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WEIGHTED OVERCOMPLETE DENOISING
Onur G. Guleryuz
[email protected] Palo Alto Laboratory
Palo Alto, CA
(Please view in full screen mode to see animations)
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
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
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.
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 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.