* apps/statues/trimesh/: New directory. * apps/statues/trimesh/misc.hh: New. * apps/statues/trimesh/README: New. * apps/statues/Makefile.am (noinst_HEADERS): New. Add trimesh/misc.hh. (EXTRA_DIST): New. Add trimesh/README. --- milena/ChangeLog | 12 + milena/apps/statues/Makefile.am | 3 + milena/apps/statues/trimesh/README | 4 + milena/apps/statues/trimesh/misc.hh | 657 +++++++++++++++++++++++++++++++++++ 4 files changed, 676 insertions(+), 0 deletions(-) create mode 100644 milena/apps/statues/trimesh/README create mode 100644 milena/apps/statues/trimesh/misc.hh diff --git a/milena/ChangeLog b/milena/ChangeLog index fae6012..f08a8d9 100644 --- a/milena/ChangeLog +++ b/milena/ChangeLog @@ -1,5 +1,17 @@ 2008-12-30 Roland Levillain &lt;roland(a)lrde.epita.fr&gt; + Port Trimesh's curvature computation routines to Milena. + + * apps/statues/trimesh/: New directory. + * apps/statues/trimesh/misc.hh: New. + * apps/statues/trimesh/README: New. + * apps/statues/Makefile.am (noinst_HEADERS): New. + Add trimesh/misc.hh. + (EXTRA_DIST): New. + Add trimesh/README. + +2008-12-30 Roland Levillain &lt;roland(a)lrde.epita.fr&gt; + Simplify the normalization of vectors. * mln/algebra/vec.hh (mln::algebra::vec<n, T>::normalize): diff --git a/milena/apps/statues/Makefile.am b/milena/apps/statues/Makefile.am index 0d1bfc1..8dd7199 100644 --- a/milena/apps/statues/Makefile.am +++ b/milena/apps/statues/Makefile.am @@ -57,6 +57,9 @@ mesh_max_curv_LDADD = $(LDADD_trimesh) ## Complex-based applications, independent of Trimesh. ## ## ---------------------------------------------------- ## +noinst_HEADERS = trimesh/misc.hh +EXTRA_DIST = trimesh/README + bin_PROGRAMS += mesh-complex-segm mesh_complex_segm_SOURCES = mesh-complex-segm.cc diff --git a/milena/apps/statues/trimesh/README b/milena/apps/statues/trimesh/README new file mode 100644 index 0000000..20fe1e7 --- /dev/null +++ b/milena/apps/statues/trimesh/README @@ -0,0 +1,4 @@ +Files from this directory are adaptation of some Trimesh code into +Milena. We might want to push them into Milena someday. + +Don't forget to add authors / copyright owners where necessary. diff --git a/milena/apps/statues/trimesh/misc.hh b/milena/apps/statues/trimesh/misc.hh new file mode 100644 index 0000000..78af913 --- /dev/null +++ b/milena/apps/statues/trimesh/misc.hh @@ -0,0 +1,657 @@ +// Copyright (C) 2008 EPITA Research and Development Laboratory (LRDE) +// +// This file is part of the Olena Library. This library is free +// software; you can redistribute it and/or modify it under the terms +// of the GNU General Public License version 2 as published by the +// Free Software Foundation. +// +// This library 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 library; see the file COPYING. If not, write to +// the Free Software Foundation, 51 Franklin Street, Fifth Floor, +// Boston, MA 02111-1307, USA. +// +// As a special exception, you may use this file as part of a free +// software library without restriction. Specifically, if other files +// instantiate templates or use macros or inline functions from this +// file, or you compile this file and link it with other files to +// produce an executable, this file does not by itself cause the +// resulting executable to be covered by the GNU General Public +// License. This exception does not however invalidate any other +// reasons why the executable file might be covered by the GNU General +// Public License. + +// FIXME: Split this file. +// FIXME: Address license-related issues? + +#ifndef MILENA_APPS_STATUES_MISC_HH +# define MILENA_APPS_STATUES_MISC_HH + +# include <algorithm> // For std::swap. +# include <utility> // For