# Mixed-Sized Biomedical Image Segmentation Based on U-Net Architectures
## 2 Background and Related Works  
### Segmentation of Medical Images  
In radiology, a CT-Scan is a diagnostic technique used for reproducing sectional and three-dimensional images of the human body. Images are obtained from the computerized analysis of the information present in the X-ray scans. Since each image is the projection of the object itself from one of multiple angles, it is possible to reconstruct three-dimensional objects.  
In general, medical Image Segmentation can be useful for multiple purposes [8], such as:  
* To diagnose conditions, including damage and injury to internal organs and bones, stroke, cancer and problems with blood flow;
* To guide tests and treatments. For example, before radiotherapy treatment, a CT scan is performed to determine the location, size and shape of the tumor to be treated;
* To monitor the evolution of patients' conditions, such as the presence and size of the tumor during and after specific treatments.  
In order to quantitatively describe radiodensity, the Hounsfield scale, or Hounsfield Unit (HU), is typically regarded as the reference unit of measurement [9]. The HU values give preliminary information on the nature of the observed tissues.  
_Image Segmentation_ is a technique of partitioning an image into distinct and meaningful parts, called segments. The purpose of this process is to simplify and change the representation of images, for identifying and extracting some objects of particular interest and making it easier to analyze individually. In fact, it is particularly useful in applications of computer vision, such as object recognition, image compression and analysis of digital image content. In the medical area, segmentation is useful for many purposes, such as to locate and identify tumor cells, measure tissue volumes, perform virtual surgery simulations and intra-surgical navigation and integrate slower and more subjective manual human labor.  
It is possible to categorize segmentation techniques into three classes: clustering, edge detection and region extraction [3]. The two macro problems to avoid are sub-segmentation, which means merging semantically different objects in the same area and over-segmentation, or the subdivision of the same object into multiple areas. In order to make algorithms autonomous, they should not assume any prior knowledge of the image to be available; otherwise, it would be difficult to ensure that results are satisfactory for any type of analysis. It is also possible to obtain the segmentation of the image in a number of regions, such that each of them is homogeneous and coherent with respect to a certain criterion and at the same time their union returns the original image. To this end, the following formal definition of the segmentation problem is given [10]: let \(X\) be the image domain and let \(P\) be a homogeneity predicate, that is a feature extracted from the image and associated with each pixel, defined on a set \(S\) of connected pixels of \(X\). The segmentation of \(X\) consists of the partition of \(X\) into a number of \(n\) sub-images or regions \(S_{i}\), with \(i=1,\cdots,n\), such that:  
\[\bigcup_{i=1}^{n}S_{i}=X \tag{1}\]  
\[S_{i}\cap S_{j}=\emptyset\qquad\forall(i,j)(i\
eq j)\]  
\[P(S_{i})=true\qquad\forall i=1,\cdots,n\]  
\[P(S_{i}\cup S_{j})=false\qquad\forall(i,j)(i\
eq j)\]  
However, the partition of an image into homogeneous regions with respect to certain characteristics does not guarantee the correct subdivision into semantic objects, especially in particularly complex images. In this regard, it is not certain that the automatic segmentation of images admits a single solution that is also robust. Nevertheless, it is possible to reach the expected result with a good approximation by implementing a multi-level convolutional model [11].