Convex optimization of measurement allocation for magnetic tracking systems

Convex optimization of measurement allocation for magnetic tracking systems Convex optimization of measurement allocation for magnetic tracking systems Oskar Talcoth1 • Gustav Risting1 • Thomas Ryland[.]

Convex optimization of measurement allocation

for magnetic tracking systems

Oskar Talcoth1•Gustav Risting1• Thomas Rylander1

Received: 30 September 2013 / Revised: 27 November 2014 / Accepted: 6 September 2016

Ó The Author(s) 2016 This article is published with open access at Springerlink.com

Abstract Magnetic tracking is a popular technique that exploits static and frequency magnetic fields for positioning of quasi-stationary objects One importantsystem design aspect, which substantially influences the performance of the trackingsystem, is how to collect as much information as possible with a given number ofmeasurements In this work, we optimize the allocation of measurements given alarge number of possible measurements of a generic magnetic tracking system thatexploits time-division multiplexing We exploit performance metrics based on theFisher information matrix In particular, the performance metrics measure worst-case or average performance in a measurement domain, i.e the domain where thetracking is to be performed An optimization problem with integer variables isformulated By relaxing the constraint that the variables should be integer, a convexoptimization problem is obtained The two performance metrics are compared forseveral realistic measurement scenarios with planar transmitter constellations Theresults show that the worst performance is obtained in the most distant parts of themeasurement domain Furthermore, measurement allocations optimized for worst-case performance require measurements in a larger area than measurement alloca-tions optimized for average performance

low-Keywords Convex optimization Design of experiments  Magnetic tracking Optimal measurements

& Thomas Rylander

1 Introduction

Magnetic tracking systems are designed to estimate the position and/or orientation

of a specially designed object by means of its interaction with static or frequency magnetic fields Given that the human body is transparent to magneticfields at these frequencies, magnetic tracking systems are popular within thebiomedical engineering community For example, magnetic tracking has been usedfor eye tracking to diagnose Me´nie`re’s disease (Plotkin et al.2010), positioning ofwireless capsule endoscopes within the gastro-intestinal tract (Yang et al 2009),real-time organ-positioning during radiotherapy of cancer tumors (Iustin et al

low-2008), catheter tracking (Krueger et al.2005; Biosense Webster2011), monitoring

of heart valve prostheses (Baldoni and Yellen 2007), tongue movement ing (Gilbert et al 2010; Wang et al 2013), tracking of lung segment move-ments (Leira et al 2012), and positioning of bone-embedded implants (Sherman

track-et al 2007) Examples of non-medical applications of magnetic tracking includehead tracking for helmet-mounted sights in military aircraft (Raab et al 1979),underground drilling guidance (Ripka et al 2012), augmented and virtual real-ity (Liu et al 2004), and tracking of the ball during an American footballgame (Arumugam et al.2011)

In general, the performance of a measurement system is improved if the number

of measurements increases because more information is collected and noise tends to

be averaged Nevertheless, this comes at the cost of more expensive hardware,lengthier measurement time and longer post-processing of the collected data.Therefore, a key issue in the design of measurement systems is how to maximize theinformation gained per measurement

This question is fundamental to the theory on the (optimal) design of experimentsthat has been extensively applied to geo-spatial sciences for problems in agriculture,geology, meteorology etc The reader is referred to Walter and Pronzato (1997),Ucin´ski (2005), Pukelsheim (2006), Atkinson et al (2007), and Pronzato andPa´zman (2013) for an introduction to the subject Joshi and Boyd (2009) studiedsensor selection by means of convex optimization without a specific application inmind Examples of electromagnetic applications include optimization of measure-ment setups for antenna measurements in the near-field (Nordebo and Gustafsson

2006), tracking of human tongue movements (Wang et al 2013), estimation ofcurrent densities in magnetic resonance imaging magnets (Begot et al.2002), andreconstruction of AC electric currents flowing in massive parallel conductors (DiRienzo and Zhang2010)

Within the magnetic tracking community, the impact of the number ofmeasurements has been studied by Schlageter et al (2001) and Plotkin and Paperno(2003) Schlageter et al (2001) found that the accuracy of their magnetic trackingsystem was improved when the number of transmitters, and thus the number ofmeasurements, was doubled Plotkin and Paperno (2003) found that using moretransmitters reduces the number of local minima present in the inverse problem Incontrast, how to obtain as much information as possible from a given number ofmeasurements has received little attention A rare example of such a study is the

work by Shafrir et al (2010) in which the positions of a fixed number oftransmitters are optimized using a two-step evolutionary algorithm However, theirapproach is devoted to a specific estimator and it requires that a positioningalgorithm is executed a large number of times to build statistics.

