Exemple #1
0
static gdouble region_area (GtsVertex * v, GtsFace * f)
{
  /* cf. Section 3.3 of [Meyer et al 2002] */
  
  if (gts_triangle_area (GTS_TRIANGLE (f)) == 0.0) return (0.0);

  if (triangle_obtuse (v, f)) {
    if (angle_obtuse (v, f))
      return (gts_triangle_area (GTS_TRIANGLE (f))/2.0);
    else
      return (gts_triangle_area (GTS_TRIANGLE (f))/4.0);
  } else {
    GtsEdge * e = gts_triangle_edge_opposite (GTS_TRIANGLE (f), v);

    return ((cotan (GTS_SEGMENT (e)->v1, v, GTS_SEGMENT (e)->v2)* 
             gts_point_distance2 (GTS_POINT (v), 
				  GTS_POINT (GTS_SEGMENT (e)->v2)) +
             cotan (GTS_SEGMENT (e)->v2, v, GTS_SEGMENT (e)->v1)* 
             gts_point_distance2 (GTS_POINT (v), 
                                  GTS_POINT (GTS_SEGMENT (e)->v1)))
            /8.0);
  }
}
Exemple #2
0
/*! gts_vertex_principal_directions:
 *  @v: a #WVertex.
 *  @s: a #GtsSurface.
 *  @Kh: mean curvature normal (a #Vec3r).
 *  @Kg: Gaussian curvature (a real).
 *  @e1: first principal curvature direction (direction of largest curvature).
 *  @e2: second principal curvature direction.
 *
 *  Computes the principal curvature directions at a point given @Kh and @Kg, the mean curvature normal and
 *  Gaussian curvatures at that point, computed with gts_vertex_mean_curvature_normal() and
 *  gts_vertex_gaussian_curvature(), respectively.
 *
 *  Note that this computation is very approximate and tends to be unstable. Smoothing of the surface or the principal
 *  directions may be necessary to achieve reasonable results.
 */
void gts_vertex_principal_directions(WVertex *v, Vec3r Kh, real Kg, Vec3r &e1, Vec3r &e2)
{
	Vec3r N;
	real normKh;

	Vec3r basis1, basis2, d, eig;
	real ve2, vdotN;
	real aterm_da, bterm_da, cterm_da, const_da;
	real aterm_db, bterm_db, cterm_db, const_db;
	real a, b, c;
	real K1, K2;
	real *weights, *kappas, *d1s, *d2s;
	int edge_count;
	real err_e1, err_e2;
	int e;
	WVertex::incoming_edge_iterator itE;

	/* compute unit normal */
	normKh = Kh.norm();

	if (normKh > 0.0) {
		Kh.normalize();
	}
	else {
		/* This vertex is a point of zero mean curvature (flat or saddle point). Compute a normal by averaging
		 * the adjacent triangles
		 */
		N[0] = N[1] = N[2] = 0.0;

		for (itE = v->incoming_edges_begin(); itE != v->incoming_edges_end(); itE++)
			N = Vec3r(N + (*itE)->GetaFace()->GetNormal());
		real normN = N.norm();
		if (normN <= 0.0)
			return;
		N.normalize();
	}

	/* construct a basis from N: */
	/* set basis1 to any component not the largest of N */
	basis1[0] =  basis1[1] =  basis1[2] = 0.0;
	if (fabs (N[0]) > fabs (N[1]))
		basis1[1] = 1.0;
	else
		basis1[0] = 1.0;

	/* make basis2 orthogonal to N */
	basis2 = (N ^ basis1);
	basis2.normalize();

	/* make basis1 orthogonal to N and basis2 */
	basis1 = (N ^ basis2);
	basis1.normalize();

	aterm_da = bterm_da = cterm_da = const_da = 0.0;
	aterm_db = bterm_db = cterm_db = const_db = 0.0;
	int nb_edges = v->GetEdges().size();

	weights = (real *)malloc(sizeof(real) * nb_edges);
	kappas = (real *)malloc(sizeof(real) * nb_edges);
	d1s = (real *)malloc(sizeof(real) * nb_edges);
	d2s = (real *)malloc(sizeof(real) * nb_edges);
	edge_count = 0;

	for (itE = v->incoming_edges_begin(); itE != v->incoming_edges_end(); itE++) {
		WOEdge *e;
		WFace *f1, *f2;
		real weight, kappa, d1, d2;
		Vec3r vec_edge;
		if (!*itE)
			continue;
		e = *itE;

		/* since this vertex passed the tests in gts_vertex_mean_curvature_normal(), this should be true. */
		//g_assert(gts_edge_face_number (e, s) == 2);

