/* An Algorithm for Automatically Fitting Digitized Curves by Philip J. Schneider from "Graphics Gems", Academic Press, 1990 */ #define TESTMODE /* fit_cubic.c */ /* Piecewise cubic fitting code */ #include "GraphicsGems.h" #include #include #include typedef Point2 *BezierCurve; /* Forward declarations */ void FitCurve(); static void FitCubic(); static double *Reparameterize(); static double NewtonRaphsonRootFind(); static Point2 BezierII(); static double B0(), B1(), B2(), B3(); static Vector2 ComputeLeftTangent(); static Vector2 ComputeRightTangent(); static Vector2 ComputeCenterTangent(); static double ComputeMaxError(); static double *ChordLengthParameterize(); static BezierCurve GenerateBezier(); static Vector2 V2AddII(); static Vector2 V2ScaleIII(); static Vector2 V2SubII(); #define MAXPOINTS 1000 /* The most points you can have */ #ifdef TESTMODE void DrawBezierCurve(n, curve) int n; BezierCurve curve; { /* You'll have to write this yourself. */ } /* * main: * Example of how to use the curve-fitting code. Given an array * of points and a tolerance (squared error between points and * fitted curve), the algorithm will generate a piecewise * cubic Bezier representation that approximates the points. * When a cubic is generated, the routine "DrawBezierCurve" * is called, which outputs the Bezier curve just created * (arguments are the degree and the control points, respectively). * Users will have to implement this function themselves * ascii output, etc. * */ void main() { static Point2 d[7] = { /* Digitized points */ { 0.0, 0.0 }, { 0.0, 0.5 }, { 1.1, 1.4 }, { 2.1, 1.6 }, { 3.2, 1.1 }, { 4.0, 0.2 }, { 4.0, 0.0 }, }; double error = 4.0; /* Squared error */ FitCurve(d, 7, error); /* Fit the Bezier curves */ } #endif /* TESTMODE */ /* * FitCurve : * Fit a Bezier curve to a set of digitized points */ void FitCurve(d, nPts, error) Point2 *d; /* Array of digitized points */ int nPts; /* Number of digitized points */ double error; /* User-defined error squared */ { 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); } /* * FitCubic : * Fit a Bezier curve to a (sub)set of digitized points */ static void FitCubic(d, first, last, tHat1, tHat2, error) Point2 *d; /* Array of digitized points */ int first, last; /* Indices of first and last pts in region */ Vector2 tHat1, tHat2; /* Unit tangent vectors at endpoints */ double error; /* User-defined error squared */ { 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 = (Point2 *)malloc(4 * sizeof(Point2)); 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); } /* * GenerateBezier : * Use least-squares method to find Bezier control points for region. * */ static BezierCurve GenerateBezier(d, first, last, uPrime, tHat1, tHat2) Point2 *d; /* Array of digitized points */ int first, last; /* Indices defining region */ double *uPrime; /* Parameter values for region */ Vector2 tHat1, tHat2; /* Unit tangents at endpoints */ { 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 */ det_C0_X, det_X_C1; double alpha_l, /* Alpha values, left and right */ alpha_r; Vector2 tmp; /* Utility variable */ BezierCurve bezCurve; /* RETURN bezier curve ctl pts */ bezCurve = (Point2 *)malloc(4 * sizeof(Point2)); 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]) * 10e-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. * */ static double *Reparameterize(d, first, last, u, bezCurve) Point2 *d; /* Array of digitized points */ int first, last; /* Indices defining region */ double *u; /* Current parameter values */ BezierCurve bezCurve; /* Current fitted curve */ { 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. */ static double