\section{Introduction}

Segmentation of anatomical structures is an important step in many computer-aided procedures, like medical image navigation and detection algorithms. Many of these methods rely on manually segmented inputs performed by clinical experts. However, this is a time-consuming task due to the large amount of information (generally volumes) that is generated. Organ segmentation in  CT or MRI slices has been a topic of research for many years. Recently, with the growth of deep learning models, many architectures have been proposed for dealing with this problem. One of these challenges is related to the similarity between organs and background yielding to misclassifications, mainly in boundary regions of the organs resulting in many false positives (FP) and false negatives (FN) regions. Such a problem hinders the model integration into clinical practice, where higher precision is required. One way to improve the model performance is by introducing a post-processing refinement step in the pipeline. 

Recent segmentation models for medical structures are based on convolutional neural networks (CNN). These models  can be composed of aggregations of multiple 2D CNN \cite{bib:Zhou17,bib:Roth17} or by 3D CNN \cite{bib:Zhu17,bib:Roth18}. Similarly, models that incorporate shape and geometric priors have been recently proposed \cite{bib:yuyin19,bib:yao19,degel2018domain}. 
Refinement strategies are typically introduced at the end of the process to improve the model's prediction. This can also be used as an intermediate processing step, where more complex strategies can use the refined results to improve the segmentation. For example, in \cite{bib:Wang18}, a set of scribbles is generated by defining a conditional random field (CRF) problem that is solved with Graph Cut methods. These results can be combined with user-defined scribbles to perform an image specific fine-tune of a CNN segmentor. In another context, given the limited availability of labeled medical data, semi-supervised learning methods define strategies to include the (most commonly) available unlabeled medical data. Such strategies include the generation of pseudo-labels for unlabeled data. Here, refinement methods, like densely connected CRF \cite{bib:Bai17} are included in the semi-supervised steps, to refine the pseudo-labels. Uncertainty has also proved to be useful as an attention mechanism in semi-supervised learning \cite{bib:Xia18} and recent works in computer vision have started to explore the capabilities of uncertainty for finding potential misclassified regions for segmentation refinement purposes \cite{bib:Dias18}. In the medical context, uncertainty has been employed as a measure of quality for the segmented output \cite{bib:Roy18}, and its ability to reflect incorrect predictions has been recently studied \cite{bib:Nair18}. A recent work, presented by \cite{bib:Yu19}, uses the uncertainty of a teacher model to select the pseudo-labels to train a student model.

Even though dense graph representations of three-dimensional data have been applied for refinement \cite{bib:Kamnitsas16}, the use of recent graph convolutional networks (GCN)  with sparse graphs representations of 3-D data has not been fully investigated. In this paper, we propose a two-step approach for the refinement of volumetric segmentation coming from a CNN. First, we perform an uncertainty analysis by applying Monte Carlo dropout (MCDO) \cite{bib:Kendall17} to the network to obtain the model's uncertainty. This is used to divide the CNN output in high confidence background, high confidence foreground and low confidence points (FP and FN candidates). The uncertainty is also used to define a 3-D shape-adapted region of interest (ROI) around the organ. With this information, we define a semi-labeled graph inside the ROI. We use this graph to train a GCN in a semi-supervised way using the high confidence predictions as labeled training nodes for the GCN. The refined segmentation is obtained by evaluating the full graph in the trained GCN.

\textit{Contributions}: To our best knowledge, this is the first time a semi-supervised GCN learning strategy is employed in the medical image segmentation task, specifically, for single organ segmentation. Also, this work presents one of the first cases of using GCN and uncertainty analysis for segmentation refinement. We provide a framework in which a per voxel segmentation refinement task is mapped into a semi-supervised graph classification problem. Thanks to the Monte-Carlo dropout estimation, voxels with high confidence are treated as labelled nodes in the graph, while the rest are treated as unlabelled nodes, and the main objective is to learn a GCN model to classify the unlabeled ones. Our framework operates on top of a CNN in inference time and it does not need to retrain the network. 