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Diffstat (limited to 'source/blender/freestyle/intern/geometry/FitCurve.cpp')
| -rw-r--r-- | source/blender/freestyle/intern/geometry/FitCurve.cpp | 602 |
1 files changed, 602 insertions, 0 deletions
diff --git a/source/blender/freestyle/intern/geometry/FitCurve.cpp b/source/blender/freestyle/intern/geometry/FitCurve.cpp new file mode 100644 index 00000000000..4eae543c9f9 --- /dev/null +++ b/source/blender/freestyle/intern/geometry/FitCurve.cpp @@ -0,0 +1,602 @@ +/* + * ***** BEGIN GPL LICENSE BLOCK ***** + * + * This program is free software; you can redistribute it and/or + * modify it under the terms of the GNU General Public License + * as published by the Free Software Foundation; either version 2 + * of the License, or (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program; if not, write to the Free Software Foundation, + * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. + * + * The Original Code is Copyright (C) 2010 Blender Foundation. + * All rights reserved. + * + * The Original Code is: all of this file. + * + * Contributor(s): none yet. + * + * ***** END GPL LICENSE BLOCK ***** + */ + +/** \file blender/freestyle/intern/geometry/FitCurve.cpp + * \ingroup freestyle + * \brief An Algorithm for Automatically Fitting Digitized Curves by Philip J. Schneider, + * \brief from "Graphics Gems", Academic Press, 1990 + * \author Stephane Grabli + * \date 06/06/2003 + */ + +#include <cstdlib> // for malloc and free +#include <stdio.h> +#include <math.h> + +#include "FitCurve.h" + +using namespace std; + +typedef Vector2 *BezierCurve; + +// XXX Do we need "#ifdef __cplusplus" at all here??? +#ifdef __cplusplus +extern "C" +{ +#endif + +/* Forward declarations */ +static double *Reparameterize(Vector2 *d, int first, int last, double *u, BezierCurve bezCurve); +static double NewtonRaphsonRootFind(BezierCurve Q, Vector2 P, double u); +static Vector2 BezierII(int degree, Vector2 *V, double t); +static double B0(double u); +static double B1(double u); +static double B2(double u); +static double B3(double u); +static Vector2 ComputeLeftTangent(Vector2 *d, int end); +static Vector2 ComputeLeftTangent(Vector2 *d, int end); +static Vector2 ComputeLeftTangent(Vector2 *d, int end); +static double ComputeMaxError(Vector2 *d, int first, int last, BezierCurve bezCurve, double *u, int *splitPoint); +static double *ChordLengthParameterize(Vector2 *d, int first, int last); +static BezierCurve GenerateBezier(Vector2 *d, int first, int last, double *uPrime, Vector2 tHat1, Vector2 tHat2); +static Vector2 V2AddII(Vector2 a, Vector2 b); +static Vector2 V2ScaleIII(Vector2 v, double s); +static Vector2 V2SubII(Vector2 a, Vector2 b); + +#define MAXPOINTS 1000 /* The most points you can have */ + +/* returns squared length of input vector */ +static double V2SquaredLength(Vector2 *a) +{ + return (((*a)[0] * (*a)[0]) + ((*a)[1] * (*a)[1])); +} + +/* returns length of input vector */ +static double V2Length(Vector2 *a) +{ + return (sqrt(V2SquaredLength(a))); +} + +static Vector2 *V2Scale(Vector2 *v, double newlen) +{ + double len = V2Length(v); + if (len != 0.0) { + (*v)[0] *= newlen / len; + (*v)[1] *= newlen / len; + } + return v; +} + +/* return the dot product of vectors a and b */ +static double V2Dot(Vector2 *a, Vector2 *b) +{ + return (((*a)[0] * (*b)[0]) + ((*a)[1] * (*b)[1])); +} + +/* return the distance between two points */ +static double V2DistanceBetween2Points(Vector2 *a, Vector2 *b) +{ + double dx = (*a)[0] - (*b)[0]; + double dy = (*a)[1] - (*b)[1]; + return (sqrt((dx * dx) + (dy * dy))); +} + +/* return vector sum c = a+b */ +static Vector2 *V2Add(Vector2 *a, Vector2 *b, Vector2 *c) +{ + (*c)[0] = (*a)[0] + (*b)[0]; + (*c)[1] = (*a)[1] + (*b)[1]; + return c; +} + +/* normalizes the input vector and returns it */ +static Vector2 *V2Normalize(Vector2 *v) +{ + double len = V2Length(v); + if (len != 0.0) { + (*v)[0] /= len; + (*v)[1] /= len; + } + return v; +} + +/* negates the input vector and returns it */ +static Vector2 *V2Negate(Vector2 *v) +{ + (*v)[0] = -(*v)[0]; + (*v)[1] = -(*v)[1]; + return v; +} + +/* GenerateBezier: + * Use least-squares method to find Bezier control points for region. + * Vector2 *d; Array of digitized points + * int first, last; Indices defining region + * double *uPrime; Parameter values for region + * Vector2 tHat1, tHat2; Unit tangents at endpoints + */ +static BezierCurve GenerateBezier(Vector2 *d, int first, int last, double *uPrime, Vector2 tHat1, Vector2 tHat2) +{ + int i; + Vector2 A[MAXPOINTS][2]; /* Precomputed rhs for eqn */ + int nPts; /* Number of pts in sub-curve */ + double C[2][2]; /* Matrix C */ + double X[2]; /* Matrix X */ + double det_C0_C1; /* Determinants of matrices */ + double det_C0_X; + double det_X_C1; + double alpha_l; /* Alpha values, left and right */ + double alpha_r; + Vector2 tmp; /* Utility variable */ + BezierCurve bezCurve; /* RETURN bezier curve ctl pts */ + + bezCurve = (Vector2*)malloc(4 * sizeof(Vector2)); + nPts = last - first + 1; + + /* Compute the A's */ + for (i = 0; i < nPts; i++) { + Vector2 v1, v2; + v1 = tHat1; + v2 = tHat2; + V2Scale(&v1, B1(uPrime[i])); + V2Scale(&v2, B2(uPrime[i])); + A[i][0] = v1; + A[i][1] = v2; + } + + /* Create the C and X matrices */ + C[0][0] = 0.0; + C[0][1] = 0.0; + C[1][0] = 0.0; + C[1][1] = 0.0; + X[0] = 0.0; + X[1] = 0.0; + for (i = 0; i < nPts; i++) { + C[0][0] += V2Dot(&A[i][0], &A[i][0]); + C[0][1] += V2Dot(&A[i][0], &A[i][1]); +// C[1][0] += V2Dot(&A[i][0], &A[i][1]); + C[1][0] = C[0][1]; + C[1][1] += V2Dot(&A[i][1], &A[i][1]); + + tmp = V2SubII(d[first + i], + V2AddII(V2ScaleIII(d[first], B0(uPrime[i])), + V2AddII(V2ScaleIII(d[first], B1(uPrime[i])), + V2AddII(V2ScaleIII(d[last], B2(uPrime[i])), + V2ScaleIII(d[last], B3(uPrime[i])) + ) + ) + ) + ); + + X[0] += V2Dot(&((A[i])[0]), &tmp); + X[1] += V2Dot(&((A[i])[1]), &tmp); + } + + /* Compute the determinants of C and X */ + det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1]; + det_C0_X = C[0][0] * X[1] - C[0][1] * X[0]; + det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1]; + + /* Finally, derive alpha values */ + if (det_C0_C1 == 0.0) { + det_C0_C1 = (C[0][0] * C[1][1]) * 10.0e-12; + } + alpha_l = det_X_C1 / det_C0_C1; + alpha_r = det_C0_X / det_C0_C1; + + + /* If alpha negative, use the Wu/Barsky heuristic (see text) (if alpha is 0, you get coincident control points + * that lead to divide by zero in any subsequent NewtonRaphsonRootFind() call). + */ + if (alpha_l < 1.0e-6 || alpha_r < 1.0e-6) { + double dist = V2DistanceBetween2Points(&d[last], &d[first]) / 3.0; + + bezCurve[0] = d[first]; + bezCurve[3] = d[last]; + V2Add(&(bezCurve[0]), V2Scale(&(tHat1), dist), &(bezCurve[1])); + V2Add(&(bezCurve[3]), V2Scale(&(tHat2), dist), &(bezCurve[2])); + return bezCurve; + } + + /* First and last control points of the Bezier curve are positioned exactly at the first and last data points + * Control points 1 and 2 are positioned an alpha distance out on the tangent vectors, left and right, respectively + */ + bezCurve[0] = d[first]; + bezCurve[3] = d[last]; + V2Add(&bezCurve[0], V2Scale(&tHat1, alpha_l), &bezCurve[1]); + V2Add(&bezCurve[3], V2Scale(&tHat2, alpha_r), &bezCurve[2]); + return (bezCurve); +} + +/* + * Reparameterize: + * Given set of points and their