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

Prague Institute of Chemical Technology - Department of Computing and Control Engineering Digital Signal & Image Processing Research Group Brunel University, London - Department of Electronics and Computer Engineering Communications & Multimedia Signal Processing Research Group

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

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  1. Prague Institute of Chemical Technology - Department of Computing and Control Engineering Digital Signal & Image Processing Research Group Brunel University, London - Department of Electronics and Computer Engineering Communications & Multimedia Signal ProcessingResearch Group IMAGE RESOLUTION ENHANCEMENT Jiří Ptáček Aleš Procházka 10th June 2002

  2. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 1. INTRODUCTION • INTRODUCTION • Image Enhancement– The improvement of digital image quality • Main aims of Magnetic Resonance (MR) Images Enhancement: • Reconstruction of missing or corrupted parts of MR Images • Image De-noising • Image Resolution Enhancement Signal Resolution – Defines the sampling period in the case of time series or the pixel distance in the case of images Signal Resolution Enhancement– Allows both global and detailed views of specific one- dimensional or two-dimensional signal components

  3. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 1. INTRODUCTION • Main problems of Magnetic Resonance (MR) Images Resolution Enhancement: • Resolution enhancement of MR images (512x 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

  4. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT 2. 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 – 1 and f1(k)=k/N , f2(l)=l/M

  5. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 2. FOURIER TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT • 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 • 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 :

  6. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 2. 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

  7. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT • 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: 3. 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

  8. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 3. 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

  9. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT

  10. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT • Definition of wavelet functions • 1. Analytical form • (a) Gaussian derivative • (b) Shannon wavelet function • (c) Morlet wavelet function • (d) Harmonic wavelet function

  11. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT • 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 • Definition of wavelet functions • 2. Numerical form – Dilation equations • Scaling function: • Wavelet function • for j=1,2,3,…

  12. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 3. WAVELET TRANSFORM IN SIGNAL RESOLUTION ENHANCEMENT • Conclusion • 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.

  13. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 4. EXAMPLES OF USING WAVELET TRANSFORM IN SIGNAL ANALYSIS 4. EXAMPLES OF USING WAVELET TRANSFORM IN SIGNAL ANALYSIS • Simulated non-stationary signal • Real EEG signal • Gas consumption

  14. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 5. BAYESIAN METHODS USED AFTER WAVELET DECOMPOSITION 5. BAYESIAN METHODS USED AFTER WAVELET DECOMPOSITION WAVELET DECOMPOSI- TION IMAGE ARTIFACTS RECONSTRUC- TION IN EACH LEVEL USING BAYESIAN MODELS BACKWARD WAVELET RECONSTRUC- TION

  15. J. Ptáček, Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group A. Procházka,Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group 6. FOLLOWING WORK 7. REFERENCES • 6. FOLLOWING WORK • AR modelling after wavelet decomposition in image reconstruction • Utilize of the probabilistic models after wavelet decomposition in image reconstruction • Edge detection • 7. REFERENCES • D. E. Newland : An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Longman Scientific & Technical, Essex, U.K., third edition, 1994 • G. Strang : Wavelets and Dilation Equations: A brief introduction, SIAM Review, 31(4):614-627, December 1998 • G. 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

  16. THANK YOU FOR YOURATTENTION THANK YOU FOR YOURATTENTION

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