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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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
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◦
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
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◦
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Introduction to DTCWT
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)
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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.
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Q-Shift Dual Tree CWT
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.
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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)
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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)
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
Denoising Results
Conclusions
Further Reading
Denoising Results
Residuals After Denoising
Axial brain CT image.
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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.
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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
CWT IN BIOMEDICALIMAGE DENOISING
E. Hošťálková, A. Procházka
Introduction
DTCWTIntroduction to DTCWT
Q-Shift DTCWT
Denoising of CT ImagesDenoising Technique
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.