std::pair. + +# include <mln/algebra/vec.hh> +# include <mln/algebra/mat.hh> + +# include <mln/norm/l2.hh> + + +/** See http://gcc.gnu.org/onlinedocs/gcc-4.3.2/gcc/Other-Builtins.html + for a definition of __builtin_expect. + + Macros likely and unlikely are for optimization purpose only (the + developer can use them to help the compiler improve the + basic-block sewing pass. */ +// FIXME: We should turn `likely' and `unlikely' into functions. +#ifndef likely +# if !defined(__GNUC__) || (__GNUC__ == 2 && __GNUC_MINOR__ < 96) +# define likely(x) (x) +# define unlikely(x) (x) +# else +# define likely(x) (__builtin_expect((x), 1)) +# define unlikely(x) (__builtin_expect((x), 0)) +# endif +#endif + +namespace mln +{ + + namespace algebra + { + + // From Trimesh. + // FIXME: Doc (input, return value, etc.). + // FIXME: Idea: Add reference to Numerical Recipes. + + /** \brief Perform LDL^T decomposition of a symmetric positive + definite matrix. Like Cholesky, but no square roots. + Overwrites lower triangle of matrix. + + From Trimesh's ldltdc routine. */ + template <unsigned N, typename T> + inline + bool + ldlt_decomp(mat<N, N, T>& A, vec<N, T>& rdiag) + { + vec<N - 1, T> v; + for (int i = 0; i < N; ++i) + { + for (int k = 0; k < i; ++k) + v[k] = A(i, k) * rdiag[k]; + for (int j = i; j < N; ++j) + { + T sum = A(i, j); + for (int k = 0; k < i; k++) + sum -= v[k] * A(j, k); + if (i == j) + { + if (unlikely(sum <= T(0))) + return false; + rdiag[i] = T(1) / sum; + } + else + A(j, i) = sum; + } + } + return true; + } + + // From Trimesh. + // FIXME: Doc (input, return value, etc.). + // FIXME: Idea: Add reference to Numerical Recipes. + + /// \brief Solve A x = B after mln::algebra::ldlt_decomp. + template <unsigned N, typename T> + inline + void + ldlt_solve(const mat<N, N, T>& A, const vec<N, T>& rdiag, + const vec<N, T>& B, vec<N, T>& x) + { + for (unsigned i = 0; i < N; ++i) + { + T sum = B[i]; + for (int k = 0; k < i; ++k) + sum -= A(i, k) * x[k]; + x[i] = sum * rdiag[i]; + } + for (int i = N - 1; i >= 0; --i) + { + T sum = 0; + for (int k = i + 1; k < N; ++k) + sum += A(k, i) * x[k]; + x[i] -= sum * rdiag[i]; + } + } + + } // end of namespace mln::algebra + +} // end of namespace mln + + +// FIXME: To be moved elsewhere (in mln/geom)? + +// Inspired from Trimesh. + +namespace mln +{ + + namespace geom + { + + // FIXME: More doc (see Trimesh's). + + /* FIXME: We should restrict attached data to vertices in return + value. */ + /** \brief Compute normals at vertices. + + Inspired from the method Trimesh::need_normals of the Trimesh + library. + \see http://www.cs.princeton.edu/gfx/proj/trimesh2/ + + For simplicity purpose, and contrary to Trimesh, this routine + only compute normals from a mesh, not from a cloud of + points. */ + inline + complex_image< 2, mln::space_2complex_geometry, algebra::vec<3, float> > + mesh_normal(const p_complex<2, space_2complex_geometry>& mesh) + { + // Shortcuts. + static const unsigned D = 2; + typedef space_2complex_geometry G; + typedef algebra::vec<3, float> vec3f; + typedef complex_image< D, G, vec3f > normal_t; + + normal_t normal(mesh); + level::fill(normal, literal::zero); + + mln::p_n_faces_fwd_piter<D, G> f(mesh, 2); + // A neighborhood where neighbors are the set of 0-faces + // transitively adjacent to the reference point. + typedef mln::complex_m_face_neighborhood<D, G> adj_vertices_nbh_t; + adj_vertices_nbh_t adj_vertices_nbh; + mln_niter_(adj_vertices_nbh_t) adj_v(adj_vertices_nbh, f); + /* FIXME: We should be able to pas this value (m) either at the + construction of the neighborhood or at the construction of + the iterator. Alas, we can't