Restoration by multilevel method, 3 levels, D2-P, 1+2+4 mvp, PSNR=35.85 dB u u u u PSNR

k k

ˆ 255 log 20 ) ˆ , (

10

− =

Peak Signal to Noise Ratio

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Example 2

Blurred and noisy image:

Gaussian blur, ν=0.1 noise band=5 sigma=3 Blur- and noise-free 512 × 512-pixel image

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Restoration using multilevel method, 5 levels, nonlinear Perona-Malik-D2 P, PSNR=27.12

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5 iterations for the 1-level method instead of 3 iterations as suggested by the discrepancy principle PSNR=23.01

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Restore image contaminated by Gaussian blur (band=5,sigma=5) and ν=0.05 noise. Then apply edge detector from gimp.

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1-level method (MR II)

Restore image contaminated by Gaussian blur (band=5,sigma=5) and ν=0.05 noise. Then apply edge detector from gimp.

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multilevel method, 3-levels, D2-PM-P

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Band=5, sigma=5, ν=0.05 noise, 3 levels, D2-PM-P

Cross-section of a restored image (black) and contamined image (red)

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Cross-section of a restored image (black) and contamined image (red)

Band=5, sigma=5, ν=0.1 noise, 3 levels, D2-PM-P

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Cross-section of a restored image (black) and contamined image (red)

Band=5, sigma=5, ν=0.1 noise, 3 levels, D2-PM-P

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Multilevel methods

W1 Wk

1 1 1 1 2

1 L

i i+1 i

: Res triction Operator

n i i i i i i i i k * i i i

R W M :W W b M b R M M M A R AR i k

+ + + + +

ℜ → → = = = ≤ <

  • averaging (replaces groups of four adjacent

by one pixel, whose value is the average of the values of the four pixels it replaces)

  • local weighted least-squares

approximation

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A non symmetric first n/2 rows are the made up of the first n/2 rows of last n/2 rows are the last n/2 rows of

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Spatially variant Gaussian blur

, A : I (T T ) I (T T ) nxn n I diagonal matrix whose first n/2 diagonal entries are , the remaining are I I I T T Toeplitz banded matrices

= ⊗ + ⊗ = = −

2 1 1 1 2 2 2 1 2 1 1 2

512 1

( ) jk

Gaussian in D ( j k ) exp( , if j k band t

  • therwise

σ σ π  − − − ≤  =   

2 2

1 1 2 2

l l l

T T

⊗

2 2

T T

⊗

1 1

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Spatially variant Gaussian blur

GMRES

Spatially variant Gaussian blur

band=5 σ1=4 σ2=4.5 noise level ν=0.05

RRGMRES

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Spatially variant Gaussian blur

GMRES

Spatially variant Gaussian blur

band=5 σ1=4 σ2=4.5 noise level ν=0.05

RRGMRES

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band=5 s1=4 s2=4.5 noise level n=0.05

band=5 σ1=4 σ2=4.5 noise level ν=0.1

Spatially variant Gaussian blur

5 steps PDE-models PSNR=20.86

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Example 3 Linear motion Blur

412x412

The PSF is represented by a line segment of length r (width)

pixels in the direction of the blur.

The latter is determined by the angle θ (in degrees) measured counter-clockwise from the positive axis. The PSF takes on the value r−1 on this segment and vanishes elsewhere.

The larger the width, the more ill-conditioned the matrix A

width= 5, θ=30 noise level ν = 0.1

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Example 3 Linear motion Blur

1 level

Multilevel cascadic restoration PSNR=21.53 LSQR PM-D 2 levels PSNR=22.10

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1-level method (LSQR) PSNR=25.44 iter=4

band=9 σ1=4.5 σ2=9 noise level ν=0.1

Spatially variant Gaussian blur

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3-levels, D2-PM-P PSNR=26.78 iter=2 1 2

Spatially variant Gaussian blur

band=9 σ1=4.5 σ2=9 noise level ν=0.1

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3-levels, D2-PM-P PSNR=26.78 iter=2 1 2

Spatially variant Gaussian blur

band=9 σ1=4.5 σ2=9 noise level ν=0.1

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TV- Prolongation operator

3-levels, TV-P PSNR=25.96 iter=2 1 2

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TV- Prolongation operator

3-levels, TV-P PSNR=25.96 iter=2 1 2

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1-level method PSNR=28.59 iter=11

band=9 σ1=4.5 σ2=9 noise level ν=0.01 GMRES

Spatially variant Gaussian blur

Original 512x512

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band=9 σ1=4.5 σ2=9 noise level ν=0.01 GMRES

Spatially variant Gaussian blur

3-levels, TV-P PSNR=29.90 iter=1 6 8 3-levels, D2-PM-P PSNR=30.92 1 5 8

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band=9 σ1=4.5 σ2=9 noise level ν=0.01 GMRES

Spatially variant Gaussian blur

Difference TV-PM Difference 1Level- PM 3Levels

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Future work

  • Gain better understanding of convergence properties
  • Gain better understanding of prolongation and restriction
  • perators
  • More investigation with different blur operators
  • Apply cascadic idea to general large-scale ill-posed

problems

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Thanks for your attention !

Thanks to the Organizers Thanks to the Organizers