Abstract. Stereo matching is an ill-posed problem for at least two principal reasons: (1) because of the random nature of match similarity measure and (2) because of structural ambiguity due to repetitive patterns. Both ambiguities require the problem to be posed in the regularization framework. Continuity is a natural choice for a prior model. But this model may fail in low signal-to-noise ratio regions. The resulting artefacts may then completely spoil the subsequent visual task. A question arises whether one could (1) find the unambiguous component of matching and, simultaneously, (2) identify the ambiguous component of the solution and then, optionally, (3) regularize the task for the ambiguous component only. Some authors have already taken this view. In this paper we define a new stability property which is a condition a set of matches must satisfy to be considered unambiguous at a given confidence level. It turns out that for a given matching problem this set is (1) unique and (...