Adaptive active contours driven by the squared Hellinger distance and local correlation features for inhomogeneous image segmentation

Abstract

Intensity inhomogeneity is an intractable issue that leads to significant challenges for image segmentation. Moreover, inhomogeneous images are very common in real life, especially the medical images generated by medical instruments. Most existing active contour models yield poor performance when they are applied to segment these images. Thus, a new image segmentation model is developed in this paper by integrating the squared Hellinger distance and multiple truncation functions that are defined by local correlation features into the level set framework. As a result, the problems of intensity inhomogeneity and dark edges can be effectively solved. Specifically, the Hellinger distance is first defined as a fitting method of the energy function and has better performance in splitting the boundary of dark areas in inhomogeneous images. More importantly, the proposed model constructs local correlation features (LCFs) to redefine the scale of the local areas and makes the division of the local information more reasonable, which effectively enhances the accuracy of inhomogeneous images. Then, to combine the global and local information of the image, an adaptive combination coefficient function defined by the evolution process is applied in our model; moreover, the length term combined with the saliency feature of the image achieves a remarkable improvement effect. Finally, all terms are integrated into a variational level set framework, and our new image segmentation model is proposed. The experimental results on medical and natural inhomogeneous images demonstrate the excellent performance of our model over most state-of-the-art active contour models in terms of accuracy and robustness.

Appendix

1.1 A1 Adaptive weight connection coefficient

Figure 14 shows the curve of evolution weight $w_l$ and $w_g$ to iteration times t, and we can find that the fitting energy term can be divided into two stages in the segmentation process. In the first stage, the global term plays a major role in energy fitting, and the evolution curve at this stage can have strong robustness to the initial region of evolution and evolve to the basic contour of the target object. In the second stage, the local term is the dominant force in energy fitting, and the evolution curve of this stage can be refined and segmented based on the basic contour of the previous stage. The phased evolution causes the model to obtain a suitable energy fitting method and effectively solve the problem of the initial region and local optimal solution.

Fig. 14

The curve of evolution weight $w_l$ and $w_g$ to iteration times t

1.2 A2 The $L^*$ of energy-optimized adaptive local term

We present an example shown in Fig. 15. $L^*$ is small in the complex limbic regions of the cerebral cortex, the novel truncation function (Gaussian kernel function) can restrict a small local area and achieve accurate segmentation of the region, $L^*$ is large in the smooth regions of the brain stem, and the novel truncation function can restrict a large local area and prevent evolution from falling into local optima. In other words, the constructed function $\Gamma _{L^*}$ can assign an appropriate local area range according to the intensity characteristics of different areas and help to improve model segmentation performance.

Fig. 15

$L^*$ of brain magnetic resonance images

1.3 A3 The comparison experiment of iteration times

We select complex images with large differences in evolution results from the above experimental images and analyze their evolution efficiency. The results are shown in Table 5. We can see that our proposed model needs a minimum number of iterations, which is based on better and more rational energy construction. Meanwhile, our model needs two convolution operations in each iteration, which is less than the RSF [19] and LGIF [34] and the same as the JDACM [14]. Therefore, the total amount of computation in the segmentation process of the proposed model is minimal. According to the above comparison experiments, the proposed model is more accurate and efficient than existing models for segmenting images with intensity inhomogeneity.

Table 5 Comparison of iteration times of different models

1.4 A4 The supplementary ablation experiment on truncation function

In Fig. 16, experimental images from the left column to the right column are the evolution results of $\sigma =0.5$, $\sigma =1$, $\sigma =3$, $\sigma =6$ and adaptive $\sigma$ based on local correlation feature in our model, the 1st and the 2nd row are two groups of the segmenting results and the blue boxes are examples of the local areas with distinct differences. The 3rd row is binary images of segmentation results. We evaluate the effectiveness of the novel truncation function in the proposed model. The experimental image was an MR image of the brain with complex internal structure and intensity inhomogeneity. When $\sigma$ is equal to 0.5, the model has a small local region division, the segmenting process is slow, and some areas are not accurate. When $\sigma$ is equal to 1, the segmentation result has better performance in some areas of the image but still has defects at the edges of the cerebral cortex. When $\sigma$ is equal to 3, the model has fallen into the local optimal solution due to the unreasonable division of local areas. When $\sigma$ is equal to 6, the local region scope of the model is large, and many details of the target object (such as marginal details of the cerebral cortex) cannot be accurately segmented. In contrast, our proposed model adopts the truncation function based on the local correlation feature (LCF), which can divide smaller local areas of the image detail part to enhance the evolutionary accuracy and divide larger local areas of the rest of the image to solve the problem of falling into the local optimal solution. The 3rd row of Fig. 16 shows binary images of the segmentation results. We can see that the segmentation of the proposed model is more refined and has smoother edges; moreover, our results have no extra small, isolated regions compared to other images. The DSC and JS values of the proposed model based on the novel truncation function and the comparison method of fixed $\sigma$ values for segmenting brain MR images are shown in Table 6.

