Field mice: Extracting hand geometry from electric field measurements

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IBM SYSTEMS JOURNAL, VOL 35, NOS 3&4, 1996 0018-8670/96/$5.00 1996 IBM SMITH

Several members of a family of techniques calledElectric Field Sensing are described. Eachsensing technique can be understood as ameasurement of the value of a differentcomponent in an effective circuit diagram thatsummarizes all possible current pathwaysinvolving the body to be sensed and the sensingelectrodes. An analytical model of the sensorresponse is presented, and then a probabilisticframework for inferring geometrical informationfrom field measurements is described. Theinference framework is demonstrated for the casesof a two-dimensional and a three-dimensionalmouse.

lectric Field Sensing has existed in some formsince a musical instrument known as the there-

min was invented circa 1917.1-3 Before that time, onlyaquatic animals had used electric fields to sense theirenvironments.4 The theremin combined analog soundsynthesis and an early form of Electric Field Sensingin a single clever circuit. It is surprising how littleresearch attention has been focused since then on theuse of electric fields for measuring the human body,because electric field sensors are safe, fast (millisec-ond time scales), high resolution (millimeters), andinexpensive (dollars per channel), and measure robustbulk properties of the body rather than ephemeral sur-face properties. Nevertheless, with the exception ofthe work of Matthews5 and Vranish,6,7 little effort hasbeen made to improve upon the early, “capacitive”form of Electric Field Sensing until recently.

There are two reasons why Electric Field Sensing haslanguished. The first is that at the time of its invention,very few devices were capable of being controlledelectronically. The physical user interface problem of

transducing voluntary human action into electronicsignals had therefore not yet become important.Today, computers and other electronic devices arebecoming increasingly ubiquitous, and in more andmore application domains the factor that is limitingthe performance of these systems is the physicalhuman interface rather than raw processing power.Devices such as the computer mouse, the trackball,and theIBM TrackPoint* were an improvement over asimple keyboard, but are inadequate for more com-plex tasks, such as interacting with three-dimensionalenvironments or animating complex articulated three-dimensional models. The DataGlove8 is a much moreexpressive input device, but is a step backward fromthe point of view of convenience, since it must beworn and interferes with other noncomputer activity.A noncontact three-dimensional mouse, in which thehandis the pointing device, is an appealing compro-mise, since it allows richer interactions than a conven-tional mouse, but with lower “transaction costs” thanthe DataGlove, since it does not have to be put on andthen removed to start and then stop using it.

In addition to the initial lack of applications for Elec-tric Field Sensing, there was also a second, technicalobstacle to its development. Since the response of thesensors is a nonlinear function of the input (for exam-ple, hand position), extracting useful informationfrom the measurements is a difficult computational

E

Field mice: Extractinghand geometryfrom electric fieldmeasurements

by J. R. Smith

SMITH IBM SYSTEMS JOURNAL, VOL 35, NOS 3&4, 1996588

problem and, until recently, a prohibitively difficultone. The theremin relied on the extraordinary abilityof humans to learn complex mappings.1 The per-former had to devote years of practice to learn to playit, like any traditional musical instrument. A computermouse that required years of practice to master wouldclearly be impractical.

In this paper, I present a noncontact three-dimensionalmouse as an introduction to Electric Field Sensingand as an illustration of mathematical techniques that

may be used to extract information about a matter dis-tribution from measurements of an electric field. Inthe first section of the paper, I introduce Electric FieldSensing, situating the recent work of the Physics andMedia Group of theMIT Media Laboratory in relationto earlier so-called capacitive-sensing techniques. Inthe second section, I present a quantitative forwardmodel of the response of a sensor to a small groundedobject. The third section shows how to use this for-ward model in conjunction with a probabilistic frame-work to solve the inverse problem of inferring theposition of an object from sensor values. The finalsection describes a field mouse, a prototype noncon-tact three-dimensional input device whose only mov-ing part is the user’s body.

Physics of Electric Field Sensing

We will use the term Electric Field Sensing to refer toa family of noncontact measurements of the humanbody that may be made with slowly varying (approxi-mately 50 kHz) electric fields. Previously, several ofthese measurements had been lumped together underthe accurate but imprecise rubric of “capacitive sens-ing.” Close attention to the current transport pathwaysreveals that what has been called capacitive sensing isactually comprised of several distinct mechanisms.Furthermore, two transport pathways have been over-looked. This has led the Physics and Media Group to

two new forms of Electric Field Sensing.9-16 For rea-sons we will explain later, we refer to the previouslyknown technique as a “loading mode” measurement.The new techniques involve previously unexplored“shunt mode” and “transmit mode” measurements.

In shunt mode, which is the version of Electric FieldSensing on which we will focus in this paper, a volt-age oscillating at low frequency is applied to a trans-mit electrode, and the displacement current induced ata receive electrode is measured with a current amp.The received displacement current may be modifiedby the body being sensed, which need not be in con-tact with either electrode. The basic arrangement isshown in Figure 1.

In the figure, T is the transmit electrode,R is thereceive electrode,H is the hand, andε is the dielectricconstant of the medium in which the hand is moving.Ri andCi are the internal resistance and capacitance ofthe body. Each capacitor represents the total flow ofdisplacement current from one of its terminalsdirectly to the other. The capacitors are variable: theirvalues depend on the geometry. In particular,C0, C1,andC2 depend on the hand position.

Later in this section, we explain the physics of Elec-tric Field Sensing in terms of an effective circuit dia-gram whose components correspond to current trans-port pathways. Both existing and new Electric FieldSensing techniques can be understood as measure-ments of component values in this diagram. Butbefore explaining the effective circuit model, wedescribe the actual sensing circuitry developed by thePhysics and Media Group.

