A non-contact, three-dimensional mouse was used to introduce the concept of Electric Field Sensing and demonstrate mathematical approaches for deriving information on matter distribution from electric field measurements. A quantitative forward model of a sensor's response to a small grounded object was used together with a probabilistic framework for solving the inverse problem of determining an object's position from values generated by the sensor.
Several members of a family of techniques called Electric Field Sensing are described. Each sensing technique can be understood as a measurement of the value of a different component in an effective circuit diagram that summarizes all possible current pathways involving the body to be sensed and the sensing electrodes. An analytical model of the sensor response is presented, and then a probabilistic framework for inferring geometrical information from field measurements is described. The inference framework is demonstrated for the cases of a two-dimensional and a three-dimensional mouse.
Electric Field Sensing has existed in some form since a musical instrument known as the theremin was invented circa 1917.(1-3) Before that time, only aquatic animals had used electric fields to sense their environments.(4) The theremin combined analog sound synthesis and an early form of Electric Field Sensing in a single clever circuit. It is surprising how little research attention has been focused since then on the use of electric fields for measuring the human body, because electric field sensors are safe, fast (millisecond time scales), high resolution (millimeters), and inexpensive (dollars per channel), and measure robust bulk properties of the body rather than ephemeral surface properties. Nevertheless, with the exception of the work of Matthews(5) and Vranish,(6,7) little effort has been made to improve upon the early, "capacitive" form of Electric Field Sensing until recently.
There are two reasons why Electric Field Sensing has languished. The first is that at the time of its invention, very few devices were capable of being controlled electronically. The physical user interface problem of transducing voluntary human action into electronic signals had therefore not yet become important. Today, computers and other electronic devices are becoming increasingly ubiquitous, and in more and more application domains the factor that is limiting the performance of these systems is the physical human interface rather than raw processing power. Devices such as the computer mouse, the trackball, and the IBM TrackPoint(*) were an improvement over a simple keyboard, but are inadequate for more complex tasks, such as interacting with three-dimensional environments or animating complex articulated three-dimensional models. The DataGlove(8) is a much more expressive input device, but is a step backward from the point of view of convenience, since it must be worn and interferes with other noncomputer activity. A noncontact three-dimensional mouse, in which the hand is the pointing device, is an appealing compromise, since it allows richer interactions than a conventional mouse, but with lower "transaction costs" than the DataGlove, since it does not have to be put on and then removed to start and then stop using it.
In addition to the initial lack of applications for Electric Field Sensing, there was also a second, technical obstacle to its development. Since the response of the sensors is a nonlinear function of the input (for example, hand position), extracting useful information from the measurements is a difficult computational problem and, until recently, a prohibitively difficult one. The theremin relied on the extraordinary ability of humans to learn complex mappings.(1) The performer had to devote years of practice to learn to play it, like any traditional musical instrument. A computer mouse that required years of practice to master would clearly be impractical.
In this paper, I present a noncontact three-dimensional mouse as an introduction to Electric Field Sensing and as an illustration of mathematical techniques that may be used to extract information about a matter distribution from measurements of an electric field. In the first section of the paper, I introduce Electric Field Sensing, situating the recent work of the Physics and Media Group of the MIT Media Laboratory in relation to earlier so-called capacitive-sensing techniques. In the second section, I present a quantitative forward model of the response of a sensor to a small grounded object. The third section shows how to use this forward model in conjunction with a probabilistic framework to solve the inverse problem of inferring the position of an object from sensor values. The final section describes a field mouse, a prototype noncontact three-dimensional input device whose only moving part is the user's body.
Physics of Electric Field Sensing
We will use the term Electric Field Sensing to refer to a family of noncontact measurements of the human body that may be made with slowly varying (approximately 50 kHz) electric fields. Previously, several of these measurements had been lumped together under the accurate but imprecise rubric of "capacitive sensing." Close attention to the current transport pathways reveals that what has been called capacitive sensing is actually comprised of several distinct mechanisms. Furthermore, two transport pathways have been overlooked. This has led the Physics and Media Group to two new forms of Electric Field Sensing.(9-16) For reasons we will explain later, we refer to the previously known 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 Field Sensing on which we will focus in this paper, a voltage oscillating at low frequency is applied to a transmit electrode, and the displacement current induced at a receive electrode is measured with a current amp. The received displacement current may be modified by the body being sensed, which need not be in contact with either electrode. The basic arrangement is shown in Figure 1.
