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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

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


Prague institute of chemical technology department of computing and control engineering

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)


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

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.


Prague institute of chemical technology department of computing and control engineering

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


Prague institute of chemical technology department of computing and control engineering

THANK YOU FOR YOURATTENTION

THANK YOU FOR YOURATTENTION

THANK YOU FOR YOURATTENTION


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