parameterization, try to find a better parameterization. + * Vector2 *d; Array of digitized points + * int first, last; Indices defining region + * double *u; Current parameter values + * BezierCurve bezCurve; Current fitted curve + */ +static double *Reparameterize(Vector2 *d, int first, int last, double *u, BezierCurve bezCurve) +{ + int nPts = last - first + 1; + int i; + double *uPrime; /* New parameter values */ + + uPrime = (double*)malloc(nPts * sizeof(double)); + for (i = first; i <= last; i++) { + uPrime[i-first] = NewtonRaphsonRootFind(bezCurve, d[i], u[i - first]); + } + return (uPrime); +} + +/* + * NewtonRaphsonRootFind: + * Use Newton-Raphson iteration to find better root. + * BezierCurve Q; Current fitted curve + * Vector2 P; Digitized point + * double u; Parameter value for "P" + */ +static double NewtonRaphsonRootFind(BezierCurve Q, Vector2 P, double u) +{ + double numerator, denominator; + Vector2 Q1[3], Q2[2]; /* Q' and Q'' */ + Vector2 Q_u, Q1_u, Q2_u; /* u evaluated at Q, Q', & Q'' */ + double uPrime; /* Improved u */ + int i; + + /* Compute Q(u) */ + Q_u = BezierII(3, Q, u); + + /* Generate control vertices for Q' */ + for (i = 0; i <= 2; i++) { + Q1[i][0] = (Q[i + 1][0] - Q[i][0]) * 3.0; + Q1[i][1] = (Q[i + 1][1] - Q[i][1]) * 3.0; + } + + /* Generate control vertices for Q'' */ + for (i = 0; i <= 1; i++) { + Q2[i][0] = (Q1[i + 1][0] - Q1[i][0]) * 2.0; + Q2[i][1] = (Q1[i + 1][1] - Q1[i][1]) * 2.0; + } + + /* Compute Q'(u) and Q''(u) */ + Q1_u = BezierII(2, Q1, u); + Q2_u = BezierII(1, Q2, u); + + /* Compute f(u)/f'(u) */ + numerator = (Q_u[0] - P[0]) * (Q1_u[0]) + (Q_u[1] - P[1]) * (Q1_u[1]); + denominator = (Q1_u[0]) * (Q1_u[0]) + (Q1_u[1]) * (Q1_u[1]) + + (Q_u[0] - P[0]) * (Q2_u[0]) + (Q_u[1] - P[1]) * (Q2_u[1]); + + /* u = u - f(u)/f'(u) */ + if (denominator == 0) // FIXME + return u; + uPrime = u - (numerator / denominator); + return uPrime; +} + +/* + * Bezier: + * Evaluate a Bezier curve at a particular parameter value + * int degree; The degree of the bezier curve + * Vector2 *V; Array of control points + * double t; Parametric value to find point for + */ +static Vector2 BezierII(int degree, Vector2 *V, double t) +{ + int i, j; + Vector2 Q; /* Point on curve at parameter t */ + Vector2 *Vtemp; /* Local copy of control points */ + + /* Copy array */ + Vtemp = (Vector2*)malloc((unsigned)((degree + 1) * sizeof (Vector2))); + for (i = 0; i <= degree; i++) { + Vtemp[i] = V[i]; + } + + /* Triangle computation */ + for (i = 1; i <= degree; i++) { + for (j = 0; j <= degree-i; j++) { + Vtemp[j][0] = (1.0 - t) * Vtemp[j][0] + t * Vtemp[j + 1][0]; + Vtemp[j][1] = (1.0 - t) * Vtemp[j][1] + t * Vtemp[j + 1][1]; + } + } + + Q = Vtemp[0]; + free((void*)Vtemp); + return Q; +} + +/* + * B0, B1, B2, B3: + * Bezier multipliers + */ +static double B0(double u) +{ + double tmp = 1.0 - u; + return (tmp * tmp * tmp); +} + + +static double B1(double u) +{ + double tmp = 1.0 - u; + return (3 * u * (tmp * tmp)); +} + +static double B2(double u) +{ + double tmp = 1.0 - u; + return (3 * u * u * tmp); +} + +static double B3(double u) +{ + return (u * u * u); +} + +/* + * ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent: + * Approximate unit tangents at endpoints and "center" of digitized curve + */ +/* Vector2 *d; Digitized points + * int end; Index to "left" end of region + */ +static Vector2 ComputeLeftTangent(Vector2 *d, int end) +{ + Vector2 tHat1; + tHat1 = V2SubII(d[end + 1], d[end]); + tHat1 = *V2Normalize(&tHat1); + return tHat1; +} + +/* Vector2 *d; Digitized points + * int end; Index to "right" end of region + */ +static Vector2 ComputeRightTangent(Vector2 *d, int end) +{ + Vector2 