Hardware. The Physics and Media Group has devel-oped an evaluation board, “The Fish,”17 for experi-menting with Electric Field Sensing. A more capable,almost all-digital successor, “The Smart Fish,” is cur-rently under development.18 The Fish consists of atransmitter that can be tuned from 20 kilohertz (kHz)to 100 kHz and four receive channels that use syn-chronous detection, a measurement technique de-scribed below. The transmitter consists of an oscilla-tor connected to an op-amp. The op-amp defines thevoltage on the transmit electrode, as specified by theoscillator, by putting out as much current as requiredto maintain the correct voltage. The amount of powerthat the user is exposed to is on the same order ofmagnitude as that received from a pair of stereo head-phones, and is several orders of magnitude below theamount permitted byFCC regulations.

Electric Field Sensing refers to afamily of noncontact measure-ments of the human body made

with slowly varying electric fields.

IBM SYSTEMS JOURNAL, VOL 35, NOS 3&4, 1996 SMITH 589

Each receive channel consists of an op-amp gainstage, a multiplier, and another op-amp used as anintegrator; these components are used to implementsynchronous detection. In the version of this tech-nique used in the Fish, the received signal is multi-plied by the original transmitted signal, and theresulting function is integrated over an interval of 60milliseconds (ms). The effect of these two operationsis to project out all of the Fourier components of thereceived signal except for the component that wastransmitted. The multiplier and integrator are comput-ing (in analog electronics) the inner product of thetransmitted signal functionst and the received signalfunctionsr, with a window function set by the integra-tion time. The sense in which the multiplier and inte-grator project out all undesirable Fourier componentsis the following: because all distinct pairs of Fouriercomponents are orthogonal, the contribution to theinner product <st,sr> from all the undesirable (i.e., dif-ferent fromst and therefore orthogonal) componentsis zero. The input stage is therefore a very sharp filterthat rejects all signals not of the proper frequency andphase. If desired, phase shifts may also be measuredby performing a second, quadrature demodulation.Synchronous detection gives very high immunity tonoise because of this sharp filtering.

It is also possible to describe the sensing circuitry interms of amplitude modulation. The transmitter maybe thought of as a carrier whose amplitude is modu-lated by the person’s body. The receive multipliermixes the carrier down to direct current (DC), and thenthe final low-pass filter rejects all signals other thanthose superimposed on the carrier.

Lumped circuit model and sensing modes. Thissubsection considers a lumped circuit model of a sin-gle transmit-receive pair with a single target object,in whose position we are interested. The various“modes” in which the Fish circuitry can be used haveclear interpretations as current paths through the cir-cuit diagram shown in Figure 1. For each sensingmode, we give a brief overview from a user’s point ofview and then explain the physics of the mode interms of this diagram.

Figure 1 depicts the model. There are four “termi-nals”: the transmitter, the receiver, the target object(shown as a hand), and ground. The six distinct inter-conductor capacitances are shown. The small resistorRi and capacitorCi represent the internal capacitanceand resistance of the body. CapacitorC5 is the targetobject’s coupling to ground. If a person is being

sensed,C5 is usually dominated by the capacitancethrough the shoes of the person. The ground terminalmay either be a groundplane in close proximity to thetransmitter and receiver, or the ambient room ground.

The sensors can be used in a variety of ways,explained below, each of which modifies these capaci-tances differently. We measure capacitance by mea-suring the current arriving at the receiver, as explainedin the earlier subsection on hardware.

Transmitter loading mode. This mode is the originalElectric Field Sensing pathway. When a handapproaches the transmitter, the capacitance betweenthe two conductors increases. In the theremin this newvalue ofC1 changes the oscillation frequency of a par-allel inductor-capacitor resonant circuit, orLC tank,which is then mixed with a constant frequency to pro-duce an audible beat. In the work of Vranish,6,7 thevalue of C1 is found by measuring the current lostthrough the transmitter. In loading mode, there is noreceiver.

Figure 1 Lumped circuit model of Electric Field Sensing

H

C1 C2

C0

C4C5 C3

Ci

Ri

GROUND

ε

T R


The Smart Fish18 has circuitry to measure the currentbeing lost at the transmit electrode.

Transmit mode. In transmit mode,11,16 the transmitelectrode is put in contact with the user’s body, whichthen becomes a transmitter, either because of directelectrical connection, or capacitive coupling throughthe clothes, which is shown as current pathC1 in thecircuit diagram.

When the hand moves, the spacing to the receiverchanges, which changes the value ofC2. This sensingtechnique is one that has been overlooked untilrecently.

When the spacing from the hand to the receiver islarge, the received signal goes roughly as 1/r 2,because the hand acts like a point object and the field

falls off as 1/r 2. By Gauss’s law, the induced chargeon the receiver also goes as 1/r 2. Since the potentialson the electrodes are defined by the Fish circuit, weknow the capacitance to beC = Q/V, and the receivedcurrentIR = 2πfCV, as explained in the previous sub-section. When the hand is very close to the receiver,C2 (typically) has the geometry of a parallel platecapacitor, and the signal goes as 1/r.

Shunt mode. The remainder of this paper will be con-cerned with shunt mode, which is the most radicaldeparture from previous practice, since it is a three-terminal measurement. With shunt mode it is possibleto extract more geometrical information per electrodethan with other modes, as we will subsequentlyexplain.

In shunt mode, neither the transmitter nor the receiveris in contact with the user’s body. When the user’sbody is out of the field, current flows from transmitterto receiver through the effective capacitanceC0.

When part of the user’s body, such as a hand, entersthe field, it functions as a third terminal, and thecapacitance matrix changes, often drastically. In par-ticular, the values ofC0, C1, andC2 shift. Since thevoltage between the transmitter and receiver is heldconstant, the change in the component values betweenthe transmitter and receiver leads to a change in thecurrent arriving at the receiver. From the amount ofcurrent that fails to arrive at the receiver, one can infersomething—what, exactly, is the question addressedin the third section of this paper—about the “amountof arm” in the vicinity of the sensor.

There is a strong sense in which shunt mode is moreinformative than loading mode: with shunt mode onecan maken2 measurements using onlyn electrodes,whereas loading mode allows onlyn measurementswith the same number of electrodes. This allows shuntmode measurements to distinguish conductivity dis-tributions that yield identical loading mode measure-ments. Figure 2 shows two distributions that yieldidentical loading-mode measurements and distinctshunt-mode measurements. We will explain the exam-ple in detail later, after we have discussed the effectivecircuit mode quantitatively.

