14-4 DTW of Type-1 and 2

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(請注意：中文版本並未隨英文版本同步更新！)

Slides: DTW, DTW for melody recognition, DTW for speech recognition

DTW 也是常用於旋律辨識的方法，其最主要優點就是辨識率高，而最大缺點則是在於計算時間較久。因此有很多研究，都是在強調如何不影響辨識率的情況下，來加速 DTW 的運算。以下是幾個 DTW 的使用範例，來說明 DTW 的特性。

有關於 DTW 的技術細節，可見有關於動態規劃的章節。以下就是示範幾個 DTW 的範例。首先，我們可以畫出 type-1 DTW 的路徑，如下：

Example 1: mrDtw1Plot01.m

% inputPitch: input pitch vector inputPitch=[48.044247 48.917323 49.836778 50.154445 50.478049 50.807818 51.143991 51.486821 51.486821 51.486821 51.143991 50.154445 50.154445 50.154445 49.218415 51.143991 51.143991 50.807818 49.524836 49.524836 49.524836 49.524836 51.143991 51.143991 51.143991 51.486821 51.836577 50.807818 51.143991 52.558029 51.486821 51.486821 51.486821 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 49.218415 50.807818 50.807818 50.154445 50.478049 48.044247 49.524836 52.193545 51.486821 51.486821 51.143991 50.807818 51.486821 51.486821 51.486821 51.486821 51.486821 55.788268 55.349958 54.922471 54.922471 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 53.699915 58.163541 59.213095 59.762739 59.762739 59.762739 59.762739 58.163541 57.661699 58.163541 58.680365 58.680365 58.680365 58.163541 55.788268 54.505286 55.349958 55.788268 55.788268 55.788268 54.922471 54.505286 56.237965 55.349958 55.349958 55.349958 55.349958 54.505286 54.505286 55.349958 48.917323 50.478049 50.807818 51.143991 51.143991 51.143991 50.807818 50.807818 50.478049 50.807818 51.486821 51.486821 51.486821 51.486821 51.486821 51.486821 52.558029 52.558029 52.558029 52.558029 52.193545 51.836577 52.193545 53.310858 53.310858 53.310858 52.930351 52.930351 53.310858 52.930351 52.558029 52.193545 52.930351 53.310858 52.930351 51.836577 52.558029 53.699915 52.930351 52.930351 52.558029 52.930351 52.930351 52.558029 52.558029 52.558029 53.310858 53.310858 53.310858 53.310858 52.930351 52.930351 52.930351 52.558029 52.930351 52.930351 52.930351 52.930351 52.930351 52.930351 52.930351 53.310858 53.310858 53.310858 52.193545 52.193545 52.193545 54.097918 52.930351 52.930351 52.930351 52.930351 52.930351 51.143991 51.143991 51.143991 48.917323 49.524836 49.524836 49.836778 49.524836 48.917323 49.524836 49.218415 48.330408 48.330408 48.330408 48.330408 48.330408 49.524836 49.836778 53.310858 53.310858 53.310858 52.930351 52.930351 52.930351 53.310858 52.930351 52.930351 52.558029 52.558029 51.143991 52.930351 49.218415 49.836778 50.154445 49.836778 49.524836 48.621378 48.621378 48.621378 49.836778 49.836778 49.836778 49.836778 46.680365 46.680365 46.680365 46.163541 45.661699 45.661699 45.910801 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 50.807818 51.486821 51.486821 51.143991]; % dbPitch: database pitch vector dbPitch =[60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 64 64 64 64 64 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 67 67 67 67 67 64 64 64 64 64 64 64 64 64 64 64 64 64 60 60 60 60 60 60 60 60 60 60 60 60 60 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 59 59 59 59 59 59 59 59 59 59 59 59 59 62 62 62 62 62 62 62 62 62 62 62 62 59 59 59 59 59 59 59 59 59 59 59 59 59 55 55 55 55 55 55 55 55 55 55 55 55 55 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 64 64 64 64 64 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 67 67 67 67 67 64 64 64 64 64 64 64 64 64 64 64 64 64 60 60 60 60 60 60 60 60 60 60 60 60 60 67 67 67 67 67 67 67 67 67 67 67 67 65 65 65 65 65 65 65 65 65 65 65 65 65 64 64 64 64 64 64 64 64 64 64 64 64 62 62 62 62 62 62 62 62 62 62 62 62 62 60 60 60 60 60 60 60 60 60 60 60 60 60]; n=length(inputPitch); meanPitch=mean(dbPitch(1:n)); inputPitch=inputPitch-mean(inputPitch)+meanPitch; % Shift input pitch to have the same mean anchorBeginning=1; % Anchor beginning anchorEnd=0; % Anchor end m=11; % Number of pitch shifts for key transposition pitchStep=linspace(-2, 2, m); dtwDist=zeros(1, m); % DTW distances for different pitch shifts for i=1:length(pitchStep) newInputPitch=inputPitch+pitchStep(i); dtwDist(i) = dtw1(newInputPitch, dbPitch, anchorBeginning, anchorEnd); end [minValue, index]=min(dtwDist); optInputPitch=inputPitch+pitchStep(index); [minDist, dtwPath, dtwTable]=dtw1(optInputPitch, dbPitch, anchorBeginning, anchorEnd); dtwPathPlot(inputPitch+pitchStep(index), dbPitch, dtwPath);

