We introduce a new algorithm for reconstructing an unknown shape from a finite number of noisy measurements of its support function. The algorithm, based on a least squares procedure, is very easy to program in standard software such as Matlab and allows, for the first time, good 3D reconstructions to be performed on an ordinary PC. Under mild conditions, theory guarantees that outputs of the algorithm will converge to the input shape as the number of measurements increases. Reconstructions may be obtained without any pre- or post-processing steps and with no restriction on the sets of measurement directions except their number, a limitation dictated only by computing time.

In addition we offer a linear program version of the new algorithm that is much faster and better, or at least comparable, in performance at low levels of noise and reasonably small numbers of measurements. Another modification of the algorithm, suitable for use in a "focus of attention" scheme, is also described.