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Pose tracking of magnetic objects
Niklas Wahlstrom
Department of Information Technology, Uppsala University, Sweden
Novmber 13, 2017
[email protected] Seminar Vi2
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Short about me
I 2005 - 2010: Applied Physics and Electrical Engineering -International, Linkoping University.
I 2007-2008: Exchange student, ETH Zurich, Swizerland
I 2010-2015 : PhD student in Automatic Control, LinkopingUniversity
I Spring 2014, Research visit, Imperial College, London, UK
I 2016- : Researcher at Department of Information Technology,Uppsala University
1 / 17 [email protected] Seminar Vi2
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My thesis and my work at Uppsala
Three areas:
I Magnetic tracking and mapping
I Extended target tracking
I Deep dynamical models for control
Two areas:
I Constrained Gaussian processes
I Deep learning and systemidentification
2 / 17 [email protected] Seminar Vi2
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My thesis and my work at Uppsala
Three areas:
I Magnetic tracking and mapping
I Extended target tracking
I Deep dynamical models for control
Two areas:
I Constrained Gaussian processes
I Deep learning and systemidentification
2 / 17 [email protected] Seminar Vi2
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Magnetometer measurement models
1. Common use: Magnetometer provides orientation headinginformation.
Assume that the magnetometer (almost) only measures thelocal (earth) magnetic field.
2. My use: Magnetometer(s) to provide position andorientation information.
I Magnetic tracking: Measure the position and orientation of aknown magnetic source.
I Magnetic mapping: Build a map of the (indoor) magneticfield.
3 / 17 [email protected] Seminar Vi2
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Magnetometer measurement models
1. Common use: Magnetometer provides orientation headinginformation.
Assume that the magnetometer (almost) only measures thelocal (earth) magnetic field.
2. My use: Magnetometer(s) to provide position andorientation information.
I Magnetic tracking: Measure the position and orientation of aknown magnetic source.
I Magnetic mapping: Build a map of the (indoor) magneticfield.
3 / 17 [email protected] Seminar Vi2
![Page 7: Niklas Wahlstrom Novmber 13, 2017 - it.uu.se thesis and my work at Uppsala Three areas: I Magnetic tracking and mapping I Extended target tracking I Deep dynamical models for control](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f263f7f8b9a8e178c632d/html5/thumbnails/7.jpg)
Magnetometer measurement models
1. Common use: Magnetometer provides orientation headinginformation.
Assume that the magnetometer (almost) only measures thelocal (earth) magnetic field.
2. My use: Magnetometer(s) to provide position andorientation information.
I Magnetic tracking: Measure the position and orientation of aknown magnetic source.
I Magnetic mapping: Build a map of the (indoor) magneticfield.
3 / 17 [email protected] Seminar Vi2
![Page 8: Niklas Wahlstrom Novmber 13, 2017 - it.uu.se thesis and my work at Uppsala Three areas: I Magnetic tracking and mapping I Extended target tracking I Deep dynamical models for control](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f263f7f8b9a8e178c632d/html5/thumbnails/8.jpg)
Sensor setup
We use a sensor network with four three-axis magnetometers todetermine the position and orientation of a magnet.
4 / 17 [email protected] Seminar Vi2
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Magnetic tracking
Advantages
I Cheap sensors
I Small sensors
I Low energy consumption
I No weather dependency
I Passive unit, requires nobatteries
5 / 17 [email protected] Seminar Vi2
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Magnetic tracking
Advantages
I Cheap sensors
I Small sensors
I Low energy consumption
I No weather dependency
I Passive unit, requires nobatteries
5 / 17 [email protected] Seminar Vi2
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Magnetic tracking
Advantages
I Cheap sensors
I Small sensors
I Low energy consumption
I No weather dependency
I Passive unit, requires nobatteries
5 / 17 [email protected] Seminar Vi2
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Magnetic tracking
Advantages
I Cheap sensors
I Small sensors
I Low energy consumption
I No weather dependency
I Passive unit, requires nobatteries
5 / 17 [email protected] Seminar Vi2
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Magnetic tracking
Advantages
I Cheap sensors
I Small sensors
I Low energy consumption
I No weather dependency
I Passive unit, requires nobatteries
5 / 17 [email protected] Seminar Vi2
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Mathematical model - dipole field
The magnetic field can be described with a dipole field.
