We propose, analyze and realize a variational multiclass segmentation scheme that partitions a given image into multiple regions exhibiting specific properties. Our method determines multiple functions that encode the segmentation regions by minimizing an energy functional combining information from different channels. Multichannel image data can be obtained by lifting the image into a higher dimensional feature space using specific multichannel filtering or may already be provided by the imaging modality under consideration, such as an RGB image or multimodal medical data. Experimental results show that the proposed method performs well in various scenarios. In particular, promising results are presented for two medical applications involving classification of brain abscess and tumor growth, respectively. As main theoretical contributions, we prove the existence of global minimizers of the proposed energy functional and show its stability and convergence with respect to noisy inputs. In particular, these results also apply to the special case of binary segmentation, and these results are also novel in this particular situation.

翻译：我们提出、分析并实现了一种变分多类分割方案，该方案将给定图像分割成多个具有特定属性的区域。本方法通过最小化一个融合多通道信息的能量泛函，确定编码分割区域的多个函数。多通道图像数据可通过将图像提升到更高维特征空间（使用特定多通道滤波）获得，或直接由所考虑的成像模态提供，例如RGB图像或多模态医学数据。实验结果表明，该方法在各种场景下均表现良好，特别是在脑脓肿分类和肿瘤生长两个医学应用中取得了令人满意的结果。作为主要理论贡献，我们证明了所提能量泛函全局极小值的存在性，并展示了其对噪声输入的稳定性与收敛性。值得注意的是，这些结果同样适用于二值分割这一特例，且在该特殊情形下亦具有创新性。