A Chan-Vese Model Based on the Markov Chain for Unsupervised Medical Image Segmentation

Superpixels are useful in a wide range of vision tasks, and we can fully take advantage of boundary information for subsequent segmentation due to the superpixel. We test the results of whether or not to use superpixels in our proposed method. In the case of not using superpixels, each node of the undirected graph corresponds to a particular pixel of the input image. As shown in Fig. 3 , in the images with intensity inhomogeneity, nodes with coarser granularity have more boundary information. Therefore, the superpixels provide more cues compared with a single pixel, and the segmentation results do not easily fall into the local optima. Hence, we can derive additional global results.

10.26599/TST.2020.9010042.F003 Fig. 3Comparison of whether to use superpixels.

In our method, we adopt the standard SLIC algorithm. Given the single-channel characteristics of medical images, we apply the algorithm on the original features rather than the XYLab features. In the choice of parameters, we test the number of superpixels K on the dataset MINI-CVIP, and the experiment shows that as the value of K decreased (the number of pixels in each superpixel increased), the segmentation performance did not fluctuate within a certain range (as shown in Fig. 4 ) and then declined rapidly. The smaller K leads to more pixels in each superpixel, ultimately reducing the boundary segmentation precision. Therefore, for experimental accuracy and computational efficiency, we keep the number of pixels in each superpixel at approximately 10; that is, the value of the parameter K is approximately $N/10$ .

10.26599/TST.2020.9010042.F004 Fig. 4Influence of different numbers of superpixels on the segmentation result.

Table 1 shows the segmentation results of different methods. To demonstrate the merits of our model, we compare K-means clustering, the Chan-Vese model, the snake model, the Local Binary Fitting (LBF) model, and our method. In addition, we compare our method with the supervised method U-net on some datasets. In all the experiments, all methods are tested on the corresponding testing set. Table 1 shows that our method achieves the highest accuracy for majority of the datasets. The K-means method focuses on global losses and does not perform well on images with intensity inhomogeneity. Compared with the snake and Chan-Vese models, our method considers local information and shows better segmentation results in all experiments. LBF, an implicit active contour model driven by local binary fitting energy, performs best on the CVIP dataset. However, each image’s processing time takes long due to the introduction of the convolution operation. Although U-net performs better on the SPINE dataset, this performance is mainly due to the fact that spines have roughly the same shape. Another reason is that an extensive training set with 87 training images is provided.

As shown in Fig. 5 , obtaining the separated bones is difficult because no clear distinction exists between the gap and bones; thus, other areas can be easily mistaken as bones. With our method, we can fit the target boundary well. In addition, the similarity between superpixels is used to enhance the local relevance to obtain a compact segmentation. To achieve a compact contour, the Chan-Vese model introduces a smoothing term, which leads to segmentation results that cannot fit the target boundary well, especially for images with intensity inhomogeneity. For the snake model, the result is often not locally smooth, and the final result may have many isolated contours.

10.26599/TST.2020.9010042.F005 Fig. 5Segmentation results of different methods. Each row represents a sample corresponding to the ground truth and the segmentation results under different methods.

Our proposed method has remarkable advantages in terms of initialization and iteration times. Compared with our model, the Chan-Vese model is more sensitive to initialization. In many cases, it does not converge to the target region. As shown in Fig. 6a , if the initial contour is around the target, then the algorithm easily obtains the desired segmentation; otherwise, as shown in Fig. 6b , the result is poor due to the failure to consider the remote pixel. In addition, as shown in Fig. 7 and Fig. 8 , our method iterates less than ten times in all experiments compared with the Chan-Vese model, which requires dozens and even hundreds of iterations to achieve convergence. Moreover, our method has a certain iterative stop condition; that is, the min-cut of the undirected graph is no longer changed. However, for the Chan-Vese model, the evolution contour produces different results due to different iteration times and stop conditions. It is irreversible for the evolution contour to cross the target contour in the iteration process.

10.26599/TST.2020.9010042.F006 Fig. 6Results of Chan-Vese model and our model under different initializations. The green line represents the initial contour, the blue line is the segmentation result of the Chan-Vese model, and the red line is the result of our model.

10.26599/TST.2020.9010042.F007 Fig. 7Result of the Chan-Vese model and our model under different numbers of iterations.

10.26599/TST.2020.9010042.F008 Fig. 8Segmentation performance under different numbers of iterations (MINI-CVIP).

We further explore the parameters of our method. In practice, the segmentation results are adjusted mainly by four parameters ( ${\lambda }_{g0},{\lambda }_{g1},{\lambda }_{l0}$ , and ${\lambda }_{l1})$ . ${\lambda }_{l0}$ and ${\lambda }_{g0}$ have the same function ( ${\lambda }_{l0}$ is mainly used for intensity inhomogeneous images) as ${\lambda }_{l1}$ and ${\lambda }_{g1}$ do. As shown in Fig. 9 , by setting different ${\lambda }_{l1}$ and ${\lambda }_{l0}$ , we can find that the larger ${\lambda }_{l0}$ is, the more uniform the interior of the foreground target becomes; conversely, the larger ${\lambda }_{l1}$ is, the more consistent the interior of the background becomes. In addition, we conduct ablation studies, as shown in Table 2 . Compared with using only global information, the weight maps we introduced result in great improvement because the weight maps mainly consider the pixels in the surrounding neighborhood. Therefore, in actual use, images with intensity inhomogeneity must be segmented by adjusting ${\lambda }_{l1}$ and ${\lambda }_{l0}$ .

10.26599/TST.2020.9010042.F009 Fig. 9Segmentation results under different parameters.