Quantitative discussion of lumped circuit model. Anexpansion in a (small) time rate parameter of the fieldgenerated by the transmitter shows that 100 kHz iscomfortably in the quasistatic regime for measure-ments on room scales (10 m) or smaller.19 By using

Figure 2 Two conductivity distributions that byconstruction yield identical loading-modemeasurements and distinct shunt-modemeasurements

1 2

3 4

1 2

3 4


the quasistatic approximation, calculating the currentreceived at a particular electrode is straightforward.

The static charge on a conductori is due to the statictermE0 in the expansion of theE field:

(1)

whereSi is the surface ofi, n is the outward normal toSi, φ0 is the potential of whichE0 is the gradient, andthe permittivityε is a function of position, since themedium is not homogeneous. This expression relatesthe macroscopic charge on a measurement electrodeto the microscopic permittivity fieldε that we are ulti-mately interested in knowing. This microscopic per-mittivity field determines the value of the collectiveproperty of capacitance, which we will find by mea-suring a current.

Using the standard definition, the capacitance of con-ductor i due to a conductorj is the ratio between thecharge onQi and the voltage betweenj and a refer-ence. If we know the capacitance and voltages for apair of electrodes, we can find the charge induced onone by the other. Because of the linearity of all theequations involved, the total charge oni induced byall the other conductors is the sum of the separatelyinduced charges20 (but note that the capacitances arenot linear functions of position):

(2)

An element of the matrixCij represents the ratiobetweenQi andVj assuming all the otherVs are zero.

The macroscopic quantity we actually want to mea-sure, because it encodes geometrical information, isthe capacitance. But it is easier to measure currentthan charge, and we can extract the same informationfrom the current. The currentI i entering receiveri isgiven by the time derivative of the charge oni: I i =dQi /dt.

(3)

The off-diagonal terms of the capacitance matrixCij

represent the ratio betweenQi andVj for i not equal to

j when all the otherVs are zero. The diagonal “self-capacitance” termsCii represent the charge oni whenit is held atVi and all the other electrodes are at zero.Thus the diagonal terms correspond to a loading-mode measurement, and the off-diagonal terms corre-

spond to shunt-mode measurements. For pure shunt-mode measurements (no contribution from transmitmode) made with identical electrodes, sensor valuesare invariant under the operation of interchanging thetransmitter and receiver, so the matrix is symmetrical.This is not the case in transmit mode, so it may bepossible to separate the contributions from shunt andtransmit mode by splitting the capacitance matrix intoits symmetric and antisymmetric components.

Having introduced the capacitance matrix, we cannow give a clearer explanation of Figure 2, the exam-ple of a pair of conductivity distributions that can bedistinguished by shunt measurements but not loadingmeasurements. For clarity we will compare the load-ing measurements that can be made with a set ofnelectrodes to the shunt-only (we assume there is notransmit mode contribution) measurements that canbe made with the same electrodes. Since the shuntmeasurements are symmetric with respect to inter-change of the transmitter and receiver, withn elec-trodes,n(n–1)/2 distinct shunt measurements may bemade, whereas justn loading-mode measurements arepossible. It is not untiln ≥ 4 that n(n–1)/2 > n, soshunt mode does not have an advantage over loadingmode in terms of the number of measurements perelectrode when there are fewer than four electrodes.

Figure 2 shows two distributions that, by construction,give the same loading measurements: the four smalldark objects that comprise the second distribution canbe moved in from infinity until the signals are thesame as those from the first distribution. Thus theloading-mode measurements on all four electrodes arethe same for the two distributions. So = , where

Qi εn ∇φ0⋅ adSi∫–=

Qi Cij V jj

∑=

I iddt----- Cij V j

j∑ Cij

dVj

dt---------

j∑= =

Cii1 Cii

2

With shunt mode it is possibleto extract more geometricalinformation per electrodethan with other modes.


the superscript indicates the distribution. Even assum-ing that the shunt measurements around the sides ofthe square formed by the electrodes yield the samevalues for the two distributions (i.e., that = =

= = = = = ), the remainingshunt-mode measurements clearly distinguish the twocases: ≠ and ≠ . In this sense shunt-mode measurements are more informative than load-ing-mode measurements.

Component values. Now that we have discussed theeffective circuit diagram qualitatively, we will presentsome typical component values. The circuit diagramcontains only one (small) resistor because the realimpedance of free space is essentially infinite, and thereal impedance of the body is nearly zero. Barber21

gives resistivity figures on the order of 10Ωm (ohm-meters), plus or minus an order of magnitude: cere-brospinal fluid has a resistivity of 0.65Ωm, wetbovine bone has 166Ωm, blood has 1.5Ωm, and ahuman arm has 2.4Ωm longitudinally and 6.75Ωmtransversely.

Zimmerman measured the capacitance between theright hand and the left foot of a living human, andfound a value of 9.1 picofarads (pF).22 A simple paral-

lel plate model of feet in shoes with 1-centimeter (cm)thick soles gives a capacitance of 35 nanofarads (nF),usingC = ε0 A/d, and takingA = 2 feet× 20 cm× 10cm andd = 1 cm. For 10-cm thick platform shoes, thevalue ofC = 3.5 nF. (We have neglected the dielectricconstant of the soles.)

Variations inC5 can cause offsets of the sensor values.The C5 of a person wearing 10-cm thick platformshoes would be one tenth that of a person wearingshoes with 1-cm thick soles. In fact, the value ofC5

can vary even more, for example, when the person isbarefoot or standing on an actual platform such as astage.23 But C5 can be measured for calibration pur-poses with a single loading-mode measurement of thecurrent out of a transmitter that the user touchesbefore operating the equipment.24

Forward problem

Having introduced the various forms of Electric FieldSensing, I now present analytical calculations of sen-sor readings given hand positions.

