The Physics and Media Group has been trying to use our electric field sensors (aka "fish" sensors) for imaging. We are working on an imaging algorithm for use with a multiplexing array of electric field sensors operating in "shunt mode." In shunt mode, one electrode is a transmitter, another is a receiver, and the object being sensed shunts displacement current away from the receiver and into ground. The measured quantity is the displacement current arriving at the receiver. In "transmit mode," the same quantity is measured, but the transmitter is in close proximity to the body being sensed, so that the body becomes a transmitter. In "loading mode," the measured quantity is the current leaving the transmitter.
With our multiplexing array (designed and built by Joe Paradiso), any of the n electrodes can be the transmitter, and any can be the receiver. An n-electrode array can therefore make n^2 measurements. These measurements can be viewed as the elements of the nxn matrix D of capacitances determined by the geometry of the electrodes and sensed object. The diagonal elements of this matrix are loading mode measurements, like those investigated by Vranish et al. We are currently exploring the hypothesis that the anti-symmetric component [1/2 (D - D-transpose)] represents transmit-mode only effects, and not shunt-mode effects.
The Swiss Cheese imaging algorithm (being developed by Joshua Smith) relies on the fact that, though the response of the EF sensors is not a linear function of position or size, within the "shunt" regime, it is (by definition) a monotonic function of both size and distance.
The figure above shows the received displacement current as a function of the distance of a unit absorber to the sensor. A unit absorber is a perfectly grounded object with the smallest area of interest in the problem. The choice of this unit area amounts to a choice of regularizer. The shunt regime is defined to be the interval between the "crossover position" and infinity. The crossover position is the object location that yields the minimum sensor reading. We use the name shunt regime because in this interval, the shunt current path from the transmitter through the object to ground dominates the coupling or transmit current path from the transmitter into the object and back out to the receiver.
In the shunt regime, moving an object closer to the sensor decreses the sensor reading. In this regime, making the object larger can only decrease a sensor reading (this can also be taken as an alternative definition of the shunt regime).
The two nested shells above---the outer one cut away to reveal the inner---represent the unit absorber ambiguity class for two readings from the same sensor. The ambiguity class for a sensor reading is the set of points---a surface---which yields the same sensor reading for a unit absorber. So if the object were known to be a unit absorber, we could infer that the object was somewhere on the inner ellipsoid if we received a the smaller sensor value, and somewhere on the outer ellipsoid if we receive the other.
What if the object is more complicated than the unit absorber? Consider the case of two unit absorbers. Assume that initially only one is present. This yields a sensor reading I. Now bring the second absorber in. We now have a new reading J, such that, because of monotonicity, J < I. This example should make it clear that the unit absorber is a limiting case. If we assume that the sensor reading from an arbitrary (but shunt-mode only) conductivity distribution is due to a unit absorber, then we will conclude that the ellipsoid E, on whose surface the unit absorber lies, is empty. Because of the monotonicty of the shunt-mode regime, this conclusion is correct, even for an arbitrary distribution. For the arbitrary distribution, an even larger region than E is empty, but the sensor makes no claim beyond "the region near me is guaranteed to be empty."
Mathematically, our image will be a (pseudo) probability distribution over three-space, not exactly the probability that there is an absorber at each point, but something close to that. Each sensed value defines an ellipsoidal unit-absorber surface. Inside the ellipsoid, the probability is close to zero; outside, it is close to one (or a constant for normalization?). To form the complete image, multiply together the probability distributions defined by each sensor. This operation embodies the rules of inference we desire: one low value times any number of high values yields a low value. (Because the same data (sensor a = 121, for example) can be explained by infinitely many theories, a sensor cannot verify a theory, but only falsify one.) A low value at a point indicates certainty that there is no feature (unit absorber) present there; a high value could be the result of a feature, or lack of knowledge.
Thus when we start, we assume that there are unit absorbers ("cheese") everywhere in the region to be imaged. Each sensor takes an ellipsoidal bite out of the cheese, by falsifying the hypothesis that "cheese" is present at each of the points inside the ellipsoid. The image consists of the remaining cheese. When we are done, we can plot low values as black and the high values as white to visualize the image. White regions do not guarantee the presence of a conductor, they simply indicate the lack of a guarantee that a conductor is not present. So as one moves out of range of the sensors, one encounters a wall of cheese beyond which the sensors cannot see.
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