In low-level image processing tasks, the circularity of an operator has been shown to be an important factor affecting its accuracy as circular differential edge operators are effective in minimising the angular error in the estimation of image gradient direction. We present a general approach to the computation of scalable near-circular low-level Laplacian image processing operators that is based on the finite element method. We use Gaussian basis functions, together with a virtual finite element mesh, to illustrate the design of operators that are scalable to near-circular neighbourhoods through the use of an explicit scale parameter. The general design technique may be applied to a range of operators. Here we illustrate the approach by discussing the implementation of a Laplacian operator, and we evaluate our approach by presenting comparative results with the Laplacian of Gaussian operators.