Various Dice Loss variants

Various Dice Loss variants

Yuque document: https://www.yuque.com/lart/idh721/gpix1i

Dice Loss is also a very common loss function in image segmentation tasks. This article is organized based on the contents of Generalized Wasserstein Dice Score for Imbalanced Multi-class Segmentation using Holistic Convolutional Networks .

hard dice score for binary segmentation

dice score is a widely used overlap measure for pairwise comparisons between binary segmentation maps S and G.

It can be expressed as a set operation or as a statistical measure:

$$D_{hard}=\frac{2|S\cap G|}{|S|+|G|}=\frac{2{Th}_{TP}}{2{Th}_{TP}+{Th}_{FP}+{Th}_{FN}}=\frac{2{Th}_{TP}}{2{Th}_{TP}+{Th}_{AE}}$$

There are several items involved here, the specific meanings are as follows:

• $S\&G$ : images to be evaluated and reference images
• ${Th}_{TP}$: the number of positive and positive samples, ie $S$ andGGThe number of positions where$G\; is\; true.$
• ${Th}_{FP}$： S S True in$S$ and GGThe number of false positions in$G.$
• ${Th}_{FN}$：$S$ is fake andGGThe number of true positions in$G.$
• ${Th}_{AE}={Th}_{FP}+{Th}_{FN}$：$S$ and$The\; number\; of\; positions\; where\; G$ is inconsistent.

soft dice score for binary segmentation

An extension to soft binary segmentation relies on an inconsistent notion of probabilistic class pairs.

for $S$ andGGPosition i ∈ X i \in \mathbf{X}in$G$$i\in$The category S i S_icorresponding to$X$$S_i$和$G_i$Can be defined as label space L = { 0 , 1 } \mathbf{L}=\{0,1\}L={ 0,1 } random variable.

Probabilistic segmentation can be represented as a label probability map, where $P\left(L\right)$ represents the collection of label probability vectors:

• p = { p i : = P ( S i = 1 ) } i ∈ X p=\{p^i:=P(S_i=1)\}_{i \in \mathbf{X}} p={ pi:=P(Si​=1)}i∈X​
• g = { g i : = P ( G i = 1 ) } i ∈ X g=\{g^i:=P(G_i=1)\}_{i \in \mathbf{X}} g={ gi:=P(Gi​=1)}i∈X​

From this, the previous statistics on the data can be ${Th}_{TP}\&{\Theta }_{AE}$Extended to the soft-segmentation case:

• ${Th}_{AE}={\sum }_{i\in \mathbf{X}}|p^i-g^i|$
• ${Th}_{TP}={\sum }_{i\in X}{g}^i(1-|p^i-g^i|)$

For gg in the general case$g$，即 ∀ i ∈ X , g i ∈ { 0 , 1 } \forall i \in \mathbf{X}, g^i \in \{0, 1\} ∀i∈X,gi∈{ 0,1 } , at this point:

• ${Th}_{AE}={\sum }_{i\in \mathbf{X}}{g}^i(1-p^i)+(1-g^i)p^i={\sum }_{i\in X}{g}^i+p^i-2g^i{p}^i$
• ${Th}_{TP}={\sum }_{i\in X}{g}^i{p}^i$

The corresponding soft dice score can be expressed as:

$$D_{soft}(p,g)=\frac{2{\sum }_i{g}^i{p}^i}{{\sum }_i(g^i+p^i)}$$

Of course, there are also variants that introduce squared forms.

soft multi-class dice score

The direct discussion above is the case of binary segmentation, but for the case of multi-classification, it is necessary to consider the integration method of different categories of calculations.

The simplest way is to directly consider the average of all categories.

It can be called mean dice score, which corresponds to $|L|$ different categories:

$$D_{mean}(p,g)=\frac{1}{|\mathbf{L}|}\sum _{l\in \mathbf{L}}\frac{2{\sum }_i{g}_l^i{p}_l^i}{{\sum }_i{g}_l^i+p_l^i}$$

The generalized form of the above formula can be introduced by introducing category weight parameter $w_l=\frac{1}{({\sum }_i{g}_l^i{)}^2},l\in L$ and get. That is, the above formula is transformed into a weighted average form. This is called a generalized soft multi-class dice score.

Finally it can be expressed as:

$$D_{generalised}(p,g)=\frac{2{\sum }_l{w}_l{\sum }_i{g}_l^i{p}_l^i}{{\sum }_l{w}_l{\sum }_i(g_l^i+p_l^i)}$$

soft multi-class wasserstein dice score

In the previous form of dice score, for $p^i$ sum$g^{}$The measure of the similarity of i can be regarded as the L1 distance, and the wasserstein distance is introduced here to naturally compare two label probability vectors in a semantically meaningful way${}^{.}$

Here first introduces the wasserstein distance.

wasserstein distance

This is also known as the earth mover's distance. Used to represent a probability vector pptransform$p$ into another probability vector qqThe minimum cost required by$q\; .$

For all $l,l^{\prime }\in L$ , from$l$ moves to$l^{\prime }$ The set of distances is defined as$l$ and$l^{\prime }$ distance matrix$M_{l,l^{\prime }}$, this matrix is ​​fixed and can be considered known.

