Prague Institute of Chemical TechnologyPrague Institute of Chemical Technology - - Department of Computing and Control EngineeringDepartment of Computing and Control Engineering

Digital Signal Digital Signal & & Image Processing Research GroupImage Processing Research Group

Brunel University, London - Department of Electronics and Computer EngineeringBrunel University, London - Department of Electronics and Computer Engineering

Communications & Multimedia Signal ProcessingCommunications & Multimedia Signal Processing Research GroupResearch Group

IIMAGE RESOLUTION MAGE RESOLUTION ENHANCEMENTENHANCEMENT

Jiří PtáčekJiří PtáčekAleAleš Procházkaš Procházka

10th June 2002

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup1. I1. INTRODUCTIONNTRODUCTION

Signal Resolution – Defines the sampling period in the case of time series or the pixel distance in the case of images

1.1. INTRODUCTION INTRODUCTION

Image Enhancement – The improvement of digital image quality

Signal Resolution Enhancement – Allows both global and detailed views of specific one- dimensional or two-dimensional signal components

Main aims of Magnetic Resonance (MR) Images Enhancement:

Reconstruction of missing or corrupted parts of MR Images

Image De-noising

Image Resolution Enhancement

Main problems of Magnetic Resonance (MR) Images Resolution Enhancement:

Resolution enhancement of MR images (512 x 512 pixels 2 times more)

Conservation of sharp edges in the image – not to obtain smooth edges

Conservation and highlighting of details – not to delete details

Designed and tested methods of image resolution enhancement:

Discrete Fourier Transform

Discrete Wavelet Transform

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup1. I1. INTRODUCTIONNTRODUCTION

2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

1-D Fourier Transform of a signal

for k=0,1,…,N/2 – 1 and f(k)=k/N

2-D Fourier Transform of a signal

for k=0,1,…,N/2 – 1 , l= 0,1,…,M/2 – 1and f1(k)=k/N , f2(l)=l/M

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

Signal enhancement can be achieved by symmetric extension of the original

sequence X(k) (for normalized frequencies) by zeros resulting in the sequence :

for even values of R>N

The IDFT of sequence Z(k) :

for n=0,1,…,R-1

Evaluating for instance values of this sequence for R=2N and even indicates only :

Comparing this result with the definition of the IDFT of sequence X(k) in the form

it is obvious that in case x(n)=2z(2n) and sequence stands for interpolated

sequence

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

This whole process applied to signals or images allows

1. Decomposition and perfect reconstruction using ext_col = 0 and ext_row = 0

2. Resolution enhancement in case of ext_col 0 and ext_row 0

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

The main benefit of WT over STFT is its multi–resolution time–scale analysis ability.

The initial function W(t) forming basis for the set of functions :

where a=2m … parameter of dilation , b=k 2m … parameter of translation

Any 1D signal can be considered as a special case of an image

having 1 column only.

One column of the image matrix is signal

Half-band low-pass filter

Corresponding high-pass filter

The 1st stage for wavelet decomposition:

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

Decomposition stage: – convolution of a given signal and the appropriate filter

– downsampling by factor D

– the same process is applied to rows

Reconstruction stage: – row upsampling by factor U and row convolution

– sum of the corresponding images

– column upsampling by factor U and column convolution, sum

The whole process can be used for:

1. Signal / image decomposition and perfect reconstruction using D=2 and U=2

2. Signal / image resolution enhancement in the case of D=1 and U=2

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

Definition of wavelet functionsDefinition of wavelet functions

1. Analytical form

(a) Gaussian derivative

(b) Shannon wavelet function

(c) Morlet wavelet function

(d) Harmonic wavelet function

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

Definition of wavelet functionsDefinition of wavelet functions

2. Numerical form – Dilation equations

Scaling function:

Wavelet function

for j=1,2,3,…

An Example: Daubechies wavelet function

of the 4th order

Prof Daubechies designed an algorithm for

calculation of the coefficients c0, c1, c2, c3

Resulting set of the coefficients is

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

CConclusiononclusion

Mean squared errors (MSE) between the magnetic resonance (MR) image of the

human brain enhanced by the discrete Fourier transform and wavelet transform

using selected wavelet functions

Comparison shows that, in each case, the image quality has been greatly

enhanced, demonstrating the success of used methods – DFT and DWT.

Problems resulting from periodic signal or image extension and boundary values

estimation, especially in case of the wavelet transform application.

Both in the case of DFT and DWT it is possible to use various methods to enhance

the resolution of one-dimensional and two-dimensional signals.

Method Method

MSEMSE MethodMethod MSEMSE

Haar WaveletHaar Wavelet 0.34020.3402 Wavelet SYM2Wavelet SYM2 0.36770.3677

Daubechies Wavelet DB3Daubechies Wavelet DB3 0.51500.5150 Wavelet SYM4Wavelet SYM4 0.09760.0976

Daubechies Wavelet DB4Daubechies Wavelet DB4 0.75150.7515 Wavelet SYM8Wavelet SYM8 0.11470.1147

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

4. EXAMPLES OF USING WAVELET TRANSFORM IN SIGNAL ANALYSIS4. EXAMPLES OF USING WAVELET TRANSFORM IN SIGNAL ANALYSIS

Simulated non-stationary signal

Real EEG signal

Gas consumption

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup4. EXAMPLES OF USING WAVELET TRANSFORM IN SIGNAL ANALYSIS4. EXAMPLES OF USING WAVELET TRANSFORM IN SIGNAL ANALYSIS

5. BAYESIAN METHODS USED AFTER WAVELET DECOMPOSITION5. BAYESIAN METHODS USED AFTER WAVELET DECOMPOSITION

WAVELETWAVELETDECOMPOSI-DECOMPOSI-TIONTION

BACKWARDBACKWARDWAVELETWAVELETRECONSTRUC-RECONSTRUC-TIONTION

IMAGEIMAGEARTIFACTSARTIFACTSRECONSTRUC-RECONSTRUC-TION IN EACHTION IN EACHLEVEL USINGLEVEL USINGBAYESIANBAYESIANMODELSMODELS

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup5. BAYESIAN METHODS USED AFTER WAVELET DECOMPOSITION5. BAYESIAN METHODS USED AFTER WAVELET DECOMPOSITION

6. FOLLOWING WORK6. FOLLOWING WORK

AR modelling after wavelet decomposition in image reconstruction

Utilize of the probabilistic models after wavelet decomposition in image reconstruction

Edge detection

7. REFERENCES7. REFERENCES

D. E. Newland : An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Longman Scientific & Technical, Essex, U.K., third edition, 1994G. Strang : Wavelets and Dilation Equations: A brief introduction, SIAM Review, 31(4):614-627, December 1998G. Strang and T. Nguyen : Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996

ELECTRONIC SOURCES:

IEEE : http://www.ieee.org

WAVELET DIGEST : http://www.wavelet.org

DSP PUBLICATIONS : http://www.dsp.rice.edu/publications

MATHWORKS : http://www.mathworks.com

JJ. . PtPtáčáčekek,, Department of Electronics and Computer Engineering, Brunel University, London, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPC&MSP Research GroupResearch GroupA. ProcházkaA. Procházka,, Department of Computing and Control EngineeringDepartment of Computing and Control Engineering, P, Prague Institute of Chemical Technolrague Institute of Chemical Technologyogy, DSP Research , DSP Research

GroupGroup6. FOLLOWING WORK 7. REFERENCES6. FOLLOWING WORK 7. REFERENCES

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