		/* identify the two triangles bordering e in s */
		f1 = e->GetaFace();
		f2 = e->GetbFace();

		/* We are solving for the values of the curvature tensor
		 *     B = [ a b ; b c ].
		 *  The computations here are from section 5 of [Meyer et al 2002].
		 *
		 *  The first step is to calculate the linear equations governing the values of (a,b,c). These can be computed
		 *  by setting the derivatives of the error E to zero (section 5.3).
		 *
		 *  Since a + c = norm(Kh), we only compute the linear equations for dE/da and dE/db. (NB: [Meyer et al 2002]
		 *  has the equation a + b = norm(Kh), but I'm almost positive this is incorrect).
		 *
		 *  Note that the w_ij (defined in section 5.2) are all scaled by (1/8*A_mixed). We drop this uniform scale
		 *  factor because the solution of the linear equations doesn't rely on it.
		 *
		 *  The terms of the linear equations are xterm_dy with x in {a,b,c} and y in {a,b}. There are also const_dy
		 *  terms that are the constant factors in the equations.
		 */

		/* find the vector from v along edge e */
		vec_edge = Vec3r(-1 * e->GetVec());

		ve2 = vec_edge.squareNorm();
		vdotN = vec_edge * N;

		/* section 5.2 - There is a typo in the computation of kappa. The edges should be x_j-x_i. */
		kappa = 2.0 * vdotN / ve2;

		/* section 5.2 */

		/* I don't like performing a minimization where some of the weights can be negative (as can be the case
		 *  if f1 or f2 are obtuse). To ensure all-positive weights, we check for obtuseness. */
		weight = 0.0;
		if (!triangle_obtuse(v, f1)) {
			weight += ve2 * cotan(f1->GetNextOEdge(e->twin())->GetbVertex(), e->GetaVertex(), e->GetbVertex()) / 8.0;
		}
		else {
			if (angle_obtuse(v, f1)) {
				weight += ve2 * f1->getArea() / 4.0;
			}
			else {
				weight += ve2 * f1->getArea() / 8.0;
			}
		}

		if (!triangle_obtuse(v, f2)) {
			weight += ve2 * cotan (f2->GetNextOEdge(e)->GetbVertex(), e->GetaVertex(), e->GetbVertex()) / 8.0;
		}
		else {
			if (angle_obtuse(v, f2)) {
				weight += ve2 * f1->getArea() / 4.0;
			}
			else {
				weight += ve2 * f1->getArea() / 8.0;
			}
		}

		/* projection of edge perpendicular to N (section 5.3) */
		d[0] = vec_edge[0] - vdotN * N[0];
		d[1] = vec_edge[1] - vdotN * N[1];
		d[2] = vec_edge[2] - vdotN * N[2];
		d.normalize();

		/* not explicit in the paper, but necessary. Move d to 2D basis. */
		d1 = d * basis1;
		d2 = d * basis2;

		/* store off the curvature, direction of edge, and weights for later use */
		weights[edge_count] = weight;
		kappas[edge_count] = kappa;
		d1s[edge_count] = d1;
		d2s[edge_count] = d2;
		edge_count++;

		/* Finally, update the linear equations */
		aterm_da += weight * d1 * d1 * d1 * d1;
		bterm_da += weight * d1 * d1 * 2 * d1 * d2;
		cterm_da += weight * d1 * d1 * d2 * d2;
		const_da += weight * d1 * d1 * (-kappa);

		aterm_db += weight * d1 * d2 * d1 * d1;
		bterm_db += weight * d1 * d2 * 2 * d1 * d2;
		cterm_db += weight * d1 * d2 * d2 * d2;
		const_db += weight * d1 * d2 * (-kappa);
	}

	/* now use the identity (Section 5.3) a + c = |Kh| = 2 * kappa_h */
	aterm_da -= cterm_da;
	const_da += cterm_da * normKh;

	aterm_db -= cterm_db;
	const_db += cterm_db * normKh;

	/* check for solvability of the linear system */
	if (((aterm_da * bterm_db - aterm_db * bterm_da) != 0.0) && ((const_da != 0.0) || (const_db != 0.0))) {
		linsolve(aterm_da, bterm_da, -const_da, aterm_db, bterm_db, -const_db, &a, &b);

		c = normKh - a;

		eigenvector(a, b, c, eig);
	}
	else {
		/* region of v is planar */
		eig[0] = 1.0;
		eig[1] = 0.0;
	}

