We present a heuristic algorithm for the choice of the wedgelet regularization parameter for the purpose of denoising in the case where the noise variance σ2 is not known. Numerical experiments comparing wavelet thresholding with wedgelet denoising, and with the related schemes quadtree approximation and platelet approximation, allow to assess the respective strengths of the different approaches. For small values of σ2, wavelets are clearly superior to wedgelets, and they are better at restoring textured regions. For large σ2, or for images of a predominantly geometric nature, wedgelets yield consistently better results. Moreover, the tests reveal that the heuristic algorithm is quite effective in choosing the regularization parameter.