Huang Jisui, Chen Ke, Alpers Andreas, Lei Na
School of Mathematical Sciences, Capital Normal University, West Third Ring Road North, Haidian District, 100048 Beijing China.
Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L697ZL Merseyside United Kingdom.
Int J Comput Vis. 2025;133(9):6491-6512. doi: 10.1007/s11263-025-02492-6. Epub 2025 Jun 15.
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.
现有的水平集模型仅基于梯度信息、一维曲率或二维曲率进行正则化。然而，对于三维图像分割，合适的基于曲率的正则化应涉及定义明确的三维曲率能量。本文首次引入一种正则化能量，该能量结合了三维标量曲率用于三维图像分割，其灵感来源于爱因斯坦 - 希尔伯特泛函。为了推导其欧拉 - 拉格朗日方程，我们采用两步梯度下降策略，交替更新水平集函数及其梯度。本文还证明了所提出模型的粘性解的存在性和唯一性。实验结果表明，我们提出的模型在三维图像分割方面优于其他现有最先进的模型。