In the above example of type-1 DTW, we need to be aware of two issues:

• For QBSH application, we have two types of comparisons:
  1. Comparison from beginning: The acoustic input corresponds to the beginning of a song in the database. This is exactly the case in the above example, so we set anchorBeginning=1 and anchorEnd=0.
  2. Comparison from anywhere: The acoustic input corresponds to the middle of a song. Under such a condition, we need to set anchorBeginning=0 and anchorEnd=0.
  In fact, if we can define the "beginning position" of a query input, then all comparisons can be viewed as "comparison from beginning" with different beginning positions. Several commonly used "beginning positions" are listed next, according to their complexities:
  1. The beginning of a song: This is not a realistic assumption since the most well-known part of a song does not necessarily starts from the beginning.
  2. The refrain (chorus or burden, 副歌) of a song: However, refrains of a song cannot be defined objectively.
  3. The beginning of each sentence: This is most likely to be the case in karaoke systems since such information is already embedded in the songs in order to display the lyrics in a sentence-by-sentence manner.
  4. The beginning of each note: This is the final catchall which requires a lot of computation.
• To deal with key transposition, we use a linear search between the interval of [-2, 2] with a resolution of 11. It is possible to use some sort of heuristic search, such as the one mentioned in section 2 of this chapter.

除了畫出最佳路徑外，我們也可以畫出兩個向量之間，點對點的對應狀態：

Example 2: mrDtw1Plot02.m

% inputPitch: input pitch vector inputPitch=[48.044247 48.917323 49.836778 50.154445 50.478049 50.807818 51.143991 51.486821 51.486821 51.486821 51.143991 50.154445 50.154445 50.154445 49.218415 51.143991 51.143991 50.807818 49.524836 49.524836 49.524836 49.524836 51.143991 51.143991 51.143991 51.486821 51.836577 50.807818 51.143991 52.558029 51.486821 51.486821 51.486821 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 49.218415 50.807818 50.807818 50.154445 50.478049 48.044247 49.524836 52.193545 51.486821 51.486821 51.143991 50.807818 51.486821 51.486821 51.486821 51.486821 51.486821 55.788268 55.349958 54.922471 54.922471 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 53.699915 58.163541 59.213095 59.762739 59.762739 59.762739 59.762739 58.163541 57.661699 58.163541 58.680365 58.680365 58.680365 58.163541 55.788268 54.505286 55.349958 55.788268 55.788268 55.788268 54.922471 54.505286 56.237965 55.349958 55.349958 55.349958 55.349958 54.505286 54.505286 55.349958 48.917323 50.478049 50.807818 51.143991 51.143991 51.143991 50.807818 50.807818 50.478049 50.807818 51.486821 51.486821 51.486821 51.486821 51.486821 51.486821 