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Meg preprocessing

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Slides from an invited talk I gave at the MEG Basics series in the winter of 2012. Covers the theory behind signal processing techniques used in magnetoencephalography (MEG), including:
- Signal Space Projection (SSP)
- Signal Space Separation (SSS)
- Temporally-extended Signal Space Separation (tSSS)
- Principle Component Analysis (PCA)
- Independent Component Analysis (ICA)

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Meg preprocessing

  1. 1. MagnetoencephalographyPreprocessing and Noise Reduction Techniques Eliezer Kanal 2/20/2012 MEG Basics Course 1
  2. 2. About Me• 2005 -! 2009!!! ! ! ! ! ! University of Pittsburgh PhD, Bioengineering• 2009 -! 2011!!! ! ! ! ! ! Carnegie Mellon University Postdoctoral fellow, CNBC• 2011 -! current! !! ! ! ! ! PNC Financial Services Quantitative Analyst, Risk Analytics 2
  3. 3. Dealing with Noisy Data• Overview of MEG Noise• Noise Reduction - Averaging, thresholding, frequency filters - SSP - SSS/tSSS• Source Extraction - PCA - ICA 3
  4. 4. MEG Noise 4
  5. 5. Breathing 5
  6. 6. Breathing6
  7. 7. Frequency7
  8. 8. Frequency8
  9. 9. Time-Frequency9
  10. 10. Biological Noise Vigário, Jousmäki, Hämäläinen, Hari, & Oja (1997) 10
  11. 11. Line Noise 50 Hz Line Noise (60 Hz in USA) Subject Empty Room 11
  12. 12. Bad Channels Find the bad one: 12
  13. 13. Bad Channels Find the bad one: 12
  14. 14. Noise from nearby construction 13
  15. 15. Noise Reduction Techniques• Averaging, thresholding, frequency filters• SSP• SSS/tSSS 14
  16. 16. Averaging• Removes non-timelocked noise• Requires: - Time-locked block paradigm design - Temporal or low-frequency analyses 15
  17. 17. Thresholding• Discarding trials/channels with maximum signal intensity greater than some user-defined value• Removes most “data blips”• Rudimentary, better technique is to simply examine each trial/channel 16
  18. 18. Frequency Filter Filter Removes… High-pass Lower frequencies Low-pass Higher frequencies Band-pass Outside specified band Notch All except specified• Very good first step, remove data you won’t analyze (don’t waste time cleaning what you won’t examine)• Use more advanced techniques for specific noise signals 17
  19. 19. 18
  20. 20. 19
  21. 21. Signal Space Projection 20
  22. 22. Signal Space Projection• Overview: SSP uses the difference between source orientations and locations to differentiate distinct sources.• Theory: Since the field pattern from a single source is 1) unique 2) time-invariant, we can differentiate sources by examining the angle between their “signal space representations”, and project noise signals out of the dataset. 21
  23. 23. 22
  24. 24. 23
  25. 25. Signal Space Projection• In general, M X m(t) = ai (t)si + n(t) i=1 24
  26. 26. Signal Space Projection • In general, M X m(t) = ai (t)si + n(t)measured i=1 signal 24
  27. 27. Signal Space Projection • In general, source i M X M = Total number of channels m(t) = ai (t)si + n(t)measured i=1 signal 24
  28. 28. Signal Space Projection source • In general, amplitude source i M X M = Total number of channels m(t) = ai (t)si + n(t)measured i=1 signal 24
  29. 29. Signal Space Projection source • In general, amplitude source i M X M = Total number of channels m(t) = ai (t)si + n(t) noisemeasured i=1 signal 24
  30. 30. Signal Space Projection source • In general, amplitude source i M X M = Total number of channels m(t) = ai (t)si + n(t) noisemeasured i=1 signal • SSP states that s can be split in two: - s‖ ! = signals from known sources - s⟂ ! = signals from unknown sources s k = Pk m s ? = P? m 24
  31. 31. Signal Space Projection source • In general, amplitude source i M X M = Total number of channels m(t) = ai (t)si + n(t) noise measured i=1 signal • SSP states that s can be split in two: - s‖ ! = signals from known sources - s⟂ ! = signals from unknown sources known s k = Pk msources MEG signal s ? = P? munknownsources Projection operators 24
