wehub-resource-sync
604 lines
22 KiB
C#
604 lines
22 KiB
C#
//
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// Author: Ryan Seghers
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//
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// Copyright (C) 2013-2014 Ryan Seghers
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//
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// Permission is hereby granted, free of charge, to any person obtaining
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// a copy of this software and associated documentation files (the
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// "Software"), to deal in the Software without restriction, including
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// without limitation the irrevocable, perpetual, worldwide, and royalty-free
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// rights to use, copy, modify, merge, publish, distribute, sublicense,
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// display, perform, create derivative works from and/or sell copies of
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// the Software, both in source and object code form, and to
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// permit persons to whom the Software is furnished to do so, subject to
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// the following conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
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// LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
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// OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
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// WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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//
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using System;
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using System.Diagnostics;
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using System.Diagnostics.CodeAnalysis;
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using System.Text;
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namespace T3.Core.Utils.Splines;
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/// <summary>
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/// Cubic spline interpolation.
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/// Call Fit (or use the corrector constructor) to compute spline coefficients, then Eval to evaluate the spline at other X coordinates.
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/// </summary>
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/// <remarks>
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/// <para>
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/// This is implemented based on the wikipedia article:
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/// http://en.wikipedia.org/wiki/Spline_interpolation
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/// I'm not sure I have the right to include a copy of the article so the equation numbers referenced in
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/// comments will end up being wrong at some point.
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/// </para>
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/// <para>
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/// This is not optimized, and is not MT safe.
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/// This can extrapolate off the ends of the splines.
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/// You must provide points in X sort order.
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/// </para>
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/// </remarks>
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[SuppressMessage("ReSharper", "MemberCanBeInternal")]
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public sealed class CubicSpline
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{
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#region Fields
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// N-1 spline coefficients for N points
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private float[] _a;
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private float[] _b;
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// Save the original x and y for Eval
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private float[] _xOrig;
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private float[] _yOrig;
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#endregion
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#region Ctor
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/// <summary>
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/// Default ctor.
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/// </summary>
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public CubicSpline()
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{
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}
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/// <summary>
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/// Construct and call Fit.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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public CubicSpline(float[] x, float[] y, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false)
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{
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Fit(x, y, startSlope, endSlope, debug);
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}
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#endregion
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#region Private Methods
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/// <summary>
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/// Throws if Fit has not been called.
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/// </summary>
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private void CheckAlreadyFitted()
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{
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if (_a == null) throw new Exception("Fit must be called before you can evaluate.");
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}
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private int _lastIndex;
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/// <summary>
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/// Find where in xOrig the specified x falls, by simultaneous traverse.
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/// This allows xs to be less than x[0] and/or greater than x[n-1]. So allows extrapolation.
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/// This keeps state, so requires that x be sorted and xs called in ascending order, and is not multi-thread safe.
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/// </summary>
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private int GetNextXIndex(float x)
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{
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if (x < _xOrig[_lastIndex])
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{
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throw new ArgumentException("The X values to evaluate must be sorted.");
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}
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while ((_lastIndex < _xOrig.Length - 2) && (x > _xOrig[_lastIndex + 1]))
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{
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_lastIndex++;
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}
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return _lastIndex;
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}
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/// <summary>
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/// Evaluate the specified x value using the specified spline.
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/// </summary>
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/// <param name="x">The x value.</param>
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/// <param name="j">Which spline to use.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The y value.</returns>
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private float EvalSpline(float x, int j, bool debug = false)
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{
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var dx = _xOrig[j + 1] - _xOrig[j];
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var t = (x - _xOrig[j]) / dx;
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var y = (1 - t) * _yOrig[j] + t * _yOrig[j + 1] + t * (1 - t) * (_a[j] * (1 - t) + _b[j] * t); // equation 9
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if (debug) Console.WriteLine("xs = {0}, j = {1}, t = {2}", x, j, t);
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return y;
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}
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#endregion
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#region Fit*
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/// <summary>
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/// Fit x,y and then eval at points xs and return the corresponding y's.
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/// This does the "natural spline" style for ends.
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/// This can extrapolate off the ends of the splines.
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/// You must provide points in X sort order.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="xs">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The computed y values for each xs.</returns>
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private float[] FitAndEval(float[] x, float[] y, float[] xs, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false)
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{
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Fit(x, y, startSlope, endSlope, debug);
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return Eval(xs, debug);
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}
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/// <summary>
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/// Compute spline coefficients for the specified x,y points.
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/// This does the "natural spline" style for ends.
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/// This can extrapolate off the ends of the splines.
