Techniques for Enhancing Digital Images Author: Supervisor: William E. Harrison III Dr. Weiping Lu A thesis submitted in fulfillment of the requirements for the degree of Master of Philosophy. in Heriot Watt University School of Engineering and Physical Sciences December 2016 “The copyright in this thesis is owned by the author. Any quotation from the thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information." Declaration ________________________________________________________________________ Declaration of Authorship I, William E Harrison, III, declare that this thesis titled, ‘Techniques for Enhancing Digital Images’, and the work presented in it is my own. I confirm that this work submitted for assessment is my own and is expressed in my own words. Any uses made within it of the works of other authors in any form (e.g., ideas, equations, figures, text, tables, programs) are properly acknowledged at any point of their use. A list of the references employed is included. Signed: _______________________________________ Date: _______________________________________ _____________________________________________________________________ i Abstract ________________________________________________________________________ ABSTRACT The images obtain from either research studies or optical instruments are often corrupted with noise. Image denoising involves the manipulation of image data to produce a visually high quality image. This thesis reviews the existing denoising algorithms and the filtering approaches available for enhancing images and/or data transmission. Spatial-domain and Transform-domain digital image filtering algorithms have been used in the past to suppress different noise models. The different noise models can be either additive or multiplicative. Selection of the denoising algorithm is application dependent. It is necessary to have knowledge about the noise present in the image so as to select the appropriated denoising algorithm. Noise models may include Gaussian noise, Salt and Pepper noise, Speckle noise and Brownian noise. The Wavelet Transform is similar to the Fourier transform with a completely different merit function. The main difference between Wavelet transform and Fourier transform is that, in the Wavelet Transform, Wavelets are localized in both time and frequency. In the standard Fourier Transform, Wavelets are only localized in frequency. Wavelet analysis consists of breaking up the signal into shifted and scales versions of the original (or mother) Wavelet. The Wiener Filter (mean squared estimation error) finds implementations as a LMS filter (least mean squares), RLS filter (recursive least squares), or Kalman filter. Quantitative measure (metrics) of the comparison of the denoising algorithms is provided by calculating the Peak Signal to Noise Ratio (PSNR), the Mean Square Error (MSE) value and the Mean Absolute Error (MAE) evaluation factors. A combination of metrics including the PSNR, MSE, and MAE are often required to clearly assess the model performance. _____________________________________________________________________ ii Acknowledgements ________________________________________________________________________ Acknowledgement I wish to express my deepest gratitude to Dr. Weiping Lu, Dr. Euan Brown, and Dr. Roy Duncan for their assistance, guidance, encouragement, and friendship throughout my graduate career at Heriot Watt University. I would also wish to thank Dr. Eaun Brown for his encouragement when I took a temporary leave due to unexpected Medical problems. _____________________________________________________________________ iii Table of Contents ________________________________________________________________________ Contents Page No. Declaration of Authorship i Abstract ii Acknowledgements iii Contents iv List of Tables viii List of Figures ix List of Images xii Chapter 1. Introduction……………………………………………… 1 1.1 Fundamentals of Digital Image………………… 1 1.2 Noise in Digital Image…………………………… 4 1.3 Gaussian Noise….………………………………… 5 1.4 Salt and Pepper Noise…………………………… . 6 1.5 Speckle Noise………………………………………. 7 1.6 Brownian Noise Speckle………………………… .. 8 1.7 Quantifying Noise…………………………………. 9 1.7.1 Mathematics 9 1.8 Objectives...………………………………………….. 10 1.9 Conclusion...………………………………………….. 12 2. Image Processing Basics………………………………. 13 _____________________________________________________________________ iv Table of Contents ________________________________________________________________________ 2.1 Digital Image……………………………………… 13 2.2 Processing Methods….…………………………... 18 2.3 Spatial Domain Methods………………………… 19 2.3.1 Mean Filter………………………………… 19 2.3.2 Median Filter……………………………… 21 2.3.3 Non-local Means Filter…………………… 23 2.4 Frequency Domain Filter………………………… 26 2.4.1 Butterworth Filter……….