and Related Geometric Methods
of Computing Source-Points of Echoes
by Douglas Moreman
Abstract: For the imaging sonar of dolphins, are discussed both 1) a possible neuronal mechanism and 2) related geometric means of computing reflecting-points of echoes. The evolving computational aspects, adjusted for serial rather than parallel computing, have been tested in computer simulations by this author for the past thirteen years. Whatever the accuracy of the neuronal model, images are, in fact, produced from simulated echoes of simulated objects; though, not all is yet perfect. It is speculated that the jaw of a dolphin has “echotrigger” sensors that are exquisitely sensitive to certain features that seem likely to exist in echoes of all clicks of dolphins of the same species. Such sensors might have evolved, for example, from touch sensors. There are said to be 2000 touch-sensors in the end of a human finger. So, it seems reasonable to speculate that the jaw of a dolphin has thousands of echotriggers. It is further speculated that each axon from an echotrigger connects directly to a "ditoa-neuron" in the brain, by-passing the cochleae altogether. The frequency-analyzing cochleae might, in sonic vision, add a low spatial-resolution analog of visual color but are not expected, by this author, to add much to imaging complexities of shape. Each ditoa-neuron receives dendrites from just two echotriggers and responds to their difference-in-time-of-arrival. Operating in small sets, called "scopions," the echotriggers, via their pairwise ditoa-neurons perform an operation analogous to "triangulation." For each click-event and for each scopion, a set of possible reflecting points in space is computed via “triconication,” an analog of triangulation. Each point computed by a scopion is the intersection of conicoids: spheres, ellipsoids and hyperboloids. Each ditoa-neuron and its pair of echotriggers can determine, mathematically speaking, an hyperboloid in the space of echoing objects. The intersection of such “ditoa-hyperboloids” together, in the case of active sonar, with one or more ellipsoids determines a possible source of a feature in an echo. It might appear that four, or even just three, sensors would suffice to determine a possible echoing point. But extra sensors seem, in simulations, to be required beyond "triangulation" for the purpose of removing phantoms - indications of point-sources of sound where none exists. The number of sensors per scopion has, through trial and error, settled down over the years of simulations, to being five or six. The limitation of features-per-available-time is such that each scopion can produce just a few prospective source-points. Producing an image in simulations, and presumably a mental image in a brain, requires a multitude of scopions. Brains, of course, can easily have a hundred million ditoa-neurons. This abundance has not been replicated in simulations and so simulated pictures remain simple. Brains, of course, process data "in parallel," and each scopion, in a dolphin, computes independently of the others. It is speculated that a very large set of scopions, each producing a few high quality points for a picture, produces mental images that allow the observed ability of dolphins to identify, by sight, complex objects “seen” earlier via sonar. Strong reasons are given for expecting these computational methods to fail. So, it is remarkable that they succeed in producing images from simulated echoes of simulated fish.