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/// You must provide points in X sort order.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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private void Fit(float[] x, float[] y, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false)
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{
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if (Single.IsInfinity(startSlope) || Single.IsInfinity(endSlope))
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{
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throw new Exception("startSlope and endSlope cannot be infinity.");
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}
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// Save x and y for eval
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_xOrig = x;
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_yOrig = y;
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var n = x.Length;
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var r = new float[n]; // the right hand side numbers: wikipedia page overloads b
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var m = new TriDiagonalMatrixF(n);
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float dx1, dx2, dy1, dy2;
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// First row is different (equation 16 from the article)
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if (float.IsNaN(startSlope))
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{
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dx1 = x[1] - x[0];
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m.C[0] = 1.0f / dx1;
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m.B[0] = 2.0f * m.C[0];
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r[0] = 3 * (y[1] - y[0]) / (dx1 * dx1);
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}
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else
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{
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m.B[0] = 1;
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r[0] = startSlope;
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}
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// Body rows (equation 15 from the article)
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for (var i = 1; i < n - 1; i++)
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{
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dx1 = x[i] - x[i - 1];
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dx2 = x[i + 1] - x[i];
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m.A[i] = 1.0f / dx1;
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m.C[i] = 1.0f / dx2;
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m.B[i] = 2.0f * (m.A[i] + m.C[i]);
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dy1 = y[i] - y[i - 1];
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dy2 = y[i + 1] - y[i];
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r[i] = 3 * (dy1 / (dx1 * dx1) + dy2 / (dx2 * dx2));
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}
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// Last row also different (equation 17 from the article)
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if (float.IsNaN(endSlope))
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{
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dx1 = x[n - 1] - x[n - 2];
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dy1 = y[n - 1] - y[n - 2];
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m.A[n - 1] = 1.0f / dx1;
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m.B[n - 1] = 2.0f * m.A[n - 1];
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r[n - 1] = 3 * (dy1 / (dx1 * dx1));
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}
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else
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{
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m.B[n - 1] = 1;
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r[n - 1] = endSlope;
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}
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// if (debug) Console.WriteLine("Tri-diagonal matrix:\n{0}", m.ToDisplayString(":0.0000", " "));
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// if (debug) Console.WriteLine("r: {0}", ArrayUtil.ToString<float>(r));
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// k is the solution to the matrix
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var k = m.Solve(r);
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// if (debug) Console.WriteLine("k = {0}", ArrayUtil.ToString<float>(k));
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// a and b are each spline's coefficients
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_a = new float[n - 1];
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_b = new float[n - 1];
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for (var i = 1; i < n; i++)
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{
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dx1 = x[i] - x[i - 1];
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dy1 = y[i] - y[i - 1];
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_a[i - 1] = k[i - 1] * dx1 - dy1; // equation 10 from the article
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_b[i - 1] = -k[i] * dx1 + dy1; // equation 11 from the article
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}
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// if (debug) Console.WriteLine("a: {0}", ArrayUtil.ToString<float>(a));
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// if (debug) Console.WriteLine("b: {0}", ArrayUtil.ToString<float>(b));
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}
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#endregion
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#region Eval*
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/// <summary>
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/// Evaluate the spline at the specified x coordinates.
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/// This can extrapolate off the ends of the splines.
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/// You must provide X's in ascending order.
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/// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
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/// </summary>
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/// <param name="x">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The computed y values for each x.</returns>
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public float[] Eval(float[] x, bool debug = false)
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{
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CheckAlreadyFitted();
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var n = x.Length;
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var y = new float[n];
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_lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls
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for (var i = 0; i < n; i++)
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{
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// Find which spline can be used to compute this x (by simultaneous traverse)
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var j = GetNextXIndex(x[i]);
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// Evaluate using j'th spline
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y[i] = EvalSpline(x[i], j, debug);
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}
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return y;
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}
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/// <summary>
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/// Evaluate (compute) the slope of the spline at the specified x coordinates.
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/// This can extrapolate off the ends of the splines.
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/// You must provide X's in ascending order.
