# Week #9 (23-04) | Mathematical model to determine the parameters for optimal point cloud density.

Following the weekly discussion with the thesis advisor and in the context of the work developed in the previous weeks, it was suggested that a mathematical model of the LIDAR laser scans should be created. This model would incorporate several parameters needed to correctly represent the problem at hand, such as height of the sensor, angles between each beam of a scan and between scans, etc. Next a filter mechanism would be applied to the model with the intent of mathematically measuring the point cloud density in a given area. The chosen approach was to create a function which has a point (xp,yp,zp) , a normal vector (a,b,c), a radius r  and the point cloud array(x,y,z) as its inputs and that returns the number of points that are inside a sphere centred in (xp,yp,zp) with radius and belong to the plane defined by the (a,b,c) normal. The figures below represent the model of the laser scan and their respective mathematical equations, in which phi represents the angle between beams of the same scan (220 beams with 0.5º between them), alpha represents the angle between scans (4 scans with 0.8º between them) and h represents the height of the sensor.

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Next a the limits of the beam where determined and represented by a blue point, and a border walk was also simulated with the collision equations implemented:

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Now based on these points the filter was applied:

Este slideshow necessita de JavaScript.  This equation system now allows to determined how many points are in the sphere and are coplanar with the plane.

After a brief analysis of the results it was observed that its almost impossible for a point from the point cloud to be coplanar with the plane, so instead of complete coplanarity (dot product between the normal vector and the vector formed by the point == 0), a “close enough” margin was implemented to allow to quantify the coplanar points. Following this the next task is to integrate these equations over time to create a real point cloud from which density data can be gathered.

Also this week began the work of writing the skeleton for the dissertation, and some research was done to see if similar approaches have already been used for road edge detection.