inherit ctors (yet), so we + can't rely on a simple mixin approach. */ + adj_v.iter().set_m(0); + + // Iterate on 2-faces (triangles). + for_all(f) + { + /// Psites of 0-faces transitively adjacent to F. + std::vector<mln_psite_(normal_t)> p; + p.reserve(3); + for_all(adj_v) + p.push_back(adj_v); + /// The should be exactly three vertices adjacent to F. + mln_assertion(p.size() == 3); + + /* FIXME: Is the explicit conversion to_site() really needed? + If not, we could provide a local shortcut like + + vec(p[0]) // Instead of p[0].to_site().front().to_vec() + + to shorten these lines. */ + // FIXME: Factor as well? + vec3f a = + p[0].to_site().front().to_vec() - p[1].to_site().front().to_vec(); + vec3f b = + p[1].to_site().front().to_vec() - p[2].to_site().front().to_vec(); + vec3f c = + p[2].to_site().front().to_vec() - p[0].to_site().front().to_vec(); + + // FIXME: Factor as well? + float l2a = norm::sqr_l2(a); + float l2b = norm::sqr_l2(b); + float l2c = norm::sqr_l2(c); + vec3f face_normal = algebra::vprod(a, b); + + normal(p[0]) += face_normal * (1.0f / (l2a * l2c)); + normal(p[1]) += face_normal * (1.0f / (l2b * l2a)); + normal(p[2]) += face_normal * (1.0f / (l2c * l2b)); + } + + // Normalize normals. We don't need to iterate on all faces, just + // 0-faces. + mln::p_n_faces_fwd_piter<D, G> v(mesh, 0); + for_all(v) + normal(v).normalize(); + + return normal; + } + + + /* FIXME: We should restrict attached data to vertices in return + value. */ + /** \brief Compute the area ``belonging'' to normals at vertices. + + Inspired from the method Trimesh::need_pointareas of the + Trimesh library. + \see http://www.cs.princeton.edu/gfx/proj/trimesh2/ + + From the documentation of Trimesh: + + ``Compute the area "belonging" to each vertex or each corner + of a triangle (defined as Voronoi area restricted to the + 1-ring of a vertex, or to the triangle).'' + */ + inline + std::pair< + complex_image< 2, mln::space_2complex_geometry, algebra::vec<3, float> >, + complex_image< 2, mln::space_2complex_geometry, float > + > + mesh_corner_point_area(const p_complex<2, space_2complex_geometry>& mesh) + { + // Shortcuts. + static const unsigned D = 2; + typedef space_2complex_geometry G; + typedef algebra::vec<3, float> vec3f; + + // Hold data for 2-faces only. + typedef complex_image< D, G, vec3f > corner_area_t; + // Hold data for 0-faces only. + typedef complex_image< D, G, float > point_area_t; + // Packed output. + typedef std::pair<corner_area_t, point_area_t> output_t; + + // Initialize output. + output_t output(mesh, mesh); + corner_area_t& corner_area = output.first; + point_area_t& point_area = output.second; + level::fill(corner_area, literal::zero); + level::fill(point_area, literal::zero); + + mln::p_n_faces_fwd_piter<D, G> f(mesh, 2); + // A neighborhood where neighbors are the set of 0-faces + // transitively adjacent to the reference point. + typedef mln::complex_m_face_neighborhood<D, G> adj_vertices_nbh_t; + adj_vertices_nbh_t adj_vertices_nbh; + mln_niter_(adj_vertices_nbh_t) adj_v(adj_vertices_nbh, f); + /* FIXME: We should be able to pas this value (m) either at the + construction of the neighborhood or at the construction of + the iterator. Alas, we can't inherit ctors (yet), so we + can't rely on a simple mixin approach. */ + adj_v.iter().set_m(0); + + for_all(f) + { + /// Psites of 0-faces transitively adjacent to F. + std::vector<mln_psite_(corner_area_t)> p; + p.reserve(3); + for_all(adj_v) + p.push_back(adj_v); + /// The should be exactly three vertices adjacent to F. + mln_assertion(p.size() == 3); + + // (Opposite) edge vectors. + algebra::vec<3, vec3f> e; + // FIXME: See