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.


1 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 $a=p_m(y_0)$.

  • $\begingroup$ it was quite hard to wrap my head around the $x = p_m(y)$ part. In the sense that the taylor series is a function of function $\endgroup$
    – calveeen
    Commented Nov 18, 2020 at 13:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.