影像壓縮是影像處理中重要的一環。在西元 1992 年時，Joint Photographic Experts
Group 組織發表了 JPEG Standard 演算法。與其有關的軟件 jpg 已經成為智慧型手機與數位
相機中最知名的靜態影像壓縮檔案格式。西元 1992 年時，他們發表了演算法 JPEG 2000，
是 JPEG Standard 一個對於壓縮比與彈性的重要優化。
兩個演算法主要都由四個步驟構成：前處理、變換、量化、編碼。在這篇論文裡，我們
研究了 JPEG 2000 演算法，著重於變換步驟中的離散小波變換。特別地，我們研究了正交
與雙正交小波 CDF 9/7 外加由 Daubechies 與 Sweldens 發展出的提升變換，它們分別形成
JPEG Standard 與 JPEG 2000 失真壓縮法中一個重要的部分。我們的研究材料來自於幾本
專書，值得注意的是 Van Fleet 的著作『Discrete Wavelet Transformations』以及論文 [4]和
Daubechies [3]與 Mallat [6] 所寫的經典。這些素材被組織成一個獨立且有系統的方式。我們
也利用 Mathematica 寫了一些程式去例證雙正交小波變換與提升變換。

Image compression is an important aspect of image processing. In 1992 the Joint Photographic
Experts Group announced the JPEG Standard algorithm. The related software jpg
has become one of the most popular file format for still image compression in smart phones and
digital cameras. In 2000, they announced the algorithm JPEG 2000, an important improvement
of JPEG Standard in terms of compression rate and flexibility.
Both algorithms consist of the four steps: Pre-processing, Transformation, Quantization
and Encoding. In this thesis, we shall study the general algorithm of JPEG 2000. Special emphasis
will be made on the discrete wavelet transform in the Transformation step. In particular,
we shall study with mathematical rigor orthogonal wavelets and biorthogonal wavelets CDF
9/7, plus a related lifting transformation, developed by Daubechies-Sweldens. They form an
important part in the lossy compression algorithm of JPEG 2000, respectively. Our material
comes from several monographs, notably the book ‘Discrete Wavelet Transformations’ by Van
Fleet, and the paper [4], with the help of the classical monographs by Daubechies [3] and Mallat
[6]. The materials are organized into a self-contained and systematic manner. We also write
some programs with Mathematica to exemplify the biorthogonal wavelet transform and lifting
transform.

1 Introduction 1
1.1 The discrete cosine transform and JPEG Standard . . . . . . . . . . . . . . . . . 2
1.2 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Orthogonal Wavelets 10
2.1 Multiresolution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Daubechies wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Matrix representation of orthogonal wavelets . . . . . . . . . . . . . . . . . . . . 31
3 Biorthogonal Wavelets and Lifting Scheme 37
3.1 Biorthogonal wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Cohen-Daubechies-Feauveau wavelets . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Lifting scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 JPEG 2000 Image Compression Method 64
4.1 2-dimensional wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 General alogorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Further discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
APPENDIX 76
A The orthogonality of the matrix representation for DCT 77
B Mathematica coding for image compression using CDF 9/7 biorthogonal
wavelet 79
C Mathematica coding for the lifting transform 86

[1] A. Boggess & F.J. Narcowich, A First Course in Wavelets with Fourier Analysis, 2nd ed., John Wiley & Sons, New Jersey, 2008.
[2] C. Chui, An Introduction to Wavelets, Academic Press, San Diego, 1992.
[3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
[4] I. Daubechies, and W. Sweldens, Factoring wavelet transforms into Lifting Steps, The Journal of Fourier Analysis and Applications, 4(3):247-269, 1998.
[5] G.B. Folland, Fourier Analysis and Its Application, Wadsworth & Brooks/Cole, Pacific Grove, 1992.
[6] S. Mallat, A Wavelet Tour of Signal Processing – The Sparse Way, 3rd ed., Academic Press, Burlington, 2009.
[7] C. Rousseau & Y. Saint-Aubin, Mathematics and Technology, Spinger Science+Business Media, New York, 2008
[8] D. Santa-Cruz, R. Grosbois and T. Ebrahimi, JPEG 2000 performance evaluation and assessment, Signal Processing: Image Communication 17(1):113-120, 2002.
[9] W. Sweldens, The lifting scheme: a construction of second generation wavelets, SIAM Journal on Mathematical Analysis, 29(2):511–546, 1998.
[10] M. Unser, and B. Thierry, Mathematical properties of the JPEG2000 wavelet filters, IEEE Trans. Image Process, 12(9):1080–1090, 2003.
[11] P.J. Van Fleet, Discrete Wavelet Transformations – An Elementary Approach with Applications, John Wiley & Sons, New Jersey, 2008.