[Figure 1 ILLUSTRATION OMITTED]
In the figure, T is the transmit electrode, R is the receive electrode, H is the hand, and [Epsilon] is the dielectric constant of the medium in which the hand is moving. [R.sub.i] and [C.sub.i] are the internal resistance and capacitance of the body. Each capacitor represents the total flow of displacement current from one of its terminals directly to the other. The capacitors are variable: their values depend on the geometry. In particular, [C.sub.0], [C.sub.1], and [C.sub.2] depend on the hand position.
Later in this section, we explain the physics of Electric Field Sensing in terms of an effective circuit diagram whose components correspond to current transport pathways. Both existing and new Electric Field Sensing techniques can be understood as measurements of component values in this diagram. But before explaining the effective circuit model, we describe the actual sensing circuitry developed by the Physics and Media Group.
Hardware. The Physics and Media Group has developed an evaluation board, "The Fish,"(17) for experimenting with Electric Field Sensing. A more capable, almost all-digital successor, "The Smart Fish," is currently under development.(18) The Fish consists of a transmitter that can be tuned from 20 kilohertz (kHz) to 100 kHz and four receive channels that use synchronous detection, a measurement technique described below. The transmitter consists of an oscillator connected to an op-amp. The op-amp defines the voltage on the transmit electrode, as specified by the oscillator, by putting out as much current as required to maintain the correct voltage. The amount of power that the user is exposed to is on the same order of magnitude as that received from a pair of stereo headphones, and is several orders of magnitude below the amount permitted by FCC regulations.
Each receive channel consists of an op-amp gain stage, a multiplier, and another op-amp used as an integrator; these components are used to implement synchronous detection. In the version of this technique used in the Fish, the received signal is multiplied by the original transmitted signal, and the resulting function is integrated over an interval of 60 milliseconds (ms). The effect of these two operations is to project out all of the Fourier components of the received signal except for the component that was transmitted. The multiplier and integrator are computing (in analog electronics) the inner product of the transmitted signal function [S.sub.t] and the received signal function [S.sub.r], with a window function set by the integration time. The sense in which the multiplier and integrator project out all undesirable Fourier components is the following: because all distinct pairs of Fourier components are orthogonal, the contribution to the inner product <[S.sub.t], [S.sub.r]> from all the undesirable (i.e., different from [S.sub.t] and therefore orthogonal) components is zero. The input stage is therefore a very sharp filter that rejects all signals not of the proper frequency and phase. If desired, phase shifts may also be measured by performing a second, quadrature demodulation. Synchronous detection gives very high immunity to noise because of this sharp filtering.
It is also possible to describe the sensing circuitry in terms of amplitude modulation. The transmitter may be thought of as a carrier whose amplitude is modulated by the person's body. The receive multiplier mixes the carrier down to direct current (DC), and then the final low-pass filter rejects all signals other than those superimposed on the carrier.
Lumped circuit model and sensing modes. This subsection considers a lumped circuit model of a single 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 have clear interpretations as current paths through the circuit diagram shown in Figure 1. For each sensing mode, we give a brief overview from a user's point of view and then explain the physics of the mode in terms of this diagram.
Figure 1 depicts the model. There are four "terminals": the transmitter, the receiver, the target object (shown as a hand), and ground. The six distinct interconductor capacitances are shown. The small resistor [R.sub.i] and capacitor [C.sub.i] represent the internal capacitance and resistance of the body. Capacitor [C.sub.5] is the target object's coupling to ground. If a person is being sensed, [C.sub.5] is usually dominated by the capacitance through the shoes of the person. The ground terminal may either be a groundplane in close proximity to the transmitter and receiver, or the ambient room ground.
The sensors can be used in a variety of ways, explained below, each of which modifies these capacitances differently. We measure capacitance by measuring the current arriving at the receiver, as explained in the earlier subsection on hardware.
Transmitter loading mode. This mode is the original Electric Field Sensing pathway. When a hand approaches the transmitter, the capacitance between the two conductors increases. In the theremin this new value of [C.sub.1] changes the oscillation frequency of a parallel inductor-capacitor resonant circuit, or LC tank, which is then mixed with a constant frequency to produce an audible beat. In the work of Vranish,(6,7) the value of [C.sub.1] is found by measuring the current lost through the transmitter. In loading mode, there is no receiver.