above. + e.set + (p[2].to_site().front().to_vec() - p[1].to_site().front().to_vec(), + p[0].to_site().front().to_vec() - p[2].to_site().front().to_vec(), + p[1].to_site().front().to_vec() - p[0].to_site().front().to_vec()); + + // Compute corner weights. + float area = 0.5f * norm::l2(algebra::vprod(e[0], e[1])); + vec3f sqr_norm; + sqr_norm.set(norm::sqr_l2(e[0]), + norm::sqr_l2(e[1]), + norm::sqr_l2(e[2])); + vec3f edge_weight; + edge_weight.set + (sqr_norm[0] * (sqr_norm[1] + sqr_norm[2] - sqr_norm[0]), + sqr_norm[1] * (sqr_norm[2] + sqr_norm[0] - sqr_norm[1]), + sqr_norm[2] * (sqr_norm[0] + sqr_norm[1] - sqr_norm[2])); + + if (edge_weight[0] <= 0.0f) + { + corner_area(f)[1] = -0.25f * sqr_norm[2] * area / (e[0] * e[2]); + corner_area(f)[2] = -0.25f * sqr_norm[1] * area / (e[0] * e[1]); + corner_area(f)[0] = area - corner_area(f)[1] - corner_area(f)[2]; + } + else if (edge_weight[1] <= 0.0f) + { + corner_area(f)[2] = -0.25f * sqr_norm[0] * area / (e[1] * e[0]); + corner_area(f)[0] = -0.25f * sqr_norm[2] * area / (e[1] * e[2]); + corner_area(f)[1] = area - corner_area(f)[2] - corner_area(f)[0]; + } + else if (edge_weight[2] <= 0.0f) + { + corner_area(f)[0] = -0.25f * sqr_norm[1] * area / (e[2] * e[1]); + corner_area(f)[1] = -0.25f * sqr_norm[0] * area / (e[2] * e[0]); + corner_area(f)[2] = area - corner_area(f)[0] - corner_area(f)[1]; + } + else + { + float ewscale = + 0.5f * area / (edge_weight[0] + edge_weight[1] + edge_weight[2]); + for (int i = 0; i < 3; ++i) + corner_area(f)[i] = + ewscale * (edge_weight[(i+1)%3] + edge_weight[(i+2)%3]); + } + + for (int i = 0; i < 3; ++i) + point_area(p[i]) += corner_area(f)[i]; + } + + return output; + } + + + namespace internal + { + + /** \brief \a i + 1 and \a i - 1 modulo 3 + + Defined as macros in Trimesh. + + #define NEXT(i) ((i)<2 ? (i)+1 : (i)-2) + #define PREV(i) ((i)>0 ? (i)-1 : (i)+2) + + According to Trimesh doc, + + ``This way of computing it tends to be faster than using %'' + + but I (Roland) just can't believe it... We should profile + some code first. + + \{ */ + static inline unsigned next(unsigned i) { return (i + 1) % 3; } + static inline unsigned prev(unsigned i) { return (i - 1) % 3; } + /** \} */ + + + /// Rotate a coordinate system to be perpendicular to the given normal + inline + void + rot_coord_sys(const algebra::vec<3, float> &old_u, + const algebra::vec<3, float> &old_v, + const algebra::vec<3, float> &new_norm, + algebra::vec<3, float> &new_u, + algebra::vec<3, float> &new_v) + { + new_u = old_u; + new_v = old_v; + algebra::vec<3, float> old_norm = vprod(old_u, old_v); + float ndot = old_norm * new_norm; + if (unlikely(ndot <= -1.0f)) + { + new_u = -new_u; + new_v = -new_v; + return; + } + algebra::vec<3, float> perp_old = new_norm - ndot * old_norm; + algebra::vec<3, float> dperp = + 1.0f / (1 + ndot) * (old_norm + new_norm); + new_u -= dperp * (new_u * perp_old); + new_v -= dperp * (new_v * perp_old); + } + + + // FIXME: Add const to formals whenever we can. + + // Reproject a curvature tensor from the basis spanned by old_u and old_v + // (which are assumed to be unit-length and perpendicular) to the + // new_u, new_v basis. + inline + void + proj_curv(const algebra::vec<3, float>& old_u, + const algebra::vec<3, float>& old_v, + float old_ku, float old_kuv, float old_kv, + const algebra::vec<3, float>& new_u, + const algebra::vec<3, float>& new_v, + float& new_ku, float& new_kuv, float& new_kv) + { + algebra::vec<3, float> r_new_u, r_new_v; + rot_coord_sys(new_u, new_v, vprod(old_u, old_v), r_new_u, r_new_v); + + float u1 = r_new_u * old_u; + float v1 = r_new_u * old_v; + float u2 = r_new_v * old_u; + float