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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?

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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 $\phi'(q)\,dq$.

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  • $\begingroup$ Alright, makes so much sense $\endgroup$
    – gfdsal
    May 12, 2020 at 11:08
  • $\begingroup$ 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? $\endgroup$
    – gfdsal
    May 15, 2020 at 14:55
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    $\begingroup$ 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 ($Q\to\Bbb R^+$). $\endgroup$
    – ə̷̶̸͇̘̜́̍͗̂̄︣͟ ♦
    May 15, 2020 at 15:40
  • $\begingroup$ I see. Quite nicely explained! A "measure" is essentially integrating of degradation function over the pdf of the partition. right? $\endgroup$
    – gfdsal
    May 15, 2020 at 16:52

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