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Motivated by the problem of dealing with incomplete or imprecise acquisitionof data in computer vision and computer graphics, we extend results concerningthe stability of persistent homology with respect to function perturbations toresults concerning the stability with respect to domain perturbations. Domainperturbations can be measured in a number of different ways. An importantmethod to compare domains is the Hausdorff distance. We show that by encodingsets using the distance function, the multidimensional matching distancebetween rank invariants of persistent homology groups is always upperly boundedby the Hausdorff distance between sets. Moreover we prove that our constructionmaintains information about the original set. Other well known methods tocompare sets are considered, such as the symmetric difference distance betweenclassical sets and the sup-distance between fuzzy sets. Also in these cases wepresent results stating that the multidimensional matching distance betweenrank invariants of persistent homology groups is upperly bounded by thesedistances. An experiment showing the potential of our approach concludes thepaper.

Stability of multidimensional persistent homology with respect to domain perturbations