484 lines
13 KiB
C++
484 lines
13 KiB
C++
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// Copyright NVIDIA Corporation 2008 -- Ignacio Castano <icastano@nvidia.com>
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#include "nvmesh.h" // pch
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#include "LeastSquaresConformalMap.h"
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#include "ParameterizationQuality.h"
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#include "Util.h"
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#include "nvmesh/halfedge/Mesh.h"
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#include "nvmesh/halfedge/Vertex.h"
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#include "nvmesh/halfedge/Face.h"
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#include "nvmath/Sparse.h"
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#include "nvmath/Solver.h"
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#include "nvmath/Vector.inl"
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#include "nvcore/Array.inl"
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using namespace nv;
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using namespace HalfEdge;
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namespace
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{
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// Test all pairs of vertices in the boundary and check distance.
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static void findDiameterVertices(HalfEdge::Mesh * mesh, HalfEdge::Vertex ** a, HalfEdge::Vertex ** b)
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{
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nvDebugCheck(mesh != NULL);
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nvDebugCheck(a != NULL);
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nvDebugCheck(b != NULL);
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const uint vertexCount = mesh->vertexCount();
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float maxLength = 0.0f;
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for (uint v0 = 1; v0 < vertexCount; v0++)
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{
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HalfEdge::Vertex * vertex0 = mesh->vertexAt(v0);
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nvDebugCheck(vertex0 != NULL);
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if (!vertex0->isBoundary()) continue;
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for (uint v1 = 0; v1 < v0; v1++)
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{
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HalfEdge::Vertex * vertex1 = mesh->vertexAt(v1);
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nvDebugCheck(vertex1 != NULL);
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if (!vertex1->isBoundary()) continue;
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float len = length(vertex0->pos - vertex1->pos);
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if (len > maxLength)
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{
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maxLength = len;
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*a = vertex0;
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*b = vertex1;
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}
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}
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}
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nvDebugCheck(*a != NULL && *b != NULL);
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}
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// Fast sweep in 3 directions
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static bool findApproximateDiameterVertices(HalfEdge::Mesh * mesh, HalfEdge::Vertex ** a, HalfEdge::Vertex ** b)
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{
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nvDebugCheck(mesh != NULL);
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nvDebugCheck(a != NULL);
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nvDebugCheck(b != NULL);
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const uint vertexCount = mesh->vertexCount();
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HalfEdge::Vertex * minVertex[3];
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HalfEdge::Vertex * maxVertex[3];
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minVertex[0] = minVertex[1] = minVertex[2] = NULL;
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maxVertex[0] = maxVertex[1] = maxVertex[2] = NULL;
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for (uint v = 1; v < vertexCount; v++)
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{
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HalfEdge::Vertex * vertex = mesh->vertexAt(v);
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nvDebugCheck(vertex != NULL);
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if (vertex->isBoundary())
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{
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minVertex[0] = minVertex[1] = minVertex[2] = vertex;
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maxVertex[0] = maxVertex[1] = maxVertex[2] = vertex;
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break;
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}
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}
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if (minVertex[0] == NULL)
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{
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// Input mesh has not boundaries.
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return false;
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}
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for (uint v = 1; v < vertexCount; v++)
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{
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HalfEdge::Vertex * vertex = mesh->vertexAt(v);
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nvDebugCheck(vertex != NULL);
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if (!vertex->isBoundary())
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{
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// Skip interior vertices.
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continue;
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}
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if (vertex->pos.x < minVertex[0]->pos.x) minVertex[0] = vertex;
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else if (vertex->pos.x > maxVertex[0]->pos.x) maxVertex[0] = vertex;
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if (vertex->pos.y < minVertex[1]->pos.y) minVertex[1] = vertex;
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else if (vertex->pos.y > maxVertex[1]->pos.y) maxVertex[1] = vertex;
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if (vertex->pos.z < minVertex[2]->pos.z) minVertex[2] = vertex;
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else if (vertex->pos.z > maxVertex[2]->pos.z) maxVertex[2] = vertex;
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}
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float lengths[3];
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for (int i = 0; i < 3; i++)
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{
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lengths[i] = length(minVertex[i]->pos - maxVertex[i]->pos);
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}
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if (lengths[0] > lengths[1] && lengths[0] > lengths[2])
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{
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*a = minVertex[0];
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*b = maxVertex[0];
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}
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else if (lengths[1] > lengths[2])
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{
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*a = minVertex[1];
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*b = maxVertex[1];
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}
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else
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{
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*a = minVertex[2];
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*b = maxVertex[2];
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}
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return true;
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}
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// Conformal relations from Bruno Levy:
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// Computes the coordinates of the vertices of a triangle
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// in a local 2D orthonormal basis of the triangle's plane.