In this work, we consider magnetic tracking systems that exploit time-divisionmultiplexing and study how to allocate measurement efforts in an optimal waygiven a large number of possible measurements We exploit the theory on theoptimal design of experiments and formulate performance metrics based on theFisher information matrix The optimization of measurement allocation yields anoptimization problem with integer variables We approximate the integer variables

by real variables, which gives us a convex optimization problem In contrast to themethod presented by Shafrir et al (2010), the proposed method is valid for allunbiased estimators and it does not require a massive amount of computation.Furthermore, the convex nature of the proposed method is very attractive because itremoves two difficulties commonly encountered in design of experimentsoptimization problems, namely, high dimensionality and presence of several localminima that are not globally optimal Also, the convexity of the method proposed inthis work makes it feasible to treat large scale problems

In this work, we optimize for a measurement domain of arbitrary shape byformulating two cost functions that improve (i) the worst-case performance(minimax approach) and (ii) the expected performance for an assumed priordistribution of the position and orientation of the object we wish to track (averageapproach) The two approaches are compared for several test cases Furthermore, weinvestigate optimal measurement allocation for a realistic measurement scenario.Finally, we study the impact of restrictions on the transmitter positions, which arecommonly encountered in practice

The paper is organized as follows The modeling of a generic magnetic trackingsystem is presented in Sect.2 Section3 presents performance metrics and theproposed solution methods The results are then presented in Sect.4and discussed

in Sect.5 Finally, the work is concluded in Sect.6

2 Modeling of the measurement system

Consider a quasi-magnetostatic tracking system operating at a single frequency Thetracking system consists of (i) one receiving coil with unknown position ~ðxr; ~yr; ~zrÞand unknown orientation ^mr¼ ðmr

x; mr

y; mr

zÞ, and (ii) Ntidentical transmitting coils(also referred to as transmitters) with known positions ð~kt; ~ykt; ~zktÞ and knownorientations ^mtk Here and in the following, a vector a¼ a^a is represented by themagnitude a and the unit vector ^a The tracking system exploits time-divisionmultiplexing to separate the signals from the different transmitters, i.e., thetransmitters are operated in sequence such that only one transmitter is transmitting

at any given time instant

The aim of the tracking system is to estimate the position and orientation of thereceiving coil, i.e to estimate

To obtain entries with identical units in the vector that we wish to estimate, thespatial coordinates are normalized with the distance d, which yields

kdenote the distance vector of length Rkfrom the transmitting coil k

to the receiving coil By modeling the transmitting and receiving coils as magneticdipoles and exploiting Faraday’s law, the scaled induced voltage in the receivingcoil generated by transmitting coil k is given by Jackson (1998)

Vk¼ jxak

V0

l04p

ð5Þ

where x is the angular frequency, l0 is the permeability of free space, and V0 is areference voltage that renders Vk unit-less and thereby independent of the unit ofmeasurement The parameter ak is assumed to be known and it describes thediameter, number of turns, and the excitation current for each transmitting coil k

We use xak=V0¼ xa=V0¼ 4:33  106 Am/Vs for all k throughout this work,which implies that all transmitting coils are identical

The gradient of the scaled induced voltage in the receiver generated bytransmitting coil k with respect to the position of the receiver rr is given by

and the gradient of the scaled voltage with respect to the magnetic dipole moment ofthe receiver mr is given by

rðm r ÞVk¼ oVk

omr x

;oVk

omr y

;oVk

omr z

¼ jx a

V0

l04p

R3 k

on time-division multiplexing For convenience, we assume that Nmeasis an integer

in the following given the nature of an actual measurement system, i.e.measurements are collected and processed as single units by standard off-the-shelfmeasurement instruments