General case. The sensor values can be determined inthe most general case by solving the Laplace equation

C121 C12

2

C231 C23

2 C341 C34

2 C411 C41

2

C141 C14

2 C231 C23

2

Figure 3 The unperturbed electric field impinging on the receiver, left, and the perturbed field, right

C5


with an inhomogeneous permittivityε

(4)

and then using Equation 1 to find the charge inducedon the receive electrodes by the field that solves Equa-tion 4. However, for a problem as simple as the three-dimensional mouse, this model is too general. We donot need to solve Laplace’s equation each time wemove the mouse. In this section an approximate for-ward model of the response of the sensor to a singlepoint-like grounded object will be presented, and itsrelevance to more general problems discussed. Amore detailed discussion of the model is presented inReference 19; for the purposes of this paper, the mainpoint is that the model is consistent with the sensordata, as shown later in Figure 4. There are two parts tothe model: the first part describes how the sensedobject interacts with any electric field, and the secondmodels the particular electric field induced by ourtransmitter and receiver.

Modeling the interaction: The point absorber. Wewant to model the effect of a small, perfect conductorh at a pointx in space, connected to ground through acapacitanceC5 and a wire whose effect on the field isnegligible. To create a simple model of the effect ofhon the displacement current arriving at the receiver,we will assume that the object affects the field only ina limited way: the field lines (of the unperturbed field)are attenuated when they intersect the object, but thepath through space traced out by any particular fieldline is unchanged.

To find the change in received displacement currentusing this approximation, we can start with a Gauss-ian surface surrounding the receiver and add a tubefrom the perturbation back to the surface, with theaxis of the tube everywhere parallel to the field linefrom the perturbation to the surface, as shown in Fig-ure 3. Since the tube is small, its surface is also paral-lel to the field, so there is no flux through any part ofthe tube except the end cap, which is adjacent to theobject and perpendicular to the field at that point. Thechange in flux through the Gaussian surface thatresults from the perturbation is therefore proportionalto the original field strength at the location of the per-turbation diminished by an attenuation factor andmultiplied by the area of the perturbation. Gauss’s lawthen gives the charge induced on the receiver. So thereceived currentIR ∝ I0 – αE(x) · dA, whereI0 is theunperturbed current,α is an attenuation factor be-

tween 0 and 1, anddA is a vector representing the areaof the cap. If the object has no orientational depen-dence (i.e., if it is spherical) and its size is fixed, thenthe area and attenuation factors may be combined intoa single constant, and the dot product replaced withthe magnitude of the field strength, givingIR ∝ I0 –E(x). Having introduced a model for how an objectchanges the received signal that depends on the fieldstrength at the location of the object, we now present amodel of the unperturbed field itself.

Modeling the field: The dipole approximation. Wewill approximate the field resulting from a pair ofsmall, identical, rectangular electrodes of dimensionb × c and displaced from one another bya along thex-axis as a dipole with the same spacing. The dipolemoment of a charge distribution is

(5)

If the surface charge density on the electrodes had auniform value ofν,25 then Equation 5 yieldspx = νabcand py = 0. The expression forpx makes sense:νbcyields the total charge on one electrode, so we couldwrite px = Qa. Thus the pair of rectangular electrodesdisplaced from one another bya and charged to +Qand –Q has the same dipole moment as a pair of pointcharges +Q and –Q displaced by the same amount.

To justify the dipole approximation more rigorously,we could solve for the charge distribution on the elec-trode surfaces and then perform a multipole expan-sion of the charge distribution. The dimensions atwhich the higher-order terms became significantwould be the limits of the approximation’s validity.

Modeling the sensor response. Using the pointabsorber model together with the dipole approxima-tion of the field geometry, we can model the sensorresponse data measured using a small grounded objectas a hand phantom. Figure 4 shows a plot of the func-tion I0 – E(x), whereI0 is a constant andE(x) is adipole field, given by the gradient of the dipole poten-tial . The dipole momentp is a constant repre-senting charge multiplied by the vector from thecenter of the transmitter to the center of the receiver.Measurements were made in a plane parallel to thedipole axis, using a grounded metal cube (one inch ona side) as a hand phantom. The electrodes were oneinch square pieces of copper foil spaced eight inchesapart. The theoretical plot is for a plane parallel to the

∇ εE0⋅– ∇ ε∇φ0( )⋅ ε∇2φ0 ∇ε ∇φ0⋅+ 0= = =

p x'ρ x'( )d3x'∫=

p r /r2⋅


Figure 4 Comparison of functional form of dipole model with experimental data

y

y

y

x

x

x

EXPERIMENTALDATA

ANALYTICALMODEL

RESIDUALS(ERROR)

D(x,y)

f(x,y)

D(x,y) – f(x,y)

–2

–2

–2

–2

–2

–2

0

0

0

0

0

0

2

2

2

0

0

–0.5

1

1

0.5

0.5

0

0.5


dipole axis, at a distance of .9a above the dipole axis,wherea is the dipole spacing.

Figure 5 shows sensor measurements along thez-axis,perpendicular to the transmit-receive axis, and thedipole response model. Sincex and y are zero, thedipole model simplifies toE(0,0,z) = 1/z3. Scale andoffset parameters for the distance (abscissa) and sen-sor value (ordinate) have been fit to the data. Thefunction plotted is shown at the top of the graph. Mea-surements were made along a line originating at thedipole origin and extending outward, perpendicular tothe dipole axis. At very short distances, transmit modestarts to dominate, and the signal rises again.

Isosignal shells. Given this field model, we can plotsurfaces of constant sensor readings. These plots arevery helpful in shaping one’s intuition about thebehavior of the sensors, and the isosignal surfacesthemselves are fundamental to understanding what

information a measurement provides. The surfaces areellipsoidal shells. The central axis of the ellipsoid isthe dipole axis. Figure 6 shows two nested isosignalshells for two different sensor readings. The outershell has been cut away to reveal the inner one. Thedipole generating the field lies along the central axisof the ellipsoid. A shell represents the ambiguity classof a measurement, that is, the set of points in modelparameter space all of which explain the data as wellas possible. A point in parameter space that is not partof the shell generated by a measurement correspondsto a setting of model parameters that does not explainthe data as well as the points that are part of the shell.Measurements made by additional electrodes generateadditional shells in parameter space. The intersectionof all the shells gives the set of points in parameterspace whose members all explain the data as well aspossible. In the case of a three-dimensional mouse,the parameters are spatial coordinates, so the parame-ter space is ordinary three-dimensional space.

