本論文提出了影像與視訊壓縮技術中，用於資料取樣和資料重建的一種新的插值補償法，稱為立方樣條插補法(cubic-spline interpolation)，簡稱CSI。這種新的CSI插補法，是根據最小平方法和立方樣條函數兩種基本理論，以及快速富立葉轉換(FFT)發展而成。在本論文中將詳細說明、推導這種簡單且快速的插值補償法。並利用電腦模擬程式，將它和線性插補法(linear interpolation)、線性樣條插補法(linear-spline interpolation)、立方迴旋插補法(cubic-convolution interpolation)、立方B-樣條插補法(cubic B-spline interpolation)等相互比較，證實此CSI插補法，是一種極為精確的插補法。而線性插補法、線性樣條插補法、立方迴旋插補法和立方B-樣條插補法等，在性能上均明顯較差。
在本論文中亦進一步提出此CSI插補法，可以利用一種更快速且有效率的計算法來改進，茲說明如下：
1. 根據資料分佈在取樣點的位置特性，提出一種帶狀區域的濾波器(zonal filter)，可以大幅減少CSI插補法在取樣過程所需的加法及乘法計算量。
2. 此CSI插補法除了可以使用FFT轉換之外，還可以改用一種更有效率的9點Winograd離散富立葉轉換(Winograd DFT)，來加速CSI插補法的演算性能。
3. 對於實際的影像處理，提出一種新的交互重疊(overlap-save)技術，可以快速且輕易解決兩個相鄰次影像區塊的邊界問題。
此外，本論文亦將此新的快速CSI插補法，與JPEG演算法結合，設計一種擁有高壓縮率的改良式JPEG影像壓縮法。並且再以電腦模擬程式實驗證明，在高壓縮率時，此新的改良式JPEG法比傳統的JPEG影像壓縮法，以及美國線上(America on Line, AOL)演算法，獲得較佳的重建影像品質，而且在編解碼過程所需的時間均較少。最後，本論文再將此快速CSI插補法，運用於JPEG 2000、MPEG-1及MPEG-2演算法中，並且經由電腦模擬程式驗證，在高壓縮率時，所獲得的重建影像品質，與JPEG 2000相近，但是在編解碼過程所需的時間大幅減少；而在高壓縮率及低位元率的視訊壓縮編碼時，比傳統的MPEG-1及MPEG-2演算法，獲得較佳的重建視訊品質。

In this dissertation, a new cubic-spline interpolation (CSI) for both one-dimensional and two-dimensional signals is developed to sub-sample signal, image and video compression data. This new interpolation scheme that is based on the least-squares method with a cubic-spline function can be implemented by the fast Fourier transform (FFT). The result is a simpler and faster interpolation scheme than can be obtained by other conventional means. It is shown by computer simulation that such a new CSI yields a very accurate algorithm for smoothing. Linear interpolation, linear-spline interpolation, cubic-convolution interpolation and cubic B-spline interpolation tend to be inferior in performance.
In addition it is shown in this dissertation that the CSI scheme can be performed by a fast and efficient computation. The proposed method uses a simpler technique in the decimation process. It requires substantially fewer additions and multiplications than the original CSI algorithm. Moreover, a new type of overlap-save scheme is utilized to solve the boundary-condition problems that occur between two neighboring subimages in the actual image. It is also shown in this dissertation that a very efficient 9-point Winograd discrete Fourier transform (Winograd DFT) can be used to replace the FFT needed to implement the CSI scheme.
Furthermore, the proposed fast new CSI scheme is used along with the Joint Photographic Experts Group (JPEG) standard to design a modified JPEG encoder- decoder for image data compression. As a consequence, for higher compression ratios the proposed modified JPEG encoder-decoder obtains a better quality of reconstructed image and also requires less computational time than both the conventional JPEG method and the America on Line (AOL) algorithm. Finally, the new fast CSI scheme is applied to the JPEG 2000, MPEG-1 and MPEG-2 algorithms, respectively. A computer simulation shows that in the encoding and decoding, the proposed modified JPEG 2000 encoder-decoder speeds up the JPEG 2000 standard, respectively, and still obtains a good quality of reconstructed image that is similar to JPEG 2000 standard for high compression ratios. Additionally, the reconstructed video using the modified MPEG encoder-decoder indicates a better quality than the conventional MPEG-1 and MPEG-2 algorithms for high compression ratios or low-bit rates.

Contents
Acknowledgments
中文摘要………………………………………………………… i
Abstract …………………………………………………………… iii
List Of Figures …………………………………………………… vii
List Of Tables ……………………………………………………… xii
1 Introduction …………………………………………………… 1
1.1 Motivation and Recent Related Research ………………… 1
1.2 Summary of the Dissertation ……………………………... 4
1.3 Organization of the Dissertation ………………………… 7
2 Interpolation Functions ……………………………………… 10
2.1 The Cubic-Convolution Interpolation …………………….. 15
2.2 The Cubic B-Spline Function …………………………….. 22
2.3 The Linear-Spline Interpolation ………………………….. 27
3 Encoding Algorithm of Cubic-Spline Interpolation ………... 32
3.1 Cubic-Spline Interpolation for the 1-D Signal ……………. 32
3.2 Cubic-Spline Interpolation for the 2-D Signal ……………. 38
4 Decoding Algorithm of Cubic-Spline Interpolation ………... 45
4.1 Decoding of the Compressed 1-D Signal ………………… 45
4.2 Decoding of the Compressed 2-D Signal ………………… 46
5 Fast Computation of Cubic-Spline Interpolation …………... 48
5.1 The Use of Zonal Filter to Compute CSI …………………. 48
5.2 Calculation of Constants ………………………………….. 52
5.3 Winograd DFT and Overlap-Save Method ……………….. 57
6 The Application of Cubic-Spline Interpolation …………….. 66
6.1 A Modified JPEG Encoder-Decoder ……………………... 66
6.2 A Modified JPEG 2000 Encoder-Decoder ……………….. 69
6.3 Low-Bit-Rate Video Coding ……………………………… 71
7 Experimental Results and Discussion ……………………….. 73
7.1 Experimental Results for CSI and FCSI ………………….. 74
7.2 Experimental Results for Modified JPEG ………………... 89
7.3 Experimental Results for Modified JPEG 2000 ………….. 94
7.4 Experimental Results for Low-Bit-Rate Video Coding ….. 97
8 Conclusions and Further Research …………………………. 108
8.1 Conclusions ………………………………………………. 108
8.2 Further Research …………………………………………. 109
Appendix A
A.1 The 9-Point Winograd Discrete Fourier Transform ………. 113
A.2 The 5-Point Winograd Discrete Fourier Transform ………. 114
Appendix B
B.1 The C Program for the 9-Point Winograd DFT …………... 116
B.2 The C Program for the 5-Point Winograd DFT …………... 121
Bibliography ……………………………………………………… 126

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