wavelet based denoising of mri and fmri imagessanja/sanja_files/fmrisymp24jan... · 2006-01-25 ·...

Wavelet based denoising of MRI and fMRI images

Aleksandra Pizurica

Ghent University

Department Telecommunications and Information Processing

Image Processing and Interpretation

fMRI Symposium, Gent, 24. January 2006 2

Interesting to note…

MRI image

Denoising by wavelet thresholdinghas originated from a biomedical application!

Weaver (1991) introduced wavelet thresholding for noise reductionin magnetic resonance images (MRI).

Theoretical development: Donoho and Johnstone


LL3 HL3

LH3 HH3

HH2

HH1LH1

LH2

HL1

HL2

Two dimensional DWT

highpass g

h

g

h

2

2

2

2

2

2

g

h

HH j+1

HL j+1

LH j+1

LL j+1

LL j

horizontal filtering

vertical filtering

lowpass


Discrete wavelet transform

DETAIL IMAGESwavelet coefficients

APPROXIMATIONscaling coefficients

Wavelet coefficient values

0

Peaks indicate image edges


Noise in the wavelet domain


Peaks indicate image edges

0

Noise-free reference


Wavelet domain noise reduction

DETAILSwavelet coefficients

APPROXIMATION‘scaling’ coefficients

Spatial context

Noisy coefficient

Noise-free estimate

‘Wavelet shrinkage’


Choosing a wavelet: Nv, support size K, symmetry

0 5 10 15-0.5

0

0.5

1

1.5

-5 0 5-1

-0.5

0

0.5

1

1.5Symmlets (Daubechies)

ϕ ψsym8

-4 -2 0 2 4-1

-0.5

0

0.5

1

-5 00

0.2

0.4

0.6

0.8

5

-5 0 5

-0.5

0

0.5

1

1.5

2

-4 -2 0 2 4

-2

-1

0

1

2

Biorthogonal wavelets

ϕ ψ

ϕ~ ψ~

1.5 2

-0.5 0 0.5 1 1.50

0.5

1

1.5

-1.5 -1 -0.5 0 0.5-1.5

-1

-0.5

0

0.5

1

1.5

db1

ϕ ψ

0 1 2 3-0.5

0

0.5

1

-1 0 1 2-2

-1

0

1

db2

ϕ ψ

0 5 10 15-0.5

0

0.5

1

1.5

-5 0 5

-1

-0.5

0

0.5

1

db8

ϕ ψ

Daubechies wavelets dbNv:

K=15

K=3

K=1

Nv - number of vanishing moments: ,0)( =ψ∫∞

∞−k

10 −≤≤ vNkdttt

K 2Nv-1≥A tradeoff:


Wavelet domain denosing

Wavelet transform

wavelet coefficients

scaling coefficients

Inverse wavelet

transform

sL

w1w2

wL

Remove noise

Estimated noise-free wavelet coefficients

Noisy input

Denoised output


Gaussian noise modelϑ+= fv

],...,[ 1 Nff=f - ideal, noise-free image;

a single index l refers to spatial position (raster scanning)

],...,[ 1 Nvv=v - observed noisy image; ],...,[ 1 Nϑϑϑ = - noise vector;

ϑϑϑ π

ϑ1

2

1

2/e

)det()2(

1)(

−−=

QT

n Qp

Covariance matrix: ]))())(([( TEEEQ ϑϑϑϑ −−=

We usually assume )(0)( TEQE ϑϑϑ =⇒=

Gaussian noise

Independent identically distributed (i.i.d.) noise: σσ =l for all l

In the wavelet domain

w = y + n; w=W{v}; y=W{f}; n=W{ϑ};


Noise model and notation

image domain: vl = fl + ϑl; ϑl ~N(0,σin)

resolutionlevel

spatial position

orientation

;,,,d

ljd

ljd

lj nyw +=wavelet domain: ),0(~,dj

dlj Nn σ

Orthogonal wavelet transform maps white noise into white noise and in

dj σσσ == for all j,d

( ) 22

indj

dj S σσ =

12

22,

−

= ∑∑

j

kk

kk

HLLHj hgS

)1(2

2

2

2

−

= ∑∑

j

kk

kk

HHj hgS


Performance evaluation

• Noise model: v = f + ϑ;

• The objective is to produce an estimate of the unknown image f, which approximates it best under given criterion

• A common criterion: minimize Mean Squared Error (MSE)

f̂

∑=

−=−=N

iii ff

NNMSE

1

22 )ˆ(1

||ˆ||1 ff

• Signal to Noise Ratio (SNR)

][/||||

log10||ˆ||

||||log10

2

102

2

10 dBMSE

NSNR

ffff

=−

=

• Peak Signal to Noise Ratio (PSNR)

][/255

log102

10 dBMSE

NPSNR =

],...,[ 1 Nff=f


Ideal coefficient selection and attenuation

• Noise model w = y + n; n~N(0,σ)

• Consider a nonlinear estimate

• What is ideal θ(y) in the MSE sense?

