COMPLEX WAVELET TRANSFORM IN [1mm] BIOMEDICAL IMAGE DENOISINGdsp.vscht.cz/hostalke/upload/TCP07_presentation.pdf · 2007-11-20 · CWT IN BIOMEDICAL IMAGE DENOISING E. Hošťálková,

CWT IN BIOMEDICALIMAGE DENOISING

E. Hošťálková, A. Procházka

Table of ContentsCOMPLEX WAVELET TRANSFORM IN

BIOMEDICAL IMAGE DENOISING

Eva Hošťálková & Aleš Procházka

Institute of Chemical Technology in PragueDept of Computing and Control Engineering

http://dsp.vscht.cz/

Technical Computing Prague 2007



Table of Contents

Table of Contents

1 Introduction

2 Dual Tree Complex Wavelet TransformIntroduction to Dual Tree Complex Wavelet TransformQ-Shift Dual Tree Complex Wavelet Transform

3 Denoising of CT ImagesDenoising TechniqueDenoising Results

4 Conclusions



Table of Contents

Table of Contents

1 Introduction



4 Conclusions



Table of Contents

Table of Contents

1 Introduction



4 Conclusions



Table of Contents

Table of Contents

1 Introduction



4 Conclusions



Introduction

DTCWTIntroduction to DTCWT

Q-Shift DTCWT

Denoising of CT ImagesDenoising Technique

Denoising Results

Conclusions

Further Reading

Introduction

Applications of the Wavelet Transform in Image processingNoise reductionImage compression and codingEdge detectionFeature extraction ⇒ segmentation & retrievalRestoration of missing or corrupted components

Limitations of the Discrete Wavelet Transform (DWT)Zero crossings of the coefficients at a singularityStrong shift dependenceAliasing ⇐ downsampling and non-ideal filtersLack of directional selectivity - unable to distinguishbetween +45◦ and −45◦



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Introduction

Applications of the Wavelet Transform in Image processingNoise reductionImage compression and codingEdge detectionFeature extraction ⇒ segmentation & retrievalRestoration of missing or corrupted components

Limitations of the Discrete Wavelet Transform (DWT)Zero crossings of the coefficients at a singularityStrong shift dependenceAliasing ⇐ downsampling and non-ideal filtersLack of directional selectivity - unable to distinguishbetween +45◦ and −45◦



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Introduction

Undecimated DWTDWT without downsampling.

ADVANTAGES

Shift independenceDISADVANTAGES

Poor directional selectivityGreat computation cost

Complex Wavelet Transform (CWT)Employs analytic complex wavelets⇒ Magnitude-phase representation

Large magnitude ⇒ presence of a singularityPhase: its position within the support of the wavelet

⇒ Shift invariance & no aliasingIn this work: Dual-Tree CWT (DTCWT) by Kingsbury,Selesnick



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Introduction

Undecimated DWTDWT without downsampling.

ADVANTAGES

Shift independenceDISADVANTAGES

Poor directional selectivityGreat computation cost

Complex Wavelet Transform (CWT)Employs analytic complex wavelets⇒ Magnitude-phase representation

Large magnitude ⇒ presence of a singularityPhase: its position within the support of the wavelet

⇒ Shift invariance & no aliasingIn this work: Dual-Tree CWT (DTCWT) by Kingsbury,Selesnick



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Table of Contents

1 Introduction



4 Conclusions



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Introduction to DTCWT

Dual Tree Complex Wavelet TransformDual tree (two DWT trees) of real filters ⇒ real andimaginary parts of each complex coefficientPerfect reconstruction (PR)Approx. analytic filters ⇒ approx. shift invarianceDirectional selectivity in 2D:DTCWT

6 directional subbands±15◦, ±45◦ and ±75◦

DWT

3 directional subbands0◦, 45◦ and 90◦

Limited redundancy 2d in d-dimensional space

Q-Shift DTCWTBy Prof. Kingsbury, used in this workQ-shift . . . quarter of a sample period shift