	/* Although the eigenvectors of B are good estimates of the principal directions, it seems that which one is
	 * attached to which curvature direction is a bit arbitrary. This may be a bug in my implementation, or just
	 * a side-effect of the inaccuracy of B due to the discrete nature of the sampling.
	 *
	 * To overcome this behavior, we'll evaluate which assignment best matches the given eigenvectors by comparing
	 * the curvature estimates computed above and the curvatures calculated from the discrete differential operators.
	 */

	gts_vertex_principal_curvatures(0.5 * normKh, Kg, &K1, &K2);

	err_e1 = err_e2 = 0.0;
	/* loop through the values previously saved */
	for (e = 0; e < edge_count; e++) {
		real weight, kappa, d1, d2;
		real temp1, temp2;
		real delta;

		weight = weights[e];
		kappa = kappas[e];
		d1 = d1s[e];
		d2 = d2s[e];

		temp1 = fabs (eig[0] * d1 + eig[1] * d2);
		temp1 = temp1 * temp1;
		temp2 = fabs (eig[1] * d1 - eig[0] * d2);
		temp2 = temp2 * temp2;

		/* err_e1 is for K1 associated with e1 */
		delta = K1 * temp1 + K2 * temp2 - kappa;
		err_e1 += weight * delta * delta;

		/* err_e2 is for K1 associated with e2 */
		delta = K2 * temp1 + K1 * temp2 - kappa;
		err_e2 += weight * delta * delta;
	}
	free (weights);
	free (kappas);
	free (d1s);
	free (d2s);

	/* rotate eig by a right angle if that would decrease the error */
	if (err_e2 < err_e1) {
		real temp = eig[0];

		eig[0] = eig[1];
		eig[1] = -temp;
	}

	e1[0] = eig[0] * basis1[0] + eig[1] * basis2[0];
	e1[1] = eig[0] * basis1[1] + eig[1] * basis2[1];
	e1[2] = eig[0] * basis1[2] + eig[1] * basis2[2];
	e1.normalize();

	/* make N,e1,e2 a right handed coordinate sytem */
	e2 =  N ^ e1;
	e2.normalize();
}
Exemple #3
0
/** 
 * gts_vertex_principal_directions:
 * @v: a #GtsVertex.  
 * @s: a #GtsSurface.
 * @Kh: mean curvature normal (a #GtsVector).
 * @Kg: Gaussian curvature (a gdouble).
 * @e1: first principal curvature direction (direction of largest curvature).
 * @e2: second principal curvature direction.
 *
 * Computes the principal curvature directions at a point given @Kh
 * and @Kg, the mean curvature normal and Gaussian curvatures at that
 * point, computed with gts_vertex_mean_curvature_normal() and
 * gts_vertex_gaussian_curvature(), respectively. 
 *
 * Note that this computation is very approximate and tends to be
 * unstable.  Smoothing of the surface or the principal directions may
 * be necessary to achieve reasonable results.  
 */
void gts_vertex_principal_directions (GtsVertex * v, GtsSurface * s,
                                      GtsVector Kh, gdouble Kg,
				      GtsVector e1, GtsVector e2)
{
  GtsVector N;
  gdouble normKh;
  GSList * i, * j;
  GtsVector basis1, basis2, d, eig;
  gdouble ve2, vdotN;
  gdouble aterm_da, bterm_da, cterm_da, const_da;
  gdouble aterm_db, bterm_db, cterm_db, const_db;
  gdouble a, b, c;
  gdouble K1, K2;
  gdouble *weights, *kappas, *d1s, *d2s;
  gint edge_count;
  gdouble err_e1, err_e2;
  int e;

  /* compute unit normal */
  normKh = sqrt (gts_vector_scalar (Kh, Kh));

  if (normKh > 0.0) {
    N[0] = Kh[0] / normKh;
    N[1] = Kh[1] / normKh;
    N[2] = Kh[2] / normKh;
  } else {
    /* This vertex is a point of zero mean curvature (flat or saddle
     * point).  Compute a normal by averaging the adjacent triangles
     */
    N[0] = N[1] = N[2] = 0.0;
    i = gts_vertex_faces (v, s, NULL);
    while (i) {
      gdouble x, y, z;
      gts_triangle_normal (GTS_TRIANGLE ((GtsFace *) i->data),
                           &x, &y, &z);
      N[0] += x;
      N[1] += y;
      N[2] += z;

      i = i->next;
    }
    g_return_if_fail (gts_vector_norm (N) > 0.0);
    gts_vector_normalize (N);
  }
    

