MILtracking目標跟蹤解析
三種概率
在傳統的機器學習中，樣本只有一個包標簽。但是在MIL中，有樣本與樣本集合的概念，樣本集合有包標簽\(y_{bag}\)（若集合含有正樣本，包標簽為1，否則0），故樣本除了具有樣本標簽\(y_{sample}\)還有包標簽\(y_{bag}\)。通過Noisy-OR模型可以由集合各元素的樣本標簽計算該集合的包標簽：
\[p\left( y_{bag}=1\left|\right.X_{n}\right)=1-\prod_{n} \left(1-p\left(y_{bag}=1\left|\right. x_{i} \right ) \right ) \]
其中\(X_{n}\)是含有n個樣本的集合\(x_{i}\)是集合中的一個元素。
樣本包標簽后驗概率推導
對於任意一個樣本\(x\)的\(p\left( y_{bag}=1\left|\right.x\right)\)，我們可以通過naive bayes導出:
\[\begin{split} p\left( y_{bag}=1\left|\right.x\right) &=\frac{p\left( x\left|\right. y_{bag}=1\right)p\left(y_{bag}=1 \right)}{p\left( x\left|\right. y_{bag}=0\right)p\left(y_{bag}=0 \right)+p\left( x\left|\right. y_{bag}=1\right)p\left(y_{bag}=1 \right)} \\ &=\frac{A}{B+A}=\frac{1}{{\frac{A}{B}}^{-1}+1}=\frac{1}{e^{-\ln\frac{A}{B}}+1}=\sigma\left( \ln\frac{A}{B}\right)\\ &=\sigma \left( \ln\frac{p\left( x\left|\right. y_{bag}=1\right)p\left(y_{bag}=1 \right)}{p\left( x\left|\right. y_{bag}=0\right)p\left(y_{bag}=0 \right)}\right)\\ \\ &又因為p\left(y_{bag}=1 \right)=p\left(y_{bag}=0 \right)\\ \\ &=\sigma \left( \ln\frac{p\left( x\left|\right. y_{bag}=1\right)}{p\left( x\left|\right. y_{bag}=0\right)}\right)\\ \\ &又因為服從naive bayes假設（觀測樣本維度獨立） \\ &=\sigma \left( \ln\frac{\prod p\left( x_{i}\left|\right. y_{bag}=1\right)}{\prod p\left( x_{i}\left|\right. y_{bag}=0\right)}\right)=\sigma \left( \ln \prod \frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}\right)\\ &=\sigma \left( \sum \ln\frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}\right) \end{split} \]
弱分類器與強分類器
設弱分類器
\[h_{i}= \ln\frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)} \]
則\(h_{1},h_{2},...,h_{n}\)級聯構成的強分類器為
\[H_{n}=\sum \ln\frac{ p\left( x_{i}\left|\right. y_{bag}=1\right)}{ p\left( x_{i}\left|\right. y_{bag}=0\right)}=\sum h_{i} \]
在上一次跟蹤結果附近某范圍采樣一個集合\(X_{near}\)，遠處采樣一個集合\(X_{far}\).假設跟蹤結果有漂移，但真實位置仍然落在\(X_{near}\),則\(X_{near}\)的包標簽為1，\(X_{far}\)的包標簽為0，即已知了包標簽。進而可求取\(p\left( x_{i}\left|\right. y_{bag}=1\right)\)與\(p\left( x_{i}\left|\right. y_{bag}=0\right)\)的分布
\[p\left( x_{i}\left|\right. y_{bag}=1\right) \sim N\left( \mu_{i}^{near},\sigma_{i}^{near}\right)， p\left( x_{i}\left|\right. y_{bag}=0\right) \sim N\left( \mu_{i}^{far},\sigma_{i}^{far}\right)\]