Traditional approaches to estimate a scale invariant spatial scope for local image descriptors, a.k.a. characteristic scale, work well for intensity images. However, they fail when it comes to deal with binary images. We address this problem and propose a new method to estimate the characteristic scale of local shape descriptors. The proposed method extends the use of the distance map transform to produce similarity maps that approximate local intensity changes in binary images. We first validated our method evaluating the consistency of characteristic scales estimated across scaled instances of images; and then by comparing its performance, with respect to traditional methods, in the tasks of Content-Based Image Retrieval and shape detection in different datasets of binary images (shapes of Maya and Chinese hieroglyphs, and generic shapes). As shown by our results, the proposed similarity map produces characteristic scales that are more robust to scale variations, and leads to competitive results.