………………… 26 2.4.2 Gaussian Filter-Convolution……………… 27 2.4.3 Wiener Filter ………………..……………… 29 2.4.3.1 Point Spread Function-Wiener Filter……. 31 2.4.3.2 Application of Wiener Filter……………. … 32 2.5 Algorithm…………………………………………….. 35 3. Wavelets…………………………………………………….. 36 3.1 Introduction………………………..………………. 36 3.2 Wavelet Transform…..…………………………… 37 3.2.1 Preview 37 3.2.2 Simplest Example of Image Transform- Walsh Transform………………………………….. 40 3.2.3 The Haar Wavelet…………………………………. 42 3.3 Image Denoising…………………………………... 44 3.4 Discrete Wavelet Transform………………………… 45 3.5 Wavelet Thresholding…………………………… …. 48 3.5.1 Visushrink ……….………………… 50 3.5.2 SureShrink……………………….……………. 51 3.5.3 Bayes Shrink………..……….………………… 52 3.6 Wavelet Filters in MATLAB…………………………. 54 3.7 VisuShrink Example…………………………………. 56 3.8 SureShrink Example…………………………………. 58 3.9 BayesShrink Example……………………………….. 59 3.10 Summary……………………………………………… 63 4. Image Processing Applications in Science and Medicine... 64 4.1 Astronomy………………………..………………… 64 4.2 Medicine…………………………………………... .. 65 _____________________________________________________________________ v Table of Contents ________________________________________________________________________ 4.2.1 Tomographic Imaging…………………………..... 66 4.2.1.1 Transmissive Mode………………………….......... 67 4.2.1.2 Emissive Mode……………………………………. 67 4.2.1.3 Reflective Mode………………………….............. 68 4.3 Experimental Results………………………......... 74 4.3.1 Performance Evaluation – Denoising Technique 74 4.3.2 Table of Results Comparison…………………….. 80 4.4 Generalization into other Scenarios 3D MRI Imaging ………………………………….. 82 4.4.1 Create Head Phantom – Two Dimensions …….. 83 4.4.2 Extension to creating 3D MRI Images ………….. 84 5. Conclusions ………………………………………………… 90 6. Scope for Future Work ......................................................... 91 6.1 Fuzzy Methods……………………..………………. 91 6.2 Vision 3-D Map……………………..………………. 92 6.3 Medical Image Computing…………..……………. 92 7. References ..…………………………………………………. 95 APPENDIX A: MATLAB FUNCTIONS………........................................ 100 APPENDIX A: Latest Versions MATLAB………………………………. 102 APPENDIX A: MATLAB Convolution Denoised……………..………. 103 APPENDIX A: MATLAB Airy Point Spread Function……………..…. 104 APPENDIX A: MATLAB Image Denoised. …………..………………. .. 105 APPENDIX B: Medical Glossary………………….………....................... 107 _____________________________________________________________________ vi Table of Contents ________________________________________________________________________ APPENDIX C: C Code………………….………........................................ 109 _____________________________________________________________________ vii Tables ________________________________________________________________________ List of Tables Table 2.1 List of Important Denoising Techniques in the Spatial and Frequency Domain …….......... 18 Table 2.2 Metrics for Denoised Lena Image with Median Filter ………… 22 Table 3.1 Performance Evaluation Metrics Wavelet db4 …………………. 60 Table 3.2 Performance Evaluation Metrics Wavelet sym4 ………………... 61 Table 3.3 Performance Evaluation Metrics Wavelet bior 6.8 ...…………… 62 Table 4.1 Quality Analyses (MRI Image) - GLOBAL THRESHOLDING … 80 Table 4.2 Quality Analyses (MRI Image) - LEVEL DEPENDENT THRESHOLDING ……………. 81 viii Techniques for Enhancing Digital Images Figures ________________________________________________________________________ List of Figures Figure 1.1 Gaussian distribution …………………………………………… 5 Figure 1.2 PDF for salt and pepper noise ………………………………….. 6 Figure 1.3 Gamma Distribution …………………………………………….. 7 Figure 1.4: Brownian noise distribution ……………………………………. 8 Figure 2.1 The discrete pixel numbering convention ……………………… 14 Figure 2.2 Using different colormaps to display the same binary image ... 14 Figure 2.3 A greyscale image ………………………………………………… 15 Figure 2.4 Default Color Maps available in Matlab ………………………… 16 Figure 2.5 The Shift-Multiply-Sum Principle ………………………………. 20 Figure 2.6 Illustrates an example calculation ……………………………… 21 Figure 3.1 Haar wavelets of various scales and translations ……………. 43 Figure 3.2 Block Diagram of Image denoising using Wavelet transform … 44 Figure 3.3 First Level Decomposition ………………………………………. 46 Figure 3.4 Second Level Subbands ………………………………………… 46 Figure 3.5 Third Level Subbands …………………………………………… 46 Figure 3.6 Hard thresholding ………………………………………………… 49 Figure 3.7 Soft thresholding …………………………………………………… 50 Figure 3.8 Wavelet Families …………………………………………………… 54 ix Techniques for Enhancing Digital Images Figures ________________________________________________________________________ Figure 3.9 Wavelet Families …………………………………………………… 54 Figure 3.10 Wavelet Families ………………………………………………… 55 Figure 4.1