  /* construct a basis from N: */
  /* set basis1 to any component not the largest of N */
  basis1[0] =  basis1[1] =  basis1[2] = 0.0;
  if (fabs (N[0]) > fabs (N[1]))
    basis1[1] = 1.0;
  else
    basis1[0] = 1.0;
    
  /* make basis2 orthogonal to N */
  gts_vector_cross (basis2, N, basis1);
  gts_vector_normalize (basis2);

  /* make basis1 orthogonal to N and basis2 */
  gts_vector_cross (basis1, N, basis2);
  gts_vector_normalize (basis1);
  
  aterm_da = bterm_da = cterm_da = const_da = 0.0;
  aterm_db = bterm_db = cterm_db = const_db = 0.0;

  weights = g_malloc (sizeof (gdouble)*g_slist_length (v->segments));
  kappas = g_malloc (sizeof (gdouble)*g_slist_length (v->segments));
  d1s = g_malloc (sizeof (gdouble)*g_slist_length (v->segments));
  d2s = g_malloc (sizeof (gdouble)*g_slist_length (v->segments));
  edge_count = 0;

  i = v->segments;
  while (i) {
    GtsEdge * e;
    GtsFace * f1, * f2;
    gdouble weight, kappa, d1, d2;
    GtsVector vec_edge;

    if (! GTS_IS_EDGE (i->data)) {
      i = i->next;
      continue;
    }

    e = i->data;

    /* since this vertex passed the tests in
     * gts_vertex_mean_curvature_normal(), this should be true. */
    g_assert (gts_edge_face_number (e, s) == 2);

    /* identify the two triangles bordering e in s */
    f1 = f2 = NULL;
    j = e->triangles;
    while (j) {
      if ((! GTS_IS_FACE (j->data)) || 
          (! gts_face_has_parent_surface (GTS_FACE (j->data), s))) {
        j = j->next;
        continue;
      }
      if (f1 == NULL)
        f1 = GTS_FACE (j->data);
      else {
        f2 = GTS_FACE (j->data);
        break;
      }
      j = j->next;
    }
    g_assert (f2 != NULL);

    /* We are solving for the values of the curvature tensor 
     *     B = [ a b ; b c ].  
     * The computations here are from section 5 of [Meyer et al 2002].  
     *
     * The first step is to calculate the linear equations governing
     * the values of (a,b,c).  These can be computed by setting the
     * derivatives of the error E to zero (section 5.3).
     * 
     * Since a + c = norm(Kh), we only compute the linear equations
     * for dE/da and dE/db.  (NB: [Meyer et al 2002] has the
     * equation a + b = norm(Kh), but I'm almost positive this is
     * incorrect.)
     *
     * Note that the w_ij (defined in section 5.2) are all scaled by
     * (1/8*A_mixed).  We drop this uniform scale factor because the
     * solution of the linear equations doesn't rely on it.
     *
     * The terms of the linear equations are xterm_dy with x in
     * {a,b,c} and y in {a,b}.  There are also const_dy terms that are
     * the constant factors in the equations.  
     */

    /* find the vector from v along edge e */
    gts_vector_init (vec_edge, GTS_POINT (v), 
                     GTS_POINT ((GTS_SEGMENT (e)->v1 == v) ? 
                                GTS_SEGMENT (e)->v2 : GTS_SEGMENT (e)->v1));
    ve2 = gts_vector_scalar (vec_edge, vec_edge);
    vdotN = gts_vector_scalar (vec_edge, N);

    /* section 5.2 - There is a typo in the computation of kappa.  The
     * edges should be x_j-x_i.
     */
    kappa = 2.0 * vdotN / ve2;

    /* section 5.2 */

    /* I don't like performing a minimization where some of the
     * weights can be negative (as can be the case if f1 or f2 are
     * obtuse).  To ensure all-positive weights, we check for
     * obtuseness and use values similar to those in region_area(). */
    weight = 0.0;
    if (! triangle_obtuse(v, f1)) {
      weight += ve2 * 
        cotan (gts_triangle_vertex_opposite (GTS_TRIANGLE (f1), e), 
               GTS_SEGMENT (e)->v1, GTS_SEGMENT (e)->v2) / 8.0;
    } else {
      if (angle_obtuse (v, f1)) {
        weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f1)) / 4.0;
      } else {
        weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f1)) / 8.0;
      }
    }

    if (! triangle_obtuse(v, f2)) {
      weight += ve2 * 
        cotan (gts_triangle_vertex_opposite (GTS_TRIANGLE (f2), e), 
               GTS_SEGMENT (e)->v1, GTS_SEGMENT (e)->v2) / 8.0;
    } else {
      if (angle_obtuse (v, f2)) {
        weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f2)) / 4.0;
      } else {
        weight += ve2 * gts_triangle_area (GTS_TRIANGLE (f2)) / 8.0;
      }
    }