Figure 5 Comparison of functional form of dipole model with experimental data

0 2 4 6 8 10 12

z

250

225

200

175

150

125

100

f(z)

255 – (0.164343 + 0.027864 z)–3


Later, we will also interpret these ellipsoidal shells asthe ambiguity class for a single sensor reading. Wewill use the term ambiguity class to mean the set ofpoints in model parameter space, all of which aremaximally likely given the data readings. Our goalwill be to use multiple sensor readings to reduce theambiguity class from a two-dimensional manifold to asingle point, the unique setting of model parametersthat explains the data.

With the isosignal shells in mind, this is a good placeto consider the generality of the point-absorbermodel. Can it be extended? What happens if there aretwo point absorbers? Though the sensor values are notlinear, they do depend monotonically on distance, atleast in the shunt regime. Furthermore, in the shuntregime, adding another point absorber can onlydecrease the received displacement current. Thismeans that in the general case of an arbitrary number

Figure 6 Nested isosignal shells for two readings returned by a sensor


of point absorbers, a sensor reading guarantees thatthe interior of the (single-absorber) shell is free ofabsorbers, and makes, in the worst case, no guaranteeswhatsoever about the exterior of the shell.

This shows that even though the capacitance is not alinear function of the absorber configuration, the pointabsorber still plays a crucial role in understanding thebehavior of the sensors in the presence of more com-plex geometries. We are presently developing an elec-tric field imaging method called the Swiss Cheesealgorithm that is based on this understanding of therelationship between the single absorber responsefunction and the response to a general distribution.

Groundplane. The dipole approximation discussedabove also turns out to be a good model of theresponse of a sensor in the presence of a groundplane.The reasons for this are discussed in Reference 19.Here we only explain the consequences of having ananalytical model for the behavior of a single transmit-receive pair in the presence of a groundplane.

Since receivers are virtual grounds, the field configu-ration produced by, say, one transmitter, should not bechanged by replacing a patch of groundplane with areceiver. In other words, the field produced by a trans-mitter surrounded by groundplane is identical to thatproduced by a transmitter surrounded by a patchworkof receivers and ground. This in turn means that, inthe presence of a groundplane, receivers do not affectone another—they are entirely independent dipoles.So we now have an approximate analytical solution tothe forward problem of predicting the sensor valuesgiven a hand location, and this solution will be veryhelpful in solving the inverse problem.

Constructing the ambiguity class

In this section we introduce a general probabilisticframework that will allow us to solve inversion prob-lems such as the framework needed to make the three-dimensional (3D) mouse. The framework will alsosuggest a means of designing maximally informativesensor geometries. This approach, as applied to imag-ing problems, is described by Jaynes,26 Herman etal.,27 Gull and Daniell,28 and Skilling and Gull.29

The essence of the approach is to view hand finding,for example, as an inference problem. We define amodel whose parameters—in the case of the3Dmouse, the three coordinates of the hand—we wish toknow, and a probability distribution over those param-

eters. As more data become available (for example, aswe consider additional sensors), the volume of thefeasible set of model parameters decreases, and theprobability distribution becomes increasingly peakedaround the “true” values of the parameters.

As mentioned in the previous section, we will definethe ambiguity class for a set of measurements andmodel parameters to be the maximally probable sub-set of model parameter values, given the measuredvalues. As we remarked in the discussion of the for-

ward model, for a single shunt-mode measurementand model (mx,my,mz), the ambiguity class is a two-dimensional manifold, an ellipsoidal isosignal (andthus isoprobability) shell. A second measurementyields a one-dimensional ambiguity class (the inter-section of the two single-measurement ambiguityclasses), and a third measurement yields a zero-dimensional ambiguity class, i.e., a single point or setof isolated points. Because each additional (nonde-generate) measurement reduces the dimensionality ofthe ambiguity class by one, each measurement allowsthe value of an additional model parameter to beinferred. Thus, with two measurements, we can inferthe values of two position parameters or one positionand one size parameter; with three measurements, wecan infer two positions and one size, or two positionsand one orientation, or three positions, and so on.Each additional measurement allows us to infer thevalue of an additional parameter characterizing thedistribution.

Ill-posed (underdetermined) problems, in which thereare more unknown parameters than measurements,can be made well-posed either by collecting addi-tional data or by specifying additionala priori con-straints on the feasible set. This is the Bayesian viewof regularization. These constraints can be encoded in

In the presence of agroundplane, receivers do not

affect one another.


the prior probability distribution that defines the ini-tial feasible set.

If enough measurements have been made to yield azero-dimensional ambiguity class, then the posteriorprobability distribution has isolated peaks at the maxi-mally probable points. In this paper, we always use aprior (a prior probability distribution) to select a sin-gle one of these peaks; this corresponds to a designchoice about the working region of the mouse. Givena peaked probability distribution, we can analyze theuncertainty of our estimate of the model parametervalues by examining the curvature of the distributionin the vicinity of the maximum.

Assuming a Gaussian approximation to the posteriorprobability distribution, the inverse curvature of apeak in a particular direction gives the uncertainty ofthe estimate of the parameter value (or linear combi-nation of parameter values) corresponding to thatdirection. The amount of information provided by ameasurement can be quantified by the change inentropy of the distribution that resulted from the mea-surement. The problem of designing sensor arraysmay be posed in terms of maximizing the expectedinformation provided by a measurement.

Since the sensors are subject to additive Gaussiannoise,19 the probability of the data given some settingof model parameters is given by

(6)

whereσ is the standard deviation,30 D is a data value,f(m) is the data value predicted by our analytical for-ward model given a model configuration (hand posi-tion) m. This distribution is normalized: if weintegrate over all values ofD, we get 1.

