A Comparative Study of DCT, LOT, and DWT-based Image Coders by Warit Wichakool Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degrees of Bachelor of Science in Electircal Engineering and Computer Science and Master of Engineering in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2001 © Warit Wichakool, MMI. All rights reserved. The author hereby grants to MIT permission to reproduce and BARKE distribute publicly paper and electronic copies of this thesis document in whole or in part. MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUL 11 2001 Author ...... LIBRARIES Department of Electrical Engineering and Computer Science May 23, 2001 Certified by..... K.S. Thyagarajan Principal Engineer VIA daiesisispupervisf Certified by... David H.itaelin _-Professor of Electrical Engineering WL.T ThesiS~uj rvisor Accepted by........... Arthur C. Smith Chairman, Department Committee on Graduate Students A Comparative Study of DCT, LOT, and DWT-based Image Coders by Warit Wichakool Submitted to the Department of Electrical Engineering and Computer Science on May 23, 2001, in partial fulfillment of the requirements for the degrees of Bachelor of Science in Electrical Engineering and Computer Science and Master of Engineering in Electrical Engineering and Computer Science Abstract Ten transform coding systems were compared in terms of their system complexity and peak signal-to-noise ratios (PSNR). The results showed that the relationship between PSNR and bit rate was affected by the combination of the quantizer and the encoder, but was not affected by the type of transform. The performance of the embedded zerotree wavelet (EZW) system was effected by the number of discrete wavelet transform (DWT) levels. In comparison with the 2-level EZW system at a given bit rate, the EZW system improved PSNR by up to 4 dB at low bit rates as the number of levels increased from two to three, and gained another 1 dB as the number of levels increased from three to four. Four of the ten systems were studied further: the discrete cosine transform (DCT) baseline JPEG, the lapped orthogonal transform (LOT) version of baseline JPEG, the visual threshold wavelet with the run-length Huffman coder, and the EZW with the adaptive Huffman coder. The PSNR values for the Lena image at 0.5 bit/pixel for the four systems were 34.56, 34.43, 34.97, and 34.52 dB, respectively. In comparison with the DCT JPEG, the LOT JPEG provided 0.5 dB better PSNR and also reduced the image blockiness, but it introduced small ringing artifacts in areas with sharp edges. The visual threshold wavelet yielded better PSNR than the DCT system at the same bit rate, but the reconstructed image suffered from blurriness. Finally, the EZW system performed comparably to the DCT system. Although the reconstructed image exhibited no blockiness, it clearly lost some details. VI-A Company Thesis Supervisor: K.S. Thyagarajan Title: Principal Engineer M.I.T. Thesis Supervisor: David H. Staelin Title: Professor of Electrical Engineering Acknowledgments I would like to take this opportunity to express my gratitude toward many people on the course of my thesis. First of all, I would like to thank K.S. Thyagarajan for his advice and guidance. I also would like to thank Professor Staelin for his guidance and comments on my thesis, and Henrique Malvar for providing the programs and references of the LOT system for my simulation. I also would like to thank all members of Digital Cinema at QUALCOMM INCORPORATED for their supports during my research at the company. In addition, I would like to thank Peter Agboh and Songpon Deechongkit for their comments on my research. In addition, I have to thank Yui (Siraprapha Sanchatjate), my family, and all my friends for all the mental support and encouragement they have been giving me through out the years at MIT. Contents 1 Introduction 12 2 Background 16 2.1 Transform ...... ........ ...... 17 2.1.1 Discrete Cosine Transform ....... 18 2.1.2 Lapped Orthogonal Transform . .... 20 2.1.3 Discrete Wavelet Transform . ..... 26 2.2 Quantization ... ............... 31 2.2.1 Optimal Uniform Quantizer ...... 33 2.2.2 JPEG Uniform Quantizer ....... 34 2.2.3 Visual Threshold Wavelet Quantizer 35 2.2.4 Embedded Zerotree Wavelet Quantizer 37 2.3 Entropy Coding . ........ ........ 40 2.3.1 Huffman Coding ..... ....... 41 2.3.2 Adaptive Huffman Coding ..... .. 41 2.3.3 Run-Length Huffman Coding ..... 41 3 Simulation Methods 44 3.1 Part I: System Complexity ...... 46 3.2 Part II: System Performance..... 46 3.2.1 Part II-A: Effect of Number of Levels on DWT Systems . 47 3.2.2 Part II-B: Effect of Transform ............... 47 3.2.3 Part II-C: Effect of Quantizer 48 4 3.2.4 Part II-D: Effect of Entropy Coder .......... ..... 49 3.2.5 Part II-E: Overall System Performance ........ ..... 49 3.3 Part III: Visual Quality ........ ............ ...... 50 3.4 Test Im ages ............ .................... 50 4 Results and Discussions 52 4.1 Part I: System Complexity ........................ 52 4.1.1 Transform ......................... .... 52 4.1.2 Quantization ........................... 54 4.1.3 Entropy Coding ........ .................. 55 4.2 Part II: System Performance ............ ........... 56 4.2.1 Part II-A: Effect of Number Levels on DWT Systems ..... 56 4.2.2 Part II-B: Effect of Transform ... ............ ... 65 4.2.3 Part II-C: Effect of Quantizer .................. 70 4.2.4 Part II-D: Effect of Entropy Coder ............... 71 4.2.5 Part II-E: Overall System Performance .... ......... 78 4.3 Part III: Visual Quality. .... ........ ........ ..... 90 5 Summary 105 A General Thesis Release Letter and Classification Review Letter 107 5 List of Figures 2-1 Basic transform coding system . ..... ... .... ...... 16 2-2 Block diagram for a separable 2-D transform .... ...... 18 2-3 Flowgraph conventions ... ..... ..... ....... .. 18 2-4 1-D DCT basis functions .. .... .... ... .......... 19 2-5 Fast DCT for M=8 .................... ..... ..... 2 1 2-6 Fast IDCT for M=8 ................ .......... 22 2-7 General structure of the LOT ... .... ... .......... 23 2-8 LOT basis functions ................ .......... 25 2-9 Type-I, fast LOT for a block size of 16 . .. .......... 26 2-10 Type-I, fast ILOT for a block size of 16 .... ....... ... 27 2-11 Type-I, fast LOT for the finite length signal . .......... 28 2-12 2-level, 2-band analysis and synthesis for the 1-D DV T ... 30 2-13 Organization of 3-level DWT coefficients ....... ..... 31 2-14 Locations of parent-descendants of the 3-level EZW . .... 39 2-15 Zig-zag scan of the run-length Huffman coder .... .... 43 2-16 Flowgraph of the run-length Huffman coder ..... ..... 43 4-1 Effect of number of DWT levels on the PSNR performance of the DWT systems using the EZW quantizer and the adaptive H uffm an coder .... ....... ....... ....... .... 64 4-2 Efficiency test for the adaptive Huffman coder ..... .... 89 4-3 bfragl: original image at 8 bpp ....... ........ .... 93 4-4 bfrag2: DCT + JPEG + Run-Length Huffman at 0.50 bpp . 94 6 4-5 bfrag3: 3-level DWT + EZW + Adaptive Huffman at 0.50 bpp 95 4-6 bfrag4: 4-level DWT + EZW + Adaptive Huffman at 0.50 bpp 96 4-7 bfrag5: 3-level DWT + Visual Threshold + Run-Length Huff- man at 0.50 bpp ....... ............................ 97 4-8 bfrag6: LOT + JPEG + Run-Length Huffman at 0.50 bpp . 98 4-9 lenal: original image at 8 bpp .................. 99 4-10 lena2: DCT + JPEG + Run-Length Huffman at 0.50 bpp . 100 4-11 lena3: 3-level DWT + EZW + Adaptive Huffman at 0.50 bpp 101 4-12 lena4: 4-level DWT + EZW + Adaptive Huffman at 0.50 bpp 102 4-13 lena5: LOT + JPEG + Run-Length Huffman at 0.50 bpp .. 103 4-14 lena6: 3-level DWT + Visual Threshold + Run-Length Huff- m an at 0.50 bpp ............................ 104 7 List of Tables 2.1 Computational costs of the fast DCT ..... ......... 21 2.2 Computational costs of the type-I, fast LOT ...... .... 29 2.3 Coefficients of 9/7-tap Villasenor biorthogonal filters .. ... 29 2.4 Computational costs of the DWT .. ..... ..... ..... 32 2.5 Basis function amplitudes for 9/7-tap biorthogonal filters . 36 2.6 Quantization levels for 9/7-tap biorthogonal filters ...... 37 3.1 List of tested transform coding systems ...... ....... 44 3.2 List of systems for comparing the effect of number of DWT levels on the PSNR performance .................. 47 3.3 List of systems for comparing the effect of transforms on the PSNR performance using the optimal uniform quantizer, the Huffman coder, and the run-length Huffman coder ...... 48 3.4 List of systems for comparing the effect of quantizers on the PSNR performance using the run-length Huffman coder .. 49 3.5 List of systems for comparing the effect of entropy coders on the PSNR performance using Huffman and run-length Huff- m an coders ... ........................ .... 49 3.6 List of selected systems for comparing the PSNR performance 50 3.7 List of test im ages ... ..... ..... ..... ..... .... 51 4.1 Computational costs of transform algorithms .......... 53 4.2 Normalized costs of transforming a 512x512 image ..... 53 8 4.3 List of systems for comparing the effect of DWT levels on the PSNR performance of the DWT systems ............. 57 4.4 Effect of DWT levels on the PSNR performance of the DWT systems using the optimal uniform quantizer and the run- length Huffman coder ............................... 58 4.5 Effect of DWT levels on the PSNR performance of the DWT systems using the optimal uniform quantizer and the run- length Huffman coder (cont.) .................... 59 4.6 Effect of DWT levels on the PSNR performance of the DWT systems using the visual threshold quantizer and the run- length Huffman coder ..............................