貝蘇章臺灣大學：電信工程學研究所高孟平Kao, Meng-PingMeng-PingKao2007-11-272018-07-052007-11-272018-07-052004http://ntur.lib.ntu.edu.tw//handle/246246/58960比例縮放（scaling）一直是數位信號處理中重要的課題，主要原因是其應用非常廣泛，舉凡圖形的放大縮小扭曲變形、信號取樣頻率的升降、以及近來討論熱烈的小波轉換等，都離不開比例縮放的範疇。 而比例縮放中最重要的參數就是比例縮放參數（scaling factor），其決定信號放大縮小的倍率。一直以來，大家對於處理有理數的比例縮放參數較有共識，相反的，對於無理數則是各持己見，沒有定論。而討論所有實數參數之比例縮放演算法我們稱之為廣義比例縮放（generalized scaling）。 在本篇論文中，我們利用信號頻域與時域的關係，在頻譜上作處理以達到廣義比例縮放的目的。相較於別人的方法，這個演算法具有線性的優點，因此可以應用在完整重建濾波器串（perfect reconstruction filter bank）的設計上，無論是一維還是二維的完整重建濾波器串皆有新突破。 此外，上述的方法也可以應用在數位信號的變形（warping）上。所謂變形與比例縮放的不同在於變形是非線性的取樣，而比例縮放則是線性的取樣。也就是說，透過這個方法，我們可以對一個數位信號做任意的扭曲變形，這在數位圖像的處理上有廣泛的應用，例如多投影機的校準便是一個例子。 論文的最後，鑑於近年來區塊離散餘弦轉換（block-based DCT）被廣泛地應用在JPEG與MPEG上，我提出一個快速轉換區塊大小的架構，這對於以區塊離散餘弦轉換為基礎的壓縮標準有快速比例縮放的效果。Scaling has always been an important topic in the field of digital signal processing, since its applications are so widespread and extensive. For example, interpolation, decimation and warping of a digital image, sampling rate conversion of a digital signal, and wavelet transform based filter banks are all closely related to scaling. The operation of scaling entirely depends on the scaling factor, which controls the ratio between the original signal and the scaled signal. In general, scaling factor can be either rational or irrational numbers. We named the operation on both rational and irrational factors, i.e. real factors, the generalized scaling. For a long time, there is no steadfast definition for generalized scaling, since everyone could justify his opinions from different points of view. In this paper, I propose a new algorithm to realize the generalized scaling using linear operations in the frequency domain. Compared to previous methods, this algorithm is entirely linear such that it can be well applied to the design of Perfect Reconstruction Filter Banks (PR FB). Furthermore, both one dimensional and two dimensional FB cases are explored in this paper. In addition, after a fine modification of the above algorithm, it can be utilized in image warping, which is a common problem in multi-projection applications. The term “warping” is different from scaling in that the former one is a non-linear operation while the later is linear. Through image warping, we can arbitrarily change the shape of a digital image. At the end of this paper, I put more emphases on DCT compressed domain manipulations, which are similar to generalized scaling in main concepts. I propose a new scheme to efficiently implement the block size conversion between different DCT’s. This scheme can be useful in image resizing for which the image is compressed in the DCT domain, such as JPEG and MPEG.Chapter 1 Introduction 01 Chapter 2 Review of Generalized Scaling and Its Applications 05 2.1 Methods for Generalized Scaling 05 2.1.1 Bilinear Transform 06 2.1.2 Partial Linear Partial Bilinear Transform 07 2.1.3 Generalized Bilinear Transform 10 2.2 Applications 12 2.2.1 Non-uniform Filter Banks 12 2.2.2 Digital Image Warping 12 2.2.3 Discrete Self-Similar Signals 13 Chapter 3 Algorithms for Generalized Scaling and Translation 15 3.1 Generalized Interpolation 15 3.2 Generalized Decimation 17 3.3 Generalized Scaling for Two-dimensional Case 18 3.4 Generalized Translation 19 Chapter 4 Application 1：Non-uniform Perfect Reconstruction Filter Bank 21 4.1 Beatty Sequence and PR FB 21 4.1.1 Review of Beatty Sequence and Its Properties 21 4.1.2 PR FB Using Beatty Sequence 23 4.1.3 Design Difficulties 24 4.2 1D Non-uniform PR FB with Irrational Down-sampling Factors 24 4.2.1 Proposed Structure 24 4.2.2 Experimental Results 26 4.3 2D Non-uniform PR FB with Irrational Down-sampling Matrices 27 4.3.1 Proposed Structure 27 4.3.2 Experimental Results 30 4.4 Practical Implementation – Window Design and Error Analysis 32 4.5 Appendix 42 Chapter 5 Application 2：Non-linear Warping of a Digital Image 45 5.1 Review of Multi-projection Scheme 45 5.2 Non-linear Warping Method 46 5.2.1 1D Warping Function 46 5.2.2 2D Warping Function 50 5.3 Fast Algorithm and Scheme for Practical Implementation 53 5.3.1 Window Function 54 5.3.2 Sample and Hold Approximation 55 5.3.3 Mirroring 56 5.4 Post-processing for Text/Bi-level Images 57 5.5 Practical Implementation and Error Evaluation 59 Chapter 6 Fast Manipulations in Compressed Domain：The DCT Case 65 6.1 Review of Fast Manipulations in DCT Domain 65 6.1.1 Generalized Discrete Fourier Transform 66 6.1.2 Symmetric-Periodic Sequences 68 6.1.3 The Relationship between DCT, SPS and GDFT 69 6.1.4 Some Basic Manipulation Functions in The DCT Domain 74 6.2 Fast Algorithm for Changing Block Size between Two DCT Sequences 80 6.2.1 Conventional Approach 82 6.2.2 Proposed Approach 84 6.2.3 Computational Complexity 87 Chapter 7 Conclusions and Future Works 91 References 932173973 bytesapplication/pdfen-US任意變形內插完整回復濾波器串快速離散餘弦轉換nonuniform filter bankinterpolationdecimationperfect reconstructiondirect DCT domain computationradix-3 dctirrational scalingthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/58960/1/ntu-93-R91942027-1.pdf