  32. 32. Signal Space Projection source • In general, amplitude source i M X M = Total number of channels m(t) = ai (t)si + n(t) noise measured i=1 signal • SSP states that s can be split in two: - s‖ ! = signals from known sources - s⟂ ! = signals from unknown sources known s k = Pk msources MEG signal s ? = P? munknownsources Projection Worth mentioning that sk + s? = s operators 24
  33. 33. Signal Space ProjectionHow find P‖ and P⟂? 25
  34. 34. Signal Space Projection How find P‖ and P⟂? • Ingenious application of the magic 1 technique of Singular Value Decomposition (SVD)1 Not really magic 25
  35. 35. Signal Space Projection How find P‖ and P⟂? • Ingenious application of the magic technique of 1 Singular Value Decomposition (SVD) a matrix of all known sources • Let K = {s , s , . . . , s } 2 s . Using SVD, we find a basis 1 2 k k for s‖, and therefore P‖.21 Not really magic 25
  36. 36. Signal Space Projection How find P‖ and P⟂? • Ingenious application of the magic technique of 1 Singular Value Decomposition (SVD) a matrix of all known sources • Let K = {s , s , . . . , s } 2 s . Using SVD, we find a basis 1 2 k k for s‖, and therefore P‖.21 Not really magic2 Let K = U⇤VT. By the properties of the SVD, the first k columns of U form an orthonormal basis for the column space of K, so we can define Pk = U k U T k since s + s = P m + P m = s k ? k ? P? = I Pk 25
  37. 37. Signal Space Projection M X• Recall m(t) = i=1 ai (t)si + n(t) . To find a(t), invert s‖: m(t) = a(t)sk a(t) = sk 1 m(t) ˆ 1 a = V⇤ ˆ UT m(t)• In practice, soften consists of known noise signals ‖ specific to a particular MEG scanner. The final step is simply to project those out of m(t), leaving only unknown (and presumably neural) sources in s. 26
  38. 38. Signal Space Projection M X• Recall m(t) = i=1 ai (t)si + n(t) . To find a(t), invert s‖: m(t) = a(t)sk a(t) = sk 1 m(t) ˆ Recall that K = {s1 , s2 , . . . , sk } 2 sk a = V⇤ 1 UT m(t) ˆ = U⇤VT | {z }• In practice, soften consists of known noise signals ‖ specific to a particular MEG scanner. The final step is simply to project those out of m(t), leaving only unknown (and presumably neural) sources in s. 26
  39. 39. Signal Space Separation (SSS) 27
  40. 40. Signal Space Separation• Overview: Separate MEG signal into sources (1) outside and (2) inside the MEG helmet• Theory: Analyzing the MEG data using a basis which expresses the magnetic field as a “gradient of the harmonic scalar potential” (defined below) allows the field to be separated into internal and external components. By simply dropping the external component, we can significantly reduce the MEG signal noise. 28
  41. 41. MEG data – raw 29
  42. 42. MEG data – SSP 30
  43. 43. MEG data – SSS 31
  44. 44. Signal Space Separation• Begin with Maxwell’s laws: ⇤⇥H=J (1) ⇤ ⇥ B = µ0 J (2) ⇤·B=0 (3) 32
  45. 45. Signal Space Separation • Begin with Maxwell’s laws: ⇤⇥H=J (1)magnetic ⇤ ⇥ B = µ0 J sources (2) field ⇤·B=0 (3) 32
  46. 46. Signal Space Separation • Begin with Maxwell’s laws: ⇤⇥H=J (1)magnetic ⇤ ⇥ B = µ0 J sources (2) field i. e., nos! ce ⇤·B=0 sour (3) • Note that on surface of sensor array, J = 0. As such, ⇥ H = 0 on array surface Taulu et al, 2005 32
  47. 47. Signal Space Separation • Begin with Maxwell’s laws: ⇤⇥H=J (1)magnetic ⇤ ⇥ B = µ0 J sources (2) field i. e., nos! ce ⇤·B=0 sour (3) • Note that on surface of sensor array, J = 0. As such, ⇥ H = 0 on array surface • Defining H = ∇Ψ, we obtain the identity ∇ × ∇Ψ = 0 in (1). This term (∇Ψ) is called the “scalar potential.” • “Scalar potential” has no physical correlate. • Often written with a negative sign (–∇Ψ) for convenience. • H = –∇Ψ → B = –μ0∇Ψ… used interchangeably Taulu et al, 2005 32
  48. 48. Signal Space Separation • Begin with Maxwell’s laws: ⇤⇥H=J (1)magnetic ⇤ ⇥ B = µ0 J sources (2) field i. e., nos! ce ⇤·B=0 sour (3) • Note that on surface of sensor array, J = 0. As such, ⇥ H = 0 on array surface • Defining H = ∇Ψ, we obtain the identity ∇ × ∇Ψ = 0 in (1). This term (∇Ψ) is called the “scalar potential.” • “Scalar potential” has no physical correlate. • Often written with a negative sign (–∇Ψ) for convenience. • H = –∇Ψ → B = –μ0∇Ψ… used interchangeably • Substituting scalar potential into (3) we obtain the Laplacian: ⇥ ·⇥ = ⇥2 =0 Taulu et al, 2005 32