J(r′)
mr
B(r)
B(r) =µ0
4π‖r‖5(
3r · rT − ‖r‖2I3)
︸ ︷︷ ︸=C(r)
m
m ,1
2
∫r′ × J(r′)d3r′
6 / 17 [email protected] Seminar Vi2
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Sensor model - single dipole
The measurements can be described with a state-space model
xk+1 = Fkxk +Gkwk, wk ∼ N (0, Q),
yk,j = hj(xk) + ek, ek ∼ N (0, R)
Point target sensor model (one dipole)
hj(xk) = C(rk − θj)mk, xk = [rTk vTk mT
k ωTk ]T
C(r) =µ0
4π‖r‖5 (3rrT − ‖r‖2I3),
Measurement from a sensor network ofmagnetometers positioned at {θj}Jj=1.
Degrees of freedom
I 3D position
I 2D orientation
7 / 17 [email protected] Seminar Vi2
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Experiment 1
0.2
0.1-0.1
-0.2
-0.05
y-coordinate [m]
0
Trajectory
0
-0.1
0.05
x-coordinate [m]
0.1
z-co
ordi
nate
[m]
0 -0.1
0.15
0.2
0.1
0.25
-0.20.2
0.3
Trajectory
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2y-coordinate [m]
-0.1
-0.05
0
0.05
z-co
ordi
nate
[m]
0.1
0.15
0.2
0.25
0.3
Theaccuracy is
8 / 17 [email protected] Seminar Vi2
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Experiment 2 - results - position
20 25 30 35 40 45 50
−0.2
−0.1
0
0.1
0.2
0.3
Time [s]
Posi
tion
[m]
Black: Ground truth position. Color: Estimated position
10 / 17 [email protected] Seminar Vi2
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Experiment - results - orientation
20 25 30 35 40 45 50−200
−100
0
100
Time [s]
Ori
enta
tion
[deg
ree]
Black: Ground truth orientation. Color: Estimated orientation
11 / 17 [email protected] Seminar Vi2
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Stylaero AB
I In February in this year acompany was startedaround this technology
I The areas the companyhas so far one employee.
I Collaborations withgaming companied andindustrial partners havebeen initiated.
16 / 17 [email protected] Seminar Vi2
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Magnetometer measurement models
1. Common use: Magnetometer provides orientation headinginformation.
Assume that the magnetometer (almost) only measures thelocal (earth) magnetic field.
2. My use: Magnetometer(s) to provide position andorientation information.
I Magnetic tracking: Measure the position and orientation of aknown magnetic source.
I Magnetic mapping: Build a map of the (indoor) magneticfield.
18 / 17 [email protected] Seminar Vi2
![Page 27: Niklas Wahlstrom Novmber 13, 2017 - it.uu.se thesis and my work at Uppsala Three areas: I Magnetic tracking and mapping I Extended target tracking I Deep dynamical models for control](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f263f7f8b9a8e178c632d/html5/thumbnails/27.jpg)
Sensor model - multi-dipole
The measurements can be described with a state-space model
xk+1 = Fkxk +Gkwk, wk ∼ N (0, Q),
yk,j = hj(xk) + ek, ek ∼ N (0, R)
Extended target sensor model (a structure of dipoles)
hj(xk) =
L∑l=1
C(rk +Rk(qk)sl − θj)mlRk(qk)bl,
xk = [rTk vTk qT
k ωTk ]T
b1
b2
s1 s2
x
y
z
Degrees of freedom
I 3D position
I 3D orientation
19 / 17 [email protected] Seminar Vi2
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Magnetic mapping
Build a map of the indoor magnetic field. This map can be usedfor localization.
We want a statistical model of the magnetic field - Gaussianprocesses!
20 / 17 [email protected] Seminar Vi2
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Gaussian processes
Gaussian processes can be seed as a distribution over functions
f(u) ∼ GP(µ(u),K(u,u′)
),
Mean function ↑ ↑ Covariance function
It is a generalization of the multivariate Gaussian distribution f(u1)...
f(uN )
∼ N (µ,K), where µ =
µ(u1)...
µ(uN )
,K =
K(u1,u1) · · · K(u1,uN )...
...K(uN ,u1) · · · K(uN ,uN )
.
