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This question was flagged by Image Analyst

The method of parameterization of data points is as follows:

Image Analyst
on 18 Jan 2021

Wow, what a load of mathematical gobbledygook. Could they make it any more obtuse? Anyway, here is a start:

s = load('data1.mat')

x = s.data1(:, 1);

y = s.data1(:, 2);

N = length(y);

plot(x, y, 'b.', 'LineWidth', 2);

grid on;

xlabel('x', 'FontSize', 20);

ylabel('y', 'FontSize', 20);

t = zeros(1, N);

denominator = abs(diff(x) .^ 2 + diff(y) .^ 2)

for k = 2 : N - 1

numerator =

t(k) = t(k - 1) + numerator / denominator;

end

t(end) = 1;

John D'Errico
on 18 Jan 2021

The idea is to compute what I like to call a connect-the-dots arc length. I've also called it a cumulative chordal relative distance. That is, you connect each point with a straight line, thus a chord. The parameter t is the cumulative accumulation of those distances between pairs of points.

The parameter t is a monotonically increasing parameter as you move from one point to the next in the sequence. And this makes it a nice scheme to do interpolation, because it can be used for any set of points, regardless of how they traverse your domain. For example, you can use it to interpolate points around the perimeter of a circle. And you can also use it to interpolate a completely general curvilinear path through any space, in any number of dimensions.

In MATLAB, this scheme is used internally by cscvn. You can see the reference to Eugene Lee's centripetal scheme in the help thereof.

help cscvn

I use the same scheme in my tools interparc, arclength, and distance2curve.

Since you have shown how to compute it already, I could add that one can compute the parametric vector t in a vectorized form, using diff and cumsum, but the looped form is a simple way to do it. (I know of course that Image Analyst knows how to do that.) But one might do:

t = cumsum(sqrt(diff(x(:)).^2 + diff(y(:).^2));

t = [0;t/t(end)];

Now a typical interpolation would involve forming spline models x(t), y(t). Sometimes pchip models may be appropriate instead. For example, a similar question was asked recently, where the OP desired to not have the curve falling outside of the box of the points. pchip models for x(t), y(t) would be a good way to achieve that result.

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