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In my scenario, I have a map of 2d points and a set of disjoint circles of radius 5 that are on the map. I'm shooting a laser beam from position (0,0) at some angle until I shoot a circle. The probability of observing the circle is 1 along its whole area. The goal is to find the probability distribution of the circle occupancy after a single shot. In other words: what is the probability that if I shoot at some position, I will hit a circle?

In the single laser distance measurement, assuming I have a perfect sensor, along the shot there is a single point of object hit and multiple points of miss. Let's assume I've hit some circle shooting from point (0,0) along the x-axis.

After a shot, I have 1 point of existence evidence and multiple points of free space evidence. Let's separate them into a "hit" point and "miss" points.

I can use my circle model (where I have a probability of hit = 1 inside the circle) to construct all probable circle center positions and make sure I multiply it by 1 - positions free of circle centers. This leaves me with the probable circle centers that could have been observed. The sum of those points should be 1.

Now using convolution with kernel as my model, I can construct all the places possibly occupied by those circles and all the free places.

The question is how to combine those two distributions. I need to end up with a probability of 0 on the left of the contact point and 1 on the right. The only way I can think of is to use:

occupancy = existance - (1-existance) * free

That gives me this result, which I can scale into 0-1. I think about it as:

For each point of the map, if the probability of existance is not 100%, the rest from 100% is either empty or unknown, which are both calculated in the other plot.

But is there any mathematical equation that I can use to prove that what I'm doing is correct? And am I correct with that reasoning?

The other question is: What if I have another shot 1 pixel below the hit spot? How can I combine those two observations? I know that I could transform it into the log odds, add the new observation, and transform it back to the probabilities, but because of 0s and 1s, I will have some -inf and inf numbers...

siadajpan
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  • What's the underlying model here? There's a single radius-5 circle and you want to estimate where it is? There's some set of radius-5 circles (known to be disjoint? possibly overlapping?) and you want to estimate their union? Something else? – Gareth McCaughan Feb 16 '24 at 14:42
  • Thanks for the comment, I've updated the description – siadajpan Feb 16 '24 at 15:07
  • That's clearer, thanks! It's still not entirely clear. E.g., a process that always just makes one circle and then stops is consistent with your description, and so is one that places a circle and then tries to place another at random and just keeps on until there is no space left where another circle could go, etc. – Gareth McCaughan Feb 16 '24 at 15:29
  • the circles are already spread in the environment. I don't know where they are. By shooting the laser I can do the observations and that's how I want to calculate the occupancy map of the existing circles. – siadajpan Feb 16 '24 at 15:38

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