580 lines
16 KiB
C++
580 lines
16 KiB
C++
/*
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* This program is free software; you can redistribute it and/or
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* modify it under the terms of the GNU General Public License
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* as published by the Free Software Foundation; either version 2
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* of the License, or (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software Foundation,
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* Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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/** \file
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* \ingroup freestyle
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* \brief An Algorithm for Automatically Fitting Digitized Curves by Philip J. Schneider,
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* \brief from "Graphics Gems", Academic Press, 1990
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*/
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#include <cmath>
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#include <cstdio>
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#include <cstdlib> // for malloc and free
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#include "FitCurve.h"
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using namespace std;
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namespace Freestyle {
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using BezierCurve = Vector2 *;
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/* Forward declarations */
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static double *Reparameterize(Vector2 *d, int first, int last, double *u, BezierCurve bezCurve);
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static double NewtonRaphsonRootFind(BezierCurve Q, Vector2 P, double u);
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static Vector2 BezierII(int degree, Vector2 *V, double t);
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static double B0(double u);
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static double B1(double u);
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static double B2(double u);
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static double B3(double u);
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static Vector2 ComputeLeftTangent(Vector2 *d, int end);
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static double ComputeMaxError(
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Vector2 *d, int first, int last, BezierCurve bezCurve, double *u, int *splitPoint);
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static double *ChordLengthParameterize(Vector2 *d, int first, int last);
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static BezierCurve GenerateBezier(
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Vector2 *d, int first, int last, double *uPrime, Vector2 tHat1, Vector2 tHat2);
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static Vector2 V2AddII(Vector2 a, Vector2 b);
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static Vector2 V2ScaleIII(Vector2 v, double s);
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static Vector2 V2SubII(Vector2 a, Vector2 b);
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/* returns squared length of input vector */
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static double V2SquaredLength(Vector2 *a)
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{
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return (((*a)[0] * (*a)[0]) + ((*a)[1] * (*a)[1]));
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}
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/* returns length of input vector */
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static double V2Length(Vector2 *a)
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{
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return (sqrt(V2SquaredLength(a)));
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}
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static Vector2 *V2Scale(Vector2 *v, double newlen)
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{
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double len = V2Length(v);
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if (len != 0.0) {
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(*v)[0] *= newlen / len;
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(*v)[1] *= newlen / len;
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}
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return v;
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}
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/* return the dot product of vectors a and b */
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static double V2Dot(Vector2 *a, Vector2 *b)
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{
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return (((*a)[0] * (*b)[0]) + ((*a)[1] * (*b)[1]));
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}
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/* return the distance between two points */
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static double V2DistanceBetween2Points(Vector2 *a, Vector2 *b)
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{
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double dx = (*a)[0] - (*b)[0];
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double dy = (*a)[1] - (*b)[1];
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return (sqrt((dx * dx) + (dy * dy)));
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}
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/* return vector sum c = a+b */
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static Vector2 *V2Add(Vector2 *a, Vector2 *b, Vector2 *c)
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{
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(*c)[0] = (*a)[0] + (*b)[0];
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(*c)[1] = (*a)[1] + (*b)[1];
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return c;
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}
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/* normalizes the input vector and returns it */
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static Vector2 *V2Normalize(Vector2 *v)
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{
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double len = V2Length(v);
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if (len != 0.0) {
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(*v)[0] /= len;
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(*v)[1] /= len;
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}
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return v;
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}
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/* negates the input vector and returns it */
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static Vector2 *V2Negate(Vector2 *v)
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{
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(*v)[0] = -(*v)[0];
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(*v)[1] = -(*v)[1];
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return v;
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}
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/* GenerateBezier:
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* Use least-squares method to find Bezier control points for region.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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* double *uPrime; Parameter values for region
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* Vector2 tHat1, tHat2; Unit tangents at endpoints
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*/
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static BezierCurve GenerateBezier(