tHat2; + tHat2 = V2SubII(d[end - 1], d[end]); + tHat2 = *V2Normalize(&tHat2); + return tHat2; +} + +/* Vector2 *d; Digitized points + * int end; Index to point inside region + */ +static Vector2 ComputeCenterTangent(Vector2 *d, int center) +{ + Vector2 V1, V2, tHatCenter; + + V1 = V2SubII(d[center - 1], d[center]); + V2 = V2SubII(d[center], d[center + 1]); + tHatCenter[0] = (V1[0] + V2[0]) / 2.0; + tHatCenter[1] = (V1[1] + V2[1]) / 2.0; + tHatCenter = *V2Normalize(&tHatCenter); + return tHatCenter; +} + +/* + * ChordLengthParameterize: + * Assign parameter values to digitized points using relative distances between points. + * Vector2 *d; Array of digitized points + * int first, last; Indices defining region + */ +static double *ChordLengthParameterize(Vector2 *d, int first, int last) +{ + int i; + double *u; /* Parameterization */ + + u = (double*)malloc((unsigned)(last - first + 1) * sizeof(double)); + + u[0] = 0.0; + for (i = first + 1; i <= last; i++) { + u[i - first] = u[i - first - 1] + V2DistanceBetween2Points(&d[i], &d[i - 1]); + } + + for (i = first + 1; i <= last; i++) { + u[i - first] = u[i - first] / u[last - first]; + } + + return u; +} + + + + +/* + * ComputeMaxError : + * Find the maximum squared distance of digitized points to fitted curve. + * Vector2 *d; Array of digitized points + * int first, last; Indices defining region + * BezierCurve bezCurve; Fitted Bezier curve + * double *u; Parameterization of points + * int *splitPoint; Point of maximum error + */ +static double ComputeMaxError(Vector2 *d, int first, int last, BezierCurve bezCurve, double *u, int *splitPoint) +{ + int i; + double maxDist; /* Maximum error */ + double dist; /* Current error */ + Vector2 P; /* Point on curve */ + Vector2 v; /* Vector from point to curve */ + + *splitPoint = (last - first + 1) / 2; + maxDist = 0.0; + for (i = first + 1; i < last; i++) { + P = BezierII(3, bezCurve, u[i - first]); + v = V2SubII(P, d[i]); + dist = V2SquaredLength(&v); + if (dist >= maxDist) { + maxDist = dist; + *splitPoint = i; + } + } + return maxDist; +} + +static Vector2 V2AddII(Vector2 a, Vector2 b) +{ + Vector2 c; + c[0] = a[0] + b[0]; + c[1] = a[1] + b[1]; + return c; +} + +static Vector2 V2ScaleIII(Vector2 v, double s) +{ + Vector2 result; + result[0] = v[0] * s; + result[1] = v[1] * s; + return result; +} + +static Vector2 V2SubII(Vector2 a, Vector2 b) +{ + Vector2 c; + c[0] = a[0] - b[0]; + c[1] = a[1] - b[1]; + return c; +} + +#ifdef __cplusplus +} +#endif + + +//------------------------- WRAPPER -----------------------------// + +FitCurveWrapper::FitCurveWrapper() +{ +} + +FitCurveWrapper::~FitCurveWrapper() +{ + _vertices.clear(); +} + +void FitCurveWrapper::DrawBezierCurve(int n, Vector2 *curve) +{ + for (int i = 0; i < n + 1; ++i) + _vertices.push_back(curve[i]); +} + +void FitCurveWrapper::FitCurve(vector<Vec2d>& data, vector<Vec2d>& oCurve, double error) +{ + int size = data.size(); + Vector2 *d = new Vector2[size]; + for (int i = 0; i < size; ++i) { + d[i][0] = data[i][0]; + d[i][1] = data[i][1]; + } + + FitCurve(d, size, error); + + // copy results + for (vector<Vector2>::iterator v = _vertices.begin(), vend = _vertices.end(); v != vend; ++v) { + oCurve.push_back(Vec2d(v->x(), v->y())) ; + } +} + +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); + bezCurve = GenerateBezier(d, first, last, uPrime, tHat1, tHat2); + maxError = ComputeMaxError(d, first, last, + bezCurve, uPrime, &splitPoint); + if (maxError < error) { + DrawBezierCurve(3, bezCurve); + free((void*)u); + free((void*)bezCurve); + return; + } + free((void*)u); + u = uPrime; + } + } + + /* 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); +} |