WAVELET DE-NOISING AND GRADIENT...
Transcript of WAVELET DE-NOISING AND GRADIENT...
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 1 / 23
WAVELET DE-NOISINGAND GRADIENT ENHANCEMENT
IN BIOMEDICAL IMAGE PROCESSING
Aleš Procházka, Eva Hošt’álková, and Oldrich Vyšata
Institute of Chemical Technology, PragueDept of Computing and Control Engineering
VIIP 2008Mallorca
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 2 / 23
Contents
1 Introduction
2 Image De-NoisingWavelet Transform AlternativesWavelet ShrinkageDe-Noising Results
3 Edge DetectionGradient MasksCanny Edge Detector
4 Conclusions
5 Selected Bibliography
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 3 / 23
Introduction
Image Edges
Most important for imageperceptionAbrupt changes of intensity(high frequencies)Problems: blurring & noise
Applications of Edge DetectionImage enhancementImage segmentation (indexingof objects)Image recognition (databases)
0 10 20 30 400
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200 (a) SHARP EDGE
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pixels
0 10 20 30 400
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100
150
200 (b) BLURRED EDGE
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nsity
pixels
0 10 20 30 400
50
100
150
200 (c) BLURRED & NOISY EDGE
inte
nsity
pixels
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 3 / 23
Introduction
Image Edges
Most important for imageperceptionAbrupt changes of intensity(high frequencies)Problems: blurring & noise
Applications of Edge DetectionImage enhancementImage segmentation (indexingof objects)Image recognition (databases)
0 10 20 30 400
50
100
150
200 (a) SHARP EDGE
inte
nsity
pixels
0 10 20 30 400
50
100
150
200 (b) BLURRED EDGE
inte
nsity
pixels
0 10 20 30 400
50
100
150
200 (c) BLURRED & NOISY EDGE
inte
nsity
pixels
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 3 / 23
Introduction
Image Edges
Most important for imageperceptionAbrupt changes of intensity(high frequencies)Problems: blurring & noise
Applications of Edge DetectionImage enhancementImage segmentation (indexingof objects)Image recognition (databases)
0 10 20 30 400
50
100
150
200 (a) SHARP EDGE
inte
nsity
pixels
0 10 20 30 400
50
100
150
200 (b) BLURRED EDGE
inte
nsity
pixels
0 10 20 30 400
50
100
150
200 (c) BLURRED & NOISY EDGE
inte
nsity
pixels
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 4 / 23
Introduction
Biomedical Image Data
Magnetic Resonance (MR) imagesComputed Tomography (CT) images
(a) MR BRAIN IMAGE (b) CT BRAIN IMAGE
Prior to Edge Detection
Noise reduction by wavelet coefficients shrinkage
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 4 / 23
Introduction
Biomedical Image Data
Magnetic Resonance (MR) imagesComputed Tomography (CT) images
(a) MR BRAIN IMAGE (b) CT BRAIN IMAGE
Prior to Edge Detection
Noise reduction by wavelet coefficients shrinkage
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImageNoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?
How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 5 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?
How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 6 / 23
Wavelet Transform Alternatives
DWT = Discrete Wavelet Transform
ORIGINAL MR IMAGELEVEL 1 LEVEL 2
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
2−LEVEL DWT
DTCWT = Dual-Tree Complex Wavelet Transform
LEVEL 1 & 2
Tree D
Tree C
Tree B
Tree A Realparts
Imaginaryparts
}}
ORIGINAL MR IMAGE 2−LEVEL DTCWT: COEFS MAGNITUDES
Employs 2d DWT trees of real-tap filters in d-dimensions
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 6 / 23
Wavelet Transform Alternatives
DWT = Discrete Wavelet Transform
ORIGINAL MR IMAGELEVEL 1 LEVEL 2
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
2−LEVEL DWT
DTCWT = Dual-Tree Complex Wavelet Transform
LEVEL 1 & 2
Tree D
Tree C
Tree B
Tree A Realparts
Imaginaryparts
}}
ORIGINAL MR IMAGE 2−LEVEL DTCWT: COEFS MAGNITUDES
Employs 2d DWT trees of real-tap filters in d-dimensions
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 7 / 23
Wavelet Transform Alternatives
DTCWTAnalytic complex wavelets ψc(t) = ψr (t) + j · ψi(t)
⇒ Correct magnitude-phase representation⇒ Shift invariance & no aliasing
Impossible for wavelets of compact support⇒ onlyapproximately analytic