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Vector2 *d, int first, int last, double *uPrime, Vector2 tHat1, Vector2 tHat2)
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{
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int i;
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Vector2 A[2]; /* rhs for eqn */
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int nPts; /* Number of pts in sub-curve */
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double C[2][2]; /* Matrix C */
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double X[2]; /* Matrix X */
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double det_C0_C1; /* Determinants of matrices */
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double det_C0_X;
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double det_X_C1;
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double alpha_l; /* Alpha values, left and right */
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double alpha_r;
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Vector2 tmp; /* Utility variable */
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BezierCurve bezCurve; /* RETURN bezier curve control points. */
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bezCurve = (Vector2 *)malloc(4 * sizeof(Vector2));
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nPts = last - first + 1;
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/* Create the C and X matrices */
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C[0][0] = 0.0;
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C[0][1] = 0.0;
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C[1][0] = 0.0;
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C[1][1] = 0.0;
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X[0] = 0.0;
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X[1] = 0.0;
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for (i = 0; i < nPts; i++) {
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/* Compute the A's */
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A[0] = tHat1;
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A[1] = tHat2;
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V2Scale(&A[0], B1(uPrime[i]));
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V2Scale(&A[1], B2(uPrime[i]));
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C[0][0] += V2Dot(&A[0], &A[0]);
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C[0][1] += V2Dot(&A[0], &A[1]);
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// C[1][0] += V2Dot(&A[0], &A[1]);
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C[1][0] = C[0][1];
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C[1][1] += V2Dot(&A[1], &A[1]);
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tmp = V2SubII(d[first + i],
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V2AddII(V2ScaleIII(d[first], B0(uPrime[i])),
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V2AddII(V2ScaleIII(d[first], B1(uPrime[i])),
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V2AddII(V2ScaleIII(d[last], B2(uPrime[i])),
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V2ScaleIII(d[last], B3(uPrime[i]))))));
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X[0] += V2Dot(&A[0], &tmp);
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X[1] += V2Dot(&A[1], &tmp);
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}
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/* Compute the determinants of C and X */
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det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
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det_C0_X = C[0][0] * X[1] - C[0][1] * X[0];
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det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1];
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/* Finally, derive alpha values */
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if (det_C0_C1 == 0.0) {
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det_C0_C1 = (C[0][0] * C[1][1]) * 10.0e-12;
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}
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alpha_l = det_X_C1 / det_C0_C1;
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alpha_r = det_C0_X / det_C0_C1;
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/* If alpha negative, use the Wu/Barsky heuristic (see text) (if alpha is 0, you get coincident
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* control points that lead to divide by zero in any subsequent NewtonRaphsonRootFind() call).
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*/
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if (alpha_l < 1.0e-6 || alpha_r < 1.0e-6) {
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double dist = V2DistanceBetween2Points(&d[last], &d[first]) / 3.0;
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bezCurve[0] = d[first];
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bezCurve[3] = d[last];
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V2Add(&(bezCurve[0]), V2Scale(&(tHat1), dist), &(bezCurve[1]));
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V2Add(&(bezCurve[3]), V2Scale(&(tHat2), dist), &(bezCurve[2]));
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return bezCurve;
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}
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/* First and last control points of the Bezier curve are positioned exactly at the first and last
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* data points Control points 1 and 2 are positioned an alpha distance out on the tangent
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* vectors, left and right, respectively
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*/
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bezCurve[0] = d[first];
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bezCurve[3] = d[last];
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V2Add(&bezCurve[0], V2Scale(&tHat1, alpha_l), &bezCurve[1]);
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V2Add(&bezCurve[3], V2Scale(&tHat2, alpha_r), &bezCurve[2]);
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return bezCurve;
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}
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/*
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* Reparameterize:
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* Given set of points and their parameterization, try to find a better parameterization.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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* double *u; Current parameter values
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* BezierCurve bezCurve; Current fitted curve
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*/
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static double *Reparameterize(Vector2 *d, int first, int last, double *u, BezierCurve bezCurve)
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{
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int nPts = last - first + 1;
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int i;
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double *uPrime; /* New parameter values */
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uPrime = (double *)malloc(nPts * sizeof(double));
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for (i = first; i <= last; i++) {
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uPrime[i - first] = NewtonRaphsonRootFind(bezCurve, d[i], u[i - first]);
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}
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return uPrime;
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}
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/*
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* NewtonRaphsonRootFind:
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* Use Newton-Raphson iteration to find better root.
