WAVELET-BASED IMAGE COMPRESSION Shetul Saksena B. E., Gujarat University, India, 2007 PROJECT Submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in ELECTRICAL & ELECTRONIC ENGINEERING at CALIFORNIA STATE UNIVERSITY, SACRAMENTO SPRING 2011 WAVELET-BASED IMAGE COMPRESSION A Project by Shetul Saksena Approved by: __________________________________, Committee Chair Jing Pang, Ph. D. __________________________________, Second Reader Fethi Belkhouche, Ph.D. ____________________________ Date ii Student: Shetul Saksena I certify that this student has met the requirements for format contained in the University format manual, and that this project is suitable for shelving in the Library and credit is to be awarded for the Project. __________________________, Graduate Coordinator ________________ B. Preetham Kumar, Ph.D. Date Department of Electrical & Electronic Engineering iii Abstract of WAVELET-BASED IMAGE COMPRESSION by Shetul Saksena Recently, compression techniques using Wavelet Transformation have received great attention for their promising compression ratio, analysis of the temporal and spectral properties of image signals, and flexible representation of non-stationary signals by taking into account the human perception system. With lossless compression, the original image is recovered immediately after decompression. Unfortunately, with images of natural scenes it is rarely possible to obtain error-free compression at a rate beyond 2:1. Much higher compression ratios can be obtained when some errors, which are usually difficult to perceive, are allowed between the decompressed image and the original image. This project investigates wavelet-based lossy compression of 2D still images. This was achieved by performing Discrete Haar Wavelet Transformation (DWT) on an image followed by Entropy Encoding techniques including Embedded Zerotree Wavelet (EZW) and Huffman encoding. __________________________, Committee Chair ________________ Jing Pang, Ph. D. Date iv ACKNOWLEDGEMENTS The process of completing this work has left me gratefully and deeply indebted to Dr. Jing Pang whose help, stimulating suggestions and encouragement have contributed endlessly to my success. Dr. Pang’s involvement kept me motivated through the formative stages of my work and her support and enthusiasm helped me establish the project in its earliest and most vulnerable stages. Our bi-weekly meetings were essential and enjoyable stepping stone to meet my goals and her informed advice was welcomed on every aspect of this project. I would like to thank Dr. Fethi Belkhouche for his insightful suggestions following a careful review of my report. The advice I have received over the course of this project has been graciously contributed by a number of my colleagues, I am grateful to all of them. I would like to thank all the faculty members of the Department of Electrical & Electronic Engineering for helping me finish my Master of Science degree at California State University, Sacramento. To all who helped me in carrying out this project successfully, I acknowledge my indebtedness and express my great appreciation. v TABLE OF CONTENTS Page Acknowledgements ............................................................................................................. v List of Tables ................................................................................................................... viii List of Figures .................................................................................................................... ix Chapter 1. INTRODUCTION ....................................................................................................... 1 1.1 What is Compression? .......................................................................................... 1 1.2 Evolution of Data Compression ............................................................................ 2 1.3 My Project ............................................................................................................. 4 1.3.1 Significance of the Project ........................................................................... 4 1.3.2 Limitations of the Project ............................................................................. 4 2. IMAGE COMPRESSION ........................................................................................... 5 2.1 Digital Images ....................................................................................................... 5 2.1.1 Color Depth, Bits Per Pixel .......................................................................... 6 2.1.2 Compression Ratio ....................................................................................... 7 2.2 Lossy and Lossless Compression .......................................................................... 8 2.3 Basic Image Compression Block Diagram ........................................................... 9 2.4 Progressive Image Compression Algorithm ....................................................... 10 2.5 Wavelet-based Image Compression Steps (Coding Approach) .......................... 12 3. WAVELET-BASED IMAGE COMPRESSION ...................................................... 15 3.1 Discrete Haar Wavelet Transform ...................................................................... 15 3.2 Embedded Zerotrees of Wavelet Transforms Encoding .................................... 23 3.2.1 EZW Encoding Algorithm ......................................................................... 25 3.2.2 Morton Scan ............................................................................................... 29 3.3 Huffman Encoding .............................................................................................. 30 3.3.1 Huffman Tree ............................................................................................. 31 4. MATLAB IMPLEMENTATION ............................................................................. 33 vi 4.1 Steps of Compression .......................................................................................... 34 4.1.1 Reading the Image to be Compressed ........................................................ 35 4.1.2 Discrete Haar Wavelet Transformation ...................................................... 35 4.1.3 EZW Encoding of Wavelet Coefficient Matrix ......................................... 35 4.1.4 Huffman Encoding ..................................................................................... 36 4.1.5 Writing Compressed Image ........................................................................ 36 4.2 Function Flow Diagram ...................................................................................... 36 4.3 Simulation Results .............................................................................................. 37 4.4 Observations ........................................................................................................ 39 4.5 Future Improvements .......................................................................................... 39 5. APPLICATIONS OF WAVELET-BASED IMAGE COMPRESSION .................. 41 6. CONCLUSION ......................................................................................................... 43 Appendix MATLAB Code ............................................................................................ 44 Bibliography .................................................................................................................. 66 vii LIST OF TABLES 1. Table 4.1: Level of Haar transformation vs. compression ratio ................................... 39 viii LIST OF FIGURES 1. Figure 2.1: Y, Cb and Cr components of an image.......................................................... 6 2. Figure 2.2: Basic image compression block diagram ..................................................... 9 3. Figure 2.3: Progressive image compression algorithm ................................................. 11 4. Figure 2.4: Region of Interest (ROI) selection ............................................................ 12 5. Figure 3.1: Pyramidal decomposition of an image ....................................................... 20 6. Figure 3.2: Single level DWT ....................................................................................... 21 7. Figure 3.3: Level 2 DWT .............................................................................................. 22 8. Figure 3.4: Level 3 DWT .............................................................................................. 22 9. Figure 3.5: Quad trees ................................................................................................... 24 10. Figure 3.6: Wavelet coefficients represented in different subbands ........................... 25 11. Figure 3.7: Wavelet coefficients coded as P, N, T, and Z ........................................... 27 12. Figure 3.8: Raster (row-wise) scan order .................................................................... 29 13. Figure 3.9: Morton (Z) scan order .............................................................................. 30 14. Figure 3.10: Huffman tree........................................................................................... 32 15. Figure 4.1: Original test image lena.jpg ..................................................................... 33 16. Figure 4.2: Steps of compression ...............................................................................