  49. 49. Signal Space Separation• Substituting the scalar potential into (3), we obtain the Laplacian: ⇥·B=0 ⇥ ·⇥ = ⇥2 =0 33
  50. 50. Signal Space Separation• Substituting the scalar potential into (3), we obtain the Laplacian: ⇥·B=0 ⇥ ·⇥ = ⇥2 = 0 |{z}  ✓ ◆ ✓ ◆ 1 @ @ @ @ 1 @2 sin ✓ r2 + sin ✓ + + K2 =0 r2 sin ✓ @r @r @✓ @✓ sin ✓ @ 2• We can express the scalar potential using spherical coordinates ( Ψ(Φ, θ, r) ), separate the variables ( Ψ(Φ,θ,r) = Φ(φ)Θ(θ)R(r) ), and solve the harmonic to obtain ⇥ l ⇥ l lm (⇥, ⌅) lm (⇥, ⌅) l B(r) = µ0 lm µ0 lm r rl+1 l=0 m= l l=0 m= l ⇥ B (r) + B (r) internal external signal signal 33
  51. 51. Signal Space Separation• Substituting the scalar potential into (3), we obtain the Laplacian: ⇥·B=0 ⇥ ·⇥ = ⇥2 = 0 |{z}  ✓ ◆ ✓ ◆ 1 @ @ @ @ 1 @2 sin ✓ r2 + sin ✓ + + K2 =0 r2 sin ✓ @r @r @✓ @✓ sin ✓ @ 2• We can express the scalar potential using spherical coordinates ( Ψ(Φ, θ, r) ), separate the variables ( Ψ(Φ,θ,r) = Φ(φ)Θ(θ)R(r) ), and solve the harmonic to obtain ⇥ l lm (⇥, ⌅) B(r) = µ0 lm internal rl+1 l=0 m= l ⇥ B (r) internal signal 33
  52. 52. Signal Space Separation 34
  53. 53. Temporally-extendedSignal Space Separation (tSSS) 35
  54. 54. Temporally-extended Signal Space SeparationConceptually very simple: 36
  55. 55. Temporally-extended Signal Space SeparationConceptually very simple:• Recall that the SSS algorithm ends with two signal components – Bα(r) and Bβ(r), or Bin(r) and Bout(r) – and we discard the Bout(r) component - Rationale: signals originating outside MEG sensor helmet cannot be brain signal 36
  56. 56. Temporally-extended Signal Space SeparationConceptually very simple:• Recall that the SSS algorithm ends with two signal components – Bα(r) and Bβ(r), or Bin(r) and Bout(r) – and we discard the Bout(r) component - Rationale: signals originating outside MEG sensor helmet cannot be brain signal• tSSS looks for correlations between B out(r) and Bin(r) and projects those correlations out of Bin(r) - Rationale: Any internal signal correlated with the external noise component must represent noise that leaked into the Bin(r) component 36
  57. 57. Temporally-extended Signal Space Separation• From the original article: 37
  58. 58. Temporally-extended Signal Space Separation• From the original article: 38
  59. 59. Temporally-extended Signal Space Separation• Without tSSS: 39
  60. 60. Temporally-extended Signal Space Separation• With tSSS: 40
  61. 61. Source Separation Algorithms 41
  62. 62. Primary Component Analysis (PCA) 42
  63. 63. • Ordinary Least Squares (OLS) regression of X to YFollowing five plots from http://stats.stackexchange.com/a/2700/2019 43
  64. 64. • Ordinary Least Squares (OLS) regression of Y to X 44
  65. 65. • Regression lines are different! 45
  66. 66. • PCA minimizes error orthogonal to the model line (Yes, this is a different dataset) 46
  67. 67. Primary Component Analysis• “Most accurate” regression line for the data (Yes, this is another different dataset) 47
  68. 68. PCA – Formal Definition 48
  69. 69. PCA – Formal Definition http://stat.ethz.ch/~maathuis/teaching/fall08/Notes3.pdf 49
  70. 70. PCA – Formal Definition http://stat.ethz.ch/~maathuis/teaching/fall08/Notes3.pdf 49
  71. 71. PCA shortcomings• Will only detect orthogonal signals “A Tutorial on Principal Component Analysis”, Jonathon Shlens, April 2009• • Cannot detect polymodal distributionsAppl. Environ. Microbiol. May 2007 vol. 73 no. 9 2878-2890 50
  72. 72. Independent Component Analysis (ICA) 51
  73. 73. Independent Component Analysis• Assumptions: Each signal is… 1. Statistically independent 2. Non-gaussian• Recall Central Limit Theorem: ! “Given independent random variables x + y = z, z is ! more gaussian than x or y.”• Theory: We can find S by iteratively identifying and extracting the most independent and non-gaussian components of X 52
  74. 74. ICA in FieldTrip package 53
  75. 75. ICA – Mixing matrix 54
  76. 76. ICA – Mixing matrix s2 s1 54
  77. 77. ICA – Mixing matrix s2 s1 x2x1 54
  78. 78. ICA – Mixing matrix x1 = a11 s1 + a12 s2 ⌘ x = As x2 = a21 s1 + a22 s2 s2 s1 x2x1 54
  79. 79. ICA – Mixing matrix x1 = a11 s1 + a12 s2 ⌘ x = As x2 = a21 s1 + a22 s2 s2 s1 x2x1 Goal: Separate s1 and s2 using information from x1 and x2 54