By Bayes’s theorem,

(7)

For the case of a two- or three-dimensional mouse, wecan choose a priorp(m) that renders the inversion

well-posed by, for example, restricting the possiblehand positions to positive coordinate values. Thisrestriction selects one of the two peaks in the posteriordistribution. A useful prior (probability) for one of themodel parametersmx is p(mx) = c/(1+ ), definedin some finite range ofmx, wherec is a normalizingconstant andβ is a sharpness parameter. This functionis a way to approximate a step function with a closedform expression. A possible advantage of using thisfunction over a hard step function is that numericaloptimization techniques are able to follow it back intothe high probability region, since it is smoothly vary-ing. The prior for our entire model is the product ofthe priors formx, my, andmz. So the posterior, with theprior for just one dimension shown, is

(8)

Apart from the prior, which we might have chosen tobe a constant over some region, the functional form ofp(mD) is identical to that ofp(Dm). The fact that thep(mD) distribution and thep(Dm) distribution,which have completely different meanings, have thesame functional form is the content of Bayes’s rule.However, the similarity in functional form is in somesense superficial. Consider the normalization ofp(mD). Rather than performing the analytically trac-table Gaussian integral overD (tractable and Gaussianbecause when we integratedp(Dm), m and thereforef(m) was fixed), we must integrate over all values ofm, which means integrating our forward model com-posed with a Gaussian. The difficulty of performingthis integration depends on the form off. This normal-ization constant, which Bayesians grandly call theevidence, is not important for finding the best setting

of the model parameters, since a scaling of thedependent variable (probability) has no effect on thelocation of maxima. However, it does become impor-tant when making any sort of comparison betweendifferent functionsf, or calculating entropies.31

Information collected by multiple sensors can easilybe fused: simply take the product of the posteriortermsp(mDi) for each separate sensori to find theposterior distribution given all the available data. (Weare assuming that each sensor makes just one mea-surement; otherwise we would need separate indicesfor the sensors and the data values.) Thus ifD nowdenotes the set ofN measurementsDi,

p D m( ) 1

2πσ--------------e

D f m( )–( )2

2σ2------------------------------–=

p m D( ) 1

2πσ--------------e

D f m( )–( )2

2σ2------------------------------–

p m( )p D( )

------------------------------------------=

e βmx–

p m D( ) eD f m( )–( )2

2σ2------------------------------– c

1 e βmx–+----------------------∝

m


(9)

Notice that, since log is a monotonically increasingfunction, if we maximize logp(mD), we will get thesamem as if we had maximizedp(mD). It will bedesirable in practice to work with log probabilitiesrather than probabilities for several reasons: we cansave computation time since exponentials disappear,and multiplication and division become addition andsubtraction. Furthermore, when we multiply manyprobabilities together, the numbers become verysmall, so that numerical precision can become a prob-lem. Using log probabilities alleviates this problemand reduces computation time, since exponentialsappear so often in each of our Gaussian probabilitydistributions. Rather than maximizing the log proba-bility, we could minimize the negative of the log prob-ability:

(10)

This quantity has the familiar interpretation of thesum of squared errors between the actual data and thedata predicted by the model, with an additional errorterm derived from the prior.

Error bars: Local uncertainty about the maxi-mum. Once the basic degeneracies have been broken,either by collecting sufficient data or imposing con-straints via a prior, so that there is a single maximumin the log probability, the uncertainty about the bestsetting of model parameters may be represented bythe inverse Hessian matrixA–1 evaluated at the maxi-mum. To see why, we will consider the Hessian andits properties. The HessianA gives the curvature,which is a measure of confidence or certainty. InA’seigenvector basis, in which it is diagonal, the diagonalelements (the eigenvalues)Aii represent the curvaturealong each of the eigenvector directions (known as theprincipal directions). The curvatures along the princi-pal directions are called the principal curvatures. Theproduct of the curvatures, the Gauss curvature, whichserves as a summary of the certainty at a point, isgiven by the determinant ofA. The average curvatureis given by 1/2 traceA = (A11 + A22)/2. Finally, the cur-

vature in a particular direction (in two dimensions)v = (cosθ, sinθ) is given by Euler’s formula:32

(11)

The inverse ofA in the eigenvector basis is the matrixwith diagonal elements 1/Aii . Thus the inverse Hessianspecifies “radii of curvature” of the probability distri-bution, which can be used as a measure of uncertainty.The determinant and trace of the inverse Hessian areindependent of coordinates, so we may use these aslocal measures of the “Gauss uncertainty” and meanuncertainty even when we are not in the eigenvectorbasis. Using Euler’s formula above, we could deter-mine the uncertainty in any desired direction. Ourinverse Hessian is ordinarily known as the variance-covariance matrix in statistics, so the geometricaldescription is certainly not the only way to understandthis quantity; however, we find the geometricaldescription helpful in the context of this problem.

Entropy: Global uncertainty and maximally infor-mative sensor geometries. The most general globalmeasure of uncertainty is the entropy. The change inentropy of thep(mD) distribution resulting from thecollection of new data measures the change in uncer-tainty about the values of the model parameters,including uncertainty due to multiple maxima, given aset of measurements. The change in total entropy∆Hof the posterior distribution resulting from a measure-mentDn+1 is

(12)

where

(13)

The expected change in entropy when we collect anew piece of data, that is, the change in entropy aver-aged over possible data values, gives a basis for com-paring sensor geometries. The expected value ofH(mD) is

(14)

where is an actual object position and ,with f being the forward model. Thus,

p m D( ) eDi f i m( )–( )2

2σ2---------------------------------– c

1 e βmj–+----------------------

j∏

i

N

∏∝

log p m D( )–Di f i m( )–( )2

2σ2---------------------------------

i

N

∑=

log 1 e βmj–+( ) logc–j

∑+

κ vT

Av κ1 θ2 κ+ 2 θ2sincos= =

H m Dn 1+( )∆ H m Dn 1+( ) H m Dn( )–=

H m Dn( ) p m Dn( ) p m Dn( )log md∫–=

I p m( )H m D( ) md∫=

m D f m( )=


(15)

and substituting in for ,

(16)

I is a measure of the quality of a sensor geometry. Byanalogy with coding theory, the best measurementprocedure (for single measurements) reduces the

entropy as much as possible. One could thereforesearch for optimal sensor geometries by minimizingI.31,33,34

Example: Two-dimensional field mouse. Here weuse this technique to construct the ambiguity class andfind the most likely model parameters given two sen-sor readings. We want to infer the position of the handin two dimensions from two sensor readings. So themodel consists simply of two numbers, representingthe position of the object purported to explain the sen-sor readings. The sensor axes are oriented perpendicu-lar to one another, and the transmit electrode isshared.

