Traditional variational models often fail to segment images in the presence of inhomogeneity or weak boundaries, partly due to their reliance on unreliable region metrics that quantify inhomogeneity based on a single mean value or a smoothed image serving as a mean function. The former, such as variance-based methods, are highly sensitive to image inhomogeneity, whereas the latter, such as local convolution-based approaches, lack a global receptive field. To address these issues, we employ an optimal transport-based data fidelity term in our segmentation objective functional. This term accounts for global differences between regions, resolving problems arising from local convolutions. It can also adaptively seek an optimized match between two probability density functions, proving more robust than relying solely on their mean values. Our proposed functional is minimized by gradually performing region merging. Experimental results demonstrate that our model outperforms state-of-the-art variational and deep learning models.

Existing level set models employ regularization based only on gradient information, 1D curvature or 2D curvature. For 3D image segmentation, however, an appropriate curvature-based regularization should involve a well-defined 3D curvature energy. This is the first paper to introduce a regularization energy that incorporates 3D scalar curvature for 3D image segmentation, inspired by the Einstein-Hilbert functional. To derive its Euler-Lagrange equation, we employ a two-step gradient descent strategy, alternately updating the level set function and its gradient. The paper also establishes the existence and uniqueness of the viscosity solution for the proposed model. Experimental results demonstrate that our proposed model outperforms other state-of-the-art models in 3D image segmentation.

3D medical images consist of voxels with points, edges, faces, and volumes. A fascinating question is how to compute the shortest basis of the first homology group of a 3D image. The fastest time complexity known for this question is O($n^{\omega}+n^{2}$g), where n is the size of voxels and $\omega \lt$ 2.3728639 is a quantity so that two n×n matrices can be multiplied in O($n^{\omega}$) time. But it is still slow in practical applications. We first construct a hexahedral mesh of an arbitrary domain of a 3D image and second propose an approximate algorithm with time complexity O($n^{\omega}$) to calculate the minimal homology basis for the 3D images. Experiments show that our approximate algorithm is very close to the exact algorithm. We demonstrate the effectiveness of our algorithm in segmenting the semicircular canals, the organ with complex topology.