Information fusion in polar coordinates

Many sensing modalities used in robotics collect information in polar coordinates. For mobile robots and autonomous vehicles these modalities include radar, sonar, and laser range finders. In the context of medical robotics, ultrasound imaging and CT both collect information in polar coordinates. Moreover, every sensing modality has associated noise. Therefore when the position of a point in space is estimated in the reference frame of the sensor, that position is replaced by a probability density expressed in polar coordinates. If the sensor moves from one location to another and the same point is sensed, then the two associated probabilities can be "fused" together to obtain a better estimate than either one individually. Here we derive the equations for this fusion process in polar coordinates. The result involves the computation of integrals of three Bessel functions. We derive new recurrence relations for the efficient computation of these Bessel-functon integrals to aid in the information-fusion process.