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Dual Tree Complex Wavelet TransformDual tree (two DWT trees) of real filters ⇒ real andimaginary parts of each complex coefficientPerfect reconstruction (PR)Approx. analytic filters ⇒ approx. shift invarianceDirectional selectivity in 2D:DTCWT

6 directional subbands±15◦, ±45◦ and ±75◦

DWT

3 directional subbands0◦, 45◦ and 90◦

Limited redundancy 2d in d-dimensional space

Q-Shift DTCWTBy Prof. Kingsbury, used in this workQ-shift . . . quarter of a sample period shift



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Directional Selectivity of 2D Wavelets

(a) REAL PARTS OF 2D Q−SHIFT COMPLEX WAVELETS

+15◦ +45◦ +75◦ −75◦ −45◦ −15◦

(b) IMAGINARY PARTS OF 2D Q−SHIFT COMPLEX WAVELETS

+15◦ +45◦ +75◦ −75◦ −45◦ −15◦

(c) 2D DB4 REAL WAVELETS

90◦ (LoHi) 45◦ (HiHi) 0◦ (HiLo)



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Table of Contents

1 Introduction



4 Conclusions



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Q-Shift Dual Tree CWT

Q-Shift DTCWT 3-level analysis scheme

X(z)

Tree a

Tree b

Ho0a(z)(1)

Ho1a(z)(0)

Ho0b(z)(0)

Ho1b(z)(1)

2

2

2

2

H0a(z)(3q)

H1a(z)(q)

H0b(z)(q)

H1b(z)(3q)

2

2

2

2

H0a(z)(3q)

H1a(z)(q)

H0b(z)(q)

H1b(z)(3q)

2

2

2

2

Level 1 Level 2 Level 3

odd

odd

even

even

even

even

Re{Lo3}

Re{Hi3}

Re{Hi2}

Re{Hi1} Im {Lo3}

Im {Hi3}

Im {Hi2}

Im {Hi1}

Q−SHIFT DUAL−TREE CWT

Red - lowpass filters, green - highpass filters, 2↓ - downsampling by 2



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Q-Shift DTCWTLevel 1: any orthogonal/biorthogonal set of filtersBeyond level 1: even-tap Q-shift filtersBoth trees - same frequency responseConjugate symmetry ⇒ linear phaseIndividual asymmetry ⇒ orthonormal PR

Orthonormal Set of Q-Shift FiltersFilters in tree b - reverse of the filters in tree a

h0b(n) = h0a(N−1−n)

Synthesis filters - reverse of the analysis filters

g0a(n) = h0a(N−1−n)

where n = 0, . . . ,N − 1 and N is the filter length



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Q-Shift DTCWTLevel 1: any orthogonal/biorthogonal set of filtersBeyond level 1: even-tap Q-shift filtersBoth trees - same frequency responseConjugate symmetry ⇒ linear phaseIndividual asymmetry ⇒ orthonormal PR

Orthonormal Set of Q-Shift FiltersFilters in tree b - reverse of the filters in tree a

h0b(n) = h0a(N−1−n)

Synthesis filters - reverse of the analysis filters

g0a(n) = h0a(N−1−n)

where n = 0, . . . ,N − 1 and N is the filter length



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Analytic WaveletsComplex wavelet ψc(t) = ψr (t) + j ·ψi(t) is analytic when

ψi(t) = HT{ψr (t)} = 1π

∫∞−∞

ψr (t)t−τ dτ = ψr (t) 1

π t

where t,τ is continuous time

Fourier transform of a Hilbert transform pair

Hi(ω) = FT{HT{ψr (t)}} = −j · sgn(ω)Hr (ω)

where ω denotes frequency and j the complex unit

ImplicationsSingle sided spectrum ⇒ no aliasing ⇒ shift invarianceImpossible with compact support! ⇒ only approximatelyanalytic