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/// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
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/// </summary>
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/// <param name="x">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The computed y values for each x.</returns>
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public float[] EvalSlope(float[] x, bool debug = false)
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{
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CheckAlreadyFitted();
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var n = x.Length;
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var qPrime = new float[n];
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_lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls
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for (var i = 0; i < n; i++)
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{
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// Find which spline can be used to compute this x (by simultaneous traverse)
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var j = GetNextXIndex(x[i]);
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// Evaluate using j'th spline
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var dx = _xOrig[j + 1] - _xOrig[j];
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var dy = _yOrig[j + 1] - _yOrig[j];
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var t = (x[i] - _xOrig[j]) / dx;
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// From equation 5 we could also compute q' (qp) which is the slope at this x
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qPrime[i] = dy / dx
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+ (1 - 2 * t) * (_a[j] * (1 - t) + _b[j] * t) / dx
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+ t * (1 - t) * (_b[j] - _a[j]) / dx;
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if (debug) Console.WriteLine("[{0}]: xs = {1}, j = {2}, t = {3}", i, x[i], j, t);
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}
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return qPrime;
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}
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#endregion
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#region Static Methods
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/// <summary>
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/// Static all-in-one method to fit the splines and evaluate at X coordinates.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="xs">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The computed y values for each xs.</returns>
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public static float[] Compute(float[] x, float[] y, float[] xs, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false)
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{
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var spline = new CubicSpline();
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return spline.FitAndEval(x, y, xs, startSlope, endSlope, debug);
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}
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/// <summary>
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/// Fit the input x,y points using the parametric approach, so that y does not have to be an explicit
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/// function of x, meaning there does not need to be a single value of y for each x.
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/// </summary>
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/// <param name="x">Input x coordinates.</param>
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/// <param name="y">Input y coordinates.</param>
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/// <param name="nOutputPoints">How many output points to create.</param>
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/// <param name="xs">Output (interpolated) x values.</param>
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/// <param name="ys">Output (interpolated) y values.</param>
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/// <param name="firstDx">Optionally specifies the first point's slope in combination with firstDy. Together they
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/// are a vector describing the direction of the parametric spline of the starting point. The vector does
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/// not need to be normalized. If either is NaN then neither is used.</param>
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/// <param name="firstDy">See description of dx0.</param>
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/// <param name="lastDx">Optionally specifies the last point's slope in combination with lastDy. Together they
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/// are a vector describing the direction of the parametric spline of the last point. The vector does
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/// not need to be normalized. If either is NaN then neither is used.</param>
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/// <param name="lastDy">See description of dxN.</param>
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public static void FitParametric(float[] x, float[] y, int nOutputPoints, out float[] xs, out float[] ys,
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float firstDx = Single.NaN, float firstDy = Single.NaN, float lastDx = Single.NaN, float lastDy = Single.NaN)
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{
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// Compute distances
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var n = x.Length;
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var dists = new float[n]; // cumulative distance
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dists[0] = 0;
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float totalDist = 0;
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for (var i = 1; i < n; i++)
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{
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var dx = x[i] - x[i - 1];
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var dy = y[i] - y[i - 1];
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var dist = (float)Math.Sqrt(dx * dx + dy * dy);
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totalDist += dist;
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dists[i] = totalDist;
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}
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// Create 'times' to interpolate to
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var dt = totalDist / (nOutputPoints - 1);
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var times = new float[nOutputPoints];
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times[0] = 0;
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for (var i = 1; i < nOutputPoints; i++)
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{
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times[i] = times[i - 1] + dt;
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}
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// Normalize the slopes, if specified
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NormalizeVector(ref firstDx, ref firstDy);
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NormalizeVector(ref lastDx, ref lastDy);
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// Spline fit both x and y to times
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var xSpline = new CubicSpline();
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xs = xSpline.FitAndEval(dists, x, times, firstDx / dt, lastDx / dt);
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var ySpline = new CubicSpline();
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ys = ySpline.FitAndEval(dists, y, times, firstDy / dt, lastDy / dt);
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}
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private static void NormalizeVector(ref float dx, ref float dy)
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{
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if (!Single.IsNaN(dx) && !Single.IsNaN(dy))
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{
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var d = (float)Math.Sqrt(dx * dx + dy * dy);
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if (d > Single.Epsilon) // probably not conservative enough, but catches the (0,0) case at least
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{
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dx = dx / d;
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dy = dy / d;
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}
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else
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{
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throw new ArgumentException("The input vector is too small to be normalized.");
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}
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}
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else
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{
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// In case one is NaN and not the other
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dx = dy = Single.NaN;
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}
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}
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#endregion
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}
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/// <summary>
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/// A tri-diagonal matrix has non-zero entries only on the main diagonal, the diagonal above the main (super), and the
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/// diagonal below the main (sub).
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/// </summary>
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/// <remarks>
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/// <para>
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/// This is based on the wikipedia article: http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
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/// </para>
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/// <para>
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/// The entries in the matrix on a particular row are A[i], B[i], and C[i] where i is the row index.
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/// B is the main diagonal, and so for an NxN matrix B is length N and all elements are used.
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/// So for row 0, the first two values are B[0] and C[0].
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/// And for row N-1, the last two values are A[N-1] and B[N-1].
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/// That means that A[0] is not actually on the matrix and is therefore never used, and same with C[N-1].
