Averaging periodic data with a variable sampling rate - matlab

Question

I have a long array of [x,y] coordinate values of a toy car running around a track for 5-6 laps. The number of data points is inconsistent per lap (the laps have anywhere between 50-60 [x,y] points). The data plotted in Matlab makes sense, and it plots the car as it moves around the track:
However I need to somehow average the noisy laps to create a more accurate single map of the track.
I have tried marking the beginning the the track in order to identify the start of a new lap and then average each corresponding point from each lap however due to the different amount of data points for each lap this leads to more errors.
I thought about sorting the [x,y] data to join all of the points into one lap, but this doesn't work since the track is circular.
Does anyone know of a way to somehow average together my data to merge the laps together?

Answer 1

One way to do this is to define the beginning of the track and then parameterize each traversal around the loop by the normalized arc-length of the path. You can then interpolate each of the curves at specific intervals along the track using this parameterization and average the result.
% Assume that the first point is the start point (t = 0)
start_point = path(1,:);
% Compute the distance to this point for all data points
distances = sqrt(sum(bsxfun(#minus, path, start_point).^2, 2));
% Find the minima of this curve (these are all the times that the car passed the start)
% We apply some smoothing to get rid of necessary noise. Really depends on your data
[~, locs] = findpeaks(smooth(-distances, 20));
% Make sure we include the first and last point
locs = [1; locs; numel(distances)];
% Desired samples for each loop
nSamples = 1000;
% Pre-allocate the outputs
xpoints = zeros(numel(locs) - 1, nSamples);
ypoints = zeros(numel(locs) - 1, nSamples);
for k = 1:(numel(locs) - 1)
% Get the coordinates recorded for this particular loop
loop_points = path(locs(k):locs(k+1),:);
% Compute the cumulative arc-length using these points
arc_length = cumsum([0; sum(diff(loop_points, [], 1).^2, 2)]);
% Normalize the arc_length between 0 and 1
arc_length = arc_length ./ arc_length(end);
% Interpolate along the curve
xpoints(k,:) = interp1(arc_length, loop_points(:,1), linspace(0, 1, nSamples));
ypoints(k,:) = interp1(arc_length, loop_points(:,2), linspace(0, 1, nSamples));
end
% Average all the x and y locations
X = mean(xpoints, 1);
Y = mean(ypoints, 1);
plot(X, Y)
We can test this by going in a perfect circle and add some noise to each circuit and change the number of samples each time
nLoops = 10;
x = [];
y = [];
for k = 1:nLoops
nSamples = randi([50, 70]);
t = linspace(0, 2*pi, nSamples + 1);
t(end) = [];
x = cat(1, x(:), cos(t(:)) + 0.1 * (rand(size(t(:))) - 0.5));
y = cat(1, y(:), sin(t(:)) + 0.1 * (rand(size(t(:))) - 0.5));
end
path = [x(:), y(:)];
NOTE: findpeaks and smooth are toolbox functions that can likely be replaced with functions from the MATLAB File Exchange. Alternately, if you know when the car passes the beginning already, you can remove the usage of findpeaks altogether.