# existence probability of an object in fusion

I want to compute an existence probability of an object in a sensor fusion on the high level (having from each sensor list of objects already filtered with e.g. Kalman Filter).

There are these formulae:

$$LR(G)_{old} = \frac{p(Ex_{out})_{old}}{1 - p(Ex_{out})_{old}}$$ $$\alpha = \frac{p(Ex_{in})_{old}}{p(Ex_{in})_{old}*(1 - p(Ex_{in})_{new})}$$ $$LR(G)_{new} = LR(G)_{old} * \alpha$$ $$p(Ex_{out})_{new} = \frac{LR(G)_{new}}{1 + LR(G)_{new}}$$

Where $p(Ex)$ is the probability of existence and $LR$ is the Likelihood Ratio.

The idea is that $p(Ex)_{in}$ is some probability existence of local object, which was fused into the global $p(Ex_{out})$, and its probability influences that global one. $old$ would mean values from previous cycle.

How do you condition that computation to avoid situations of dividing by zero, obtaining NaN, or Inf? Also, if $p(Ex_{in})_{new}$ is almost 1, then $\alpha$ will be huge, increasing output, and increasing it enormously in each later cycle, so that the object will live forever. How to prevent it?

• Where did "this formula" come from? Perhaps if you go back to the original paper, book, or whatever, it would make these things clear? Jul 13 '13 at 0:57

## 1 Answer

There's not enough information there for me to comment on the validity of your formula.

However, if you substitute the first three equations into the last on, you will find two things.

1. the expression will be very messy, and
2. the factors $1 - p\left(Ex_{out}\right)_{old}$ and $1 - p\left(Ex_{in}\right)_{new}$ occur in both the numerator and denominator, and can be canceled out, making the expression more opaque, but numerically more tractable.