The Smart Fish(18) has circuitry to measure the current being lost at the transmit electrode.
Transmit mode. In transmit mode,(11,16) the transmit electrode is put in contact with the user's body, which then becomes a transmitter, either because of direct electrical connection, or capacitive coupling through the clothes, which is shown as current path [C.sub.1] in the circuit diagram.
When the hand moves, the spacing to the receiver changes, which changes the value of [C.sub.2]. This sensing technique is one that has been overlooked until recently.
When the spacing from the hand to the receiver is large, the received signal goes roughly as 1/[r.sup.2], because the hand acts like a point object and the field falls off as 1/[r.sup.2]. By Gauss's law, the induced charge on the receiver also goes as 1/[r.sup.2]. Since the potentials on the electrodes are defined by the Fish circuit, we know the capacitance to be C = Q/V, and the received current [I.sub.R] = 2[Pi]fCV, as explained in the previous subsection. When the hand is very close to the receiver, [C.sub.2] (typically) has the geometry of a parallel plate capacitor, and the signal goes as 1/r.
Shunt mode. The remainder of this paper will be concerned with shunt mode, which is the most radical departure from previous practice, since it is a three-terminal measurement. With shunt mode it is possible to extract more geometrical information per electrode than with other modes, as we will subsequently explain.
In shunt mode, neither the transmitter nor the receiver is in contact with the user's body. When the user's body is out of the field, current flows from transmitter to receiver through the effective capacitance [C.sub.0].
When part of the user's body, such as a hand, enters the field, it functions as a third terminal, and the capacitance matrix changes, often drastically. In particular, the values of [C.sub.0], [C.sub.1], and [C.sub.2] shift. Since the voltage between the transmitter and receiver is held constant, the change in the component values between the transmitter and receiver leads to a change in the current arriving at the receiver. From the amount of current that fails to arrive at the receiver, one can infer something--what, exactly, is the question addressed in the third section of this paper--about the "amount of arm" in the vicinity of the sensor.
There is a strong sense in which shunt mode is more informative than loading mode: with shunt mode one can make [n.sup.2] measurements using only n electrodes, whereas loading mode allows only n measurements with the same number of electrodes. This allows shunt mode measurements to distinguish conductivity distributions that yield identical loading mode measurements. Figure 2 shows two distributions that yield identical loading-mode measurements and distinct shunt-mode measurements. We will explain the example in detail later, after we have discussed the effective circuit mode quantitatively.
[Figure 2 ILLUSTRATION OMITTED]
Quantitative discussion of lumped circuit model. An expansion in a (small) time rate parameter of the field generated by the transmitter shows that 100 kHz is comfortably in the quasistatic regime for measurements on room scales (10 m) or smaller.(19) By using the quasistatic approximation, calculating the current received at a particular electrode is straightforward.
The static charge on a conductor i is due to the static term [E.sub.0] in the expansion of the E field:
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [S.sub.i] is the surface of i, n is the outward normal to [S.sub.i], [[Phi].sub.0] is the potential of which [E.sub.0] is the gradient, and the permittivity [Epsilon] is a function of position, since the medium is not homogeneous. This expression relates the macroscopic charge on a measurement electrode to the microscopic permittivity field [Epsilon] that we are ultimately interested in knowing. This microscopic permittivity field determines the value of the collective property of capacitance, which we will find by measuring a current.
Using the standard definition, the capacitance of conductor i due to a conductor j is the ratio between the charge on [Q.sub.i] and the voltage between j and a reference. If we know the capacitance and voltages for a pair of electrodes, we can find the charge induced on one by the other. Because of the linearity of all the equations involved, the total charge on i induced by all the other conductors is the sum of the separately induced charges(20) (but note that the capacitances are not linear functions of position):
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
An element of the matrix [C.sub.ij] represents the ratio between [Q.sub.i] and [V.sub.j] assuming all the other Vs are zero.
The macroscopic quantity we actually want to measure, because it encodes geometrical information, is the capacitance. But it is easier to measure current than charge, and we can extract the same information from the current. The current [I.sub.i] entering receiver i is given by the time derivative of the charge on i: [I.sub.i] = d[Q.sub.i]/dt.