21 / 17 [email protected] Seminar Vi2
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Gaussian process regression
Objective: Estimate f(u) from noisy observations yk = f(uk) + ek
−3 −2 −1 0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
u
f(u)
22 / 17 [email protected] Seminar Vi2
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Gaussian process regression
Objective: Estimate f(u) from noisy observations yk = f(uk) + ek
−3 −2 −1 0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
u
f(u)
22 / 17 [email protected] Seminar Vi2
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Gaussian process regression
Objective: Estimate f(u) from noisy observations yk = f(uk) + ek
−3 −2 −1 0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
u
f(u)
22 / 17 [email protected] Seminar Vi2
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Gaussian process regression
Objective: Estimate f(u) from noisy observations yk = f(uk) + ek
−3 −2 −1 0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
u
f(u)
22 / 17 [email protected] Seminar Vi2
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Gaussian process regression
Objective: Estimate f(u) from noisy observations yk = f(uk) + ek
−3 −2 −1 0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
u
f(u)
22 / 17 [email protected] Seminar Vi2
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Modeling the magnetic field
I The magnetic field H is curl-free, i.e. ∇×H = 0 [1]
yk = f(xk) + εk
f(x) ∼ GP(0, σ2const.I3 +Kcurl(x,x′))
I If a vector-field is curl-free, a scalar potential ϕ existsH = −∇ϕ [2]
yk = −∇ϕ(x)∣∣x=xi
+ εk
ϕ(x) ∼ GP(0, klin.(x,x
′) + kSE(x,x′))
[1] Niklas Wahlstrom, Manon Kok, Thomas B. Schon and Fredrik Gustafsson, Modeling magnetic fields usingGaussian processes The 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP),Vancouver, Canada, May 2013.
[2] Arno Solin, Manon Kok, Niklas Wahlstrom, Thomas B. Schon and Simo Sarkka, Modeling and interpolation ofthe ambient magnetic field by Gaussian processes IEEE Transactions on Robotics, 2017. Accepted.
23 / 17 [email protected] Seminar Vi2
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Modeling the magnetic field
I The magnetic field H is curl-free, i.e. ∇×H = 0 [1]
yk = f(xk) + εk
f(x) ∼ GP(0, σ2const.I3 +Kcurl(x,x′))
I If a vector-field is curl-free, a scalar potential ϕ existsH = −∇ϕ [2]
yk = −∇ϕ(x)∣∣x=xi
+ εk
ϕ(x) ∼ GP(0, klin.(x,x
′) + kSE(x,x′))
[1] Niklas Wahlstrom, Manon Kok, Thomas B. Schon and Fredrik Gustafsson, Modeling magnetic fields usingGaussian processes The 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP),Vancouver, Canada, May 2013.[2] Arno Solin, Manon Kok, Niklas Wahlstrom, Thomas B. Schon and Simo Sarkka, Modeling and interpolation ofthe ambient magnetic field by Gaussian processes IEEE Transactions on Robotics, 2017. Accepted.
23 / 17 [email protected] Seminar Vi2
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Experiment
Build a map of the indoor magnetic field using Gaussian processes.
https://www.youtube.com/watch?v=enlMiUqPVJo
24 / 17 [email protected] Seminar Vi2
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Experiment
Build a map of the indoor magnetic field using Gaussian processes.
Optical marker Smartphone
Trivisio IMUInvensense IMU
DiddyBorg robot board
24 / 17 [email protected] Seminar Vi2
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Experiment
Build a map of the indoor magnetic field using Gaussian processes.
https://www.youtube.com/watch?v=enlMiUqPVJo
24 / 17 [email protected] Seminar Vi2
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Reduced-Rank GPR
Hilbert-space approximation of the covariance operator in terms ofan eigenfunction expansion of the Laplace operator in a compactsubset of Rd.
I Assume that the measurements are confined to a certaindomain.
I Approximate the covariance using the spectral density and anumber of eigenvalues and eigenfunctions. For d = 1:
k(x, x′) ≈m∑j=1
S(λj)φj(x)φj(x′)
φj(x) = 1√L
sin(πnj(x+L)
2L
), λj = πj
2L ,
I Converges to the true GP when the number of basis functionsand the size of the domain goes to infinity.
Hilbert Space Methods for Reduced-Rank Gaussian Process Regression – Arno Solin and Simo Sarkka(http://arxiv.org/pdf/1401.5508v1.pdf)
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Reduced-Rank GPR
Consequences for our problem:
Original formulation:50 or 100 Hz magnetometer data (in 3D)⇒ Size of the matrix to invert grows very quickly
with each additional second of data⇒ Downsampling needed and large buildings
become infeasible
Reduced-rank formulation:Possible to use all data⇒ Size of the problem does not grow for longer
data sets
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Sequential updating
Initialize µ0 = 0 and Σ0 = Λθ (from the GP prior). For each newobservation i = 1, 2, . . . , n update the estimate according to
Si = ∇ΦiΣi−1[∇Φi]T + σ2noise I3,
Ki = Σi−1[∇Φi]TS−1i ,
µi = µi−1 +Ki(yi −∇Φiµi−1),
Σi = Σi−1 −KiSiKTi .
