123MagneTrack: Magnetic Field Separation Method for Continuous andSimultaneous 1-DOF Tracking of Two-magnets

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ANONYMOUS AUTHOR(S)∗78

ACM Reference Format:

9Anonymous Author(s). 2021. MagneTrack: Magnetic Field Separation Method for Continuous and Simultaneous 1-DOF Tracking of

10Two-magnets. 1, 1 (December 2021), 2 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn

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1APPENDIX

This appendix shows the derivation process of a magnetic field separation which is omitted in 3.3 separation process.

151617181920211.1

Separation Process

We obtained the magnetic field difference generated from each magnet 𝑌𝜙 as the observation data, as follows:Õ𝑌𝜙 =𝑀𝑖,𝜙 + 𝜖𝜙 .

(1)

𝑖

22

As the given magnetic field value is real, we separated the observation data and the basis vector into positive and23

negative components applied to NMF:

24(+)

26(+)

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(−)

𝑋𝑖,𝜙 = max(0, 𝑋𝑖,𝜙 )25

𝑌𝜙𝑋𝑖,𝜙 = min(0, 𝑋𝑖,𝜙 )(−)= max(0, 𝑌𝜙 )𝑌𝜙= min(0, 𝑌𝜙 )

(2)
(3)

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Therefore, when the label of a sign is defined as 𝑠 ∈ {(+), (−)}, the magnetic field difference model and the observation30

data can be transcribed to3133
343536(4)(5)𝑠𝑠𝑀𝑖,𝜙Õ= 𝐻𝑖 𝑋𝑖,𝜙𝑠𝑌𝜙𝑠 =𝑀𝑖,𝜙+ 𝜖𝜙𝑠 ,32𝑖respectively. In theory, the error in the observation data 𝜖𝜙𝑠 is zero. The minimization problem of the error is addressedby minimizing
J(𝑌,𝑀)≡3738

ÕD(𝑌𝜙 ,𝑀𝜙 )+2𝜆L

Õ

|𝐻𝑖 |

(6)𝑖

𝜙39

with respect to 𝐻𝑖 . Here, the second term is the regularization parameter that prevents 𝐻𝑖 from being overfitted when40

an empirical value is assigned to 𝜆L . The first term indicates 𝛽 divergence. Even though the 𝛽 divergence is a convex

4142434445464748
49505152

function whose shape can be transformed variedly, we adopted Euclidean distance in the case of 𝛽 = 2:ÕÕ𝑠 2
D(𝑌𝜙 , 𝑀𝜙 ) =(𝑌𝜙𝑠 −𝑀𝑖,𝜙) .
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Manuscript submitted to ACM
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1

2Anon.

53

Based on the auxiliary function method, the auxiliary function of the cost function given by Eq. (6) is derived using the54following pair of inequality expressions.

5556(𝑌𝜙𝑠 −5758
5960
616263

Õ𝐽 + (𝑌,𝑀)≤𝑠 −𝐻 𝑋 𝑠 ) 2Õ (𝑌ˆ𝑖,𝜙𝑖 𝑖,𝜙66

𝑠,𝑖,𝜙67

then J(𝑌, 𝑀) ≤ 𝐽 + (𝑌, 𝑀), so that in the case of69
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84+𝜆L (|𝐻ˆ𝑖 | −1 𝐻𝑖2 +|𝐻ˆ𝑖 |),

(10)

𝑠𝑠𝑠𝑠
= 𝐻𝑖 𝑋𝑖,𝜙+ 𝛽𝑖,𝜙(𝑌𝜙𝑠 − 𝑀𝑖,𝜙)𝑌ˆ𝑖,𝜙(11)𝐻ˆ𝑖 = 𝐻𝑖(12)𝑠𝛽𝑖,𝜙= Í𝑠𝐻𝑖 𝑋𝑖,𝜙𝑠𝑛 𝐻𝑛 𝑋𝑛,𝜙under the above stated conditions. The update function can be obtained using 𝜕𝐽 + /𝜕𝐻𝑖 = 0 as follows:Í𝑠ˆ𝑠 𝑠𝑠,𝜙 (𝑌𝑖,𝜙 𝑋𝑖,𝜙 /𝛽𝑖,𝜙 )𝐻𝑖 = Í𝑠 2 𝑠
ˆ −1𝑠,𝜙 ((𝑋𝑖,𝜙 ) /𝛽𝑖,𝜙 ) + 𝜆L |𝐻𝑖 |(13)(14)

By iteratively updating Eq. (4), (11), (12), (13) and (14), the solution will converge to a stationary point. In our study, weset zero to 𝜆L .

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𝑠𝛽𝑖,𝜙effectively by assigning

7678(8)𝐽 + (𝑌 , 𝑀) fulfills the definition of auxiliary function because J(𝑌, 𝑀) = 𝐽 + (𝑌, 𝑀). Note that the solution can be obtained

7577𝑠𝛽𝑖,𝜙ˆ| 𝐻𝑖 |𝑖|𝐻𝑖 | 2|𝐻𝑖 | ≤𝐻 + |𝐻ˆ𝑖 | −(9)2 𝑖2Í ˆ𝑠𝑠𝑠𝑠Note that Eq. (8) holds under the conditions that 𝑖 𝑌𝑖,𝜙 = 𝑌𝜙 , and 𝛽𝑖,𝜙 is an arbitrary value satisfying 0 ≤ 𝛽𝑖,𝜙 ≤ 1 andÍ 𝑠𝑖 𝛽𝑖,𝜙 = 1. Thus, assuming that6572𝑠 − 𝐻 𝑋 𝑠 )2Õ (𝑌ˆ𝑖,𝜙𝑖 𝑖,𝜙𝑖64

68𝑠 2𝑀𝑖,𝜙) ≤Manuscript submitted to ACM