52.558029 52.558029 52.558029 52.558029 52.193545 51.836577 52.193545 53.310858 53.310858 53.310858 52.930351 52.930351 53.310858 52.930351 52.558029 52.193545 52.930351 53.310858 52.930351 51.836577 52.558029 53.699915 52.930351 52.930351 52.558029 52.930351 52.930351 52.558029 52.558029 52.558029 53.310858 53.310858 53.310858 53.310858 52.930351 52.930351 52.930351 52.558029 52.930351 52.930351 52.930351 52.930351 52.930351 52.930351 52.930351 53.310858 53.310858 53.310858 52.193545 52.193545 52.193545 54.097918 52.930351 52.930351 52.930351 52.930351 52.930351 51.143991 51.143991 51.143991 48.917323 49.524836 49.524836 49.836778 49.524836 48.917323 49.524836 49.218415 48.330408 48.330408 48.330408 48.330408 48.330408 49.524836 49.836778 53.310858 53.310858 53.310858 52.930351 52.930351 52.930351 53.310858 52.930351 52.930351 52.558029 52.558029 51.143991 52.930351 49.218415 49.836778 50.154445 49.836778 49.524836 48.621378 48.621378 48.621378 49.836778 49.836778 49.836778 49.836778 46.680365 46.680365 46.680365 46.163541 45.661699 45.661699 45.910801 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 50.807818 51.486821 51.486821 51.143991]; % dbPitch: database pitch vector dbPitch =[60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 64 64 64 64 64 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 67 67 67 67 67 64 64 64 64 64 64 64 64 64 64 64 64 64 60 60 60 60 60 60 60 60 60 60 60 60 60 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 59 59 59 59 59 59 59 59 59 59 59 59 59 62 62 62 62 62 62 62 62 62 62 62 62 59 59 59 59 59 59 59 59 59 59 59 59 59 55 55 55 55 55 55 55 55 55 55 55 55 55 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 64 64 64 64 64 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 67 67 67 67 67 64 64 64 64 64 64 64 64 64 64 64 64 64 60 60 60 60 60 60 60 60 60 60 60 60 60 67 67 67 67 67 67 67 67 67 67 67 67 65 65 65 65 65 65 65 65 65 65 65 65 65 64 64 64 64 64 64 64 64 64 64 64 64 62 62 62 62 62 62 62 62 62 62 62 62 62 60 60 60 60 60 60 60 60 60 60 60 60 60]; n=length(inputPitch); meanPitch=mean(dbPitch(1:n)); inputPitch=inputPitch-mean(inputPitch)+meanPitch; % Shift input pitch to have the same mean anchorBeginning=1; % Anchor beginning anchorEnd=0; % Anchor end m=11; % Number of pitch shifts for key transposition pitchStep=linspace(-2, 2, m); dtwDist=zeros(1, m); % DTW distances for different pitch shifts for i=1:length(pitchStep) newInputPitch=inputPitch+pitchStep(i); dtwDist(i) = dtw1(newInputPitch, dbPitch, anchorBeginning, anchorEnd); end [minValue, index]=min(dtwDist); optInputPitch=inputPitch+pitchStep(index); [minDist, dtwPath, dtwTable]=dtw1(optInputPitch, dbPitch, anchorBeginning, anchorEnd); dtwBridgePlot(inputPitch+pitchStep(index), dbPitch, dtwPath, '2d');

