The estimation of multiple orientations in multidimensional signals is a strongly non-linear problem to which a two-step solution is here presented. First, the problem is linearized by introducing the so-called mixedorientation parameters as a unique, albeit implicit, descriptor of the orientations. Second, the non-linearities are decomposed such as to find the individual orientations. For two-dimensional signals, e.g., images, this decomposition step is solved by simply determining the roots of a polynomial. For multi-dimensional signals, the nD decomposition problem is solved by reducing it to a cascade of 2D decomposition problems. In this way, a full solution for the estimation of any number of orientations in any dimension is achieved for the first time.