论文题目：High-Speed Tracking with Kernelized Correlation Filters
作者主页：ht://wtpww.robots.ox.ac.uk/~joao/circulant/
参考博文：https://blog.csdn.net/discoverer100/article/details/53835507
1. 从岭回归开始推导
online training process can be written:
while, w is the cofficience of filter, the symbol φ(·) means feature extraction operator, and y is landmark we selected which can be generated by gaussian heatmap.
Generally speaking, format above can be understood by variances and bias. When we using ridge regression, we should pay more attention to parameter λ， if λ sets too high, model's bias would expand, which is bad for our classifier.
Then, we make a deviation for above from w perspective, that is:
so, the close-form solution for ridge regression is:
This close-form solution is deduced in spatial space, which can also be represented in Fourier space.
^ 是离散傅里叶变换； H表示共轭转置矩阵。
2. 判别式相关滤波部分
DCF中我们已经限定了前提，X是循环矩阵，而循环矩阵又拥有如下特性：
其中，^表示样本的离散傅里叶变换；F是离散傅里叶变换矩阵，是一个常量； x是生成向量，用于表示人们感兴趣的图像块;X是根据上述x生成的循环矩阵;H表示X的共轭转置矩阵，即先进行共轭再进行转置;
其中，*表示复共轭。因为对角矩阵之间乘积运算就是元素级运算，element-wise，即位置相同的各元素分别相乘。所以上式可以表达为：
将循环卷积的构建方程改写成傅里叶反变换形式，即：
所以可以继续改写为：
下面利用循环卷积的性质：
所以，又可以继续改写为：
所以，DCF中滤波器的close-form solution是：
