A generalization of linear Gaussian scale-space theory for scalar images is proposed, based on a particular type of metric transform preserving the intrinsic properties of the spatial domain. The existence of such a transformation defines an equivalence class. As such, a generalized scale-space can be considered as a conventional scale-space ‘in disguise’, representing the scale-space data in a format that is more convenient for specific applications. Conventional, linear scale-space is a convenient representative of the equivalence class for general purposes. The equivalence constraint is shown to yield a particular class of (linear or nonlinear) diffusion equations. The emphasis is on nonlinear scale-spaces, although the principle of equivalence can be used within the linear context as well. Examples are included to illustrate the theory both in the linear as well as in the nonlinear sector.