22

2

)(σ

θ+

=y

yyideal

• The estimator θideal(y) that minimizes this error is

ideal attenuator

( ){ } ( ) 22222 )()(1)( yyyywyE θσθθ +−=−• One can show that

• If we restrict }1,0{)( ∈yθ

≤>

=σσ

θ||if,0

||if,1)(./

y

yyselideal oracle thresholding

this becomes

wyy )(ˆ θ=


Denoising by wavelet thresholding


0T

-T0

Thresholded values

Hard thresholding “keep or kill” Soft thresholding “shrink or kill”

thresholded thresholded

input input


Denoising by wavelet thresholding

Hard thresholding “keep or kill” Soft thresholding “shrink or kill”

thresholded thresholded

input input

w = y + n

≥<

=Tww

Twyht

||,

||,0ˆ

≥−<

=TwTww

Twyst

||),|)(|sgn(

||,0ˆ

-T T -T T

T

w – noisy coefficient;

y – noise-free coefficient; n – i.i.d. Gaussian noise


Choice of the threshold

• Universal threshold [Donoho&Johnstone, 1994]

• SURE threshold [Donoho&Johnstone, 1995]

– Minimizes Stein’s unbiased risk estimate

• Cross-validation threshold– Threshold selection based on cross-validation [Nason, 1994] and

generalized cross-validation [Jansen, 1997], [Weyrich, 1998]

• Optimal threshold in the mean squared error sense under a Laplacian prior for noise free coefficients [Chang2000]

;)log(2ˆ nTuniv σ=

A central problem in wavelet threshoding, studied by many researchers. Several notable examples:

yBayesT σσ ˆ/ˆ 2=

σ - noise standard dev.; n - number of samples

- standard dev. of the signalyσ


T=Tuniv

Denoising by hard thresholding

T=1.5Tuniv


Denoising by soft thresholding

Hard thresholdingSoft thresholding Noisy


Denoising by singularity detection

w1 wavelet coefficients

w2

w3

w1

Input signal Rate of increase of the modulus of the wavelet transform across scales is proportional to thelocal Lipschitz regularity

Reconstruction from multiscale edges [Mallat&Hwang, 1992]


Interscale Correlation Method

T?

Coarser, processed detail

A noisy detailMask

[Xu et al, 1994; Bao et al, 2004]

• Find masks of important coefficoients by comparing interscale products to a threshold

• Perform hard-thresholding based on the detected masks


Soft thresholdingNoisy Interscale correlation

Denoising by interscale correlation

Interscale correlation

Different noise estimates


Noise in MRI

- observed noisy image;

- ideal, noise-free image;],...,[ 1 Nff=f

],...,[ 1 Ndd=d

2Im,

2Re, )sin()cos( lllll nfnfd +++= θθ

low SNR)1,0( 2 == nf σ

m

high SNR

f1

)1,( 21 == nff σ

p(m)The magnitude ml

2Im,

2Re, )sin()cos(|| llllll nfnfdm +++== θθ

is Rician distributed


Noise in MRI

high SNR ( f =f1 )

f1

low SNR (f=0)

m

p(m)

noise-free f

noisy m

magnitu

de

co

ntr

ast

SNR


Removing bias

• Due to the linearity of the wavelet transform: input bias bias in both the wavelet and scaling coefficients

• Mean values of the wavelet and scaling coefficients of the magnitude MRI image differ from the means of their noise-free components

• Bias is signal dependent

• Solution: denoise SQUARED magnitude image: wavelet coefficients – not biased; scaling coefficients - a constant bias (proportional to the input noise standard deviation) [Nowak, 1999]


Wavelet domain denosing of MRI images

Wavelet transform

wavelet coefficients

scaling coefficients

Inverse wavelet

transform

sL

w1w2

wL

Remove noise

Estimated noise-free wavelet coefficients

Noisy input Denoised output

Remove constant

bias

square square root


Data adaptive MRI image denoising

[Nowak, 1999]


OBSERVATION

ESTIMATE

Our approach

• Hard- and soft-thresholding are examples of wavelet shrinkage

• Shrinkage can result from Bayesian methods too

Subband-adaptive shrinkage

OBSERVATION

ESTIMATE

Spatial context

Spatially adaptive shrinkage


Spatially adaptive wavelet shrinkage

)|(

)|(

0

1

Hwf

Hwf

l

ll

=ξ)|(

)|(

0

1

Hzf

Hzf

l

ll

=η )(

)(

0

1

HP

HP=ρ

zlwl

f(w|H1)

f(w|H0)f(z|H0)

f(z|H1) P(H1)

P(H0)

LSAInoisy coefficient subband statistics

H0: |y| < T signal of interest absent

H1: |y| > T signal of interest presentw = y + n

lll

llllll

wwzwHPyρηξ

ρηξ+

==1

),|(ˆ 1

LSAI (local spatial activity indicator)


T?

Coarser, processed detail

A noisy detailMask

Empirical estimation of data distributions

f(z|1)

histogramsf(z|0)

f(m|1)

f(m|0)

0 100 200 300 400-10

0

10

20

0 200 400 600-10

0

10

20( )logf(z|0)

f(z|1)

( )logf(m|0)

f(m|1)


Results

Data dependent (Nowak) ProposedInterscale correlation (Xu)Noisy input


Noisy image Denoised image

Results


Denoising and segmentation

Denoised Segmentation by Segmentation by Proposed MethodProposed Method

Noisy


Denoising and segmentation (2)

Sample Image from Phantom

Segmentation by Segmentation by Proposed MethodProposed Method

Ground truth


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

FP FN Kappa

CSFGMWM

Denoising and segmentation (3)


Application to fMRI

BOLD imagesideal

activated regionactivated region


Results

input denoised


idealideal

Results

noisy BayesShrink Proposed


Evaluation on real data

wavelet based denoising of mri and fmri imagessanja/sanja_files/fmrisymp24jan... · 2006-01-25 ·...

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