# Reconstruction of a grounded object in an electrostatic halfspace with an indicator function

Van Berkel, C. and Lionheart, W. R. B. (2007) Reconstruction of a grounded object in an electrostatic halfspace with an indicator function. Inverse Problems in Science and Engineering, 15 (6). pp. 585-600. ISSN 1741-5985

This article explores the use of capacitance measurements made between electrodes embedded in or around a display surface, to detect the position, orientation and shape of hands and fingers. This is of interest for unobtrusive 3D gesture input for interactive displays, so called touch-less interaction. The hand is assumed to be grounded and formally the problem is a Cauchy problem for the Laplace equation in which Cauchy data on the boundary $\partial H$ (the display surface) is used to reconstruct the zero potential contour of the unknown object $D$ (the hand). The problem is solved with the so-called factorisation method developed for acoustic scattering and electrostatic problems. In the factorisation method, a test function $g_z$ is used to characterise points $z \in D \Longleftrightarrow g_z \in \mathcal{R}( \Lambda_D^{1/2} )$, in which $\Lambda_D : L^2(\partial H) \rightarrow L^2(\partial H)$ is the Dirichlet to Neumann map on the display surface. We demonstrate a suitable test function $g_z$ appropriate to the boundary conditions present here. In the application, $\Lambda_D$ is obtained from measurements at finite precision as a finite matrix and the calculation of $|| \Lambda_D^{-1/2} g_z||^2$ is implicitly regularised. The resulting level set $P(z)$ is finite and differentiable everywhere. The level representing the object $\partial D$ is found through minimising the cost function. Numerical simulations demonstrate that for realistic electrode layouts and noise levels the method provides good reconstruction. The application of explicit regularisation filters can be beneficial and allows a trade-off between resolution and stability.