WEIGHTED OVERCOMPLETE DENOISING

Onur G. Guleryuz

oguleryuz@erd.epson.comEpson Palo Alto Laboratory

Palo Alto, CA

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Overview•Signal in additive, i.i.d., Gaussian noise scenario.•Consider standard denoising with overcomplete transforms and thresholding.

•Denoised estimates are suboptimally combined to form an average.

•Optimal combination as an adaptive linear estimation problem for each pixel.

•Simulation results with DCTs and wavelets.•Conclusion.

Definition, assumptions, why it works…

Three solutions ( no explicit statistics required ! )One solution is based on number of zero coefficients.Form of equivalent adaptive linear denoising filters.

Notation

x )1( N

wxy

: N-dimensional signal

: signal corrupted with additive noise

x̂ : estimate of given x y

Denoising with Overcomplete Transforms and Thresholding

iH )( NN ),...,1( Mi

,yc ii H

),(ˆ ii cc

,ˆˆ 1iii cx H

: linear transformthi

M

iix

Mx

1

ˆ1

ˆ

: transform step

: thresholding step

: denoised estimatethi

extensive literature

: combination step this paper

1.

2.

3. ),...,1( Mi

Basic Principles for a Single Transform and Hard-Thresholding

0 N-1k

Transform Domain

x(n) X(k)

0 N-1k

+ W(k)c(k) = +w(n)y(n)=

Signal Domain

x(n) X(k)

Main Assumption:Sparse Decomposition

+T

-T 0 N-1k

c(k)^

Denoised

Overcomplete Transforms

M

iix

Mx

1

ˆ1

ˆ

1x̂2x̂

3x̂

4x̂

Denoise

…

Basic Idea of this Paper - 1

)(ˆ4

1)(ˆ

4

1e

iie nxnx

)(ˆ)()(ˆ4

1e

iieie nxnnx

Give more weight to sparse blocks (I will determine weights optimally).

Basic Idea of this Paper - 2

)(ˆ)()(ˆ4

1s

iisis nxnnx

Suppose only DCT-DC terms remain after thresholding.

Optimally determine weights everywhere.

Main Derivation

)(ˆ1

)(ˆ1

nxM

nxM

ii

)(ˆ)()(ˆ1

nxnnxM

iii

Assumption: Thresholding works! Denoising removes mostly noise.

ii wxx ~ˆ

,yc ii H

),(ˆ ii cc

,ˆˆ 1iii cx H

wxy

Optimally determine for n=1,…,N to minimize conditional mse.)(ni

...]||)(ˆ)([| 2nxnxE

•Solutions will be independent of explicit statistics.•No covariance/modeling assumptions, etc.

Reminder box

Main Derivation with Hard-Thresholding

,yc ii H),(ˆ ii cc

,ˆˆ iT

ii cx H

wxy

Tkc

Tkckckc

i

iii

|)(|

|)(|

,

,

0

)()(ˆ

TkckV ii |)(||

],...,||)(ˆ)([| 12

MVVnxnxE

Thresholding works! Sparse decompositions (signal only hascomponents in significant sets.)

Significant sets.

Reminder boxOptimization:

Main Derivation with Hard-Thresholding

(n)W(n))

11

11

)((

1

1

)( 22

nn xx

)(

)(1

n

n

M

(n)

Main Derivation with Hard-Thresholding

p,qw

nqqTqpp

Tp

Tnw

nqqTq

Tpp

Tp

Tn

Mqpp,q

uu

uwwEu

VVnwnwE

G(n)

HSHHSH

HSHHSH

W(n)

2

2

1

][

],...,|)()([

(n)W(n))

11

11

)((

1

1

)( 22

nn xx

)( iVk otherwise

Tkcandlklk i

i

|)(|,

,

,

0

1),(S

Main Derivation with Hard-Thresholding

(n)G(n)

1

1

Solution is only a function of and the significant sets iViH

Solutions

(n)G(n)

1

1

(n)D(n)

1

1

(n)(n)D

~

1

1

: only the diagonal terms of D(n) G(n)

needs basis functions of the transforms and cross scalar products depending on the iV

needs basis functions of the transforms

only needs the spatial support of the basis functions of the transforms

for block transforms, diagonal entries are 1/(number of nonzero coefficients in each block)

fullsolution

diagonalsolution

significant-only

solution

Properties of Solutions

fully overcomplete 8x8 DCTs

(256x256)

voronoi

)5( w

•The full solution is sensitive to model failures.

standard full

diagonal signif.-only

standard diagonal

Equivalent Adaptive Linear Filters

)(ˆ)()(ˆ1

nxnnxM

iii

ynLnx T)()(ˆ

,yc ii H),(ˆ ii cc

,ˆˆ iT

ii cx H

wxy Reminder boxpixel 1

pixel 2

pixel 3

•At each pixel we adaptively determine a linear filter for denoising.

Equivalent Adaptive Linear Filters)(nL pixel 1 pixel 2 pixel 3

full solution

diagonal solution

standard solution

Equivalent Adaptive Linear Filters)(nL

pixel 1 pixel 2 pixel 3

full solution

diagonal solution

standard solution

Simulation Results

fully overcomplete 8x8 DCTs

(1280x960)

teapot

)10( w

standard

diagonal

signif.-only

Simulation Results

fully overcomplete 3 level wavelets(Daubechies orthonormal D8 bank)

(1280x960)

teapot

)10( w

Simulation Results

fully overcomplete 3 level wavelets(Daubechies orthonormal D8 bank)

)10( wLena (512x512)

)10( wBarbara (512x512)

Conclusion

M

iix

Mx

1

ˆ1

ˆ )(ˆ)()(ˆ1

nxnnxM

iii

•Instead of blindly combining denoised estimates, form the optimal combination.

Also true for denoising with naturally overcomplete transforms like complex wavelets.

•Statistically “clean” formulation, no dependence on explicit statistics.

•Easily generalized to other forms of thresholding (additional degree of freedom).

•Better, more sophisticated thresholding is expected to improve results.