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The off-diagonal terms of the capacitance matrix [C.sub.ij] represent the ratio between [Q.sub.i] and [V.sub.j] for i not equal to j when all the other Vs are zero. The diagonal "self-capacitance" terms [C.sub.ij] represent the charge on i when it is held at [V.sub.i] and all the other electrodes are at zero. Thus the diagonal terms correspond to a loading-mode measurement, and the off-diagonal terms correspond to shunt-mode measurements. For pure shunt-mode measurements (no contribution from transmit mode) made with identical electrodes, sensor values are invariant under the operation of interchanging the transmitter and receiver, so the matrix is symmetrical. This is not the case in transmit mode, so it may be possible to separate the contributions from shunt and transmit mode by splitting the capacitance matrix into its symmetric and antisymmetric components.
Having introduced the capacitance matrix, we can now give a clearer explanation of Figure 2, the example of a pair of conductivity distributions that can be distinguished by shunt measurements but not loading measurements. For clarity we will compare the loading measurements that can be made with a set of n electrodes to the shunt-only (we assume there is no transmit mode contribution) measurements that can be made with the same electrodes. Since the shunt measurements are symmetric with respect to interchange of the transmitter and receiver, with n electrodes, n(n -1)/2 distinct shunt measurements may be made, whereas just n loading-mode measurements are possible. It is not until n [is greater than or equal to] 4 that n(n - 1)/2 [is greater than] n, so shunt mode does not have an advantage over loading mode in terms of the number of measurements per electrode when there are fewer than four electrodes.
Figure 2 shows two distributions that, by construction, give the same loading measurements: the four small dark objects that comprise the second distribution can be moved in from infinity until the signals are the same as those from the first distribution. Thus the loading-mode measurements on all four electrodes are the same for the two distributions. So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where the superscript indicates the distribution. Even assuming that the shunt measurements around the sides of the square formed by the electrodes yield the same values for the two distributions (i.e., that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), the remaining shunt-mode measurements clearly distinguish the two cases: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this sense shunt-mode measurements are more informative than loading-mode measurements.
Component values. Now that we have discussed the effective circuit diagram qualitatively, we will present some typical component values. The circuit diagram contains only one (small) resistor because the real impedance of free space is essentially infinite, and the real impedance of the body is nearly zero. Barber(21) gives resistivity figures on the order of 10 [Omega]m (ohmmeters), plus or minus an order of magnitude: cerebrospinal fluid has a resistivity of 0.65 [Omega]m, wet bovine bone has 166 [Omega]m, blood has 1.5 [Omega]m, and a human arm has 2.4 [Omega]m longitudinally and 6.75 [Omega]m transversely.
Zimmerman measured the capacitance between the right hand and the left foot of a living human, and found a value of 9.1 picofarads (pF).(22) A simple parallel plate model of feet in shoes with 1-centimeter (cm) thick soles gives a capacitance of 35 nanofarads (nF), using C = [[Epsilon].sub.0] A/d, and taking A = 2 feet x 20 cm x 10 cm and d = 1 cm. For 10-cm thick platform shoes, the value of C = 3.5 nF. (We have neglected the dielectric constant of the soles.)
Variations in [C.sub.5] can cause offsets of the sensor values. The [C.sub.5] of a person wearing 10-cm thick platform shoes would be one tenth that of a person wearing shoes with 1-cm thick soles. In fact, the value of [C.sub.5] can vary even more, for example, when the person is barefoot or standing on an actual platform such as a stage.(23) But [C.sub.5] can be measured for calibration purposes with a single loading-mode measurement of the current out of a transmitter that the user touches before operating the equipment.(24)
Forward problem
Having introduced the various forms of Electric Field Sensing, I now present analytical calculations of sensor readings given hand positions.
