How can I fit a Bézier curve to a set of data?
I have similar problem and I have found "An algorithm for automatically fitting digitized curves" from Graphics Gems (1990) about Bezier curve fitting.Additionally to that I have found source code for that article.
Unfortunately it is written in C which I don't know very well. Also, the algorithm is quite hard to understand (at least for me). I am trying to translate it into C# code. If I will be successful, I will try to share it.
File GGVecLib.c in the same folder as FitCurves.c contains basic vectors manipulation functions.
I have found a similar Stack Overflow question, Smoothing a hand-drawn curve. The approved answer provide C# code for a curve fitting algorithm from Graphic Gems.
What's missing in a lot of these answers is that you may not want to fit a single cubic Bézier curve to your data. More generally, you would like to fit a sequence of cubic Bézier curves, i.e., a piecewise cubic Bézier fit, to an arbitrary set of data.
There's a nice thesis dating from 1995, complete with MATLAB code, that does this:
% Lane, Edward J. Fitting Data Using Piecewise G1 Cubic Bezier Curves.% Thesis, NAVAL POSTGRADUATE SCHOOL MONTEREY CA, 1995http://www.dtic.mil/dtic/tr/fulltext/u2/a298091.pdf
To use this, you must, at minimum, specify the number of knot points, i.e., the number data points that will be used by the optimization routines to make this fit. Optionally, you can specify the knot points themselves, which increases the reliability of a fit. The thesis shows some pretty tough examples. Note that Lane's approach guarantees G1 continuity (directions of adjacent tangent vectors are identical) between the cubic Bézier segments, i.e., smooth joints. However, there can be discontinuities in curvature (changes in direction of second derivative).
I have reimplemented the code, updating it to modern MATLAB (R2015b). Contact me if you would like it.
Here's an example of using just three knot points (chosen automatically by the code) the fits two cubic Bézier segments to a Lissajous figure.
If most of the data fits the model you could try RANSAC. It would be easy enough to pick 4 points and random and fit a bezier curve from those. I'm not sure off the top of my head how expensive it would be to evaluate the curve against all the other points (part of the RANSAC algorithm). But it would be a linear solution and RANSAC is really easy to write (and there are probably open source algorithms out there).