c++ - Finding curvature from a noisy set of data points using 2d/3dsplines? (C++)

Question

I am trying to extract the curvature of a pulse along its profile (see the picture below). The pulse is calculated on a grid of length and height: 150 x 100 cells by using Finite Differences, implemented in C++.

I extracted all the points with the same value (contour/ level set) and marked them as the red continuous line in the picture below. The other colors are negligible.

Then I tried to find the curvature from this already noisy (due to grid discretization) contour line by the following means:

(moving average already applied)

1) Curvature via Tangents

The curvature of the line at point P is defined by:

So the curvature is the limes of angle delta over the arclength between P and N. Since my points have a certain distance between them, I could not approximate the limes enough, so that the curvature was not calculated correctly. I tested it with a circle, which naturally has a constant curvature. But I could not reproduce this (only 1 significant digit was correct).

2) Second derivative of the line parametrized by arclength

I calculated the first derivative of the line with respect to arclength, smoothed with a moving average and then took the derivative again (2nd derivative). But here I also got only 1 significant digit correct. Unfortunately taking a derivative multiplies the already inherent noise to larger levels.

3) Approximating the line locally with a circle

Since the reciprocal of the circle radius is the curvature I used the following approach:

This worked best so far (2 correct significant digits), but I need to refine even further. So my new idea is the following:

Instead of using the values at the discrete points to determine the curvature, I want to approximate the pulse profile with a 3 dimensional spline surface. Then I extract the level set of a certain value from it to gain a smooth line of points, which I can find a nice curvature from.

So far I could not find a C++ library which can generate such a Bezier spline surface. Could you maybe point me to any?

Also do you think this approach is worth giving a shot, or will I lose too much accuracy in my curvature?

Do you know of any other approach?

With very kind regards, Jan

edit: It seems I can not post pictures as a new user, so I removed all of them from my question, even though I find them important to explain my issue. Is there any way I can still show them?

edit2: ok, done :)

Answer 1

ALGLIB支持各种类型的插值：

• 多项式插值
• 有理插值
• 样条插值
• 最小二乘拟合（线性/非线性）
• 双线性和双三次样条插值
• 快速 RBF 插值/拟合

我不知道它是否满足您的所有要求。我个人还没有使用过这个库，但我相信三次样条插值可能是你正在寻找的（两次可微）。

为了防止对您的嘈杂输入点进行过度拟合，您应该应用某种平滑机制，例如，您可以尝试移动窗口平均/高斯/FIR 滤波器之类的东西是否适用。另请查看（三次）平滑样条线。