中文摘要
近年來，由於科技的進步，音樂的取得越來越容易，音樂也變得越來越普遍化。在我們周圍，各種各樣的音樂變得更加複雜及大量。音樂的爆炸性增加，迫使我們需要新的技術和工具，可以智慧地且自動地將音樂轉換成有用的資訊，並且將音樂分類(classification)到正確的類別。節奏(rhythm)查詢是音樂類別分類的基本技術，這也是重要的多媒體應用。最近，Christodoulakis 等人提出針對音樂時間序列(music duration sequence)以節奏來分類的CIRS 演算法。在CIRS 運算法的觀念中，一段節奏可視為一串由快速(Q)和緩慢(S)的字元組成序列，其涵意為音樂中音符的時間序列。在比對過程中，兩個連續的快速(Q)字元所組成的序列，可以組合成一個緩慢(S)字元，但是一個緩慢(S)字元不能分解成兩個連續的快速(Q)字元，這點也造成了演算法設計的困難。為了要以節奏來將音樂分類，CIRS 演算法找出MaxCover—在音樂時間序列中，被查詢之節奏所連續覆蓋(重疊地或相連地)的最長子序列。在CIRS 演算法中，可分為4 個主要步驟，以重複執行步驟2、3、4 的方式，在音樂時間序列中，針對每個不同的值掃描輸入的時間序列，找出所有的區域性MaxCover。最後以區域性MaxCover，計算出全域性MaxCover，這也是演算法最後的結果。但是，我們觀察到，CIRS 演算法，在步驟2、3、4 中產生出非必要的步驟及結果，由此會浪費演算法的執行時間。為了避免產生以上情形，我們提出JMC(Jumping by MaxCover)演算法，可以解決與CIRS 演算法相同的問題，漸增地產生MaxCover，並且提供一個刪除方法來降低執行花費。事實上，我們是從以相異時間值(different duration vale)X 所找出來的MaxCover MX，以及以相異時間值X 所截斷(cut)的時間序列，利用它們之間的關係，來減少以其他相異時間值Y 來執行演算法的花費，其Y < X。善用這項特性來降低執行時間，我們提出一個cut-sequence 的結構，並且漸增地更新它來計算出最後的全域MaxCover。透過上述方法，我們可以跳過許多步驟並且找出與CIRS 演算法相同的結果。最後，經由我們模擬結果，我們顯示出JMC 演算法在執行時間上比CIRS 演算法更快。而且當最大的相異時間值是均勻地分布在時間序列上，其執行時間會大幅縮減，這也是我們所提出JMC 演算法的最佳情形。

In recent years, the music has become more popular due to the evolution of the technology. Various kinds of music around us become more complexity and huge. This explosive growth in the music has generated the urgent need for new techniques and tools that can intelligently and automatically transform the music into useful information, and classify the music into correct music groups precisely. The rhythm query is the fundamental technique in music genre classification and content-based retrieval, which are crucial to multimedia applications. Recently, Christodoulakis et al. has proposed the CIRS algorithm that can be used to classify music duration sequences according to rhythms. In the CIRS algorithm, a rhythm is represented by a sequence of “Quick” (Q) and “Slow” (S) symbols, which corresponds to the (relative) duration of notes, such that S = 2Q. In order to classify music by rhythms, the CIRS algorithm locates the MaxCover which is the maximum-length substring of the music duration sequence, which can be covered (overlapping or consecutively) by the rhythm query continuously. During the matching step, one S symbol in the rhythm query can be regarded as two consecutive Q symbols in the duration sequence, but the two consecutive Q symbols in the rhythm query can not be combined as one S symbol in the duration sequence. This definition causes the difficulty for designing the algorithm. The CIRS algorithm contains four steps and repeat Steps 2, 3, and 4 to get local MaxCover for each different duration value of the music duration sequence. Finally, the global MaxCover is computed. We observe that it will generate unnecessary results repeatedly among Steps 2, 3, and 4. Therefore, in this thesis, to avoid repeatedly processing Steps 2, 3, and 4 for each different duration value, we propose the JMC (Jumping-by-MaxCover) algorithm which provides a pruning strategy to find the MaxCover incrementally, resulting in the reducing of the processing cost. In fact, we can make use of the relationship between the MaxCover MX founded by a different duration value X, and use the duration sequences cut by such a different duration value X to reduce the unnecessary process for the other different duration value Y , where Y < X. To make use of this property to reduce the processing time, we propose a cut-sequence structure and update it incrementally to compute the final global MaxCover. In this way, we can skip many steps and find the same answer of the CIRS algorithm. From our simulation results, we show that the running time of the JMC algorithm could be shorter than that of the CIRS algorithm. When the largest different duration value is uniformly distributed in the duration sequence, the running time can be reduced hugely, which is the best case of our proposed JMC algorithm.

TABLE OF CONTENTS
Page
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Music Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Temporal Features . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Music Information Retrieval . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Pitch Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Rhythm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Query by Humming . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 The CIRS Algorithm . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. A Survey of Music Information Retrieval . . . . . . . . . . . . . . . 19
2.1 The CIRS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 L-tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 The Dynamic Time Warping . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 The Hidden Markov Model . . . . . . . . . . . . . . . . . . . . . . . . 28
3. The Jumping-By-MaxCover Algorithm . . . . . . . . . . . . . . . . 32
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 Duration Sequences . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.2 The Rhythm Representation . . . . . . . . . . . . . . . . . . . 33
3.1.3 q-Match . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.4 q-Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.5 The Maximal Coverability Problem . . . . . . . . . . . . . . . 37
3.2 The Proposed JMC Algorithm . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Step 1: Finding All Occurrence of S . . . . . . . . . . . . . . . . . . . 39
3.4 Step 2: Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Step 3: Finding Matchings . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Step 4: Finding MaxCover . . . . . . . . . . . . . . . . . . . . . . . . 49
3.7 Step 5: Updating Cut-Sequence . . . . . . . . . . . . . . . . . . . . . 50
3.8 A Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Generation of Synthetic Data . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Simulation Results of Synthetic Data . . . . . . . . . . . . . . . . . . 60
4.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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