NewtonRaphsonRootFind(Q, P, u) BezierCurve Q; /* Current fitted curve */ Point2 P; /* Digitized point */ double u; /* Parameter value for "P" */ { double numerator, denominator; Point2 Q1[3], Q2[2]; /* Q' and Q'' */ Point2 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].x = (Q[i+1].x - Q[i].x) * 3.0; Q1[i].y = (Q[i+1].y - Q[i].y) * 3.0; } /* Generate control vertices for Q'' */ for (i = 0; i <= 1; i++) { Q2[i].x = (Q1[i+1].x - Q1[i].x) * 2.0; Q2[i].y = (Q1[i+1].y - Q1[i].y) * 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.x - P.x) * (Q1_u.x) + (Q_u.y - P.y) * (Q1_u.y); denominator = (Q1_u.x) * (Q1_u.x) + (Q1_u.y) * (Q1_u.y) + (Q_u.x - P.x) * (Q2_u.x) + (Q_u.y - P.y) * (Q2_u.y); /* u = u - f(u)/f'(u) */ uPrime = u - (numerator/denominator); return (uPrime); } /* * Bezier : * Evaluate a Bezier curve at a particular parameter value * */ static Point2 BezierII(degree, V, t) int degree; /* The degree of the bezier curve */ Point2 *V; /* Array of control points */ double t; /* Parametric value to find point for */ { int i, j; Point2 Q; /* Point on curve at parameter t */ Point2 *Vtemp; /* Local copy of control points */ /* Copy array */ Vtemp = (Point2 *)malloc((unsigned)((degree+1) * sizeof (Point2))); 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].x = (1.0 - t) * Vtemp[j].x + t * Vtemp[j+1].x; Vtemp[j].y = (1.0 - t) * Vtemp[j].y + t * Vtemp[j+1].y; } } Q = Vtemp[0]; free((void *)Vtemp); return Q; } /* * B0, B1, B2, B3 : * Bezier multipliers */ static double B0(u) double u; { double tmp = 1.0 - u; return (tmp * tmp * tmp); } static double B1(u) double u; { double tmp = 1.0 - u; return (3 * u * (tmp * tmp)); } static double B2(u) double u; { double tmp = 1.0 - u; return (3 * u * u * tmp); } static double B3(u) double u; { return (u * u * u); } /* * ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent : *Approximate unit tangents at endpoints and "center" of digitized curve */ static Vector2 ComputeLeftTangent(d, end) Point2 *d; /* Digitized points*/ int end; /* Index to "left" end of region */ { Vector2 tHat1; tHat1 = V2SubII(d[end+1], d[end]); tHat1 = *V2Normalize(&tHat1); return tHat1; } static Vector2 ComputeRightTangent(d, end) Point2 *d; /* Digitized points */ int end; /* Index to "right" end of region */ { Vector2 tHat2; tHat2 = V2SubII(d[end-1], d[end]); tHat2 = *V2Normalize(&tHat2); return tHat2; } static Vector2 ComputeCenterTangent(d, center) Point2 *d; /* Digitized points */ int center; /* Index to point inside region */ { Vector2 V1, V2, tHatCenter; V1 = V2SubII(d[center-1], d[center]); V2 = V2SubII(d[center], d[center+1]); tHatCenter.x = (V1.x + V2.x)/2.0; tHatCenter.y = (V1.y + V2.y)/2.0; tHatCenter = *V2Normalize(&tHatCenter); return tHatCenter; } /* * ChordLengthParameterize : * Assign parameter values to digitized points * using relative distances between points. */ static double *ChordLengthParameterize(d, first, last) Point2 *d; /* Array of digitized points */ int first, last; /* Indices defining region */ { 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. */ static double ComputeMaxError(d, first, last, bezCurve, u, splitPoint) Point2 *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 */ { int i; double maxDist; /* Maximum error */ double dist; /* Current error */ Point2 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(a, b) Vector2 a, b; { Vector2 c; c.x = a.x + b.x; c.y = a.y + b.y; return (c); } static Vector2 V2ScaleIII(v, s) Vector2 v; double s; { Vector2 result; result.x = v.x * s; result.y = v.y * s; return (result); } static Vector2 V2SubII(a, b) Vector2 a, b; { Vector2 c; c.x = a.x - b.x; c.y = a.y - b.y; return (c); }