Figures 7 and 8 show the posterior distributionsp(mx, myD1) and p(mx, myD2) for the two sensors,oriented perpendicular to one another. To make thefigure easier to view, the noise has been exaggerateddramatically. If the actual noise levels for the sensorshad been used, the features of the surfaces would beso minute that the contour plot routine would havevery little to display.

Figure 9 shows the posterior probability distributionp(mx, myD1, D2) = p(mx, myD1)p(mx, myD2) with auniform prior. The arrows show the principal compo-nents of the inverse Hessian evaluated at each peak.

The larger error bar on the less sharply defined peakhas been scaled down by 1/3 to fit it on the page. Theusable region of the mouse is the upper right quadrantof the region shown; in practice a nonuniform priorwould be used to eliminate the ambiguity by remov-ing the structure in the lower quadrants.

The “principal uncertainties,” or error bars, are alsoshown superimposed on the maxima. The three small-est arrows have been scaled up by a factor of 10 tomake them more visible. The larger arrow on the lesssharply defined peak has only been scaled up by 3 1/3,so that it fits on the page.

Application: 3D field mouse

In this section we describe another application of thediscussion from the previous section: a three-dimen-sional mouse. We choose a sensor geometry and con-struct its ambiguity class for an example handposition. It is possible to check the suitability andquality of a sensor layout and prior by examining theambiguity class: if there are multiple maxima, theinversion is ill-posed, and if the peak is not sharp (ifthe maximum has high radii of curvature, that is, ahigh value of the determinant of the inverse Hessianmatrix), the value is very uncertain.

Figure 10 shows the layout we selected. The electrodelabeled T is the transmitter, and R1–R3 are the receiv-ers. The surrounding square represents a groundplane.

Earlier we discussed criteria for optimal sensordesign. Evaluating the entropy integrals, and averag-ing over all possible data values, represents a substan-tial practical challenge. Efficient means of doing sowill probably require sophisticated numerical tech-niques, except in special cases.

Therefore, we will simply satisfy ourselves that thislayout does not lead to ill-posed inversion problemsby examining its ambiguity class. Figure 11 shows theambiguity classes for three single sensor measure-ments made using this layout. The object being mea-sured is at (x,y,z) = (0.8, 0.6, 1.05), using units of thesmallest transmit-receive spacing. The ambiguityshells intersect at just one point in the region of inter-est. The ambiguity class for the joint measurement ofall three sensor values is this single intersection point.Figure 12 shows the posterior distribution for this sen-sor layout and the object once again at (x,y,z) = (0.8,0.6, 1.05). Each image shows a slice through thethree-dimensional posterior probability distribution,

I p m( )H m f m( )( ) md∫=

H m f m( )( )

I p m( )[ p m f m( )( ) p m f m( )( )log( ) md∫– ] md∫=

The suitability and quality of asensor layout and prior can be

checked by examining theambiguity class.


Figure 7 Posterior probability p (x,y,0.9D1) where D1, the measurement on sensor 1, is given by D1 = f1(0.9, 0.6, 0.9); thevalue of z is constrained to be 0.9.

p(x,yD1)

x

y

1.5

1

0.5

0

–0.5

–1

–1.5

–1.5 –1 –0.5 0 0.5 1 1.5


Figure 8 Posterior probability p (x,y,0.9D2) where D2, the measurement on sensor 2, is given by D2 = f2(0.9, 0.6, 0.9); thevalue of z is constrained to be 0.9.

p(x,yD2)

x

y

1.5

1

0.5

0

–0.5

–1

–1.5

–1.5 –1 –0.5 0 0.5 1 1.5


Figure 9 Posterior probability distribution with error bars: p (x,y,0.9D1,D2) for sensors 1 and 2 given measurementsD1 = f1(0.9, 0.6, 0.9) and D2 = f2(0.9, 0.6, 0.9); the value of z is constrained to be 0.9.

p(x,yD1, D2)

x

y

1.5

1

0.5

0

–0.5

–1

–1.5

–1.5 –1 –0.5 0 0.5 1 1.5


parallel to theX-Y plane. The maximum of the poste-rior is at (0.8, 0.6, 1.05), as expected.

In the figure, the variance of the noise assumed on thesensors was increased 400 times over the observednoise to make the figure readable. Thus the figureillustrates the relative uncertainty of the position esti-mate, but not the absolute uncertainty. The figureshows that the uncertainty in theY direction is least;the most uncertainty is in the joint estimate ofX andZ.

The most important feature of the plot is that there is asingle maximum. It gives an indication of the geome-try of the uncertainty isosurfaces. We have scaled thedistribution so that the maximum value is 1. Thewhite at location (0.8, 0.6, 1.05) corresponds to 1, andthe black elsewhere corresponds to values near zero.

To invert the signals we maximize the log probability,which corresponds to minimizing a prior term plus thesum of squared error between the measured value andthat predicted by the current estimate of the handposition.

