Mobile Sensor Placement for Optimal Coverage

I have come across the paper that deals with spatial positioning of mobile sensors to optimally detect sound source, or position mobile cellphone towers to maximize the coverage.

The region $$Q$$ is partitioned into mutually exclusive $$n$$ Voronoi polytopes $$W=\{ W_1,..,W_n \}$$. A function $$\phi :Q \to \mathbb{R}_+$$ assigns probability density that a certain event (here sound source) has happened over $$Q$$. There are $$n$$ sensors to be distributed $$P=(p_1,..p_n)$$ over each Voronoi partition that satisfies the following equation:

$$H(P,W)=\min\sum_{i=1}^n \int_{W_i} f(\| q-p_i\|)\,d \phi(q)$$

The quality of observation at point $$q$$ from sensor $$p_i$$ is the distance of the point $$q$$ from the sensor $$f(\| q-p_i\|)$$ which makes sense.

Can anyone explain why are we multiplying $$(q)$$ after $$d\phi$$ in the objective formula?

Note that $$\phi$$ is a function of $$q$$ so $$d\phi(q)$$ is interpreted as w.r.t. $$\phi(q)$$, which is the same as $$d\phi$$. This is just shorthand for $$\phi'(q)\,dq$$.
• I have a small question that is related more to a concept then the notation. Why is the author minimizing the cost function: uncertanity term $f(\| q-p_i\|)$ multiplied by importance term (high density means high importance) $d\phi(q)$. Shouldnt the importance term $d\phi(q)$ require us to maximize the cost function so we place sensors where the density is high? May 15 '20 at 14:55
• What $H$ aims to do is to minimise the sum of the degradation function $f$ over the partitions; we are not multiplying by the distribution density function but are rather integrating over it like a measure ($Q\to\Bbb R^+$). May 15 '20 at 15:40