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* BezierCurve Q; Current fitted curve
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* Vector2 P; Digitized point
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* double u; Parameter value for "P"
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*/
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static double NewtonRaphsonRootFind(BezierCurve Q, Vector2 P, double u)
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{
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double numerator, denominator;
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Vector2 Q1[3], Q2[2]; /* Q' and Q'' */
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Vector2 Q_u, Q1_u, Q2_u; /* u evaluated at Q, Q', & Q'' */
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double uPrime; /* Improved u */
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int i;
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/* Compute Q(u) */
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Q_u = BezierII(3, Q, u);
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/* Generate control vertices for Q' */
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for (i = 0; i <= 2; i++) {
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Q1[i][0] = (Q[i + 1][0] - Q[i][0]) * 3.0;
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Q1[i][1] = (Q[i + 1][1] - Q[i][1]) * 3.0;
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}
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/* Generate control vertices for Q'' */
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for (i = 0; i <= 1; i++) {
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Q2[i][0] = (Q1[i + 1][0] - Q1[i][0]) * 2.0;
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Q2[i][1] = (Q1[i + 1][1] - Q1[i][1]) * 2.0;
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}
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/* Compute Q'(u) and Q''(u) */
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Q1_u = BezierII(2, Q1, u);
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Q2_u = BezierII(1, Q2, u);
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/* Compute f(u)/f'(u) */
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numerator = (Q_u[0] - P[0]) * (Q1_u[0]) + (Q_u[1] - P[1]) * (Q1_u[1]);
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denominator = (Q1_u[0]) * (Q1_u[0]) + (Q1_u[1]) * (Q1_u[1]) + (Q_u[0] - P[0]) * (Q2_u[0]) +
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(Q_u[1] - P[1]) * (Q2_u[1]);
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/* u = u - f(u)/f'(u) */
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if (denominator == 0) { // FIXME
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return u;
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}
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uPrime = u - (numerator / denominator);
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return uPrime;
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}
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/*
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* Bezier:
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* Evaluate a Bezier curve at a particular parameter value
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* int degree; The degree of the bezier curve
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* Vector2 *V; Array of control points
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* double t; Parametric value to find point for
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*/
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static Vector2 BezierII(int degree, Vector2 *V, double t)
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{
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int i, j;
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Vector2 Q; /* Point on curve at parameter t */
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Vector2 *Vtemp; /* Local copy of control points */
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/* Copy array */
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Vtemp = (Vector2 *)malloc((unsigned)((degree + 1) * sizeof(Vector2)));
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for (i = 0; i <= degree; i++) {
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Vtemp[i] = V[i];
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}
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/* Triangle computation */
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for (i = 1; i <= degree; i++) {
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for (j = 0; j <= degree - i; j++) {
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Vtemp[j][0] = (1.0 - t) * Vtemp[j][0] + t * Vtemp[j + 1][0];
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Vtemp[j][1] = (1.0 - t) * Vtemp[j][1] + t * Vtemp[j + 1][1];
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}
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}
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Q = Vtemp[0];
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free((void *)Vtemp);
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return Q;
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}
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/*
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* B0, B1, B2, B3:
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* Bezier multipliers
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*/
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static double B0(double u)
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{
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double tmp = 1.0 - u;
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return (tmp * tmp * tmp);
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}
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static double B1(double u)
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{
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double tmp = 1.0 - u;
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return (3 * u * (tmp * tmp));
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}
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static double B2(double u)
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{
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double tmp = 1.0 - u;
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return (3 * u * u * tmp);
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}
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static double B3(double u)
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{
