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)
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. 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
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
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. 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. 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
24
30. 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
s k = Pk m
s ? = P? m
24
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 m
sources MEG signal
s ? = P? m
unknown
sources
Projection
operators
24
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 m
sources MEG signal
s ? = P? m
unknown
sources
Projection Worth mentioning that sk + s? = s
operators
24
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. 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‖.2
1 Not really magic
25
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‖.2
1 Not really magic
2 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. 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. 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
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
45. Signal Space Separation
• Begin with Maxwell’s laws:
⇤⇥H=J (1)
magnetic ⇤ ⇥ B = µ0 J sources (2)
field
⇤·B=0 (3)
32
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. 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. 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. Signal Space Separation
• Substituting the scalar potential into (3), we obtain the
Laplacian:
⇥·B=0
⇥ ·⇥ = ⇥2 =0
33
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. 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
55. Temporally-extended Signal Space Separation
Conceptually 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. Temporally-extended Signal Space Separation
Conceptually 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
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
79. ICA – Mixing matrix
x1 = a11 s1 + a12 s2
⌘ x = As
x2 = a21 s1 + a22 s2
s2
s1
x2
x1 Goal: Separate s1 and s2 using
information from x1 and x2
54
80. Independent Component Analysis
• Consider the general mixing equation:
9
x1 = a11 s1 + . . . + a1n sn >
=
.
. .
. = .
. >
⌘ x = As
;
xn = an1 s1 + . . . + ann sn
55
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. 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. 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
w
Some ro-1
from A
55
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
w
Some ro-1
from A
55
86. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing matrix Some row fr -1
T
z=A w
56
87. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing matrix z = A w Some row fr
-1
T
• So, y = w x
T
= wT As
= zT s
56
88. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing matrix z = A w Some row fr
-1
T
• So, y = w x
T
One of = wT As
the ICs = zT s
56
89. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing matrix z = A w Some row fr
-1
T
• So, y = w x
T
One of = wT As
the ICs = zT s
56
90. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing matrix z = A w Some row fr
-1
T
• So, y = w x
T
One of = wT As
the ICs = zT s
56
91. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing matrix z = A w Some row fr
-1
T
• So, y = w x
T
One of = wT As
the ICs = zT s
56
92. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing 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. Independent Component Analysis
X
T
w x= w i xi = y
• Working through the math… om A
let x = As
i
mixing 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. 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 of
One of = wT As wTx, ensuring that wTx = zTs
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
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
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
