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/*
* ***** 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.
*
* ***** 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;
namespace Freestyle {
typedef Vector2 *BezierCurve;
/* 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 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);
/* 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[2]; /* 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;
/* 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++) {
/* Compute the A's */
A[0] = tHat1;
A[1] = tHat2;
V2Scale(&A[0], B1(uPrime[i]));
V2Scale(&A[1], B2(uPrime[i]));
C[0][0] += V2Dot(&A[0], &A[0]);
C[0][1] += V2Dot(&A[0], &A[1]);
// C[1][0] += V2Dot(&A[0], &A[1]);
C[1][0] = C[0][1];
C[1][1] += V2Dot(&A[1], &A[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[0], &tmp);
X[1] += V2Dot(&A[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;
}
//------------------------- 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);
}
} /* namespace Freestyle */
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