MILtracking目标跟踪解析
MILtracking目标跟踪解析
三种概率
在传统的机器学习中，样本只有一个包标签。但是在MIL中，有样本与样本集合的概念，样本集合有包标签\(y_{bag}\)（若集合含有正样本，包标签为1，否则0），故样本除了具有样本标签\(y_{sample}\)还有包标签\(y_{bag}\)。通过Noisy-OR模型可以由集合各元素的样本标签计算该集合的包标签：
\[p\left( y_{bag}=1\left|\right.X_{n}\right)=1-\prod_{n} \left(1-p\left(y_{bag}=1\left|\right. x_{i} \right ) \right )
\]
其中\(X_{n}\)是含有n个样本的集合\(x_{i}\)是集合中的一个元素。
样本包标签后验概率推导
对于任意一个样本\(x\)的\(p\left( y_{bag}=1\left|\right.x\right)\)，我们可以通过naive bayes导出:
\[\begin{split}
p\left( y_{bag}=1\left|\right.x\right) &=\frac{p\left( x\left|\right. y_{bag}=1\right)p\left(y_{bag}=1 \right)}{p\left( x\left|\right. y_{bag}=0\right)p\left(y_{bag}=0 \right)+p\left( x\left|\right. y_{bag}=1\right)p\left(y_{bag}=1 \right)} \\
&=\frac{A}{B+A}=\frac{1}{{\frac{A}{B}}^{-1}+1}=\frac{1}{e^{-\ln\frac{A}{B}}+1}=\sigma\left( \ln\frac{A}{B}\right)\\
&=\sigma \left( \ln\frac{p\left( x\left|\right. y_{bag}=1\right)p\left(y_{bag}=1 \right)}{p\left( x\left|\right. y_{bag}=0\right)p\left(y_{bag}=0 \right)}\right)\\
\\
&又因为p\left(y_{bag}=1 \right)=p\left(y_{bag}=0 \right)\\
\\
&=\sigma \left( \ln\frac{p\left( x\left|\right. y_{bag}=1\right)}{p\left( x\left|\right. y_{bag}=0\right)}\right)\\
\\
&又因为服从naive bayes假设（观测样本维度独立）
\\
&=\sigma \left( \ln\frac{\prod p\left( x_{i}\left|\right. y_{bag}=1\right)}{\prod p\left( x_{i}\left|\right. y_{bag}=0\right)}\right)=\sigma \left( \ln \prod \frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}\right)\\
&=\sigma \left( \sum \ln\frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}\right)
\end{split}
\]
弱分类器与强分类器
设弱分类器
\[h_{i}= \ln\frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}
\]
则\(h_{1},h_{2},...,h_{n}\)级联构成的强分类器为
\[H_{n}=\sum \ln\frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}=\sum h_{i}
\]
在上一次跟踪结果附近某范围采样一个集合\(X_{near}\)，远处采样一个集合\(X_{far}\).假设跟踪结果有漂移，但真实位置仍然落在\(X_{near}\),则\(X_{near}\)的包标签为1，\(X_{far}\)的包标签为0，即已知了包标签。进而可求取\(p\left( x_{i}\left|\right. y_{bag}=1\right)\)与\(p\left( x_{i}\left|\right. y_{bag}=0\right)\)的分布
\[p\left( x_{i}\left|\right. y_{bag}=1\right) \sim N\left( \mu_{i}^{near},\sigma_{i}^{near}\right)，
p\left( x_{i}\left|\right. y_{bag}=0\right) \sim N\left( \mu_{i}^{far},\sigma_{i}^{far}\right)\]