  80. 80. Independent Component Analysis• Consider the general mixing equation: 9 x1 = a11 s1 + . . . + a1n sn > = . . . . = . . > ⌘ x = As ; xn = an1 s1 + . . . + ann sn 55
  81. 81. Independent Component Analysis• Consider the general mixing equation: 9 mixing x1 = a11 s1 + . . . + a1n sn > matrix = . . . . = . . > ⌘ x = As ; sources xn = an1 s1 + . . . + ann sn sensors 55
  82. 82. Independent Component Analysis• Consider the general mixing equation: 9 mixing x1 = a11 s1 + . . . + a1n sn > matrix = . . . . = . . > ⌘ x = As ; sources xn = an1 s1 + . . . + ann sn sensors• If we could find one of the rows of A (let’s call that -1 vector w), we could reconstruct a row of s. Mathematically: X T w x= w i xi = y i 55
  83. 83. Independent Component Analysis • Consider the general mixing equation: 9 mixing x1 = a11 s1 + . . . + a1n sn > matrix = . . . . = . . > ⌘ x = As ; sources xn = an1 s1 + . . . + ann sn sensors • If we could find one of the rows of A (let’s call that -1 vector w), we could reconstruct a row of s. Mathematically: X T w x= w i xi = y i wSome ro-1 from A 55
  84. 84. Independent Component Analysis • Consider the general mixing equation: 9 mixing x1 = a11 s1 + . . . + a1n sn > matrix = . . . . = . . > ⌘ x = As ; sources xn = an1 s1 + . . . + ann sn sensors • If we could find one of the rows of A (let’s call that -1 vector w), we could reconstruct a row of s. Mathematically: e ICs X One of th mponents) t co wT x = w i xi = y ( independen ake up S i that m wSome ro-1 from A 55
  85. 85. Independent Component Analysis X T w x= w i xi = y• Working through the math… let x = As i z = AT w 56
  86. 86. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix Some row fr -1 T z=A w 56
  87. 87. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T = wT As = zT s 56
  88. 88. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T One of = wT As the ICs = zT s 56
  89. 89. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T One of = wT As the ICs = zT s 56
  90. 90. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T One of = wT As the ICs = zT s 56
  91. 91. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T One of = wT As the ICs = zT s 56
  92. 92. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T One of = wT As the ICs = zT s • y (an IC) is a linear combination of s, with weights z .T 56
  93. 93. Independent Component Analysis X T w x= w i xi = y • Working through the math… om A let x = As imixing matrix z = A w Some row fr -1 T • So, y = w x T One of = wT As the ICs = zT s • y (an IC) is a linear combination of s, with weights z . T • Recall Central Limit Theorem: ! “Given independent random variables x + y = z, z is ! more gaussian than x or y.” zT is more gaussian than any of si, and is least gaussian when equal to one of the si. 56
  94. 94. Independent Component Analysis X T w x= w i xi = y • Working through the math… let T x = As i z=A w • So, y = w xT We want to take w as a vector that T maximizes the nongaussianity ofOne of = wT As wTx, ensuring that wTx = zTsthe ICs = zT s • y (an IC) is a linear combination of s, with weights z . T • Recall Central Limit Theorem: ! “Given independent random variables x + y = z, z is ! more gaussian than x or y.” zT is more gaussian than any of si, and is least gaussian when equal to one of the si. 56
  95. 95. Independent Component Analysis• How can we find w Tso as to maximize the nongaussianity of wTx?• Numerous methods: - Kurtosis - Negentropy - Approximations of Negentropy• Once find, similar to PCA… find w , remove, find next T best wT, remove, repeat until no more sensors available. 57
  96. 96. ICA in Fieldtrip (2) 58
  97. 97. Mantini, Franciotti, Romani, & Pizzella (2007) 59
  98. 98. Mantini, Franciotti, Romani, & Pizzella (2007) 1
  99. 99. Mantini, Franciotti, Romani, & Pizzella (2007) 61
  100. 100. ICA – Method Comparison Zavala-Fernández, Sander, Burghoff, Orglmeister, & Trahms (2006) 62
  101. 101. Summary• Examine your data in as many ways as possible• Use SSS & tSSS to best clean data• Use ICA to find specific artifacts• Always check your data! 63
  102. 102. Questions? 64

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