This is a way to convert L \mathbf{L}The distance matrix MMon$L$$M$ (usually also called the ground distance matrix) maps toP ( L ) P(\mathbf{L})The way of distance on$P$$($$L$$)$ , here we use about L \mathbf{L}Prior knowledge of$L.$

在L \mathbf{L}When$L$$p,q\in P\left(L\right)$ , both about$The\; wasserstein\; distance\; of\; M$ can be defined as the solution of a linear programming problem.

$$\begin{array}{rl}{W}^M(p,q)& =\underset{T_{l,l^{\prime }}}{\min }\sum _{l,l^{\prime }\in L}{T}_{l,l^{\prime }}{M}_{l,l^{\prime }}\\ \text{subject to }\forall l\in \mathbf{L},\sum _{l^{\prime }\in \mathbf{L}}{T}_{l,l^{\prime }}& =p_l,\\ \text{ and }\forall l^{\prime }\in L,\sum _{l\in \mathbf{L}}{T}_{l,l^{\prime }}& =q_{l^{\prime }}\end{array}$$

Here $T=(T_{l,l^{\prime }}{)}_{l,l^{\prime }\in \mathbf{L}}$is $(p,q)$ joint probability distribution with boundary distribution$p$ and$q$。

The minimum T ^ \hat{T} of the above formula$\hat{T}$ is called for the distance matrix$M$ in$between\; p$ and$The\; optimal\; transmission\; of\; q$ .

An explanation of the wasserstein distance can be read:

• 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

Here, the wasserstein distance is used to expand the difference measure between the label probability vector pairs, so as to obtain the following extended form:

• ${Th}_{AE}={\sum }_{i\in \mathbf{X}}{W}^M(p^i,g^i)$
• Θ T P l = ∑ i ∈ X g l i ( M l . b − W M ( p i , g i ) ) , ∀ l ∈ L ∖ { b } \Theta^l_{TP}=\sum_{i \in \mathbf{X}}g^i_l(M_{l.b}-W^M(p^i,g^i)), \forall l \in \mathbf{L} \setminus \{b\} ThTPl​=∑i∈X​gli​(Ml.b​−WM(pi,gi)),∀l∈L∖{ b}

$M$ is selected such that the background class$b$ is always the furthest case from the other classes.

${Th}_{TP}={\sum }_{i\in \mathbf{X}}{a}_l{Th}_{TP}^l$

Here, the statistical results of each category are also combined in a weighted manner.

By choosing $a_l=W^M(l,b)=M_{l,b}$to make the background position not right ${Th}_{TP}$Play a role.

Finally, about MMThe wasserstein dice score of$M\; can\; be\; defined\; as:$

$$D^M(p,q)=\frac{2{\sum }_l{M}_{l,b}{\sum }_i{g}_l^i(M_{l,b}-W^M(p^i,g^i))}{2{\sum }_l{M}_{l,b}{\sum }_i{g}_l^i(M_{l,b}-W^M(p^i,g^i))+{\sum }_i{W}^M(p^i,g^i)}$$

For the binary case, you can set:

$$M=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$

From this there are

$$W^M(p^i,g^i)=|p^i-g^i|,M_{l,b}\to l\ne b$$

At this point, the wasserstein dice score degenerates into a soft binary dice score:

$$\begin{array}{rl}{D}^M(p,q)& =\frac{2{\sum }_i{g}^i(1-|p^i-g^i|)}{2{\sum }_i{g}^i(1-|p^i-g^i|)+{\sum }_i|p^i-g^i|}\\ & =\frac{2{\sum }_i{p}^i{g}^i}{2{\sum }_i{p}^i{g}^i+{\sum }_i[p^i(1-g^i)+(1-p^i)g^i]}\\ & =\frac{2{\sum }_i{g}^i{p}^i}{{\sum }_i(g^i+p^i)}\end{array}$$

Previous wasserstein distance-based losses were limited by their computational cost, however, for the segmentation case mainly considered here, a closed-form solution to the optimization problem exists.

For∀ $\forall l,l^{\prime }\in L$ , the optimal transmission is$T_{l,l^{\prime }}=p_l^i{g}_{l^{\prime }}^i$, and thus the wasserstein distance can be simplified to:

$$W^M(p^i,g^i)=\sum _{l,l^{\prime }\in \mathbf{L}}{M}_{l,l^{\prime }}{p}_l^i{g}_{l^{\prime }}^i$$

Wasserstein dice loss

MM- based$M$ can be defined as:

$$L_{D^M}:=1-D^M$$

reference

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