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return (u * u * u);
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}
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/*
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* ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent:
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* Approximate unit tangents at endpoints and "center" of digitized curve
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*/
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/* Vector2 *d; Digitized points
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* int end; Index to "left" end of region
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*/
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static Vector2 ComputeLeftTangent(Vector2 *d, int end)
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{
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Vector2 tHat1;
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tHat1 = V2SubII(d[end + 1], d[end]);
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tHat1 = *V2Normalize(&tHat1);
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return tHat1;
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}
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/* Vector2 *d; Digitized points
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* int end; Index to "right" end of region
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*/
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static Vector2 ComputeRightTangent(Vector2 *d, int end)
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{
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Vector2 tHat2;
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tHat2 = V2SubII(d[end - 1], d[end]);
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tHat2 = *V2Normalize(&tHat2);
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return tHat2;
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}
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/* Vector2 *d; Digitized points
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* int end; Index to point inside region
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*/
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static Vector2 ComputeCenterTangent(Vector2 *d, int center)
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{
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Vector2 V1, V2, tHatCenter;
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V1 = V2SubII(d[center - 1], d[center]);
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V2 = V2SubII(d[center], d[center + 1]);
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tHatCenter[0] = (V1[0] + V2[0]) / 2.0;
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tHatCenter[1] = (V1[1] + V2[1]) / 2.0;
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tHatCenter = *V2Normalize(&tHatCenter);
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/* avoid numerical singularity in the special case when V1 == -V2 */
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if (V2Length(&tHatCenter) < M_EPSILON) {
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tHatCenter = *V2Normalize(&V1);
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}
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return tHatCenter;
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}
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/*
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* ChordLengthParameterize:
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* Assign parameter values to digitized points using relative distances between points.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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*/
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static double *ChordLengthParameterize(Vector2 *d, int first, int last)
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{
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int i;
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double *u; /* Parameterization */
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u = (double *)malloc((unsigned)(last - first + 1) * sizeof(double));
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u[0] = 0.0;
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for (i = first + 1; i <= last; i++) {
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u[i - first] = u[i - first - 1] + V2DistanceBetween2Points(&d[i], &d[i - 1]);
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}
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for (i = first + 1; i <= last; i++) {
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u[i - first] = u[i - first] / u[last - first];
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}
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return u;
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}
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/*
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* ComputeMaxError :
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* Find the maximum squared distance of digitized points to fitted curve.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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* BezierCurve bezCurve; Fitted Bezier curve
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* double *u; Parameterization of points
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* int *splitPoint; Point of maximum error
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*/
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static double ComputeMaxError(
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Vector2 *d, int first, int last, BezierCurve bezCurve, double *u, int *splitPoint)
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{
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int i;
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double maxDist; /* Maximum error */
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double dist; /* Current error */
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Vector2 P; /* Point on curve */
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Vector2 v; /* Vector from point to curve */
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*splitPoint = (last - first + 1) / 2;
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maxDist = 0.0;
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for (i = first + 1; i < last; i++) {
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P = BezierII(3, bezCurve, u[i - first]);
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v = V2SubII(P, d[i]);
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dist = V2SquaredLength(&v);
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if (dist >= maxDist) {