v2 = r_new_v * old_v; + new_ku = old_ku * u1*u1 + old_kuv * (2.0f * u1*v1) + old_kv * v1*v1; + new_kuv = old_ku * u1*u2 + old_kuv * (u1*v2 + u2*v1) + old_kv * v1*v2; + new_kv = old_ku * u2*u2 + old_kuv * (2.0f * u2*v2) + old_kv * v2*v2; + } + + /** Given a curvature tensor, find principal directions and + curvatures. Makes sure that pdir1 and pdir2 are + perpendicular to normal. */ + inline + void + diagonalize_curv(const algebra::vec<3, float>& old_u, + const algebra::vec<3, float>& old_v, + float ku, float kuv, float kv, + const algebra::vec<3, float>& new_norm, + algebra::vec<3, float>& pdir1, + algebra::vec<3, float>& pdir2, + float& k1, float& k2) + { + algebra::vec<3, float> r_old_u, r_old_v; + rot_coord_sys(old_u, old_v, new_norm, r_old_u, r_old_v); + + float c = 1, s = 0, tt = 0; + if (likely(kuv != 0.0f)) + { + // Jacobi rotation to diagonalize. + float h = 0.5f * (kv - ku) / kuv; + tt = (h < 0.0f) ? + 1.0f / (h - sqrt(1.0f + h*h)) : + 1.0f / (h + sqrt(1.0f + h*h)); + c = 1.0f / sqrt(1.0f + tt*tt); + s = tt * c; + } + + k1 = ku - tt * kuv; + k2 = kv + tt * kuv; + + if (fabs(k1) >= fabs(k2)) + pdir1 = c*r_old_u - s*r_old_v; + else + { + std::swap(k1, k2); + pdir1 = s*r_old_u + c*r_old_v; + } + pdir2 = vprod(new_norm, pdir1); + } + + } // end of namespace mln::geom::internal + + + /** \brief Compute the principal curvatures of a surface at + vertices. + + These princpal curvatures are names kappa_1 and kappa_2 in + + Sylvie Philipp-Foliguet, Michel Jordan Laurent Najman and + Jean Cousty. Artwork 3D Model Database Indexing and + Classification. + + \param[in] ima The surface (triangle mesh) on which the + curvature is to be computed. */ + /* FIXME: We should restrict attached data to vertices in return + value. */ + /* FIXME: Have a typedef for this return type? */ + inline + std::pair< + complex_image< 2, mln::space_2complex_geometry, float >, + complex_image< 2, mln::space_2complex_geometry, float > + > + mesh_curvature(const p_complex<2, space_2complex_geometry>& mesh) + { + // Shortcuts. + static const unsigned D = 2; + typedef space_2complex_geometry G; + typedef algebra::vec<3, float> vec3f; + typedef algebra::mat<3, 3, float> mat3f; + + typedef complex_image< D, G, vec3f > corner_area_t; + typedef complex_image< D, G, float > point_area_t; + + // Normals at vertices. + typedef complex_image< 2, mln::space_2complex_geometry, vec3f > normal_t; + normal_t normal = mesh_normal(mesh); + + // Areas ``belonging'' to a normal. + typedef complex_image< D, G, vec3f > corner_area_t; + typedef complex_image< D, G, float > point_area_t; + typedef std::pair<corner_area_t, point_area_t> corner_point_area_t; + corner_point_area_t corner_point_area = mesh_corner_point_area(mesh); + // Shortcuts. + corner_area_t& corner_area = corner_point_area.first; + point_area_t& point_area = corner_point_area.second; + + // Curvatures at each vertex. + typedef complex_image< 2, mln::space_2complex_geometry, float > curv_t; + typedef std::pair<curv_t, curv_t> output_t; + output_t output(mesh, mesh); + curv_t& curv1 = output.first; + curv_t& curv2 = output.second; + // ?? + complex_image< 2, mln::space_2complex_geometry, float > curv12(mesh); + // Principal directions at each vertex. + complex_image< 2, mln::space_2complex_geometry, vec3f > pdir1(mesh); + complex_image< 2, mln::space_2complex_geometry, vec3f > pdir2(mesh); + + /*-------------------------------------------------. + | Set up an initial coordinate system per vertex. | + `-------------------------------------------------*/ + + mln::p_n_faces_fwd_piter<D, G> f(mesh, 2); + // A neighborhood where neighbors are the set of 0-faces + // transitively