How is it a perspective function?

Question

In one of the paper (I have shown the image also) it is written that $$f(t_0,t_i)=\frac{t_i}{2}\log\left(1+\frac{a\frac{t_0}{t_i}}{b\frac{t_0}{t_i}+c}\right)$$ is a perspective function of $$f(t_0)=\frac{1}{2}\log\left(1+\frac{at_0}{bt_0+c}\right)$$ where $a>0,b>0,c>0$ and $t_0$,$t_i$ are the optimization variables. I can understand that if we divide every $t_0$ by $t_i$ then we can get perspective function but I do not understand $\frac{t_i}{2}$ term before the logrithm function. Can anybody explain how it has appeared. Thanks in advance.

Answer 1

The perspective of a function $f: \mathbb{R}^n \to \mathbb{R}$ is the function $g:\mathbb{R}^{n} \times \mathbb{R} \to \mathbb{R}$

$$g(x,t)=\color{blue}tf(x/t), \operatorname{dom}(g)=\{(x,t)|x/t \in \operatorname{dom}(f), t>0\}$$

Hence, starting from $f(t_0)$, we replace $t_0$ by $t_0/t$ and then multiply the function by $t_0$ will give us the perspective function.