Figure 13 shows a screen shot of the mouse. Theuser’s hand motion is mapped onto the motion of thehand icon. The hand can pick up the small cubeshown, move it around the space, and set it back onthe floor. Because we cannot yet extract hand size, we

have used a “sticky hand, sticky floor” protocol forgrasping and releasing the cube. The small cube startson the floor. When the hand first touches the cube, thehand closes, and the cube “sticks” to the hand andmoves with it until the hand returns to the floor, atwhich point the hand opens and the cube sticks to thefloor, where it remains until the hand returns. Moreinformation on the3D mouse is available on the WorldWide Web.35

Conclusion

The main technical obstacle to the use of ElectricField Sensing, the computational burden associatedwith inverting the signals, appears to be tractable. Weare currently working on the problem of inferringparameters beyond position (for example, size andorientation) in order to make a “Gloveless Data-Glove,” and also on the problem of extracting low-res-olution three-dimensional images from electric fieldmeasurements, a process we call Electric FieldTomography.19

Electric Field Sensing could profoundly affect peo-ple’s mode of interaction with machines, and theirexpectations about the properties of objects generally.A common first reaction to a table with embeddedelectric field sensors is amazement, since it appears tobe magic. However, the interaction soon begins to feeltransparent, natural, and ordinary, rather than magi-cal—why should you not be able to indicate yourintentions to an object by waving your hands at it?Hume, in his famous argument on the impossibility ofmiracles,36 says that the scope of the ordinary or natu-ral can always be enlarged to subsume “magical” phe-nomena. Our experience with Electric Field Sensingseems mainly to support his position, but we havefound there is an enjoyable transient period, beforethe user’s experience has become routinized, in whichmagic is possible.

Acknowledgments

The 3D graphics code used to visualize the3D mousewas written by Barrett Comisky. I thank the othermembers of the Physics and Media Group whoworked on the development of the Fish board: JoeParadiso, Tom Zimmerman, and Neil Gershenfeld.For helpful comments on earlier versions of thispaper, I thank Neil Gershenfeld, Seth Lloyd, StephenA. Benton, and the anonymous reviewers. I thankDavid MacKay and Steve Gull for teaching me thestatistical techniques used in this paper. This work

Figure 10 Sensor geometry for 3D mouse

T

R3R2

R1


Figure 11 The ambiguity classes for three single measurements made using the sensor geometry from the previousfigure

yx

z

0

0.5

1

1.5

0.5

1

1.5

1

0.5

0

0


Figure 12 Posterior distribution over model parameters ( x,y,z) for sensor geometry from the previous two figures:p (x,y,zD1,D2,D3) for sensors 1, 2, and 3 given measurements D1 = f1(0.8, 0.6, 1.05), D2 = f2(0.8, 0.6, 1.05), andD3 = f3(0.8, 0.6, 1.05)

x

y

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

x

y

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

x

y

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

x

y

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

x

y

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

x

y

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

z = 0.97 z = 1.01

z = 1.05 z = 1.09

z = 1.13 z = 1.17


Figure 13 3D mouse


was supported in part by the Hewlett-Packard Corpo-ration, Festo Corporation, Microsoft Corporation,Compaq Computer Corporation, the Media Lab’sNews in the Future and Things That Think consortia,and a Motorola fellowship.

*Trademark or registered trademark of International BusinessMachines Corporation.

Cited references and notes

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13. T. G. Zimmerman, J. R. Smith, J. A. Paradiso, D. Allport, andN. Gershenfeld, “Applying Electric Field Sensing to Human-Computer Interfaces,”CHI’95 Human Factors in ComputingSystems, Denver, CO (1995), pp. 280–287.

14. J. A. Paradiso and N. Gershenfeld, “Musical Applications ofElectric Field Sensing,” Computer Music Journal, forthcoming.

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17. It is called “Fish” because electric fish use similar mechanismsto sense their environments, and because we hope that TheFish, which navigates in three dimensions, might be the succes-sor input device to the mouse, which only navigates in two.

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19. J. R. Smith,Toward Electric Field Tomography, master’sdegree thesis, MIT Media Laboratory, Cambridge, MA (1995).

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22. T. G. Zimmerman,Personal Area Networks (PAN): Near-FieldIntra-Body Communication, master's degree thesis, MIT MediaLaboratory, Cambridge, MA (1995).

23. J. A. Paradiso, personal communication.24. J. A. Paradiso,Gesture Wall Electronics, technical report, Brain

Opera, MIT Media Laboratory, Cambridge, MA (1996).25. The surface charge distribution is not in fact uniform, although

the surface is an equipotential.26. E. T. Jaynes, “Prior Information and Ambiguity in Inverse

Problems,”SIAM AMS Proceedings14, 151–166 (1983).27. G. T. Herman, H. Hurwitz, and A. Lent, “A Bayesian Analysis

of Image Reconstruction,” G. T. Herman, Editor,Image Recon-struction from Projections: Implementations and Applications,Springer-Verlag, Berlin (1979), pp. 85–103.

28. S. F. Gull and G. J. Daniell, “Image Reconstruction fromIncomplete and Noisy Data,”Nature272, 686–690 (1978).

29. J. Skilling and S. F. Gull, “The Entropy of an Image,”SIAMAMS Proceedings14, 167–188 (1983).

30. It will not represent conductivity in this section.31. D. J. C. MacKay,Bayesian Methods for Adaptive Models,

Ph.D. thesis, California Institute of Technology, Pasadena, CA(1991).

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Accepted for publication May 31, 1996.

Joshua R. Smith MIT Media Laboratory, 20 Ames Street, Cam-bridge, Massachusetts 02139-4307 (electronic mail: [email protected]). Mr. Smith is a Ph.D. candidate and research assis-tant in the Physics and Media Group of the MIT Media Laboratory,and is interested in the physics of information and computation. Hehas an M.S. from the Media Laboratory, a B.A. Hons. in natural sci-ences (physics and theoretical physics) from Cambridge University,and a double B.A. in computer science and philosophy from Wil-liams College. He has had internships at the Santa Fe Institute, LosAlamos National Laboratory, the Yale University Department ofComputer Science, the Williams College SMALL GeometryResearch Group, and NASA’s Goddard Institute for Space Studies.

Reprint Order No. G321-5626.

Documents

Field mice: Extracting hand geometry from electric field measurements