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maxDist = dist;
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*splitPoint = i;
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}
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}
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return maxDist;
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}
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static Vector2 V2AddII(Vector2 a, Vector2 b)
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{
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Vector2 c;
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c[0] = a[0] + b[0];
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c[1] = a[1] + b[1];
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return c;
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}
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static Vector2 V2ScaleIII(Vector2 v, double s)
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{
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Vector2 result;
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result[0] = v[0] * s;
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result[1] = v[1] * s;
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return result;
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}
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static Vector2 V2SubII(Vector2 a, Vector2 b)
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{
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Vector2 c;
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c[0] = a[0] - b[0];
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c[1] = a[1] - b[1];
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return c;
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}
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//------------------------- WRAPPER -----------------------------//
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FitCurveWrapper::~FitCurveWrapper()
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{
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_vertices.clear();
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}
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void FitCurveWrapper::DrawBezierCurve(int n, Vector2 *curve)
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{
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for (int i = 0; i <= n; ++i) {
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_vertices.push_back(curve[i]);
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}
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}
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void FitCurveWrapper::FitCurve(vector<Vec2d> &data, vector<Vec2d> &oCurve, double error)
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{
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int size = data.size();
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Vector2 *d = new Vector2[size];
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for (int i = 0; i < size; ++i) {
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|
d[i][0] = data[i][0];
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|
d[i][1] = data[i][1];
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|
}
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|
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|
FitCurve(d, size, error);
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|
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|
delete[] d;
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|
|
|
// copy results
|
|
for (vector<Vector2>::iterator v = _vertices.begin(), vend = _vertices.end(); v != vend; ++v) {
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|
oCurve.emplace_back(v->x(), v->y());
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|
}
|
|
}
|
|
|
|
void FitCurveWrapper::FitCurve(Vector2 *d, int nPts, double error)
|
|
{
|
|
Vector2 tHat1, tHat2; /* Unit tangent vectors at endpoints */
|
|
|
|
tHat1 = ComputeLeftTangent(d, 0);
|
|
tHat2 = ComputeRightTangent(d, nPts - 1);
|
|
FitCubic(d, 0, nPts - 1, tHat1, tHat2, error);
|
|
}
|
|
|
|
void FitCurveWrapper::FitCubic(
|
|
Vector2 *d, int first, int last, Vector2 tHat1, Vector2 tHat2, double error)
|
|
{
|
|
BezierCurve bezCurve; /* Control points of fitted Bezier curve */
|
|
double *u; /* Parameter values for point */
|
|
double *uPrime; /* Improved parameter values */
|
|
double maxError; /* Maximum fitting error */
|
|
int splitPoint; /* Point to split point set at */
|
|
int nPts; /* Number of points in subset */
|
|
double iterationError; /* Error below which you try iterating */
|
|
int maxIterations = 4; /* Max times to try iterating */
|
|
Vector2 tHatCenter; /* Unit tangent vector at splitPoint */
|
|
int i;
|
|
|
|
iterationError = error * error;
|
|
nPts = last - first + 1;
|
|
|
|
/* Use heuristic if region only has two points in it */
|
|
if (nPts == 2) {
|
|
double dist = V2DistanceBetween2Points(&d[last], &d[first]) / 3.0;
|
|
|
|
bezCurve = (Vector2 *)malloc(4 * sizeof(Vector2));
|
|
bezCurve[0] = d[first];
|
|
bezCurve[3] = d[last];
|
|
V2Add(&bezCurve[0], V2Scale(&tHat1, dist), &bezCurve[1]);
|
|
V2Add(&bezCurve[3], V2Scale(&tHat2, dist), &bezCurve[2]);
|
|
DrawBezierCurve(3, bezCurve);
|
|
free((void *)bezCurve);
|
|
return;
|
|
}
|
|
|
|
/* Parameterize points, and attempt to fit curve */
|
|
u = ChordLengthParameterize(d, first, last);
|
|
bezCurve = GenerateBezier(d, first, last, u, tHat1, tHat2);
|
|
|
|
/* Find max deviation of points to fitted curve */
|
|
maxError = ComputeMaxError(d, first, last, bezCurve, u, &splitPoint);
|
|
if (maxError < error) {
|
|
DrawBezierCurve(3, bezCurve);
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
return;
|
|
}
|
|
|
|
/* If error not too large, try some reparameterization and iteration */
|
|
if (maxError < iterationError) {
|
|
for (i = 0; i < maxIterations; i++) {
|
|
uPrime = Reparameterize(d, first, last, u, bezCurve);
|
|
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
u = uPrime;
|
|
|
|
bezCurve = GenerateBezier(d, first, last, u, tHat1, tHat2);
|
|
maxError = ComputeMaxError(d, first, last, bezCurve, u, &splitPoint);
|
|
|
|
if (maxError < error) {
|
|
DrawBezierCurve(3, bezCurve);
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
return;
|
|
}
|
|
}
|
|
}
|
|
|
|
/* Fitting failed -- split at max error point and fit recursively */
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
tHatCenter = ComputeCenterTangent(d, splitPoint);
|
|
FitCubic(d, first, splitPoint, tHat1, tHatCenter, error);
|
|
V2Negate(&tHatCenter);
|
|
FitCubic(d, splitPoint, last, tHatCenter, tHat2, error);
|
|
}
|
|
|
|
} /* namespace Freestyle */
|