3 Optimization problem

In this work, we seek to improve the performance of the tracking system byallocating the Nmeasmeasurements among Nt candidate transmitters in an optimalway

Let wk2 N be the number of measurements performed with transmitter k.Clearly, it is advantageous to perform as many measurements as possible during the

particularly true for more complicated measurement scenarios that may requireparameter studies that involve the solution of many optimization problems.Now, let kk¼ wk=Nmeasdenote the fraction of the total number of measurementsthat are performed with transmitter k Thus, kk2 0; 1=Nf meas; 2=Nmeas; ; 1g Weuse the approximation kk2 0; 1½  because Nmeas is large as discussed in Sect.2above By using the notation K¼ k½ 1;k2; ;kN tT, we obtain the relaxedoptimization problem

which is a good approximation to the problem in (8) The feasible domain dictated

by the constraints is convex Thus, if the cost function J is convex with respect to K,the entire optimization problem is convex and can be readily solved

In the following subsections, we introduce a performance metric, present the costfunctions that are used, and present the method to solve the correspondingoptimization problems

of orientation on the unit sphere inR3 Noise that is caused by, for example, thermalnoise in amplifiers can degrade the performance of the positioning system.Therefore, we model the measured signal as the true signal Vkðp0Þ corrupted withadditive Gaussian noise as

where the noise terms nk N ð0; r2Þ are independent and identically distributed and

N ðl; r2Þ denotes the Gaussian distribution with mean l and variance r2 Below, wedenote the gradient of VkðpÞ with respect to the parameters in p at the point p0 by

rpVkðp0Þ

A metric for the performance of the parameter estimation is provided by theFisher information matrix M2 Rpp(Kay1993) given by

that yields a so-called D-optimal (Determinant-optimal) solution If the model VkðpÞ

is linear in p, the D-optimal solution minimizes the volume of the lower bound forthe confidence ellipsoid described by M1in (12) The volume of the b-confidenceellipsoid is given by Pronzato and Pa´zman (2013)

3.1.2 Local and non-local designs

The Crame´r-Rao inequality in (12) yields a lower bound for the covariance of theestimated parameters Given that the covariance is a measure of the linearrelationship between the estimated parameters, it does not capture their truerelationship when the functions VkðpÞ are non-linear in p In addition, M1 is afunction of p0 because of this non-linearity This is the reason why an optimalexperiment design based on (11) and (12) for VkðpÞ non-linear in p is referred to as

a local design (Walter and Pronzato1997) The region of validity of a local designdepends on the size of the region where the linearization VkðpÞ ffi Vkðp0Þ þ

rpVkðp0Þðp  p0Þ is a good approximation to the true non-linear VkðpÞ

In contrast to local designs, it is often desired to optimize the performance of themeasurement system not just for one point in the parameter spaceRpbut rather for ameasurement domain Xp p In this work, we optimize the measurementperformance in Xp for (i) average optimality and (ii) minimax optimality To thisaim, we exploit a discrete set of linearization points Xlin¼ fpigNlin

constitutes a sufficiently dense discretization of Xp

Average optimality In this case, we assign a prior probability distribution ppðpÞfor the parameters that are to be estimated We then find the so-called ELD-optimal(Expectation of Log Determinant-optimal) experiment design (Walter and Pronzato

1997) by minimizing the cost function

fea-optimization problem In addition, it preserves symmetries with respect to the mr

xmr

zand mr

-ymr

z-planes Furthermore, the value of det M is unaffected if ^mr is multiplied

by -1, which is exploited by performing the quadrature for the half sphere mr

only Notice that minimization of the continuous cost function in (17) and its cretized version in (20) are equivalent within the accuracy of the quadraturescheme Also, notice that there are other more efficient quadrature schemes forevaluating the expectation in (17) than the one presented in Quadrature Anexample of such a scheme is the one given by Gotwalt et al (2009) and Gotwalt(2010) that, however, includes negative weights in situations where more than 7parameters are to be estimated

