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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

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


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)


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


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


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


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


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


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


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


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


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.


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


THANK YOU FOR YOURATTENTION

THANK YOU FOR YOURATTENTION

THANK YOU FOR YOURATTENTION


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