Download Presentation

Loading in 3 Seconds

This presentation is the property of its rightful owner.

X

Sponsored Links

- 50 Views
- Uploaded on
- Presentation posted in: General

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

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

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

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

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

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

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

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.

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