General case. The sensor values can be determined in the most general case by solving the Laplace equation with an inhomogeneous permittivity [Epsilon]
(4) -[del] [multiplied by] [Epsilon][E.sub.0] = [del] [multiplied by] ([Epsilon][del][[Phi].sub.0]) = [Epsilon][[del].sup.2][[Phi].sub.0] + [del][Epsilon] [multiplied by] [del][[Phi].sub.0] = 0
and then using Equation 1 to find the charge induced on the receive electrodes by the field that solves Equation 4. However, for a problem as simple as the three-dimensional mouse, this model is too general. We do not need to solve Laplace's equation each time we move the mouse. In this section an approximate forward model of the response of the sensor to a single point-like grounded object will be presented, and its relevance to more general problems discussed. A more detailed discussion of the model is presented in Reference 19; for the purposes of this paper, the main point is that the model is consistent with the sensor data, as shown later in Figure 4. There are two parts to the model: the first part describes how the sensed object interacts with any electric field, and the second models the particular electric field induced by our transmitter and receiver.
[Figure 4 ILLUSTRATION OMITTED]
Modeling the interaction: The point absorber. We want to model the effect of a small, perfect conductor h at a point x in space, connected to ground through a capacitance [C.sub.5] and a wire whose effect on the field is negligible. To create a simple model of the effect of h on the displacement current arriving at the receiver, we will assume that the object affects the field only in a limited way: the field lines (of the unperturbed field) are attenuated when they intersect the object, but the path through space traced out by any particular field line is unchanged.
To find the change in received displacement current using this approximation, we can start with a Gaussian surface surrounding the receiver and add a tube from the perturbation back to the surface, with the axis of the tube everywhere parallel to the field line from the perturbation to the surface, as shown in Figure 3. Since the tube is small, its surface is also parallel to the field, so there is no flux through any part of the tube except the end cap, which is adjacent to the object and perpendicular to the field at that point. The change in flux through the Gaussian surface that results from the perturbation is therefore proportional to the original field strength at the location of the perturbation diminished by an attenuation factor and multiplied by the area of the perturbation. Gauss's law then gives the charge induced on the receiver. So the received current [I.sub.R] [varies] [I.sub.0] - [Alpha]E(x) [multiplied by] dA, where [I.sub.0] is the unperturbed current, [Alpha] is an attenuation factor between 0 and 1, and dA is a vector representing the area of the cap. If the object has no orientational dependence (i.e., if it is spherical) and its size is fixed, then the area and attenuation factors may be combined into a single constant, and the dot product replaced with the magnitude of the field strength, giving [I.sub.R] [varies] [I.sub.0] |E(x)|. Having introduced a model for how an object changes the received signal that depends on the field strength at the location of the object, we now present a model of the unperturbed field itself.
[Figure 3 ILLUSTRATION OMITTED]
Modeling the field: The dipole approximation. We will approximate the field resulting from a pair of small, identical, rectangular electrodes of dimension b x c and displaced from one another by a along the x-axis as a dipole with the same spacing. The dipole moment of a charge distribution is
(5) p = [integral of]x'[Rho](x')[d.sup.3]x'
If the surface charge density on the electrodes had a uniform value of v,(25) then Equation 5 yields [p.sub.x] = vabc and [p.sub.y] = 0. The expression for [p.sub.x] makes sense: vbc yields the total charge on one electrode, so we could write [p.sub.x] = Qa. Thus the pair of rectangular electrodes displaced from one another by a and charged to +Q and -Q has the same dipole moment as a pair of point charges +Q and -Q displaced by the same amount.
To justify the dipole approximation more rigorously, we could solve for the charge distribution on the electrode surfaces and then perform a multipole expansion of the charge distribution. The dimensions at which the higher-order terms became significant would be the limits of the approximation's validity.
Modeling the sensor response. Using the point absorber model together with the dipole approximation of the field geometry, we can model the sensor response data measured using a small grounded object as a hand phantom. Figure 4 shows a plot of the function [I.sub.0] - |E(x)|, where [I.sub.0] is a constant and E(x) is a dipole field, given by the gradient of the dipole potential p [multiplied by] r/[r.sup.2]. The dipole moment p is a constant representing charge multiplied by the vector from the center of the transmitter to the center of the receiver. Measurements were made in a plane parallel to the dipole axis, using a grounded metal cube (one inch on a side) as a hand phantom. The electrodes were one inch square pieces of copper foil spaced eight inches apart. The theoretical plot is for a plane parallel to the dipole axis, at a distance of .9a above the dipole axis, where a is the dipole spacing.
