Spacing a set of points

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I have a shapefile of a set of points that I want to space out. I DO NOT want to do this using line interpolation as I run the risk of placing a point where there wasn't one before.

I can load this into Postgis or QGIS and would like to make sure that no point is closer than 100 metres to its neighbour.

I guess I need to calculate the KNN value for each point to its neighbours as a starting point but am not sure what strategy to use to remove points to achieve the desired result.

The graphic shows a sample of the data, I am trying to overlay it onto the map to show cycle routes as points on top of the road network. At the moment the points are too dense and as such are obscuring the road below. Snapping to roads IS NOT an option as the tracks cover all sorts of different tracks/roads/bridleways etc..





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