Dice Score

本文主要基于Generalised Wasserstein Dice Score for Imbalanced Multi-class Segmentation using Holistic Convolutional Networks进行整理。

hard dice score for binary segmentation

dice score被广泛使用的针对二值分割图S和G之间成对比较的重叠度量方式。
其可以表示为集合操作或统计性度量的形式：

这里涉及到几项，具体含义如下：

• ：待评估图像和参考图像
• ：正阳性样本的数量，即和都为真的位置的数量。
• ：%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-53%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=S&id=Oj0U9)中真而中为假的位置的数量。
• ：%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-53%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=S&id=U6gwG)中假而中为真的位置的数量。

• ：和不一致的位置的数量。

  soft dice score for binary segmentation

  对于软二值分割的扩展依赖于概率分类对的不一致概念。
  对于和中的位置对应的类别和可以被定义为标签空间上的随机变量。
  概率分割可以被表示为标签概率图，其中表示标签概率向量的集合：

• 

• 

由此可以将前面的关于数据的统计量扩展到软分割情形：

对于一般情形中的，即，此时有：

对应的soft dice score可以表示为：

当然，也有引入平方形式的变体。

soft multi-class dice score

前面直接讨论的是二值分割的情形，而对于多分类情况则需要考虑不同类别计算的整合方式。
最简单的方式就是直接考虑所有类别的平均。
可以称为mean dice score，这里对应包含个不同的类别：

上式的推广形式可以通过引入类别权重参数而得到。即从而将上式转化为加权平均的形式。这被称为generalised soft multi-class dice score。
最终可以表示为：

soft multi-class wasserstein dice score

前面的dice score的形式中，对于和的相似性的度量方式可以看做是L1距离，而这里将wasserstein distance引入来自然地以一种语义上有意义的方式比较两个标签概率向量。
这里首先介绍wasserstein distance。

wasserstein distance

这也被称为earth mover’s distance。用于表示将一个概率向量变换为另一个概率向量所需要的最小成本。
对于所有的，从移动到的距离的集合定义为和之间的距离矩阵，这一矩阵是固定的，可以认为是已知的。
这是一种将上的距离矩阵（通常亦可以称为ground distance matrix）映射为上的距离的方式，这里用了关于的先验知识。
在为有限集合的情况下，对于，二者关于的wasserstein distance可以被定义为一个线性规划问题的解。

这里的是的联合概率分布，且有着边界分布和。
上式最小的被称作对于距离矩阵在和之间的最优传输。
关于wasserstein distance的解释可以阅读：

• Wasserstein GAN and the Kantorovich-Rubinstein Duality

• https://chih-sheng-huang821.medium.com/%E9%82%84%E7%9C%8B%E4%B8%8D%E6%87%82wasserstein-distance%E5%97%8E-%E7%9C%8B%E7%9C%8B%E9%80%99%E7%AF%87-b3c33d4b942

  soft multi-class wasserstein dice score

  这里使用wasserstein distance来扩展标签概率向量对之间的差异性度量，从而得到如下扩展形式：
  
  
  选择为使得背景类别总是离其他类最远的情况。
  
  这里同样使用加权的方式对各个类别的统计结果进行了组合。
  通过选择来使得背景位置并不对发挥作用。
  最终，关于的wasserstein dice score可以定义为：
  
  对于二值情况，可以设置，由此有，此时wasserstein dice score就退化为了soft binary dice score：
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  曾经的基于wasserstein distance的损失受限于其计算成本，然而，对于这里主要考虑的分割情形中，优化问题的闭式解存在。对于，最优传输为，并且因此wasserstein distance可以简化成：
  

  wasserstein dice loss

  基于可以定义为：
  。

  参考

• 代码：https://github.com/LucasFidon/GeneralizedWassersteinDiceLoss/blob/master/generalized_wasserstein_dice_loss/loss.py