Figure 5 shows sensor measurements along the z-axis, perpendicular to the transmit-receive axis, and the dipole response model. Since x and y are zero, the dipole model simplifies to E(0,0,z) = 1/[z.sup.3]. Scale and offset parameters for the distance (abscissa) and sensor value (ordinate) have been fit to the data. The function plotted is shown at the top of the graph. Measurements were made along a line originating at the dipole origin and extending outward, perpendicular to the dipole axis. At very short distances, transmit mode starts to dominate, and the signal rises again.
[Figure 5 ILLUSTRATION OMITTED]
Isosignal shells. Given this field model, we can plot surfaces of constant sensor readings. These plots are very helpful in shaping one's intuition about the behavior of the sensors, and the isosignal surfaces themselves are fundamental to understanding what information a measurement provides. The surfaces are ellipsoidal shells. The central axis of the ellipsoid is the dipole axis. Figure 6 shows two nested isosignal shells for two different sensor readings. The outer shell has been cut away to reveal the inner one. The dipole generating the field lies along the central axis of the ellipsoid. A shell represents the ambiguity class of a measurement, that is, the set of points in model parameter space all of which explain the data as well as possible. A point in parameter space that is not part of the shell generated by a measurement corresponds to a setting of model parameters that does not explain the data as well as the points that are part of the shell. Measurements made by additional electrodes generate additional shells in parameter space. The intersection of all the shells gives the set of points in parameter space whose members all explain the data as well as possible. In the case of a three-dimensional mouse, the parameters are spatial coordinates, so the parameter space is ordinary three-dimensional space.
[Figure 6 ILLUSTRATION OMITTED]
Later, we will also interpret these ellipsoidal shells as the ambiguity class for a single sensor reading. We will use the term ambiguity class to mean the set of points in model parameter space, all of which are maximally likely given the data readings. Our goal will be to use multiple sensor readings to reduce the ambiguity class from a two-dimensional manifold to a single point, the unique setting of model parameters that explains the data.
With the isosignal shells in mind, this is a good place to consider the generality of the point-absorber model. Can it be extended? What happens if there are two point absorbers? Though the sensor values are not linear, they do depend monotonically on distance, at least in the shunt regime. Furthermore, in the shunt regime, adding another point absorber can only decrease the received displacement current. This means that in the general case of an arbitrary number of point absorbers. a sensor reading guarantees that the interior of the (single-absorber) shell is free of absorbers, and makes, in the worst case, no guarantees whatsoever about the exterior of the shell.
This shows that even though the capacitance is not a linear function of the absorber configuration, the point absorber still plays a crucial role in understanding the behavior of the sensors in the presence of more complex geometries. We are presently developing an electric field imaging method called the Swiss Cheese algorithm that is based on this understanding of the relationship between the single absorber response function and the response to a general distribution.
Groundplane. The dipole approximation discussed above also turns out to be a good model of the response 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 an analytical model for the behavior of a single transmitreceive pair in the presence of a groundplane.
Since receivers are virtual grounds, the field configuration produced by, say, one transmitter, should not be changed by replacing a patch of groundplane with a receiver. In other words, the field produced by a transmitter surrounded by groundplane is identical to that produced by a transmitter surrounded by a patchwork of receivers and ground. This in turn means that, in the presence of a groundplane, receivers do not affect one another--they are entirely independent dipoles. So we now have an approximate analytical solution to the forward problem of predicting the sensor values given a hand location, and this solution will be very helpful in solving the inverse problem.
Constructing the ambiguity class
In this section we introduce a general probabilistic framework that will allow us to solve inversion problems such as the framework needed to make the three-dimensional (3D) mouse. The framework will also suggest a means of designing maximally informative sensor geometries. This approach, as applied to imaging problems, is described by Jaynes,[26] Herman et al.,[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 a model whose parameters--in the case of the 3D mouse, the three coordinates of the hand--we wish to know, and a probability distribution over those parameters. As more data become available (for example, as we consider additional sensors), the volume of the feasible set of model parameters decreases, and the probability distribution becomes increasingly peaked around the "true" values of the parameters.
