ARCHITECTURES FOR THE 3-D DISCRETE WAVELET TRANSFORM A Dissertation Presented to The Graduate Faculty of The University of Southwestern Louisiana In Fulfillment of the Requirements for the Degree Doctor of Philosophy Michael Clark Weeks Spring 1998 ARCHITECTURES FOR THE 3-D DISCRETE WAVELET TRANSFORM Michael Weeks APPROVED: ____________________________ ____________________________ Magdy A. Bayoumi, Chair C. Henry Chu Edmiston Professor of Computer Engineering Associate Professor of Computer Engineering ____________________________ ____________________________ Kimon Valavanis Guna Seetharaman Professor of Computer Engineering Associate Professor of Computer Engineering ____________________________ ____________________________ Gui-Liang Feng Lewis Pyenson Associate Professor of Computer Engineering Dean, Graduate School ii © Michael Weeks 1998 All Rights Reserved iii ACKNOWLEDGMENTS The author would like to recognize the following people for their help in this work: Dr. Magdy Bayoumi, for serving as advisor and chair of the dissertation committee; Dr. Henry Chu who served on the committee and provided the sample MRI data; Dr. Kimon Valavanis, Dr. Guna Seetharaman, and Dr. Gui-Liang Feng who served on the committee; and Dr. Jimmy Limqueco and Dr. Paul Shipley for their useful discussions. The author expresses appreciation for funding from grant LEQSF(1996-99)-RD-B-13, "Interoperable Underground Wavelet Transients Munitions Classification". iv TABLE OF CONTENTS 1. INTRODUCTION..............................................................................................................................................1 1.1 WHAT ARE WAVELETS?............................................................................................................................1 1.2 WAVELETS VS. FOURIER............................................................................................................................4 1.3 JPEG AND MPEG STANDARDS..................................................................................................................5 1.4 WAVELET APPLICATIONS ........................................................................................................................7 1.5 SOFTWARE AND HARDWARE FOR THE WAVELET TRANSFORM..................................................10 1.6 THE BASIC 1-D DWT..................................................................................................................................12 1.7 THE 2-D DISCRETE WAVELET TRANSFORM.......................................................................................15 1.8 WAVELET PACKETS.................................................................................................................................19 1.9 THE 3-D DISCRETE WAVELET TRANSFORM.......................................................................................21 1.10 THE FOCUS OF THIS WORK ................................................................................................................24 2. COMPRESSION..............................................................................................................................................26 2.1 COMPRESSION INTRODUCTION............................................................................................................26 2.2 QUANTIZATION.........................................................................................................................................27 2.3 ENTROPY CODING....................................................................................................................................29 2.4 TRANSCODING ..........................................................................................................................................30 2.5 COMPRESSION PERFORMANCE.............................................................................................................30 2.6 2-D COMPRESSION....................................................................................................................................32 2.7 COMPRESSION OF 3-D DATA ..................................................................................................................35 3. DWT ARCHITECTURES ..............................................................................................................................43 3.1 INTRODUCTION.........................................................................................................................................43 3.2 ARCHITECTURE CONSIDERATIONS .....................................................................................................48 3.3 ARCHITECTURES FOR THE 1-D DWT ....................................................................................................52 3.4 ARCHITECTURES FOR THE 2-D DWT ....................................................................................................65 3.5 OTHER ARCHITECTURES ........................................................................................................................71 3.6 CONCLUSIONS...........................................................................................................................................75 4. PROPOSED 3-D DWT ARCHITECTURES ................................................................................................76 4.1 THE NEED FOR A 3-D DWT ......................................................................................................................76 4.2 3-D DWT ARCHITECTURES .....................................................................................................................79 4.3 THE 3D-I ARCHITECTURE........................................................................................................................83 4.4 THE 3D-II ARCHITECTURE ......................................................................................................................85 5. ARCHITECTURE EVALUATION...............................................................................................................97 5.1 WAVELET COEFFICIENTS.......................................................................................................................97 5.2 FIXED POINT VERSUS FLOATING POINT ................................................................................................97 5.3 PRECISION ..................................................................................................................................................99 5.4 LOWER OCTAVE SCHEDULING ...........................................................................................................100 5.5 3D-I MEMORY REQUIREMENTS...........................................................................................................101 5.6 3D-II MEMORY REQUIREMENTS..........................................................................................................105 5.7 SERIAL VERSUS PARALLEL OPERATION .............................................................................................107 5.8 SIMULATION RESULTS..........................................................................................................................107 5.9 3D-I CLOCK CYCLE ANALYSIS.............................................................................................................109 5.10 3D-II CLOCK CYCLE ANALYSIS ......................................................................................................110 5.11 SPEED....................................................................................................................................................117 5.12 MEMORY REQUIREMENTS...............................................................................................................123 5.13 FOLDING ANALYSIS..........................................................................................................................125 5.14 COMPRESSION RESULTS..................................................................................................................127 6. CONCLUSIONS ............................................................................................................................................131 v LIST OF FIGURES Figure 1.1 - A 1-octave, 4 tap DWT............................. 2 Figure 1.2 - Three level multiresolution ....................... 2 Figure 1.3 - A 2-Dimensional, 1-octave DWT .................... 17 Figure 1.4 - The 3-D DWT ...................................... 22 Figure 2.1 - The 2-D decomposition of an image into 2 octaves .. 34 Figure 3.1 - An example FIR filter [Quarmby85] ................ 43 Figure 3.2 - Fridman’s DWT PE array ........................... 55 Figure 3.3 - Architecture to compute 1-D DWT by Syed et al..... 56 Figure 3.4 - Knowles’ 1-D pipelined architecture .............. 57 Figure 3.5 - Aware’s 1-D architecture ......................... 58 Figure 3.6 - Parhi and Nishitani’s folded 1-D architecture..... 59 Figure 3.7 - Parhi and Nishitani’s digit-serial 1-D architecture 60 Figure 3.8 - Vishwanath’s first 1-D architecture .............. 62 Figure 3.9 - Vishwanath’s second 1-D architecture ............. 62 Figure 3.10 - Vishwanath’s third 1-D architecture ............. 63 Figure 3.11 - Vishwanath’s 2-D architecture ................... 68 Figure 3.12 - Acharya’s systolic architecture ................. 69 Figure 3.13 - Limqueco’s 2-D architecture ..................... 70 Figure 4.1 - Sample MRI images................................ 78 Figure 4.2 -