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Spatio-temporal modeling
Model the scalar potential magnetic field instead as
ϕ(x, t) ∼ GP(0, κlin.(x, x′) + κSE(x, x′)κexp(t, t′)),
with
κexp(t, t′) = exp
(− |t− t
′|`time
).
The scalar potential can then sequentially be estimated by addinga time update to the measurement update from before as
µi = Ai−1µi−1,
Σi = Ai−1Σi−1ATi−1 +Qi−1.
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Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I
I
I Use a continuum of dipoles!
I Spatial correlation
29 / 17 [email protected] Seminar Vi2
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Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I Use the dipole model?
I Use multiple dipoles?
I Use a continuum of dipoles!
I Spatial correlation
29 / 17 [email protected] Seminar Vi2
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Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I Use the dipole model?
I Use multiple dipoles?
I Use a continuum of dipoles!
I Spatial correlation
← Parametric models
29 / 17 [email protected] Seminar Vi2
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Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I Use the dipole model?
I Use multiple dipoles?
I Use a continuum of dipoles!
I Spatial correlation
← Parametric models
29 / 17 [email protected] Seminar Vi2
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Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I Use the dipole model?
I Use multiple dipoles?
I Use a continuum of dipoles!
I Spatial correlation
← Parametric models
← Nonparametric models!
29 / 17 [email protected] Seminar Vi2
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Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I Use the dipole model?
I Use multiple dipoles?
I Use a continuum of dipoles!
I Spatial correlation
← Parametric models
← Nonparametric models!
29 / 17 [email protected] Seminar Vi2
![Page 50: Niklas Wahlstrom Novmber 13, 2017 - it.uu.se thesis and my work at Uppsala Three areas: I Magnetic tracking and mapping I Extended target tracking I Deep dynamical models for control](https://reader031.fdocuments.us/reader031/viewer/2022030413/5a9f263f7f8b9a8e178c632d/html5/thumbnails/50.jpg)
Problem formulation
We want to finda magnetic map ofthis object!
How should the map be modeled?
I Use the dipole model?
I Use multiple dipoles?
I Use a continuum of dipoles!
I Spatial correlation
← Parametric models
← Nonparametric models!
← Gaussian processes!
29 / 17 [email protected] Seminar Vi2
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Real world experiment
I Measurements have been collected with a magnetometer
I An optical reference system (Vicon) has been used fordetermining the position and orientation of the sensor
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
00.511.5
x−ax
is [m
]
y−axis [m]
The magnetic environment Training data
30 / 17 [email protected] Seminar Vi2
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Real world experiment
I Measurements have been collected with a magnetometer
I An optical reference system (Vicon) has been used fordetermining the position and orientation of the sensor
The magnetic environment Estimated magnetic content
30 / 17 [email protected] Seminar Vi2
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Modeling the magnetic field
I The magnetic field H is curl-free, i.e. ∇×H = 0 [1]
yk = f(xk) + εk
f(x) ∼ GP(0, σ2const.I3 +Kcurl(x,x′))
I If a vector-field is curl-free, a scalar potential ϕ existsH = −∇ϕ [2]
yk = −∇ϕ(x)∣∣x=xi
+ εk
ϕ(x) ∼ GP(0, klin.(x,x
′) + kSE(x,x′))
[1] Niklas Wahlstrom, Manon Kok, Thomas B. Schon and Fredrik Gustafsson, Modeling magnetic fields usingGaussian processes The 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP),Vancouver, Canada, May 2013.
[2] Arno Solin, Manon Kok, Niklas Wahlstrom, Thomas B. Schon and Simo Sarkka, Modeling and interpolation ofthe ambient magnetic field by Gaussian processes ArXiv e-prints, September 2015. arXiv:1509.04634.
31 / 17 [email protected] Seminar Vi2
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Modeling the magnetic field
I The magnetic field H is curl-free, i.e. ∇×H = 0 [1]
yk = f(xk) + εk
f(x) ∼ GP(0, σ2const.I3 +Kcurl(x,x′))
I If a vector-field is curl-free, a scalar potential ϕ existsH = −∇ϕ [2]
yk = −∇ϕ(x)∣∣x=xi
+ εk
ϕ(x) ∼ GP(0, klin.(x,x
′) + kSE(x,x′))
[1] Niklas Wahlstrom, Manon Kok, Thomas B. Schon and Fredrik Gustafsson, Modeling magnetic fields usingGaussian processes The 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP),Vancouver, Canada, May 2013.[2] Arno Solin, Manon Kok, Niklas Wahlstrom, Thomas B. Schon and Simo Sarkka, Modeling and interpolation ofthe ambient magnetic field by Gaussian processes ArXiv e-prints, September 2015. arXiv:1509.04634.
