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Prague Institute of Chemical Technology - Department of Computing and Control Engineering Digital Signal & Image

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

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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 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ENHANCEMENT OF BIOMEDICAL IMAGES Jiří Ptáček 8th April 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 • Image Processing– Most signals are converted into a form tractable by digital hardware, and can then be treated by Digital Signal Processing(for one-dimensional signals) or by Digital Image Processing (in two dimensions). Image Enhancement– The improvement of digital image quality Image Restoration– Removes or minimizes some known degradations in images – Special kind of Image Enhancement Image Reconstruction– Completion of missing or corrupted parts (artifacts) of images with unknown model of degradations – Special kind of Image Enhancement

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Main aims of Magnetic Resonance (MR) Images Enhancement: • Reconstruction of missing or corrupted parts of MR Images • Image De-noising • Image Resolution Enhancement • 2. RECONSTRUCTION OF MR IMAGES • Reasons of the Reconstruction: • Distortion or damaged parts or whole image (for example caused by lens) • Missing parts of an image (shining points in MR images, missing value one or more measuring stations of air pollution in air pollution images etc.) • Criteria of Image Reconstruction: • objective–sum of squared errors between pixels of an original image and a reconstructed image(It is necessary to haveundamaged image) • subjective– approximate knowledge of the image – aestetical notion (suppression of jamming defects of the image)

  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. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ Used 2D signals: – simulated2D signal (112x112 pixels) – real MR image (512x452 pixels) reconstructed part: 20 x 20 pixelsreconstructed part: 20 x 20 pixels • Designed and tested methods of image reconstruction: • Bilinear Interpolation – BLI • Predictive Image Modelling – PIM • Triangular Surface Interpolation – TSI • Matrix Moving Average – MMA

  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. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Simulated signal (missing part: 20x20 pixels), SSE = 41.61 • Real MR Image (missing part: 20x20 pixels), SSE = 0.6704 • 2.1Bilinear Interpolation • Linear interpolation applied row by row by the following relation: • between known boundary values {z(m,n)} for row m and p=1,2,…,l-k-1 followed by • asimilar procedure columns by columns

  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. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Matrixcontains known samples as well as estimated samples: • in case that the matrix and the vector in eq. (1) are separated into two matrices and • ... contains only known values • ... contains only estimated values •  (2) • The resultis possible to write using the Toeplitzmatrix A: • , let us define the vector • 2.2 Predictive image modelling using AR model • 2.2.1 Prediction of an 1D signal • AR model: • for i=l, l+1, l+2,...., l+M-1 • matrix notation: (1) • where

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • For obtaining solution of the eqs. (3) and (4) for forward and backward prediction we require: • The estimation has to minimize the error of the east-west model: • , where and • Similar to the forward prediction we can construct the AR model for the backward (east) prediction using following samples after the part with the missing samples: • for i=l+M-1, l+M-2,..., lin matrixnotation: • Then(4) • Then from eq. (2) we obtain the system of linear equations for the forward (west) prediction: (3) • In order to minimize the error, we set the derivative of  equal to zero: • (5)

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Sum of squared errors • between thereconstructed and • the original samples: 2.2.2 Prediction of an 2D signal • Calculation of the vector ofthe estimated valuesis applied row by row for the whole missing part of an image  we obtain the matrix • This algorithm isthen applied for the north–south prediction  we obtain the matrix • The result matrix of the interpolated values is given as the arithmetic mean of , • After the substitution to eq. (5), calculation, and many treatments we obtain the system of linear equations for the vector of the estimated values: • , where

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Simulated signal – 2D sinwave (missing part: 20x20 pixels), SSE = 0 • Real signal – magnetic resonance image (missing part: 20x20 pixels) Part 1 SSE1=0.4547 Order of the AR model: 4 Part 2 SSE2=0.6313 Order of the AR model: 3

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • 2.3 Triangular Surface Interpolation • Design of a linear or another surface through 3 known points • Calculation of points’ 3rd coordinate situated on the designed • surface • Simulated signal – 2D sin (missing part: 20x20 pixels) ... (1) • Real signal – MR image (missing part: 20x20 pixels) ... (2) (1) SSE1=32.5 (2) SSE2=0.68

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Simulated signal (missing part: 20x20 pixels), SSE = 24.14 • Real signal (missing part: 20x20 pixels), SSE = 0.4844 • 2.4Matrix Moving Average • Calculation of the pixel value in the missing part as arithmetic mean of 5 neighbouring pixels by the relation: • The algorithm is applied row by row for the whole missing part

  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 2. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • 2.5 Conclusion • Comparison of the designed methods using the aesthetical notion (subjective valuation) andthe value of SSEbetween the original and the reconstructed part • Sum of squared errors SSE for each method applied to the simulated signal and 2 variousmissing areas of the MR image: • MMA – low SSE,not edge sensitive, the structure in reconstructed part is erased • PIM – more difficult, more edge sensitive, it saves structure and period. components • 2.6 The next development in Image Reconstruction – visions • Development of nonlinear methodsusing wavelet transform • Utilize of the Bayesian models • 3D interpolation in models of a human brain, purpose: acquisition of the 3D model out of a finite number of 2D horizontal brain MR scans

  13. 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. RECONSTRUCTION OF MR IMAGES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Comparison of the Matrix Moving Average method (left picture) and the Predictive Image Modelling using the AR Model (right picture) • 2.7 Interests • Shielding of the whole coin

  14. 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. IMAGE DE-NOISING ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • 3. IMAGE DE-NOISING • Signal de-noising using wavelet transform: • Signal decomposition using a selected wavelet function up to the given level and evaluation of wavelet transform coefficients • The choice of threshold limits for each decomposition level and modification of its coefficients • Signal reconstruction from modified wavelet transform coefficients

  15. 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. IMAGE DE-NOISING ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • Image de-noising using wavelet transform: • Utilize the same principles as for signal decomposition and de-noising. • Each column of an image matrix is convolved with high-pass and low-pass filter followed by downsampling. • The same process is applied to image matrix rows. • The choice of threshold limits  for each decomposition level and modification of its coefficients for k=0, 1, … N-1 • Backward image reconstruction out of modified wavelet transform coefficients

  16. 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. IMAGE RESOLUTION ENHANCEMENT ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • 4. IMAGE RESOLUTION ENHANCEMENT • Using interpolation property of Fourier transform and zero padded method

  17. 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. FOLLOWING WORK 6. REFERENCES ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ • 5. FOLLOWING WORK • Bayesian methods in image reconstruction • Utilize the probabilistic models after wavelet decomposition • Image resolution enhancement using wavelet filters • Image resolution enhancement (interpolation) and edge detection • 6. 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

  18. THANK YOU FOR YOURATTENTION

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