Decision trees are the most popular weak classifiers used in boosting schemes. Often the simplest decision trees with only a single split node per tree (called stumps) are sufficient.
-The boosted model is based on $N$ training examples ${(x_i,y_i)}1N$ with $x_i \in{R^K}$ and $y_i \in{-1, +1}$. $x_i$ is a $K$-component vector. Each component encodes a feature relevant for the learning task at hand. The desired two-class output is encoded as −1 and +1.
+The boosted model is based on $N$ training examples ${(x_i,y_i)}1N$ with $x_i \in{R^K}$ and $y_i \in{-1, +1}$. $x_i$ is a $K$-component vector. Each component encodes a feature relevant for the learning task at hand. The desired two-class output is encoded as -1 and +1.
Different variants of boosting are known such as Discrete Adaboost, Real AdaBoost, LogitBoost, and Gentle AdaBoost \cross{FHT98}. All of them are very similar in their overall structure. Therefore, we will look only at the standard two-class Discrete AdaBoost algorithm as shown in the box below. Each sample is initially assigned the same weight (step 2). Next a weak classifier $f_{m(x)}$ is trained on the weighted training data (step 3a). Its weighted training error and scaling factor $c_m$ is computed (step 3b). The weights are increased for training samples, which have been misclassified (step 3c). All weights are then normalized, and the process of finding the next weak classifier continues for another $M$-1 times. The final classifier $F(x)$ is the sign of the weighted sum over the individual weak classifiers (step 4).
\begin{itemize}
\item Fit the classifier $f_m(x) \in{-1,1}$, using weights $w_i$ on the training data.
\item Compute $err_m = E_w [1_{(y =\neq f_m(x))}], c_m = log((1 - err_m)/err_m)$.
-\item Set $w_i \Leftarrow w_i exp[c_m 1_{(y_i \neq f_m(x_i))}], i = 1,2,…,N,$ and renormalize so that $\Sigma i w_i = 1$.
+\item Set $w_i \Leftarrow w_i exp[c_m 1_{(y_i \neq f_m(x_i))}], i = 1,2,...,N,$ and renormalize so that $\Sigma i w_i = 1$.
\item Output the classifier sign$[\Sigma m = 1M c_m f_m(x)]$.
\end{itemize}
\end{itemize}
\end{lstlisting}
The class represents a 2D point, specified by its coordinates $x$ and $y$.
-Instance of the class is interchangeable with С structures \texttt{CvPoint} and \texttt{CvPoint2D32f}. There is also cast operator to convert point coordinates to the specified type. The conversion from floating-point coordinates to integer coordinates is done by rounding; in general case the conversion uses \hyperref[saturatecast]{saturate\_cast} operation on each of the coordinates. Besides the class members listed in the declaration above, the following operations on points are implemented:
+Instance of the class is interchangeable with C structures \texttt{CvPoint} and \texttt{CvPoint2D32f}. There is also cast operator to convert point coordinates to the specified type. The conversion from floating-point coordinates to integer coordinates is done by rounding; in general case the conversion uses \hyperref[saturatecast]{saturate\_cast} operation on each of the coordinates. Besides the class members listed in the declaration above, the following operations on points are implemented:
\begin{lstlisting}
pt1 = pt2 + pt3;
\end{lstlisting}
The class represents a 3D point, specified by its coordinates $x$, $y$ and $z$.
-Instance of the class is interchangeable with С structure \texttt{CvPoint2D32f}. Similarly to \texttt{Point\_}, the 3D points' coordinates can be converted to another type, and the vector arithmetic and comparison operations are also supported.
+Instance of the class is interchangeable with C structure \texttt{CvPoint2D32f}. Similarly to \texttt{Point\_}, the 3D points' coordinates can be converted to another type, and the vector arithmetic and comparison operations are also supported.
The following type aliases are available:
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