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading



ψi(t) = HT{ψr (t)} = 1π

∫∞−∞


π t








Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading



ψi(t) = HT{ψr (t)} = 1π

∫∞−∞


π t








Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Frequency Spectra of a Real and an Analytic Wavelet

50 100 150 200

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

(a) Db−7 REAL WAVELET

time−0.5 0 0.5

0.2

0.4

0.6

0.8

1(b) Db−7 WAVELET: DFT

Mag

nitu

de

ω/2π

50 100 150 200

−0.1

0

0.1

0.2

(c) Q−SHIFT COMPLEX WAVELET

time−0.5 0 0.5

2

4

6

8

(d) Q−SHIFT WAVELET: DFT

Mag

nitu

de

ω/2π

ψ(t)

|ψc(t)|=|ψ

r(t)+jψ

i(t)| |Ψ

c(ω)|=|Ψ

r(ω)+jΨ

i(ω)|

ψr(t) ψ

i(t)

Ψ(ω)

4 levels, 14-tap filters: Daubechies for DWT and q-shift for DTCWT.



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Half-Sample Delay ConditionHalf of a sample period difference between filters in tree aand b ⇒ analyticIn the Fourier domain:

MAGNITUDE |H0b(ejω)| = |H0a(ejω)|

PHASE ∠H0b(ejω) = ∠H0a(ejω)− 0.5ω

Q-Shift Filters DesignFulfill the phase condition only approximately⇒ Only approx. shift independentGroup delays ' 1

4 and 34 of a sample period



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Half-Sample Delay ConditionHalf of a sample period difference between filters in tree aand b ⇒ analyticIn the Fourier domain:

MAGNITUDE |H0b(ejω)| = |H0a(ejω)|

PHASE ∠H0b(ejω) = ∠H0a(ejω)− 0.5ω

Q-Shift Filters DesignFulfill the phase condition only approximately⇒ Only approx. shift independentGroup delays ' 1

4 and 34 of a sample period



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading


Approximate Shift Invariance(a) ORIGINAL IMAGE (b) AFTER THE SHIFT (c) CHANGE OF SUBBAND ENERGY

DTCWT DWT

level 1 0 % 0 %

level 2 4.5 % 34.1 %

level 3 5.7 % 118.8 %

level 4 6.2 % 77.6 %

level 4

(LoLo)

27.7 % 95.7 %

(d) 2−LEVEL DTCWT AFTER THE SHIFT

+0%

+0%+0%

+0%

+0% +0%

+25.6% +5.3%

+0.4%+5.3%

+25.6%+5.9%

+4.1% +5.9%

(e) DWT AFTER THE SHIFT

+0%

+0% +0%

−44% +49.8%

+49.8% +2.7%

Percentual changes of subband energy.



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Table of Contents

1 Introduction



4 Conclusions



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Technique

DenoisingComputed Tomography(CT) imagesSuppressing lower energywavelet coefficients (noise)Soft universal waveletshrinkage

Soft ThresholdingSOFT THRESHOLDING

δ(s)

−δ(s)

−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)

−2 δ(s)

−δ(s)

0

δ(s)

2 δ(s)

3 δ(s)

before thr.after thr.

Thresholding magnitudes of complex coefficientsVary slowlyNot distorted by aliasing

Signal to noise ratio [dB]SNR = 20 · log10

Imax−Iminσ̂n

Imax , Imin . . . max. and min. pixel value, resp.σ̂n . . . noise standard deviation estimate (from areas - no imagecomponent)



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Technique

DenoisingComputed Tomography(CT) imagesSuppressing lower energywavelet coefficients (noise)Soft universal waveletshrinkage

Soft ThresholdingSOFT THRESHOLDING

δ(s)

−δ(s)

−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)

−2 δ(s)

−δ(s)

0

δ(s)

2 δ(s)

3 δ(s)

before thr.after thr.

