Before revision:   To derive the proposed representation, Varadhan's formula is exploited to approximate the non-linear Eikonal partial differential equation (PDE) of the Euclidean distance field~(EDF ) by the logarithm of the screen Poisson equation that is linear .
After revision:  Our key contribution is the realisation that the regularised Eikonal equation can be simply solved by applying the logarithmic transformation to a GPIS formulation to recover the accurate Euclidean distance field (EDF) and, at the same time, the implicit surface. To derive the proposed representation, Varadhan's formula is exploited to approximate the non-linear Eikonal partial differential equation (PDE) of the Euclidean distance field~(EDF ) by the logarithm of the screen Poisson equation that is linear .