Directional Selectivity of 2D Wavelets
DTCWT: 6 subbands
REAL PARTS OF Q−SHIFT WAVELETS
+15◦ +45◦ +75◦ −75◦ −45◦ −15◦
IMAGINARY PARTS OF Q−SHIFT WAVELETS
+15◦ +45◦ +75◦ −75◦ −45◦ −15◦
DWT: 3 subbands
DB4 REAL WAVELETS
90◦ 45◦ 0◦
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 7 / 23
Wavelet Transform Alternatives
DTCWTAnalytic complex wavelets ψc(t) = ψr (t) + j · ψi(t)
⇒ Correct magnitude-phase representation⇒ Shift invariance & no aliasing
Impossible for wavelets of compact support⇒ onlyapproximately analytic
Directional Selectivity of 2D Wavelets
DTCWT: 6 subbandsREAL PARTS OF Q−SHIFT WAVELETS
+15◦ +45◦ +75◦ −75◦ −45◦ −15◦
IMAGINARY PARTS OF Q−SHIFT WAVELETS
+15◦ +45◦ +75◦ −75◦ −45◦ −15◦
DWT: 3 subbands
DB4 REAL WAVELETS
90◦ 45◦ 0◦
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 8 / 23
Wavelet Transform Alternatives
DWT versus DTCWT
DWT- Zero-crossings at a
singularity- Strongly shift dependent- Aliasing- Lack of directional
selectivity (±45◦)+ Critically decimated
DTCWT+ Large magnitudes at a
singularity+ Approx. shift independent+ Approx. no aliasing+ Improved directional
selectivity- Moderately redundant
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 8 / 23
Wavelet Transform Alternatives
DWT versus DTCWT
DWT- Zero-crossings at a
singularity- Strongly shift dependent- Aliasing- Lack of directional
selectivity (±45◦)+ Critically decimated
DTCWT+ Large magnitudes at a
singularity+ Approx. shift independent+ Approx. no aliasing+ Improved directional
selectivity- Moderately redundant
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 8 / 23
Wavelet Transform Alternatives
DWT versus DTCWT
DWT- Zero-crossings at a
singularity- Strongly shift dependent- Aliasing- Lack of directional
selectivity (±45◦)+ Critically decimated
DTCWT+ Large magnitudes at a
singularity+ Approx. shift independent+ Approx. no aliasing+ Improved directional
selectivity- Moderately redundant
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 8 / 23
Wavelet Transform Alternatives
DWT versus DTCWT
DWT- Zero-crossings at a
singularity- Strongly shift dependent- Aliasing- Lack of directional
selectivity (±45◦)+ Critically decimated
DTCWT+ Large magnitudes at a
singularity+ Approx. shift independent+ Approx. no aliasing+ Improved directional
selectivity- Moderately redundant
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 8 / 23
Wavelet Transform Alternatives
DWT versus DTCWT
DWT- Zero-crossings at a
singularity- Strongly shift dependent- Aliasing- Lack of directional
selectivity (±45◦)+ Critically decimated
DTCWT+ Large magnitudes at a
singularity+ Approx. shift independent+ Approx. no aliasing+ Improved directional
selectivity- Moderately redundant
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 9 / 23
Image De-Noising
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
ORIGINAL MR IMAGE
WaveletDecomposition
WaveletDecomposition
WaveletDecomposition
Image
HiHiHiLoLoHiLoLo
HiHiHiLoLoHiLoLo
Thresholdingof wavelet
coefficients
Thresholdingof wavelet
coefficients
2−LEVEL DWT
How to threshold the wavelet coefficients?
How to threshold the wavelet coefficients?
WaveletReconstruction
WaveletReconstruction
WaveletReconstruction
Image
LoLo
LoLoLoHiHiLoHiHi
LoHiHiLoHiHi
Which wavelet transform to use?
DenoisedImage
DenoisedImage
DE−NOISED IMAGE
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 10 / 23
Wavelet Shrinkage
Wavelet Shrinkage Principles
Suppressing lower energy wavelet coefficients (noise)Wavelet coefficients: thresholded (their magnitudes)Scaling coefficients: left unchanged
Soft Universal Shrinkage
For wavelet coefficients {ck}M−1k=0 of all levels:
c(s)k =
{sgn(ck )·(|ck|−δ(s)) for |ck|> δ(s)
0 otherwise
SOFT 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)
ck
c(s)k
before thr.after thr.
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 10 / 23
Wavelet Shrinkage
Wavelet Shrinkage Principles
Suppressing lower energy wavelet coefficients (noise)Wavelet coefficients: thresholded (their magnitudes)Scaling coefficients: left unchanged
Soft Universal Shrinkage
For wavelet coefficients {ck}M−1k=0 of all levels:
c(s)k =
{sgn(ck )·(|ck|−δ(s)) for |ck|> δ(s)
0 otherwise
SOFT 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)
ck
c(s)k
before thr.after thr.