adjacent to the reference point. + typedef mln::complex_m_face_neighborhood<D, G> adj_vertices_nbh_t; + adj_vertices_nbh_t adj_vertices_nbh; + mln_niter_(adj_vertices_nbh_t) adj_v(adj_vertices_nbh, f); + /* FIXME: We should be able to pas this value (m) either at the + construction of the neighborhood or at the construction of + the iterator. Alas, we can't inherit ctors (yet), so we + can't rely on a simple mixin approach. */ + adj_v.iter().set_m(0); + + for_all(f) + { + /// Psites of 0-faces transitively adjacent to F. + std::vector<mln_psite_(curv_t)> p; + p.reserve(3); + for_all(adj_v) + p.push_back(adj_v); + /// The should be exactly three vertices adjacent to F. + mln_assertion(p.size() == 3); + + // FIXME: See above. + pdir1(p[0]) = + p[1].to_site().front().to_vec() - p[0].to_site().front().to_vec(); + pdir1(p[1]) = + p[2].to_site().front().to_vec() - p[1].to_site().front().to_vec(); + pdir1(p[2]) = + p[0].to_site().front().to_vec() - p[2].to_site().front().to_vec(); + } + + mln::p_n_faces_fwd_piter<D, G> v(mesh, 0); + for_all(v) + { + pdir1(v) = algebra::vprod(pdir1(v), normal(v)); + pdir1(v).normalize(); + pdir2(v) = algebra::vprod(normal(v), pdir1(v)); + } + + /*-----------------------------. + | Compute curvature per-face. | + `-----------------------------*/ + + for_all(f) + { + std::vector<mln_psite_(curv_t)> p; + p.reserve(3); + for_all(adj_v) + p.push_back(adj_v); + mln_assertion(p.size() == 3); + + // (Opposite) edge vectors. + algebra::vec<3, vec3f> e; + // FIXME: See above. + e.set + (p[2].to_site().front().to_vec() - p[1].to_site().front().to_vec(), + p[0].to_site().front().to_vec() - p[2].to_site().front().to_vec(), + p[1].to_site().front().to_vec() - p[0].to_site().front().to_vec()); + + // N-T-B coordinate system per face. + vec3f t = e[0]; + t.normalize(); + vec3f n = algebra::vprod(e[0], e[1]); + /* FIXME: Why don't we normalize N? Is it a normal vector by + construction? Or maybe we don't need it to be normal? */ + vec3f b = algebra::vprod(n, t); + b.normalize(); + + // Estimate curvature based on variation of normals along edges. + vec3f m = literal::zero; + mat3f w = literal::zero; + for (int j = 0; j < 3; ++j) + { + float u = e[j] * t; + float v = e[j] * b; + w(0, 0) += u * u; + w(0, 1) += u * v; + /* FIXME: Those two lines were commented in Trimesh's + original code. + + //w(1, 1) += v*v + u*u; + //w(1, 2) += u*v; + + */ + w(2, 2) += v * v; + vec3f dn = + normal(p[internal::prev(j)]) - normal(p[internal::next(j)]); + float dnu = dn * t; + float dnv = dn * b; + m[0] += dnu*u; + m[1] += dnu*v + dnv*u; + m[2] += dnv*v; + } + w(1, 1) = w(0, 0) + w(2, 2); + w(1, 2) = w(0, 1); + + // Least squares solution. + vec3f diag; + bool ldlt_decomp_sucess_p = algebra::ldlt_decomp(w, diag); + mln_assertion(ldlt_decomp_sucess_p); + algebra::ldlt_solve(w, diag, m, m); + + // Push it back out to the vertices + for (int j = 0; j < 3; ++j) + { + float c1, c12, c2; + internal::proj_curv(t, b, m[0], m[1], m[2], + pdir1(p[j]), pdir2(p[j]), c1, c12, c2); + float wt = corner_area(f)[j] / point_area(p[j]); + curv1(p[j]) += wt * c1; + curv12(p[j]) += wt * c12; + curv2(p[j]) += wt * c2; + } + } + + /*-------------------------------------------------------------. + | Compute principal directions and curvatures at each vertex. | + `-------------------------------------------------------------*/ + + for_all(v) + internal::diagonalize_curv(pdir1(v), pdir2(v), + curv1(v), curv12(v), curv2(v), + normal(v), pdir1(v), pdir2(v), + curv1(v), curv2(v)); + + return output; + } + + } // end of namespace mln::geom + +} // end of namespace mln + +#endif // ! MILENA_APPS_STATUES_MISC_HH -- 1.6.0.4