How to compute the density of a 3D point cloud?

I'm trying to write a program that operates on 3D point clouds (.ply, .pcd), I need to know the density of the given point cloud file and compare it with a threshold or certain percentage to decide for some operations. How can I do that?

I have found this for 3D point clouds on the web:

Two methods can be used to compute the density:

• either 'Precise': the density is estimated by counting for each point the number of neighbors N (inside a sphere of radius R)
• or 'Approximate': the density is simply estimated by determining the distance to the nearest neighbor (which is generally much faster). This distance is considered as being equivalent to the above spherical neighborhood radius R (and N = 1).

However I'm not sure how to put this into matlab code, I guess I should use pcfitplane for sphere fitting and findNearestNeighbors. Would appreciate a step by step example.

Answer 1

[Thiago Henrique Gomes Lobato]

You can just directly apply the definition you gave considering that the density is N/Volume. The easiest is the second one, which would be:

AllPoints = % your points
K = 1;
for idx=1:length(AllPoints)
[~,r] = findNearestNeighbors(AllPoints,AllPoints(idx,:),K);
density(idx) = 1/(4*pi*r.^3/3);
end

The first one is a little more complicated but at the same time not so much:

AllPoints = % your points
R = 1; % depends on your data
for idx=1:length(AllPoints)
Distances = sqrt( sum( (AllPoints-AllPoints(idx,:)).^2 ,2) );
Ninside   = length( find(Distances<=R) );
density(idx) = Ninside/(4*pi*R.^3/3);
end

[Ali]

Thank you for your quick reply @Thiago I managed to get the following piece of code working thanks to your help, the 'Approximate' method (your first block of code) produces an 1xN array (N being the total point count in the cloud) with all elements of "Inf", however the 'Precise' method produces the same array size with elements of different numbers ranging from 0 to 4 with some trailing fractions.

This may sound noob, but if I want to get a single number representing the whole point cloud's density what should I do with this array? Should I divide the sum of all values by the number of total points existing in the cloud or should I calculate the mean of this array?

Also the radius variable 'R' being equal to 1 in our example, is there a way to compute it for different given point clouds before running the for loop like finding the center of the point cloud and calculating the radius to the largest possible direction so that it covers the whole point cloud?

clc
clear 
close all
% input point cloud file gui
[FileName,PathName] = uigetfile({'*.ply';'*.pcd'},'Input point cloud');
ptCloud=pcread([PathName,FileName]);

AllPoints = ptCloud.Location;

% Approximate method
K = 1;
for idx=1:length(AllPoints)
[~,r] = findNearestNeighbors(ptCloud,AllPoints(idx,:),K);
densityApprox(idx) = 1/(4*pi*r.^3/3);
end

fprintf('\n Density (Approx.): %d \n', densityApprox);

% Precise method
R = 1; % depends on your data
for idx=1:length(AllPoints)
Distances = sqrt( sum( (AllPoints-AllPoints(idx,:)).^2 ,2) );
Ninside   = length( find(Distances<=R) );
densityPrecise(idx) = Ninside/(4*pi*R.^3/3);
end

fprintf('\n Density (Precise): %d \n', densityPrecise);

[Thiago Henrique Gomes Lobato]

The first method is inf probably because the value r is too small (points are too close from each other), although I can't be entirely sure without looking at your data. The answer of your second question is your third question. For a single density value you need to find the center of your cloud, calculate the radius for the fartest point and the the density (doing this you make the implicity assumption that the density is relative homogeneous for all your domain).

Center = mean(AllPoints);
Distances = sqrt( sum( (AllPoints-Center).^2 ,2) );
R         = max(Distances);
densitySingle = size(AllPoints,1)/(4*pi*R.^3/3);

This should give actually a similar value as mean(densityPrecise), considering the assumption holds.

[Ali]

@Thiago the data is attached in my question as a .ply file. I tried this new piece of code and the results are different, mean(densityPrecise) gives '2.922395e+00' while this new method gives '7.780270e-02 ' which I guess is because the data is not homogeneous for all the domain.

Thank you for your time and help, you're a lifesaver. :-)

[Ali]

@Thiago I stumbled upon this information on the same page explaining the methods above:

The density output can be:

• the number of neighbors N (only available in 'Precise' mode)
• a surface density: number of neighbors divided by the neighborhood surface = N / (Pi.R2)
• a volume density: number of neighbors divided by the neighborhood volume = N / (4/3.Pi.R3)

It made wonder, shouldn't we reorder the (4*pi*R.^3/3) part for the volume density? Like (4/3*pi*R.^3) ?

Also from the same page:

Notes:

• a point with no neighbor in the spherical neighborhood will have an invalid (NaN) density.
• the central point is always used for computing the density (even when the output is the 'number of neighbors' as we consider the number of neighbors of the 'position in space' of each point). Therefore the density will be actually equal to kNN + 1 where kNN is the number of neighbors of each point)

[Thiago Henrique Gomes Lobato]

4*pi*R.^3/3 is the same as 4/3*pi*R.^3 since exponent has priority over division in matlab. For the notes the first point is important only when there's a limit to define neighborhood, and the second is already taken into account for all methods.

[Ali]

I did try my suggested order and indeed your statement is true, no difference in results. I appreciate your time and help.