Thresholding magnitudes of complex coefficientsVary slowlyNot distorted by aliasing

Signal to noise ratio [dB]SNR = 20 · log10

Imax−Iminσ̂n

Imax , Imin . . . max. and min. pixel value, resp.σ̂n . . . noise standard deviation estimate (from areas - no imagecomponent)



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Technique

Median Absolute Deviation (MAD)Noise standard deviation estimator (for 1D signal)

σ̂(mad) =median{|W1,0|,|W1,1|,...,|W1,N/2−1|}

0.6745

where W1,l . . . l-th wavelet coefficient of level 1Smallest scale w. coefficients - noise dominatedFor independent identically distributed Gaussian noiseRobust against large deviations ⇒ noise variance

Donoho ThresholdDonoho soft universal threshold

δ(s) =√

2 σ̂2(mad) log(N)

where N is no. coefficients



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Technique

Median Absolute Deviation (MAD)Noise standard deviation estimator (for 1D signal)

σ̂(mad) =median{|W1,0|,|W1,1|,...,|W1,N/2−1|}

0.6745

where W1,l . . . l-th wavelet coefficient of level 1Smallest scale w. coefficients - noise dominatedFor independent identically distributed Gaussian noiseRobust against large deviations ⇒ noise variance

Donoho ThresholdDonoho soft universal threshold

δ(s) =√

2 σ̂2(mad) log(N)

where N is no. coefficients



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Table of Contents

1 Introduction



4 Conclusions



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Results

Histograms of Wavelet Coefficients

0 2 4 6 8 100

2000

4000

6000

8000

10000

12000

14000

(a) DTCWT: |LoHi|, −75°, level 1

thr

−10 −5 0 5 100

0.5

1

1.5

2x 10

4 (b) DWT: LoHi, level 1

+thr−thr

4 levels, 14-tap filters: Daubechies for DWT and q-shift for DTCWT



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Results

Residuals After Denoising

Axial brain CT image.



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Denoising Results

CT Image Denoising (Cuts)

(a) ORIGINAL 44.15 dB

(b) DTCWT DEN. 46.38 dB (c) DTCWT: RESIDUALS

(d) DWT DEN. 46.1 dB (e) DWT: RESIDUALS

Donoho universal soft threshold with MAD estimate.Similar SNR results for 2- and 3-level decomposition.



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Conclusions

DTCWT surpasses DWTShift dependence (reduced aliasing)Directional selectivityNo zero-crossing of the coefficients at a singularity

Future StudyDTCWT in biomedical image denoising and enhancementProbability distribution of noise and of its waveletcoefficients in these imagesWavelet shrinkage techniques



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Conclusions

DTCWT surpasses DWTShift dependence (reduced aliasing)Directional selectivityNo zero-crossing of the coefficients at a singularity

Future StudyDTCWT in biomedical image denoising and enhancementProbability distribution of noise and of its waveletcoefficients in these imagesWavelet shrinkage techniques



Introduction


Q-Shift DTCWT


Denoising Results

Conclusions

Further Reading

Further Reading

I. W. Selesnick and R. G. Baraniuk and N. G. Kingsbury.The Dual-Tree Complex Wavelet Transform.IEEE Signal Processing Magazine, 22(6): 123–151, IEEE, 2005.

N. G. Kingsbury.A Dual-Tree Complex Wavelet Transform with ImprovedOrthogonality and Symmetry Properties.In Proceedings of the IEEE International Conf. on Image Processing,Vancouver, pages 375–378. IEEE, 2000.

P. D. Shukla.Complex Wavelet Transforms and Their Applications.PhD Thesis, The University of Strathclyde in Glasgow, U.K., 2003.

D. B. Percival and A. T. Walden.Wavelet Methods for Time Series Analysis.Cambridge Series in Statistical and Probabilistic Mathematics.Cambridge University Press, U.S.A., 2006.

Documents

COMPLEX WAVELET TRANSFORM IN [1mm] BIOMEDICAL IMAGE DENOISINGdsp.vscht.cz/hostalke/upload/TCP07_presentation.pdf · 2007-11-20 · CWT IN BIOMEDICAL IMAGE DENOISING E. Hošťálková,