Sparse component decomposition (SCA) and its applications to EEG and fMRI

Sparse component decompositSparse component decomposition (SCA) and its applications tion (SCA) and its applications t

o EEG and fMRI o EEG and fMRI

XuPeng, YaoDeZhong, ChenHuaFuSchool of Life Science and Technology, University of Electronic Science and

Technology of China

ContentsContents

What is SCA?

Approaches for SCA

Applications

What is SCA

信号稀疏性的度量（ Sparsity measurement)

信号的模

信号的模

信号分量或变换系数的下降率

0l

pl

What is SCAWhat is SCA

When a signal satisfies the above three conditions, this signal is said to be sparse or have sparse expression in the corresponding transformation domain.


Fundamental hypothesis for SCA

there must be some transformation domai

n on which the signal expression is sparse(not strictly proved)

What is SCA

Fundamental hypothesis for SCA

When the transformation dictionary is big

enough compared with the signal size, there must be sparse expression for

signal in this transformation domain ；


Overcomplete dictionary 过完备库中的分量不一定满足正交性，

称为原子 (atom) ；原子个数远大于信号的维数；在过完备库中，信号的表达是不唯一的

(Underdetermined system)


How to select and construct overcomplete dictionary

Different overcomplete dictionary for different signal and purpose


Overcomplete dictionary kinds

Pure overcomplete

Mixed overcomplete

Special overcomplete …….

Approaches for SCAApproaches for SCA

概率框架模型 (Probability Framework ， PM. Zibulevsky , et al,2001)

最佳正交基法 (Best Orthogonal Basis,BOB.

Coifman, et al ,1992) Focuss 算法 (Gorodnitsky ,et al,1997) 拉格朗日乘子法匹配追踪法 (Matching Pursuit,MP)

拉格朗日乘子法拉格朗日乘子法

目标函数

CS

CC

pmin

拉格朗日乘子法拉格朗日乘子法采用拉格朗日乘子形式，目标函数变为，

pc

ccs ||||min2

2

拉格朗日乘子法拉格朗日乘子法采用牛顿法求解拉格朗日约束问题步骤： 1. 先设初始点 , 常用欠定系统的最小模解 2. 求解出本次迭代的牛顿下降方向和在该方向

下的最优步长，更新迭代向量

3. 若不满足终止条件，转到 2 继续迭代

)0(x

)( )(ixn

)( )()()1( iii xnxx

MPMP

Mallat 和 Zhang 采用了匹配追踪 (Matching pursuit,MP)在小波库中来对信号逐步的分解，其思想是直接从信号分解的库出发，在使均方误差下降最快的方向上（选取相关系数最大的原子），通过逐步的迭代来分解信号，其每迭代一次，就抽取出过完备库中的一个适当的“原子”，通过多次迭代，在满足收敛条件时，就可以得到信号的稀疏表达。

MP MP 的迭代步骤的迭代步骤

SCA ApplicationsSCA Applications

Signal higher ratio compression

EP(ERP) estimation

FMRI artifacts removing

EEG Inverse problem

SCA applicationSCA application

Signal higher ratio compression

主要利用信号在足够大的过完备库中一

定有稀疏表达形式，通过构造小波过完备库，在过完备库中求解信号的稀疏表达形式，使信号能量更加的集中，完成对信号的压缩和恢复。

SCA application--SCA application-- Signal higher ratio compression Signal higher ratio compression

小波过完备库的构造

)()(, s

bttW bs

SCA application--SCA application-- Signal higher ratio compression Signal higher ratio compression

Three atoms in the symmlet wavelet overcomplete dictionary

脉冲方波信号的分解系数脉冲方波信号的分解系数

Reconstruction resultsReconstruction results

Reconstruction errors with different atoms and coefficientsReconstruction errors with different atoms and coefficients

EEG reconstructed by 35 atoms with 7.00% errorEEG reconstructed by 35 atoms with 7.00% error

SCA applicationSCA application --EP(ERP) estimation --EP(ERP) estimation

设观测到的脑电信号有如下的形式，

NES


相干平均方法

假设对同一种实验模式下，被试对刺激的响应是一致的；

自发脑电和背景噪声是随机的信号；

多次刺激信号的叠加平均提取 EP

SCA application–SCA application– EP(ERP) estimation EP(ERP) estimation

ERP 和自发脑电等噪声背景的特点

ERP 在时域具有：瞬态性和局部性，较强的非平稳性

背景噪声：在整个测量时间段表现的较为平稳，无明显的衰减和局部性

ERPERP 和和 EEGEEG

SCA application–SCA application– EP(ERP) estimation EP(ERP) estimation

针对 ERP 和自发脑电的上述特性，分别构造具有瞬时特性和平稳特性的变换，使信号 E 和噪声 N 在相应的变换库中体现出来 :

22112211 ,, CNCECCS

SCA application –SCA application – EP(ERP) estimation EP(ERP) estimation

混合过完备库的选择小波过完备库：很好地体现 ERP 的时域瞬态和局部特性

四类 DCT 变换库：背景噪声视为不同相位的余弦信号的组合

Atoms in mixed overcomplete dictionaryAtoms in mixed overcomplete dictionary


Separation of steady oscillatory and transient signals


Separation of EP and steady oscillatory signal


Separation of EEG and transient signal


Separation of EP and noisy background


Separation of EEG and EP


EP estimation by averaging


Peak latency contour on the recordings


EP estimations by time-delay-correction averaging

SCA applicationSCA application -- FMRI artifacts removing -- FMRI artifacts removing

EEG +fmri 技术

EEG ：极高的时间分辨率 ( 微秒级），较差的空间分辨率

FMRI ：较好的空间分辨率，时间分辨率较低


EEG +fmri 技术

分时测量的数据的融合

同时测量数据的融合


EEG受 FMRI 成象梯度磁场的影响，混入和磁场有一致周期性变化的强 MR伪迹信号


EEG 和 MR 伪迹的特性差别 EEG 相对属于非平稳的弱信号；

MR 伪迹是有明显周期性的强信号，幅度一般为 EEG 的几倍；


MR 伪迹去除效果


EEG 信号在去伪迹前和后的频谱估计结果。粗线为去伪迹后的频谱；细线为含有 MR 伪迹的频谱

SCA applicationSCA application -- --EEG Inverse problemEEG Inverse problem

EEG inverse problem

Y=DX

Y ----Recording Matrix of Mx1,

D----Delivery Matrix of MxN,

X----Source Matrix of Nx1

N>>M Underdeterminded system


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大脑网格图

x

y

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头皮网格图

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Sparseness in brain response brain neurons show sparseness in the discharge acti

vities

For certain task, only several corresponding brain f

unction area responds.

Thanks

Sparse component decomposition (SCA) and its applications to EEG and fMRI

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