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 11 / 23
Wavelet Shrinkage
Donoho EstimatorSoft universal threshold value:
δ(s) =√
2 σn 2 log(N)
σn . . . noise std. deviation estimate; N . . . image size
Median Absolute Deviation (MAD) EstimatorEstimate of std. deviation for i.i.d. Gaussian noise:
σ(MAD)n =
median{ |chh1 (k)| }N/4−1
k=00.6745
chh1 . . . HiHi wavelet coefficient of level 1 (noise dominated)
Robust against large deviations of noise variance
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 11 / 23
Wavelet Shrinkage
Donoho EstimatorSoft universal threshold value:
δ(s) =√
2 σn 2 log(N)
σn . . . noise std. deviation estimate; N . . . image size
Median Absolute Deviation (MAD) EstimatorEstimate of std. deviation for i.i.d. Gaussian noise:
σ(MAD)n =
median{ |chh1 (k)| }N/4−1
k=00.6745
chh1 . . . HiHi wavelet coefficient of level 1 (noise dominated)
Robust against large deviations of noise variance
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 12 / 23
De-Noising Results
Statistics of High-Frequency Image Components
Mean STD 0
1
2 (a) BRAIN (MR)
Mean STD 0
0.2
0.4 (b) BRAIN (CT)
Mean STD 0
5
10 (c) SPINE (MR)
Mean STD 0
0.5
1 (d) KIDNEY (MR)
DWT−db4DWT−db8DWT−haarDWT−sym3DTCWT
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 13 / 23
Gradient Masks
Gradient Edge Detectors
Filters approximating the intensity gradient2D convolution between the filter and the image
Sobel Filter
By rotation: detection of 0◦,+45◦,+90◦,−45◦ edges 1 2 10 0 0−1 −2 −1
0 1 2−1 0 1−2 −1 0
−1 0 1−2 0 2−1 0 1
−2 −1 0−1 0 1
0 1 2
For every root pixel - choose the rotation variant withthe absolute maximum value of convolution
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 13 / 23
Gradient Masks
Gradient Edge Detectors
Filters approximating the intensity gradient2D convolution between the filter and the image
Sobel Filter
By rotation: detection of 0◦,+45◦,+90◦,−45◦ edges 1 2 10 0 0−1 −2 −1
0 1 2−1 0 1−2 −1 0
−1 0 1−2 0 2−1 0 1
−2 −1 0−1 0 1
0 1 2
For every root pixel - choose the rotation variant withthe absolute maximum value of convolution
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 14 / 23
Gradient Masks
Application of the Sobel Filter
(a) ORIGINAL IMAGE (b) ORIGINAL IMAGE + SOBEL
(c) DWT DENOISING + SOBEL (d) DTCWT DENOISING + SOBEL
MR brain image de-noised using the DWT and DTCWT
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 15 / 23
Gradient Masks
Drawbacks of Gradient Masks
Short filters:Too sensitive to noiseand blurring
Longer filters:More robust against noiseBlur the originally sharpedges
Canny Edge Detector
Robust against noiseOperates at various scales
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 15 / 23
Gradient Masks
Drawbacks of Gradient Masks
Short filters:Too sensitive to noiseand blurring
Longer filters:More robust against noiseBlur the originally sharpedges
Canny Edge Detector
Robust against noiseOperates at various scales
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 16 / 23
Canny Edge Detector
Canny Edge Detector
Approximates the derivative of a 2D Gaussian in thedirection of the gradientRobust against noise
⇐ Gaussian smoothing filter prior to edge detection⇐Weak edges pixels identification algorithm
Adjustable value of the scale σ (the standarddeviation in the Gaussian)
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 17 / 23
Canny Edge Detector
Canny Algorithm
1 Convolution with 1D Gaussian masks in x and y -direction
Gσ,0(x) =1√2πσ
· exp(− x2
2σ2
)(1)
2 Convolution with the derivatives of the 2D Gaussian inx-direction (and also in y -direction)
∂Gσ,0(x , y)
∂x= − x√
2πσ3· exp
(− (x2 + y2)
2σ2
)(2)
3 Combining of these two matrices
4 Strong edges: pels value above the upper threshold
5 Weak edges:
Pels value above the lower thresholdThe gradient ≡ the direction of the strong edges inthe neighborhood
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 17 / 23
Canny Edge Detector
Canny Algorithm
1 Convolution with 1D Gaussian masks in x and y -direction
Gσ,0(x) =1√2πσ
· exp(− x2
2σ2
)(1)
2 Convolution with the derivatives of the 2D Gaussian inx-direction (and also in y -direction)
∂Gσ,0(x , y)
∂x= − x√
2πσ3· exp
(− (x2 + y2)
2σ2
)(2)
3 Combining of these two matrices
4 Strong edges: pels value above the upper threshold
5 Weak edges:
Pels value above the lower thresholdThe gradient ≡ the direction of the strong edges inthe neighborhood
