In this article, we present a novel approach devoted to robustly compute the Reeb graph of a digital binary image, possibly altered by noise. We first employ a skeletonization algorithm, named DECS (Discrete Euclidean Connected Skeleton), to calculate a discrete structure centered within the object. By means of an iterative process, valid with respect to Morse theory, we finally obtain the Reeb graph of the input object. Our various experiments show that our methodology is capable of computing the Reeb graph of images with a high impact of noise, and is applicable in concrete contexts related to medical image analysis.