31 / 17 [email protected] Seminar Vi2
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Building magnetic field maps (2)
I Encode physical knowledge in the kernel.
I Use reduced-rank GP regression based onthe method from [1].
I Use a Kalman filter formulation to allowfor sequential updating.
I Use a spatio-temporal model to allow forchanges in the magnetic field.
x-component
y-component
z-component
−40
−20
0
20
[1] Hilbert Space Methods for Reduced-Rank Gaussian Process Regression – A. Solin, S. Sarkka.
32 / 17 [email protected] Seminar Vi2
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Building magnetic field maps (2)
I Encode physical knowledge in the kernel.
I Use reduced-rank GP regression based onthe method from [1].
I Use a Kalman filter formulation to allowfor sequential updating.
I Use a spatio-temporal model to allow forchanges in the magnetic field.
x-component
y-component
z-component
−40
−20
0
20
[1] Hilbert Space Methods for Reduced-Rank Gaussian Process Regression – A. Solin, S. Sarkka.
32 / 17 [email protected] Seminar Vi2
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Building magnetic field maps (2)
I Encode physical knowledge in the kernel.
I Use reduced-rank GP regression based onthe method from [1].
I Use a Kalman filter formulation to allowfor sequential updating.
I Use a spatio-temporal model to allow forchanges in the magnetic field.
x-component
y-component
z-component
−40
−20
0
20
[1] Hilbert Space Methods for Reduced-Rank Gaussian Process Regression – A. Solin, S. Sarkka.
32 / 17 [email protected] Seminar Vi2
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Building magnetic field maps (2)
I Encode physical knowledge in the kernel.
I Use reduced-rank GP regression based onthe method from [1].
I Use a Kalman filter formulation to allowfor sequential updating.
I Use a spatio-temporal model to allow forchanges in the magnetic field.
x-component
y-component
z-component
−40
−20
0
20
[1] Hilbert Space Methods for Reduced-Rank Gaussian Process Regression – A. Solin, S. Sarkka.
32 / 17 [email protected] Seminar Vi2
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Magnetic fields
We use a slightly different version of the magnetostatic equations
∇ ·B = 0,1
µ0B−H = M,
∇×H = 0
Example
- =
1µ0B - H = M
33 / 17 [email protected] Seminar Vi2
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Gaussian process + magnetic fields
I The animation illustrated regression for one scalar functionf : R→ R
I We want to learn three different vector fields f : R3 → R3 Inaddition, these fields should obey
I
I
I 1µ0B−H = M
34 / 17 [email protected] Seminar Vi2
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Gaussian process + magnetic fields
I The animation illustrated regression for one scalar functionf : R→ R
I We want to learn three different vector fields f : R3 → R3 Inaddition, these fields should obey
I
I
I 1µ0B−H = M
34 / 17 [email protected] Seminar Vi2
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Gaussian process + magnetic fields
I The animation illustrated regression for one scalar functionf : R→ R
I We want to learn three different vector fields f : R3 → R3 Inaddition, these fields should obey
I ∇ ·B = 0 (divergence free)
I ∇×H = 0 (curl free)
I 1µ0B−H = M
34 / 17 [email protected] Seminar Vi2
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Gaussian process + magnetic fields
I The animation illustrated regression for one scalar functionf : R→ R
I We want to learn three different vector fields f : R3 → R3 Inaddition, these fields should obey
I ∇ ·B = 0 (divergence free)
I ∇×H = 0 (curl free)
I 1µ0B−H = M
← There exist covariancefunctions for this!
34 / 17 [email protected] Seminar Vi2
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Gaussian process + magnetic fields
I The animation illustrated regression for one scalar functionf : R→ R
I We want to learn three different vector fields f : R3 → R3 Inaddition, these fields should obey
I ∇ ·B = 0 (divergence free)
I ∇×H = 0 (curl free)
I 1µ0B−H = M
← There exist covariancefunctions for this!
34 / 17 [email protected] Seminar Vi2
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Simulation
Tra
inin
gd
ata
Pre
dic
tion
s
Divergence free Curl free
- =
1µ0B - H = M
35 / 17 [email protected] Seminar Vi2