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 17 / 23
Canny Edge Detector
Canny Algorithm
1 Convolution with 1D Gaussian masks in x and y -direction
Gσ,0(x) =1√2πσ
· exp(− x2
2σ2
)(1)
2 Convolution with the derivatives of the 2D Gaussian inx-direction (and also in y -direction)
∂Gσ,0(x , y)
∂x= − x√
2πσ3· exp
(− (x2 + y2)
2σ2
)(2)
3 Combining of these two matrices
4 Strong edges: pels value above the upper threshold
5 Weak edges:
Pels value above the lower thresholdThe gradient ≡ the direction of the strong edges inthe neighborhood
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 17 / 23
Canny Edge Detector
Canny Algorithm
1 Convolution with 1D Gaussian masks in x and y -direction
Gσ,0(x) =1√2πσ
· exp(− x2
2σ2
)(1)
2 Convolution with the derivatives of the 2D Gaussian inx-direction (and also in y -direction)
∂Gσ,0(x , y)
∂x= − x√
2πσ3· exp
(− (x2 + y2)
2σ2
)(2)
3 Combining of these two matrices
4 Strong edges: pels value above the upper threshold
5 Weak edges:
Pels value above the lower thresholdThe gradient ≡ the direction of the strong edges inthe neighborhood
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 17 / 23
Canny Edge Detector
Canny Algorithm
1 Convolution with 1D Gaussian masks in x and y -direction
Gσ,0(x) =1√2πσ
· exp(− x2
2σ2
)(1)
2 Convolution with the derivatives of the 2D Gaussian inx-direction (and also in y -direction)
∂Gσ,0(x , y)
∂x= − x√
2πσ3· exp
(− (x2 + y2)
2σ2
)(2)
3 Combining of these two matrices
4 Strong edges: pels value above the upper threshold
5 Weak edges:
Pels value above the lower thresholdThe gradient ≡ the direction of the strong edges inthe neighborhood
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 18 / 23
Canny Edge Detector
Canny Method for the De-Noised CT Image (σ = 1.8) (a) ORIGINAL IMAGE (b) DWT DENOISING + CANNY (c) DTCWT DENOISING + CANNY
Denoising by wavelet shrinkage:DWT: 16-tap symlet filters, 4 levelsDTCWT: 16-tap q-shift filters, 4 levels
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 19 / 23
Canny Edge Detector
Edges of the De-Noised CT Image (σ = 1.8, 1) (a) EDGES: CANNY (σ=1.8) (b) DWT DEN. & CANNY (σ=1.8) (c) DTCWT DEN. & CANNY (σ=1.8)
(d) EDGES: CANNY (σ=1) (e) DWT DEN. & CANNY (σ=1) (f) DTCWT DEN. & CANNY (σ=1)
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 20 / 23
Canny Edge Detector
Canny Method for the De-Noised MR Image (σ = 2.5) (a) ORIGINAL IMAGE (b) DWT DENOISING + CANNY (c) DTCWT DENOISING + CANNY
Denoising by wavelet shrinkage:DWT: 14-tap symlet filters, 3 levelsDTCWT: 14-tap q-shift filters, 3 levels
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 21 / 23
Conclusions
Denoising by Wavelet Shrinkage
DTCWT outperformes the DWTApproximate shift invarianceSteady values of the magnitude across scalePhase representation of edges orientationImproved directional selectivity in higher dimensions
Edge Detection Methods Used
Short gradient filters:Insufficient for blurred or noisy images
Canny detector:More robust against noiseOperating at various scales
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 21 / 23
Conclusions
Denoising by Wavelet Shrinkage
DTCWT outperformes the DWTApproximate shift invarianceSteady values of the magnitude across scalePhase representation of edges orientationImproved directional selectivity in higher dimensions
Edge Detection Methods Used
Short gradient filters:Insufficient for blurred or noisy images
Canny detector:More robust against noiseOperating at various scales
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 22 / 23
Further Reading
I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury.The Dual-Tree Complex Wavelet Transform.IEEE Signal Processing Magazine, 22(6): 123–151,IEEE, 2005.
M. Petrou and P. Bosdogiann. Image Processing.John Wiley & Sons, 2000.
R. M. Rangayyan. Biomedical Image Analysis.Biomedical Engineering Series, CRC Pres, USA,2005.
D. B. Percival and A. T. Walden. Wavelet Methods forTime Series Analysis.Cambridge Series in Statistical and ProbabilisticMathematics. Cambridge University Press, USA,2006.
DE-NOISINGAND GRADIENTENHANCEMENT
DSP Group
ICT Prague
Introduction
Image DenoisingWT Alternatives
Wavelet Shrinkage
De-Noising Results
Edge DetectionGradient Masks
Canny Detector
Conclusions
Bibliography
ISCCSP 2008 23 / 23
Prague DSP Research Group
Any questions?
http://dsp.vscht.cz