My idea is as follows: fit line to x & y data, next fit to x(1:end-1) & y(1:end-1), next x(1:end-2) & etc. Then check to which point the data behave in linear way.
For example:
% generate example data
x = linspace(0,1,100);
y = smooth(interp1([0 .4 1],[0 .9 1],x),20);
x = x(:);
y = y(:);
plot(x,y)
% fit coefficients
hold on
nrm = zeros(length(x)-1,1);
coef = zeros(length(x)-1,2);
for k = length(x):-1:2
coef(k-1,:) = ([x(1:k) ones(k,1)]\y(1:k))';
nrm(k-1) = norm(x(1:k)*coef(k-1,1)+coef(k-1,2)-y(1:k));
end
% plot nrm vector
figure
plot(nrm)
% define critical value
crit = 5e-2;
% find the value that satisfies the given condition
idx = find(nrm<crit,1,'last');
% plot result
figure(1)
plot(x(1:idx),x(1:idx)*coef(idx,1)+coef(idx,2),'r')
