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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

200 (a) SHARP EDGE

pixels

0 10 20 30 400

200 (b) BLURRED EDGE

pixels

0 10 20 30 400

200 (c) BLURRED & NOISY EDGE

pixels

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 3 / 23

Introduction

Image Edges

0 10 20 30 400

200 (a) SHARP EDGE

pixels

0 10 20 30 400

pixels

0 10 20 30 400

pixels

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 3 / 23

Introduction

Image Edges

0 10 20 30 400

200 (a) SHARP EDGE

pixels

0 10 20 30 400

pixels

0 10 20 30 400

pixels

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

Wavelet Shrinkage Algorithm

NoisyImage

ORIGINAL MR IMAGE

WaveletDecomposition

HiHiHiLoLoHiLoLo

Thresholdingof wavelet

coefficients

2−LEVEL DWT

How to threshold the wavelet coefficients?How to threshold the wavelet coefficients?

WaveletReconstruction

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

Which wavelet transform to use?

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

NoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

NoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

NoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

NoisyImageNoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

NoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

How to threshold the wavelet coefficients?

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 5 / 23

Image De-Noising

NoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 6 / 23

Wavelet Transform Alternatives

DWT = Discrete Wavelet Transform

ORIGINAL MR IMAGELEVEL 1 LEVEL 2

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 6 / 23

DWT = Discrete Wavelet Transform

ORIGINAL MR IMAGELEVEL 1 LEVEL 2

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 7 / 23

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◦

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 7 / 23

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◦

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 8 / 23

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 8 / 23

DWT versus DTCWT

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 8 / 23

DWT versus DTCWT

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 8 / 23

DWT versus DTCWT

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 8 / 23

DWT versus DTCWT

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 9 / 23

Image De-Noising

NoisyImage

ORIGINAL MR IMAGE

HiHiHiLoLoHiLoLo

coefficients

2−LEVEL DWT

LoLoLoHiHiLoHiHi

LoHiHiLoHiHi

DenoisedImage

DE−NOISED IMAGE

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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)

−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)

2 δ(s)

3 δ(s)

before thr.after thr.

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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)

−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)

2 δ(s)

3 δ(s)

before thr.after thr.

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 12 / 23

De-Noising Results

Statistics of High-Frequency Image Components

Mean STD 0

2 (a) BRAIN (MR)

Mean STD 0

0.4 (b) BRAIN (CT)

Mean STD 0

10 (c) SPINE (MR)

Mean STD 0

1 (d) KIDNEY (MR)

DWT−db4DWT−db8DWT−haarDWT−sym3DTCWT

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

For every root pixel - choose the rotation variant withthe absolute maximum value of convolution

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

For every root pixel - choose the rotation variant withthe absolute maximum value of convolution

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 16 / 23

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)

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

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)

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 17 / 23

Canny Edge Detector

Canny Algorithm

Gσ,0(x) =1√2πσ

· exp(− x2

∂Gσ,0(x , y)

∂x= − x√

2πσ3· exp

(− (x2 + y2)

5 Weak edges:

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 17 / 23

Canny Edge Detector

Canny Algorithm

Gσ,0(x) =1√2πσ

· exp(− x2

∂Gσ,0(x , y)

∂x= − x√

2πσ3· exp

(− (x2 + y2)

5 Weak edges:

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 17 / 23

Canny Edge Detector

Canny Algorithm

Gσ,0(x) =1√2πσ

· exp(− x2

∂Gσ,0(x , y)

∂x= − x√

2πσ3· exp

(− (x2 + y2)

5 Weak edges:

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 17 / 23

Canny Edge Detector

Canny Algorithm

Gσ,0(x) =1√2πσ

· exp(− x2

∂Gσ,0(x , y)

∂x= − x√

2πσ3· exp

(− (x2 + y2)

5 Weak edges:

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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)

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

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

DSP Group

ICT Prague

Introduction

Wavelet Shrinkage

De-Noising Results

Canny Detector

Conclusions

Bibliography

ISCCSP 2008 22 / 23

wavelet de-noising and gradient...

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