图像分割是图像处理中非常基础与关键的研究领域，近年来，基于深度学习的方法在图像分割领域取得了巨大的成功，然而一些基于深度学习的图像分割方法是有监督的，需要大量高质量人工标注的真实标签与图像数据一起进行训练，而在医学图像、遥感图像等很多领域中，往往没有充足的高质量有标注数据。为了利用大量无标签数据，基于无标签数据潜在的标签分布应与有标签数据的真实标签分布一致这一先验，一些可以缩小两个分布之间差异从而实现分布对齐的对抗生成式方法在该问题中被广泛应用。这些方法对齐标签的全局分布，一方面全局分布之间的差异较大并且结构更加复杂，不利于分布对齐的计算；另一方面，即便全局分布对齐地很好，不同类之间的样本也可能被错误的混淆对应，类之间的样本不平衡时，也会出现频率高的类之上的对齐效果好于频率低的类。针对上述全局分布对齐的问题，为了利用无标签数据及其各类特征所形成的潜在子分布先验，本文提出的基于光滑无穷范数的对抗式半监督分割网（∞）对标签的多维子分布进行对齐，并利用光滑无穷范数的性质实现对多
维子分布施加自适应权重的对齐，从而有效利用无标签数据进行训练。∞利用数据集之中蕴含的有助于分割的一些强先验信息，从图像真实标签之中提取一些对分割有帮助的特征，并将这些特征嵌入回真实标签所在的空间，得到与各个特征相对应的多维子分布。算法通过有标签数据反复迭代生成无标签数的潜在各类特征的多维子分布。通过子分布一致性先验，∞ 利用光滑无穷范数的对偶形式构造了一个衡量多维子分布之间差异的度量，该度量利用光滑无穷范数计算最大值函数的光滑近似，实现根据各个子分布的重要性施加自适应的权重。通过优化这些子分布的差异，算法推导出一个鞍点问题，通过交替极小化的方式近似计算鞍点问题得到一个对抗性损失。利用对抗性损失结合有标签数据的分割损失，对抗式地交替训练分割网络以及两个对偶函数网络，实现多维子分布的对齐。其中子分布的权重由其中一个对偶函数网络学习得到因此是自适应的。本文在医学与遥感数据集上进行了一系列数值实验，评估了∞ 方法在半监督分割问题中的有效性，通过消融实验验证了多维子分布对齐相比全局分布对齐方法的优势，并进行了 ∞ 方法与多个独立判别器
的其他对抗生成类方法的对比实验，验证了本文自适应权重对齐方法的优势。

Image segmentation is a fundamental and critical research field in image processing. In recent years, deep learning-based methods have achieved tremendous success in the field of image segmentation. However, some deep learning-based image segmentation methods are supervised, requiring a large amount of high-quality annotated ground truth labels to be trained together with image data. In many domains such as medical imaging and remote sensing, there is often a lack of sufficient high-quality annotated data. In order to leverage a large amount of unlabeled data, based on a prior that the potential label distribution of unlabeled data should align with the true label distribution of labeled data, adversarial generative methods have been widely applied in this problem, which can reduce the discrepancy between the two distributions and achieve distribution alignment. These methods align the global distributions of labels. On the one hand, the significant differences and complex structures between global distributions make it challenging to compute distribution alignment. On the other hand, even if global distribution alignment is achieved, samples from different classes may still be erroneously confused. Moreover, alignment may perform better on classes with higher frequencies than on those with lower frequencies when there is class imbalance. To address the aforementioned issues with global distribution alignment, and to leverage the consistency prior of sub-distributions formed by unlabeled data and their features, we propose an adversarial semi-supervised segmentation network based on the smooth ∞ norm (∞). This network aligns the multidimensional subdistributions of labels and utilizes the smooth ∞ norm to apply adaptive weights to the alignment of multidimensional sub-distributions, effectively utilizing unlabeled data for training. ∞ utilizes certain strong prior information inherent in the dataset that aids segmentation to extract some helpful features from the true labels of the images and embeds these features into the space of the true labels. In this way, we get multidimensional sub-distributions corresponding to each feature. The algorithm iteratively generates multidimensional sub-distributions of the latent features for unlabeled data by iteratively processing labeled data. Based on the prior of sub-distribution consistency, ∞ constructs a measure of the difference between multidimensional sub-distributions using the dual form of the smooth ∞ norm. This measure utilizes the property of calculating softmax values using the smooth ∞ norm and applies adaptive weights based on the importance of each sub-distribution. By optimizing the differences between these sub-distributions, the algorithm derives a saddle point problem, and derives an adversarial loss by alternately minimizing the saddle point problem. Combined the adversarial loss with the segmentation loss from the labeled data, the segmentation network and two dual function networks are adversarially trained alternately to achieve alignment of multidimensional sub-distributions. The weights of the subdistributions are learned by one of the dual function networks, making them adaptive. This paper conducts a series of numerical experiments on medical and remote sensing datasets to assess the effectiveness of the ∞ method in semi-supervised segmentation problems. Through ablation experiments, the advantages of multidimensional sub-distribution alignment over global distribution alignment methods is verified. Additionally, comparative experiments are conducted between the ∞ method and other adversarial generative methods with multiple independent discriminators, demonstrating the advantages of the proposed adaptive weight alignment approach.