## 2 Theoretical Background  
A number of algorithms and techniques for image segmentation have been developed and implemented over the years, and a large amount of literature papers about this non-trivial task were proposed. The aim of this section is to present a brief overview about theoretical notions of literature approaches from which this work drawn inspiration. For this reason, we decided for a comprehensive description, to provide the reader with all the technical information needed to understand the functioning of each of the investigated algorithms, as well as the implemented workflow.  
The segmentation involves the image partitioning into homogeneous and meaningful sub-regions. By a formal point of view, the segmentation of an image \(\mathcal{I}\) involves the identification of a finite set of regions \(\mathcal{R}_{1},\mathcal{R}_{2},\ldots,\mathcal{R}_{N}\) as in Equation (1):  
\[\bigcup_{i=1}^{N}\mathcal{R}_{i}=\mathcal{I}^{\prime} \tag{1}\]  
with the following constraints:  
\[\mathcal{R}_{i}\cap\mathcal{R}_{j}=\emptyset,\quad\text{for}\quad i\
eq j, \tag{2}\]  
\[\mathsf{P}(\mathcal{R}_{i})=\mathsf{TRUE},\quad\text{for}\quad i=1,2,\ldots,N, \tag{3}\]  
\[\mathsf{P}(\mathcal{R}_{i}\cup\mathcal{R}_{j})=\mathsf{FALSE},\quad\text{for }\quad i\
eq j. \tag{4}\]  
With more details, \(\mathsf{P}\) is an appropriate logical predicate leading the segmentation process. Equation (1) states that the union of all the sub-regions resulting from segmentation process. Equation (2) points out that the intersection of two different sub-regions is the empty set: this means that the segmented sub-regions do not overlap each other. According to Equation (3), the result of the logical predicate \(\mathsf{P}\) on all the pixels belonging to the same sub-region is always TRUE; in other words, all the pixels belonging to the same region share the same characteristics according to the predicate \(\mathsf{P}\). As a consequence, Equation (4) states that the result of \(\mathsf{P}\) on the union of two distinct sub-regions is FALSE.  
It is important to clarify that each type of medical image has a specific set of features reflecting its own properties: in fact, each image is the result of a complex interaction between the human body and the scanner (i.e., X-rays for CT, magnetic fields for MRI, radioactive decay for nuclear medicine exams) [19]. As a consequence, not all the segmentation techniques obtain the same results on all image types, but some algorithms yield better results when applied on a specific kind of image. Furthermore, it is necessary to point out that, typically, an approach that works very well with one type of image does not mean that it continues to perform well even on different images. As a matter of fact, bioimages exhibit a very high variability, as they depend on various factors, both intrinsic (e.g., patients) and extrinsic (e.g., imaging modalities, acquisition parameters). Accordingly,ad hoc modifications might be required at the level of both the segmentation approach and the pre-/post-processing phases.  
In what follows, we outline the techniques investigated and compared for contrast-enhancing mass segmentation on DCE-MRI.