How can I fit a Bézier curve to a set of data?

问题:

I have a set of data points (which I can thin out) that I need to fit with a Bézier curve. I need speed over accuracy, but the fit should be decent enough to be recognizable. I'm also looking for an algorithm I can use that doesn't make much use of libraries (specifically NumPy).

I've read several research papers, but none has enough detail to fully implement. Are there any open-source examples?

回答1:

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.

回答2:

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, 1995

http://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.

回答3:

You can setup the problem as least squares fitting to noisy data.

See http://nbviewer.ipython.org/5688579

Note that once you figure out the equations the actual computation is fairly simple. The actual computations are summing over the data and then inverting and multiplying a 4x4 matrix.

I know this thread is long dead by now, but I found this to be an interesting problem in case anyone else stumbles on it.

See http://www.embedded.com/electrical-engineer-community/industry-blog/4027019/1/Why-all-the-math- for an excellent tutorial on least squares.

回答4:

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).

回答5:

Here you find a python implementation of the referred C code. https://github.com/volkerp/fitCurves

回答6:

First of all, make sure what you ask for is actually what you want. Fitting the points to a Bezier curve will place them in the hull of the points. Using a spline will make sure your curve goes through all points.

That said, creating the function that draws either is not complicated at all. Wikipedia has a nice article that will explain the basics, Bézier curve.

回答7:

I had a MATLAB solution for this problem. I encountered the same problem, but my code is written in MATLAB. I hope its not too hard to translate it into Python.

You can find the control points by this code FindBezierControlPointsND.m For some reason, it does not have function "ChordLengthNormND" in its archive, but it is called at line 45.

I replaced it by following lines:

[arclen,seglen] = arclength(p(:,1),p(:,2),'sp');
t = zeros(size(p,1),1);
sums = seglen(1);
for i = 2:size(p,1)-1
    t(i) = sums / arclen;
    sums = sums + seglen(i);
end
t(end) = 1;

arclength's MATLAB code can be obtained here.

After that, we have control points for Bezier curve, and there are lots of implementation of building Bezier curve by control points on the web.