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

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 BAYESIAN METHODS AND WAVELET TRANSFORM IN IMAGE COMPONENTS RECONSTRUCTION Jiří Ptáček 12th August 2002 Supervisors:Prof. Aleš Procházka Prof. Saeed Vaseghi

2. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 1. INTRODUCTION • INTRODUCTION • Main aims of Magnetic Resonance (MR) Images Enhancement: • Reconstruction of missing or corrupted parts of MR Images • Image Denoising • Image Resolution Enhancement Image Reconstruction– Completion of missing or corrupted parts (artifacts) of images with unknown model of degradations –Special kind of Image Enhancement • Criteria of Image Reconstruction: • objective–sum of squared errors between pixels of an original image and reconstructed image(It is necessary to havean undamaged image) • subjective – approximate knowledge of the image – aestetical notion (suppression of jamming defects of the image)

3. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 1. INTRODUCTION • Methodsof Image Reconstruction already designed and tested: • Bilinear Interpolation • Autoregressive Modelling • Triangular Surface Interpolation (Delauny’s triangulation) • Matrix Moving Average • Image Subregions Feature Extraction and Classification • New tools of Image Reconstruction: • Bayesian models • Wavelet transform • Combination of these two methods

4. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 2. BAYESIAN INTERPOLATION METHOD • Using Bayes’ rule the posterior PDF of unknown samples xUk is • For the given vector xKn , fX(xKn) is a constant, that’s why the maximum a posterior estimation can be expressed as 2. BAYESIAN INTERPOLATION METHOD • The signal vector x can be written as • xKn=[xKn1 xKn2] … known samples • xUk … unknown samples

5. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 2. BAYESIAN INTERPOLATION METHOD • Substitution of the previous equation in equation for the conditional PDF of the unknown signal xUk given a number of samples xKn yields • After a few treatments it is possible • to obtain an expression for • the vector of unknown samples • The given signal x=K xKn +UxUk is from a zero-mean Gaussian process. The PDF of this signal is given by

6. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 2. BAYESIAN INTERPOLATION METHOD • Results obtained for a real 2D signal – MR Image 1 2 3

7. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 3. BAYESIAN INTERPOLATION METHOD APPLIED AFTER WAVELET DECOMPOSITION OF THE CORRUPTED MR IMAGE • Half-band low-pass filter • Corresponding high-pass filter • The 1st stage for wavelet decomposition: 3. BAYESIAN INTERPOLATION METHOD APPLIED AFTER WAVELET DECOMPOSITION OF THE CORRUPTED MR IMAGE • Decomposition stage: – convolution of a given signal and the appropriate filter • – downsampling by factor D=2 • – the same process is applied to rows • Interpolation stage: – Bayesian interpolation method • Reconstruction stage: – row upsampling by factor U=2 and row convolution • – sum of the corresponding images • – column upsampling by factor U=2 and column convolution, sum

8. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 3. BAYESIAN INTERPOLATION METHOD APPLIED AFTER WAVELET DECOMPOSITION OF THE CORRUPTED MR IMAGE

9. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 3. BAYESIAN INTERPOLATION METHOD APPLIED AFTER WAVELET DECOMPOSITION OF THE CORRUPTED MR IMAGE • Results obtained using Bayesian interpolation method applied after wavelet decomposition 1 2 3 • Used Wavelet functions: • – Haar • – Daubechies 4

10. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 3. BAYESIAN INTERPOLATION METHOD APPLIED AFTER WAVELET DECOMPOSITION OF THE CORRUPTED MR IMAGE • Haar scaling function • Haar wavelet function • Used parameters of the wavelet transform • – Number of decomposition levels : 1 • – Used wavelet and scaling functions : Daubechies 4 , Haar • Daubechies 4 scaling function • Daubechies 4 wavelet function • for j=1,2,3,… • where

11. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 4. CONCLUSION – COMPARISON OF THE USED METHODS 4. CONCLUSION – COMPARISON OF THE USED METHODS • SSE between uncorrupted and reconstructed image is lower in case of use of the Bayesian method after the wavelet decomposition. • The reason is that the interpolation of half number of pixels is easier than the interpolation without use of the wavelet decomposition (i.e. 2 times more interp. pixels). • Better results would be possible to obtain using interpolation after edge detection and wavelet decomposition. • Interpolation calculated along the edge would save the edge more than interpolation across the edge.

12. Jiri Ptacek , Department of Computing and Control Engineering, Prague Institute of Chemical Technology, DSP Research Group Department of Electronics and Computer Engineering, Brunel University, London, C&MSPResearch Group 5. WORK FINISHED AT BRUNEL UNIVERSITY 6. FOLLOWING WORK • 5. WORK FINISHED AT BRUNEL UNIVERSITY • Bayesian methods in Image Components Reconstruction • Image subregions feature extraction and classification • Image resolution enhancement using • – Fourier transform • – Wavelet transform • Bayesian methods after wavelet decomposition • 3 conference papers, 1 journal paper, 3 seminars • 6. FOLLOWING WORK • Writing of my Ph.D. thesis • Edge detection • AR modelling after wavelet decomposition in image component reconstruction

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