An iterative method for electrostatic object reconstruction in a half space

Abstract

Sensing electrodes arranged in or around a display can provide input function for interactive displays. Commercially this is interesting because the sensing electrodes and electronics can be made in the same manufacturing process as that of the display itself thus reducing cost. In engineering terms, the electrodes measure capacitance changes resulting from the presence and movement of objects such as hands and fingers in front of the display. At the quasi-static frequencies used (100 kHz) the human body is conductive and the hands or fingers provide a screen between the capacitive electrodes. There is no need to touch the actual display and the overall system constitutes a touchless gesture input system. Determining the shape of the hand or fingers is a boundary condition reconstruction problem of finding the boundary of an earthed conductive object D from electrostatic measurements. This is the ill-posed problem of recovering the zero surface of a solution to Laplace's equation from Cauchy data on part of the boundary of a domain. The problem has similarities with object reconstruction in EIT or inverse scattering but is complicated because only a partial Dirichlet–Neumann map is available as experimental data. We suggest an algorithm where at each iteration we have an approximation ∂Dk to ∂D on which we calculate approximate Cauchy data by solving a Tikhonov regularized linear system. These data are used to modify ∂Dk by extrapolation towards the zero surface giving the next approximation ∂Dk+1. We implemented the algorithm in two and three space dimensions using the boundary element method for discretization. Numerical results using simulated data with added noise show that simply connected but not necessarily convex objects can be reconstructed with reasonable positional accuracy and approximate shape, but as might be expected the shape is more accurately determined near the plane of measurements.