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图像分割方案，该方案将给定图像分割成多个具有特定性质的区域。我们的方法通过最小化一个融合多通道信息的能量泛函，确定多个编码分割区域的函数。多通道图像数据可以通过使用特定多通道滤波将图像提升到更高维特征空间获得，或由所考虑的成像模态直接提供（如RGB图像或多模态医学数据）。实验结果表明，所提方法在多种场景下表现良好：特别是在脑脓肿分类与肿瘤生长两个医学应用中展现了有前景的结果。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，并证明了其针对噪声输入的稳定性与收敛性。值得注意的是，这些结论同样适用于二值分割这一特例，且在该特例中亦具有新颖性。