Fig. 16

Ablation experiment on truncation function

Table 6 Dice Similarity Coefficient and Jaccard Similarity of ablation experiment on truncation function shown in Fig. 16

We know from the data that the proposed model using the $\sigma$ based on the local correlation feature gives higher DSC and JS values than fixed $\sigma$, which indicates that the novel truncation function effectively enhances the applicability and segmentation accuracy of the model.

1.5 A5 The comparison experiment on $\mu$

We construct a comparative experiment to analyze the effect of variable $\mu$ on segmentation results. $\mu$ is used to constrain the weight of the length term in the model. The greater its value, the smoother the segmentation curve, which can effectively reduce the segmentation error. The previous model [19] usually sets $\mu = 0.001\times 255^2$, and our proposed model has segmentation errors due to the change in the length term by integrating image salience, as shown in Fig. 17. Therefore, we increase $\mu = 0.002\times 255^2$ to constrain the length of the segmentation curve and reduce errors. The experimental results show that the method has made effective improvement, but when we continue to increase $\mu = 0.003\times 255^2$, the length term has negative impacts on results. Finally, we believe that the optimal selection of $\mu$ is $0.002\times 255^2$.

Fig. 17

The comparison experiment on $\mu$

1.6 A6 The comparison experiment on $\kappa$

We construct a comparative experiment to analyze the effect of variable $\kappa$ on segmentation results. $\kappa$ is the coefficient of the adaptive scale kernel, which controls the local areas. In experiments, we select 5, 7, 10, 15, and 20 for $\kappa$. The results are shown in Fig. 18. We can see that the small value of $\kappa$ such as 5 and 7 limits the evolution rate and makes the segmentation results partially missing. The large value of $\kappa$ such as 15 and 20 causes errors other than objects in the resulting segmentation. Therefore, we choose 10 as the optimal value of $\kappa$.

Fig. 18

The comparison experiment on $\kappa$

1.7 A7 The comparison experiment on $\alpha$ and $\beta$

Furthermore, we design experiments to analyze the effects of $\alpha$ and $\beta$ on segmentation results. $\alpha$ and $\beta$ are the weight coefficients of the global energy-optimized term and adaptive local energy-optimized term in the proposed model. How to adjust and select the values of two parameters is important for the evolution of the active contour model. As shown in Fig. 19, we use the same initial evolution curve and gradually change the values of the two parameters through the single variable method. The experimental results are shown in Fig. 20. We can observe that the smaller values of $\alpha$ and $\beta$ make the energy-optimized terms (global and local) account for a smaller proportion of the entire active contour model. The length term plays a major role in the model and makes the segmentation curve evolve into a small smooth curve. We gradually increase the values of $\alpha$ and $\beta$, which improves the segmentation performance. In particular, changing the value of $\alpha$ directly affects the global term, and the model tends to evolve the overall contour of the object. If the value of $\alpha$ is too large such as 400, there are defects of insufficient evolution accuracy and evolution errors. Changing the value of $\beta$ directly affects the local term. The model pays more attention to the details of the objects with the growing of $\beta$, but a large value of $\beta$ such as 2000 makes the evolution curve contain many detail errors. Finally, we consider all the results of the experiment comprehensively and found that when $\alpha$ is equal to 40 and $\beta$ is equal to 400, the evolution result is the closest to the ground truth. Therefore, we reasonably select the best value of the two coefficients.

Fig. 19

The initial box and ground truth of the comparison experiment

Fig. 20

The comparison experiment results on $\alpha$ and $\beta$