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

Sparse component decomposition (SCA) and its applications to EEG and fMRI. XuPeng, YaoDeZhong, ChenHuaFu School of Life Science and Technology, University of Electronic Science and Technology of China. Contents. What is SCA? Approaches for SCA Applications. What is SCA.

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

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  1. Sparse componentdecomposition (SCA) and its applications to EEG and fMRI XuPeng, YaoDeZhong, ChenHuaFu School of Life Science and Technology, University of Electronic Science and Technology of China

  2. Contents • What is SCA? • Approaches for SCA • Applications

  3. 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

  4. What 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.

  5. What is SCA • Fundamental hypothesis for SCA • there must be some transformation domain on which the signal expression is sparse(not strictly proved)

  6. 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；

  7. What is SCA • 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)

  8. What is SCA • How to select and construct overcomplete dictionary • Different overcomplete dictionary for different signal and purpose

  9. What is SCA • Overcomplete dictionary kinds • Pure overcomplete • Mixed overcomplete • Special overcomplete • …….

  10. Approaches 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)

  11. 拉格朗日乘子法 • 目标函数

  12. 拉格朗日乘子法 • 采用拉格朗日乘子形式，目标函数变为，

  13. 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

  14. MP • 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。

  15. MP 的迭代步骤

  16. SCA Applications • Signal higher ratio compression • EP(ERP) estimation • FMRI artifacts removing • EEG Inverse problem

  17. SCA 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。

  18. SCA application-- Signal higher ratio compression • 小波过完备库的构造

  19. SCA application-- Signal higher ratio compression • Three atoms in the symmlet wavelet overcomplete dictionary

  20. 脉冲方波信号的分解系数

  21. Reconstruction results

  22. Reconstruction results

  23. Reconstruction results

  24. Reconstruction errors with different atoms and coefficients

  25. EEG reconstructed by 35 atoms with 7.00% error

  26. SCA application --EP(ERP) estimation 设观测到的脑电信号有如下的形式，

  27. SCA application --EP(ERP) estimation • 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

  28. SCA application– 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

  29. ERP和EEG

  30. SCA application– 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:

  31. SCA application – 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

  32. Atoms in mixed overcomplete dictionary

  33. SCA application --EP(ERP) estimation Separation of steady oscillatory and transient signals

  34. SCA application --EP(ERP) estimation Separation of EP and steady oscillatory signal

  35. SCA application --EP(ERP) estimation Separation of EEG and transient signal

  36. SCA application --EP(ERP) estimation Separation of EP and noisy background

  37. SCA application --EP(ERP) estimation Separation of EEG and EP

  38. SCA application --EP(ERP) estimation EP estimation by averaging

  39. SCA application --EP(ERP) estimation Peak latency contour on the recordings

  40. SCA application --EP(ERP) estimation EP estimations by time-delay-correction averaging

  41. SCA application --EP(ERP) estimation

  42. SCA application -- 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

  43. SCA application -- FMRI artifacts removing • EEG +fmri技术 • 分时测量的数据的融合 • 同时测量数据的融合

  44. SCA application -- FMRI artifacts removing • 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

  45. SCA application -- FMRI artifacts removing • 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；

  46. SCA application -- FMRI artifacts removing MR伪迹去除效果

  47. SCA application -- FMRI artifacts removing 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

  48. SCA application --EEG 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

  49. SCA application --EEG Inverse problem

  50. SCA application --EEG Inverse problem • Sparseness in brain response • brain neurons show sparseness in the discharge activities • For certain task, only several corresponding brain function area responds.

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