Consider a two-dimensional domain containing a medium with unit electrical conductivity and one or more non-conducting objects. The problem considered here is that of identifying shape and position of the objects on the sole basis of measurements on the external boundary of the domain. An iterative technique is presented in which a sequence of solutions of the direct problem is generated by a boundary element method on the basis of assumed positions and shapes of the objects. The key new aspect of the approach is that the boundary of each object is represented in terms of Fourier coefficients rather than a point-wise discretization. These Fourier coefficients generate the fundamental "shapes" mentioned in the title in terms of which the object shape is decomposed. The iterative procedure consists in the successive updating of the Fourier coefficients at every step by means of the Levenberg-Marquardt algorithm. It is shown that the Fourier decomposition-which, essentially, amounts to a form of image compression-enables the algorithm to image the embedded objects with unprecedented accuracy and clarity. In a separate paper, the method has also been extended to three dimensions with equally good results.