我們也可以使用三維繪圖，畫出類似的對應狀態：

Example 3: mrDtw1Plot03.m

% inputPitch: input pitch vector inputPitch=[48.044247 48.917323 49.836778 50.154445 50.478049 50.807818 51.143991 51.486821 51.486821 51.486821 51.143991 50.154445 50.154445 50.154445 49.218415 51.143991 51.143991 50.807818 49.524836 49.524836 49.524836 49.524836 51.143991 51.143991 51.143991 51.486821 51.836577 50.807818 51.143991 52.558029 51.486821 51.486821 51.486821 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 51.143991 49.218415 50.807818 50.807818 50.154445 50.478049 48.044247 49.524836 52.193545 51.486821 51.486821 51.143991 50.807818 51.486821 51.486821 51.486821 51.486821 51.486821 55.788268 55.349958 54.922471 54.922471 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 55.349958 53.699915 58.163541 59.213095 59.762739 59.762739 59.762739 59.762739 58.163541 57.661699 58.163541 58.680365 58.680365 58.680365 58.163541 55.788268 54.505286 55.349958 55.788268 55.788268 55.788268 54.922471 54.505286 56.237965 55.349958 55.349958 55.349958 55.349958 54.505286 54.505286 55.349958 48.917323 50.478049 50.807818 51.143991 51.143991 51.143991 50.807818 50.807818 50.478049 50.807818 51.486821 51.486821 51.486821 51.486821 51.486821 51.486821 52.558029 52.558029 52.558029 52.558029 52.193545 51.836577 52.193545 53.310858 53.310858 53.310858 52.930351 52.930351 53.310858 52.930351 52.558029 52.193545 52.930351 53.310858 52.930351 51.836577 52.558029 53.699915 52.930351 52.930351 52.558029 52.930351 52.930351 52.558029 52.558029 52.558029 53.310858 53.310858 53.310858 53.310858 52.930351 52.930351 52.930351 52.558029 52.930351 52.930351 52.930351 52.930351 52.930351 52.930351 52.930351 53.310858 53.310858 53.310858 52.193545 52.193545 52.193545 54.097918 52.930351 52.930351 52.930351 52.930351 52.930351 51.143991 51.143991 51.143991 48.917323 49.524836 49.524836 49.836778 49.524836 48.917323 49.524836 49.218415 48.330408 48.330408 48.330408 48.330408 48.330408 49.524836 49.836778 53.310858 53.310858 53.310858 52.930351 52.930351 52.930351 53.310858 52.930351 52.930351 52.558029 52.558029 51.143991 52.930351 49.218415 49.836778 50.154445 49.836778 49.524836 48.621378 48.621378 48.621378 49.836778 49.836778 49.836778 49.836778 46.680365 46.680365 46.680365 46.163541 45.661699 45.661699 45.910801 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 46.163541 50.807818 51.486821 51.486821 51.143991]; % dbPitch: database pitch vector dbPitch =[60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 64 64 64 64 64 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 67 67 67 67 67 64 64 64 64 64 64 64 64 64 64 64 64 64 60 60 60 60 60 60 60 60 60 60 60 60 60 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 59 59 59 59 59 59 59 59 59 59 59 59 59 62 62 62 62 62 62 62 62 62 62 62 62 59 59 59 59 59 59 59 59 59 59 59 59 59 55 55 55 55 55 55 55 55 55 55 55 55 55 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 64 64 64 64 64 64 64 64 64 64 64 64 64 67 67 67 67 67 67 67 67 67 67 67 67 64 64 64 64 64 64 64 64 64 64 64 64 64 60 60 60 60 60 60 60 60 60 60 60 60 60 67 67 67 67 67 67 67 67 67 67 67 67 65 65 65 65 65 65 65 65 65 65 65 65 65 64 64 64 64 64 64 64 64 64 64 64 64 62 62 62 62 62 62 62 62 62 62 62 62 62 60 60 60 60 60 60 60 60 60 60 60 60 60]; n=length(inputPitch); meanPitch=mean(dbPitch(1:n)); inputPitch=inputPitch-mean(inputPitch)+meanPitch; % Shift input pitch to have the same mean anchorBeginning=1; % Anchor beginning anchorEnd=0; % Anchor end m=11; % Number of pitch shifts for key transposition pitchStep=linspace(-2, 2, m); dtwDist=zeros(1, m); % DTW distances for different pitch shifts for i=1:length(pitchStep) newInputPitch=inputPitch+pitchStep(i); dtwDist(i) = dtw1(newInputPitch, dbPitch, anchorBeginning, anchorEnd); end [minValue, index]=min(dtwDist); optInputPitch=inputPitch+pitchStep(index); [minDist, dtwPath, dtwTable]=dtw1(optInputPitch, dbPitch, anchorBeginning, anchorEnd); dtwBridgePlot(inputPitch+pitchStep(index), dbPitch, dtwPath, '3d');

Hint

在上述範例中，你可以在圖上使用滑鼠點選並滑動，以改變視角，得到更好的目視效果。

在上述範例中，我們均可將 "dtw1" 改成 "dtw2"，就可以得到 type-2 DTW 的對應結果。在實作上，type-1 DTW 和 type-2 DTW 的效能高低通常會隨著資料庫的不同而改變，因此我們並無學理上的根據來決定哪一種方法的效果較佳。

DTW 的變形很多，直要掌握其 DP 的特性，我們就可以根據實際應用的需求，發展出各種 DTW 的變形。

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Audio Signal Processing and Recognition (音訊處理與辨識)