Taylor series expansion in mean shift tracking

Question

I am trying to understand the first order Taylor series expansion when deriving the math behind mean shift tracking. Given the Battacharya coefficient between the target descriptor and candidate descriptor $\rho[\textbf{p}(y),\textbf{q}]$, how does $$\rho[\textbf{p}(y),\textbf{q}] \approx \frac{1}{2}\sum_m\sqrt{p_m(y_0)q_m} + \frac{1}{2}\sum_mp_m(y)\sqrt{\frac{q_m}{p_m(y_0)}}$$

I am trying to see how each term can be related to the taylor series expansion about a point $a$ given by $$f(x) \approx f(a) + f'(a)(x-a)$$

What does $f$, $x$ and $a$ correspond to in the mean shift problem ?

Would appreciate any tips and pointers regarding this. Many of the lectures online do not really explain the details.

Answer 1

For $y\approx y_0$ you have

$$\begin{align}\sqrt{p_m(y)q_m}&\approx\sqrt{p_m(y_0)q_m}+\frac{p_m(y)-p_m(y_0)}{2\sqrt{p_m(y_0)}}\sqrt{q_m}\\&=\frac12\sqrt{p_m(y_0)q_m}+\frac12 p_m(y)\sqrt{\frac{q_m}{p_m(y_0)}}\tag{1}\end{align}$$

So the function you're approximating is $f(x)=\sqrt{x}$, with $x=p_m(y)$, and you do this in the vicinity of the value $a=p_m(y_0)$.