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/// </para>
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/// </remarks>
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internal sealed class TriDiagonalMatrixF
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{
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/// <summary>
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/// The values for the sub-diagonal. A[0] is never used.
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/// </summary>
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public readonly float[] A;
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/// <summary>
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/// The values for the main diagonal.
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/// </summary>
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public readonly float[] B;
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/// <summary>
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/// The values for the super-diagonal. C[C.Length-1] is never used.
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/// </summary>
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public readonly float[] C;
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/// <summary>
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/// The width and height of this matrix.
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/// </summary>
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private int N => A?.Length ?? 0;
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/// <summary>
|
|
/// Indexer. Setter throws an exception if you try to set any not on the super, main, or sub diagonals.
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|
/// </summary>
|
|
public float this[int row, int col]
|
|
{
|
|
get
|
|
{
|
|
var di = row - col;
|
|
|
|
switch (di)
|
|
{
|
|
case 0:
|
|
return B[row];
|
|
case -1:
|
|
Debug.Assert(row < N - 1);
|
|
return C[row];
|
|
case 1:
|
|
Debug.Assert(row > 0);
|
|
return A[row];
|
|
default:
|
|
return 0;
|
|
}
|
|
}
|
|
set
|
|
{
|
|
int di = row - col;
|
|
|
|
if (di == 0)
|
|
{
|
|
B[row] = value;
|
|
}
|
|
else if (di == -1)
|
|
{
|
|
Debug.Assert(row < N - 1);
|
|
C[row] = value;
|
|
}
|
|
else if (di == 1)
|
|
{
|
|
Debug.Assert(row > 0);
|
|
A[row] = value;
|
|
}
|
|
else
|
|
{
|
|
throw new ArgumentException("Only the main, super, and sub diagonals can be set.");
|
|
}
|
|
}
|
|
}
|
|
|
|
/// <summary>
|
|
/// Construct an NxN matrix.
|
|
/// </summary>
|
|
public TriDiagonalMatrixF(int n)
|
|
{
|
|
A = new float[n];
|
|
B = new float[n];
|
|
C = new float[n];
|
|
}
|
|
|
|
/// <summary>
|
|
/// Produce a string representation of the contents of this matrix.
|
|
/// </summary>
|
|
/// <param name="fmt">Optional. For String.Format. Must include the colon. Examples are ':0.000' and ',5:0.00' </param>
|
|
/// <param name="prefix">Optional. Per-line indentation prefix.</param>
|
|
public string ToString(string fmt = "", string prefix = "")
|
|
{
|
|
if (N <= 0)
|
|
return prefix + "0x0 Matrix";
|
|
|
|
var s = new StringBuilder();
|
|
var formatString = "{0" + fmt + "}";
|
|
|
|
for (int r = 0; r < N; r++)
|
|
{
|
|
s.Append(prefix);
|
|
|
|
for (int c = 0; c < N; c++)
|
|
{
|
|
s.AppendFormat(formatString, this[r, c]);
|
|
if (c < N - 1) s.Append(", ");
|
|
}
|
|
|
|
s.AppendLine();
|
|
}
|
|
|
|
return s.ToString();
|
|
|
|
}
|
|
|
|
/// <summary>
|
|
/// Solve the system of equations this*x=d given the specified d.
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Uses the Thomas algorithm described in the wikipedia article: http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
|
|
/// Not optimized. Not destructive.
|
|
/// </remarks>
|
|
/// <param name="d">Right side of the equation.</param>
|
|
public float[] Solve(float[] d)
|
|
{
|
|
int n = N;
|
|
|
|
if (d.Length != n)
|
|
{
|
|
throw new ArgumentException("The input d is not the same size as this matrix.");
|
|
}
|
|
|
|
// cPrime
|
|
float[] cPrime = new float[n];
|
|
cPrime[0] = C[0] / B[0];
|
|
|
|
for (int i = 1; i < n; i++)
|
|
{
|
|
cPrime[i] = C[i] / (B[i] - cPrime[i-1] * A[i]);
|
|
}
|
|
|
|
// dPrime
|
|
float[] dPrime = new float[n];
|
|
dPrime[0] = d[0] / B[0];
|
|
|
|
for (int i = 1; i < n; i++)
|
|
{
|
|
dPrime[i] = (d[i] - dPrime[i-1]*A[i]) / (B[i] - cPrime[i - 1] * A[i]);
|
|
}
|
|
|
|
// Back substitution
|
|
float[] x = new float[n];
|
|
x[n - 1] = dPrime[n - 1];
|
|
|
|
for (int i = n-2; i >= 0; i--)
|
|
{
|
|
x[i] = dPrime[i] - cPrime[i] * x[i + 1];
|
|
}
|
|
|
|
return x;
|
|
}
|
|
} |