以希爾伯-黃轉換為基礎的去噪音方法以及從腦電儀資料辨識電流源的應用

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蘇致瑋(Chih-wei Su) 查詢紙本館藏

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物理學系

論文名稱

以希爾伯-黃轉換為基礎的去噪音方法以及從腦電儀資料辨識電流源的應用
(A Denoising Method based on the Hilbert-Huang Transform and an application to Current Source Identification from EEG Data)

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摘要(中)

我們提出一個方法能夠減少或是去除摻雜在資料中的噪音，例如一個
時間序列。這個方法是根據Hilbert-Huang Transform (HHT)的一部
分，實驗模分解(Empirical Mode Decomposition, EMD)。分解資料後，模 (modes) 可被配分為噪音及訊號兩個集，並計算其標準訊噪比
(Signal-to-Noise Ratio, SNR)。透過數值實驗，我們改變不同的噪音強度干擾測試訊號，研究計算的訊噪比和真正資料的訊噪比的關係，發展一個尋找最佳配分方式的演算法。實驗結果顯示，在非病理學上的條件下，當資料的訊噪比大於-10 分貝(dB)時，這個演算方法能夠可靠的估計資料的訊噪比，並且對於具有相當複雜性的訊號工作良好。
我們將這個去除噪音的演算法應用到所謂的電流源密度(Current
Source-Density, CSD)方法上。電流源密度方法常用於從腦電儀
(Electroencephalogram, EEG) 資料中建構電流源結構上。這個方法需要利用二次微分近似式計算以等空間間距記錄的離散資料。對於每個空間點，都有一個時間序列資料。雖然能夠依靠泰勒展開式(Taylor
Expansion)，利用正確的數學過程來提高二次微分精確度，但面對遭
受噪音干擾的資料卻會失敗。因此有個廣泛應用的去噪音方法，即在
計算空間二次微分之前先在空間上平滑資料。然而，有個不幸的缺點
是，在空間上平均會造成過度平滑資料的訊息。此外，這個方法也欠
缺計算二次微分的精確度。我們證明，利用我們的方法在時間方向上
去除腦電儀資料的噪音，我們能夠：(1) 估計出這組腦電儀資料的
訊噪比約為35分貝；(2) 去除腦電儀資料的噪音，再計算空間二次
微分，這結果將會在泰勒展開式中收斂；(3) 證明在空間上平滑資
料不但是不必要，還會定性上的改變結果。

摘要(英)

We present a method for reducing or removing noise in a data set, such as a times series. The method is based on the empirical mode decomposition (EMD) of a data set, which is itself a by-product of the Hilbert-Huang transform. After decomposition the modes are partitioned into a signal set
and a noise set and the standard signal-to-noise ratio (SNR) is computed. By experimenting on sets of data composed of signal and noises of a variety of intensities and studying the behavior of computed SNR in relation to the actual SNR, an algorithm for finding the best partition is derived. It
is shown that under conditions that are not pathological, the algorithm is capable of giving a reliable estimation of the SNR and works well for signals of considerable complexity, provided the true SNR is greater than -10.
We apply the denoising algorithm to the so-called current source-density (CSD) method used in the construction of current source from electroencephalogram (EEG) data. The method requires a discrete form of the second derivative to be taken on data simultaneously collected at a small set
of equally spaced points. At each point the data is a time series. Although there is a mathematically correct procedure for taking increasingly accurate discrete second derivative based on Taylor’s expansion, the procedure breaks down when the data is contaminated with noise. This has led to the wide-spread practice of spatially smoothing the data — for noise removal — before computing the derivative. However, this method has the unfortunate drawback of the act of spatial smoothing itself compromising the very meaning
of the second derivative. The method also lacks a provision for accuracy estimation. We show that by applying our denoising method to the times series of the EEG data, we are able to: (1) Estimate the SNR in the data to be of the order of 35; (2) Denoise the EEG and obtain second spatial
derivatives of the data that converge in the Taylor expansion; (3) Show that spatial smoothing of the data is not only unnecessary, but also that when it is done, qualitatively alters the results.

關鍵字(中)

★ 希爾伯-黃轉換
★ 實驗模分解
★ 電流源
★ 腦電儀資料
★ 去噪音

關鍵字(英)

★ Hilbert-Huang Transform
★ EEG
★ Empirical Mode Decomposition
★ Denoising
★ Current Source

論文目次

1 Introduction 1
2 Current Source-Density Method 5
2.1 Physical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Practical Formulas . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Some Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 A Method for Approximating Second-Order Derivatives Based
on Taylor Expansion 8
3.1 Method based on Taylor Expansion . . . . . . . . . . . . . . . 8
3.2 Numerical Result and Analysis . . . . . . . . . . . . . . . . . 10
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Denoising Method 14
4.1 Background of The Empirical Mode Decomposition . . . . . . 14
4.2 Denoising Methodology . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . 16
4.3.1 Characteristics of Gaussian white noise . . . . . . . . . 17
4.3.2 The distibution of Rk for a simple test signal . . . . . . 17
4.3.3 The distibution of Rk for a complicated test signal . . . 21
4.4 Short Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Method for Current Source Identification 27
5.1 The Improved CSD Method . . . . . . . . . . . . . . . . . . . 27
5.2 Study of A Test Signal . . . . . . . . . . . . . . . . . . . . . . 28
5.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2.2 Result and Analysis . . . . . . . . . . . . . . . . . . . . 29
5.3 Short Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 Example: Application to EEG Data of the Rat’s Primary
Somatosensory Cortex 35
6.1 Background of The Experiment . . . . . . . . . . . . . . . . . 35
6.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3 Result And Analysis . . . . . . . . . . . . . . . . . . . . . . . 36
6.4 Discussion and Comparison . . . . . . . . . . . . . . . . . . . 39
7 Conclusions 42
Bibliography 44
A Derivation of CSD Equation 47

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指導教授

李弘謙(Hoong-chien Lee)

審核日期

2007-7-21

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