Noei_PNAS_2022/Aux_functions/elbow_finder.m

function [idxOfBestPoint] = elbow_finder(curve,verbose)

if nargin <2
    verbose = 0;
end

%# get coordinates of all the points
nPoints = length(curve);
allCoord = [1:nPoints;curve]';              %'# SO formatting

%# pull out first point
firstPoint = allCoord(1,:);

%# get vector between first and last point - this is the line
lineVec = allCoord(end,:) - firstPoint;

%# normalize the line vector
lineVecN = lineVec / sqrt(sum(lineVec.^2));

%# find the distance from each point to the line:
%# vector between all points and first point
vecFromFirst = bsxfun(@minus, allCoord, firstPoint);

%# To calculate the distance to the line, we split vecFromFirst into two 
%# components, one that is parallel to the line and one that is perpendicular 
%# Then, we take the norm of the part that is perpendicular to the line and 
%# get the distance.
%# We find the vector parallel to the line by projecting vecFromFirst onto 
%# the line. The perpendicular vector is vecFromFirst - vecFromFirstParallel
%# We project vecFromFirst by taking the scalar product of the vector with 
%# the unit vector that points in the direction of the line (this gives us 
%# the length of the projection of vecFromFirst onto the line). If we 
%# multiply the scalar product by the unit vector, we have vecFromFirstParallel
scalarProduct = dot(vecFromFirst, repmat(lineVecN,nPoints,1), 2);
vecFromFirstParallel = scalarProduct * lineVecN;
vecToLine = vecFromFirst - vecFromFirstParallel;

%# distance to line is the norm of vecToLine
distToLine = sqrt(sum(vecToLine.^2,2));
%# now all you need is to find the maximum
[~,idxOfBestPoint] = max(distToLine);

%# plot the distance to the line
if verbose
    figure('Name','distance from curve to line'), plot(distToLine)
    %# plot
    figure, plot(curve)
    hold on
    plot(allCoord(idxOfBestPoint,1), allCoord(idxOfBestPoint,2), 'or')
end

end