dis-Thus, we obtain the relaxed average optimality problem

3.2 Solution method

In this work, we seek to allocate a limited number of measurements given a largenumber of possible candidate measurements in an optimal way Apart from theinformation on the measurement allocation, the solutions to our optimizationproblems also inform us of the number of transmitters to use and their positions Alower bound on the number of transmitters is given by the number of parameters pthat are to be estimated The number of measurements to use is limited by theconstraintPN t

k¼1kk¼ 1 in (21) and (24) and this constraint shows strong similaritywith penalty terms encountered in compressed sensing and related prob-lems (Bruckstein et al.2009) Such a penalty term typically involves the L1-norm

of the solution vector and it is added with a weight to the cost function that should

be minimized In the context of compressed sensing and related problems, thepenalty term favors a sparse solution with only a few non-zero entries, should such asolution be consistent with the rest of the problem statement Here, we find that theoptimized measurement allocation vectors Kcomputed from (21) and (24) featureonly a few non-zero entries in comparison to the number of transmitter candidates,which is confirmed by the results presented in this article

3.2.1 Thresholding and clustering of weights

The weights kk that are obtained in the solutions of the convex problems (21) and(24) above can, for the examples we have studied in this work, be grouped asfollows: (i) a handful of the weights are large ( [ 103); (ii) many are zero; (iii)several are nearly zero (\109) That the weights of the last group are not zero isdue to the finite precision arithmetic and termination criteria tolerances of theexploited numerical solver In addition, the weights of the last group are severalorders of magnitude smaller than the weights of the first group Therefore, we usethe threshold kth¼ 106 and set all weights kk\kth equal to zero We refer toweights kk kth as non-zero

Furthermore, the finite resolution of a Cartesian grid of transmitter candidatesmay cause several neighboring transmitters to obtain non-zero weights kk Wereplace such a cluster Xclof non-zero weights kk; k2 Xclwith only one weight kkcl

3.2.2 Evaluation of derivatives

To solve the optimization problems (21) and (24), the Fisher information matrix for

a given receiver position and orientation pi2 Xlin must be computed Thus, thederivatives with respect to the two degrees of freedom given by

rðm r ÞV The ui- and vi-components ofrðm r ÞV are then used in the computation ofthe Fisher information matrix (The wi-component ofrðm r ÞV is always zero because

of the constraint in (26).) The cost functions that are exploited in this work areunaffected by a rotation of ^uiand ^viaround ^wibecause determinants are invariant torotations

3.2.3 Solver

The relaxed average optimality problem in (21) and the relaxed minimax optimalityproblem in (24) are solved directly with the routine SNOPT (Gill et al 2005)provided in the TOMLAB (Tomlab Optimization AB2012) package of optimizationalgorithms The SNOPT-routine is an implementation of the sequential quadraticprogramming algorithm All gradients that are needed are computed analytically bySNOPT

4 Results

Planar transmitter constellations have become increasingly popular, see forexample (Iustin et al 2008; Plotkin et al 2010) In this work, we thereforeconsider only planar constellations of transmitters More specifically, we considerconstellations where all transmitters lie in the plane z¼ 0 with dipole momentsoriented along the z-axis, i.e ztk¼ 0 and ^mtk¼ ^zfor all k (It should be noted that theproposed method can handle any geometry of the transmitter constellation.Furthermore, transmitters with different orientations can also be considered withthe method, should this be desired.) In particular, we consider two types of planartransmitter constellations based on (i) a Cartesian grid of transmitter candidates and(ii) a polar grid of transmitter candidates These transmitter constellations arereferred to as Cartesian arrays and polar arrays, respectively, in the following.Examples of the two transmitter array types are shown in Fig.1 The transmitters

in a Cartesian array are placed on a Cartesian grid withjxt

an inter-transmitter distance h in both the x- and y-directions The transmitters in a

3.2.2 Evaluation of derivatives

To solve the optimization problems (21) and (24), the Fisher information matrix for

a given receiver position and orientation... degrees of freedom given by

rðm r ÞV The ui- and vi-components ofrðm r ÞV are then used in the computation ofthe Fisher information... (Gill et al 2005)provided in the TOMLAB (Tomlab Optimization AB2012) package of optimizationalgorithms The SNOPT-routine is an implementation of the sequential quadraticprogramming algorithm