    /* projection of edge perpendicular to N (section 5.3) */
    d[0] = vec_edge[0] - vdotN * N[0];
    d[1] = vec_edge[1] - vdotN * N[1];
    d[2] = vec_edge[2] - vdotN * N[2];
    gts_vector_normalize (d);
    
    /* not explicit in the paper, but necessary.  Move d to 2D basis. */
    d1 = gts_vector_scalar (d, basis1);
    d2 = gts_vector_scalar (d, basis2);

    /* store off the curvature, direction of edge, and weights for later use */
    weights[edge_count] = weight;
    kappas[edge_count] = kappa;
    d1s[edge_count] = d1;
    d2s[edge_count] = d2;
    edge_count++;

    /* Finally, update the linear equations */
    aterm_da += weight * d1 * d1 * d1 * d1;
    bterm_da += weight * d1 * d1 * 2 * d1 * d2;
    cterm_da += weight * d1 * d1 * d2 * d2;
    const_da += weight * d1 * d1 * (- kappa);

    aterm_db += weight * d1 * d2 * d1 * d1;
    bterm_db += weight * d1 * d2 * 2 * d1 * d2;
    cterm_db += weight * d1 * d2 * d2 * d2;
    const_db += weight * d1 * d2 * (- kappa);

    i = i->next;
  }

  /* now use the identity (Section 5.3) a + c = |Kh| = 2 * kappa_h */
  aterm_da -= cterm_da;
  const_da += cterm_da * normKh;

  aterm_db -= cterm_db;
  const_db += cterm_db * normKh;
  
  /* check for solvability of the linear system */
  if (((aterm_da * bterm_db - aterm_db * bterm_da) != 0.0) &&
      ((const_da != 0.0) || (const_db != 0.0))) {
    linsolve (aterm_da, bterm_da, -const_da,
              aterm_db, bterm_db, -const_db,
              &a, &b);

    c = normKh - a;

    eigenvector (a, b, c, eig);
  } else {
    /* region of v is planar */
    eig[0] = 1.0;
    eig[1] = 0.0;
  }

  /* Although the eigenvectors of B are good estimates of the
   * principal directions, it seems that which one is attached to
   * which curvature direction is a bit arbitrary.  This may be a bug
   * in my implementation, or just a side-effect of the inaccuracy of
   * B due to the discrete nature of the sampling.
   *
   * To overcome this behavior, we'll evaluate which assignment best
   * matches the given eigenvectors by comparing the curvature
   * estimates computed above and the curvatures calculated from the
   * discrete differential operators.  */

  gts_vertex_principal_curvatures (0.5 * normKh, Kg, &K1, &K2);
  
  err_e1 = err_e2 = 0.0;
  /* loop through the values previously saved */
  for (e = 0; e < edge_count; e++) {
    gdouble weight, kappa, d1, d2;
    gdouble temp1, temp2;
    gdouble delta;

    weight = weights[e];
    kappa = kappas[e];
    d1 = d1s[e];
    d2 = d2s[e];

    temp1 = fabs (eig[0] * d1 + eig[1] * d2);
    temp1 = temp1 * temp1;
    temp2 = fabs (eig[1] * d1 - eig[0] * d2);
    temp2 = temp2 * temp2;

    /* err_e1 is for K1 associated with e1 */
    delta = K1 * temp1 + K2 * temp2 - kappa;
    err_e1 += weight * delta * delta;

    /* err_e2 is for K1 associated with e2 */
    delta = K2 * temp1 + K1 * temp2 - kappa;
    err_e2 += weight * delta * delta;
  }
  g_free (weights);
  g_free (kappas);
  g_free (d1s);
  g_free (d2s);

  /* rotate eig by a right angle if that would decrease the error */
  if (err_e2 < err_e1) {
    gdouble temp = eig[0];

    eig[0] = eig[1];
    eig[1] = -temp;
  }

  e1[0] = eig[0] * basis1[0] + eig[1] * basis2[0];
  e1[1] = eig[0] * basis1[1] + eig[1] * basis2[1];
  e1[2] = eig[0] * basis1[2] + eig[1] * basis2[2];
  gts_vector_normalize (e1);

  /* make N,e1,e2 a right handed coordinate sytem */
  gts_vector_cross (e2, N, e1);
  gts_vector_normalize (e2);
}