As mentioned in the previous section, we will define the ambiguity class for a set of measurements and model parameters to be the maximally probable subset of model parameter values, given the measured values. As we remarked in the discussion of the forward model, for a single shunt-mode measurement and model ([m.sub.x],[m.sub.y],[m.sub.z]), the ambiguity class is a two-dimensional manifold, an ellipsoidal isosignal (and thus isoprobability) shell. A second measurement yields a one-dimensional ambiguity class (the intersection of the two single-measurement ambiguity classes), and a third measurement yields a zero-dimensional ambiguity class, i.e., a single point or set of isolated points. Because each additional (nondegenerate) measurement reduces the dimensionality of the ambiguity class by one, each measurement allows the value of an additional model parameter to be inferred. Thus, with two measurements, we can infer the values of two position parameters or one position and one size parameter; with three measurements, we can infer two positions and one size, or two positions and one orientation, or three positions, and so on. Each additional measurement allows us to infer the value of an additional parameter characterizing the distribution.
Ill-posed (underdetermined) problems, in which there are more unknown parameters than measurements, can be made well-posed either by collecting additional data or by specifying additional a priori constraints on the feasible set. This is the Bayesian view of regularization. These constraints can be encoded in the prior probability distribution that defines the initial feasible set.
If enough measurements have been made to yield a zero-dimensional ambiguity class, then the posterior probability distribution has isolated peaks at the maximally probable points. In this paper, we always use a prior (a prior probability distribution) to select a single one of these peaks; this corresponds to a design choice about the working region of the mouse. Given a peaked probability distribution, we can analyze the uncertainty of our estimate of the model parameter values by examining the curvature of the distribution in the vicinity of the maximum.
Assuming a Gaussian approximation to the posterior probability distribution, the inverse curvature of a peak in a particular direction gives the uncertainty of the estimate of the parameter value (or linear combination of parameter values) corresponding to that direction. The amount of information provided by a measurement can be quantified by the change in entropy of the distribution that resulted from the measurement. The problem of designing sensor arrays may be posed in terms of maximizing the expected information provided by a measurement.
Since the sensors are subject to additive Gaussian noise,[19] the probability of the data given some setting of model parameters is given by
(6) p(D|m) = 1/[square root of 2 [Phi] [Sigma]] e - (D- f[(m)).sup.2]/
2 [[Sigma].sup.2]
where [Sigma] is the standard deviation,[30] D is a data value, f(m) is the data value predicted by our analytical forward model given a model configuration (hand position) m. This distribution is normalized: if we integrate over all values of D, we get 1.
By Bayes's theorem,
(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For the case of a two- or three-dimensional mouse, we can choose a prior p(m) that renders the inversion well-posed by, for example, restricting the possible hand positions to positive coordinate values. This restriction selects one of the two peaks in the posterior distribution. A useful prior (probability) for one of the model parameters [m.sub.x] is p([m.sub.x]) = c/(1 + [e.sup.-[Beta][m.sub.x]), defined in some finite range of [m.sub.x], where c is a normalizing constant and [Beta] is a sharpness parameter. This function is a way to approximate a step function with a closed form expression. A possible advantage of using this function over a hard step function is that numerical optimization techniques are able to follow it back into the high probability region, since it is smoothly varying. The prior for our entire model is the product of the priors for [m.sub.x], [m.sub.y], and [m.sub.z]. So the posterior, with the prior for just one dimension shown, is
(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Apart from the prior, which we might have chosen to be a constant over some region, the functional form of p(m|D) is identical to that of p(D|m). The fact that the p(m|D) distribution and the p(D|m) distribution, which have completely different meanings, have the same functional form is the content of Bayes's rule. However, the similarity in functional form is in some sense superficial. Consider the normalization of p(m|D). Rather than performing the analytically tractable Gaussian integral over D (tractable and Gaussian because when we integrated p(D|m), m and therefore f(m) was fixed), we must integrate over all values of m, which means integrating our forward model composed with a Gaussian. The difficulty of performing this integration depends on the form off. This normalization constant, which Bayesians grandly call the evidence, is not important for finding the best setting m of the model parameters, since a scaling of the dependent variable (probability) has no effect on the location of maxima. However, it does become important when making any sort of comparison between different functions f, or calculating entropies.[31]
Information collected by multiple sensors can easily be fused: simply take the product of the posterior terms p(m|[D.sub.i]) for each separate sensor i to find the posterior distribution given all the available data. (We are assuming that each sensor makes just one measurement; ot