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static void project_triangle(Vector3::Arg p0, Vector3::Arg p1, Vector3::Arg p2, Vector2 * z0, Vector2 * z1, Vector2 * z2)
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{
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Vector3 X = normalize(p1 - p0, 0.0f);
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Vector3 Z = normalize(cross(X, (p2 - p0)), 0.0f);
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Vector3 Y = normalize(cross(Z, X), 0.0f);
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float x0 = 0.0f;
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float y0 = 0.0f;
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float x1 = length(p1 - p0);
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float y1 = 0.0f;
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float x2 = dot((p2 - p0), X);
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float y2 = dot((p2 - p0), Y);
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*z0 = Vector2(x0, y0);
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*z1 = Vector2(x1, y1);
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*z2 = Vector2(x2, y2);
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}
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// LSCM equation, geometric form :
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// (Z1 - Z0)(U2 - U0) = (Z2 - Z0)(U1 - U0)
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// Where Uk = uk + i.vk is the complex number
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// corresponding to (u,v) coords
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// Zk = xk + i.yk is the complex number
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// corresponding to local (x,y) coords
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// cool: no divide with this expression,
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// makes it more numerically stable in
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// the presence of degenerate triangles.
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static void setup_conformal_map_relations(SparseMatrix & A, int row, const HalfEdge::Vertex * v0, const HalfEdge::Vertex * v1, const HalfEdge::Vertex * v2)
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{
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int id0 = v0->id;
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int id1 = v1->id;
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int id2 = v2->id;
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Vector3 p0 = v0->pos;
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Vector3 p1 = v1->pos;
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Vector3 p2 = v2->pos;
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Vector2 z0, z1, z2;
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project_triangle(p0, p1, p2, &z0, &z1, &z2);
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Vector2 z01 = z1 - z0;
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Vector2 z02 = z2 - z0;
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float a = z01.x;
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float b = z01.y;
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float c = z02.x;
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float d = z02.y;
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nvCheck(b == 0.0f);
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// Note : 2*id + 0 --> u
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// 2*id + 1 --> v
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int u0_id = 2 * id0 + 0;
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int v0_id = 2 * id0 + 1;
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int u1_id = 2 * id1 + 0;
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int v1_id = 2 * id1 + 1;
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int u2_id = 2 * id2 + 0;
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int v2_id = 2 * id2 + 1;
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// Note : b = 0
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// Real part
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A.setCoefficient(u0_id, 2 * row + 0, -a+c);
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A.setCoefficient(v0_id, 2 * row + 0, b-d);
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A.setCoefficient(u1_id, 2 * row + 0, -c);
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A.setCoefficient(v1_id, 2 * row + 0, d);
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A.setCoefficient(u2_id, 2 * row + 0, a);
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// Imaginary part
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A.setCoefficient(u0_id, 2 * row + 1, -b+d);
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A.setCoefficient(v0_id, 2 * row + 1, -a+c);
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A.setCoefficient(u1_id, 2 * row + 1, -d);
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A.setCoefficient(v1_id, 2 * row + 1, -c);
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A.setCoefficient(v2_id, 2 * row + 1, a);
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}
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// Conformal relations from Brecht Van Lommel (based on ABF):
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static float vec_angle_cos(Vector3::Arg v1, Vector3::Arg v2, Vector3::Arg v3)
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{
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Vector3 d1 = v1 - v2;
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Vector3 d2 = v3 - v2;
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return clamp(dot(d1, d2) / (length(d1) * length(d2)), -1.0f, 1.0f);
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}
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static float vec_angle(Vector3::Arg v1, Vector3::Arg v2, Vector3::Arg v3)
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{
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float dot = vec_angle_cos(v1, v2, v3);
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return acosf(dot);
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}
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static void triangle_angles(Vector3::Arg v1, Vector3::Arg v2, Vector3::Arg v3, float *a1, float *a2, float *a3)
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{
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*a1 = vec_angle(v3, v1, v2);
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*a2 = vec_angle(v1, v2, v3);
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*a3 = PI - *a2 - *a1;
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}
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static void triangle_cosines(Vector3::Arg v1, Vector3::Arg v2, Vector3::Arg v3, float *a1, float *a2, float *a3)
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{
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*a1 = vec_angle_cos(v3, v1, v2);
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*a2 = vec_angle_cos(v1, v2, v3);
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*a3 = vec_angle_cos(v2, v3, v1);
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}
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static void setup_abf_relations(SparseMatrix & A, int row, const HalfEdge::Vertex * v0, const HalfEdge::Vertex * v1, const HalfEdge::Vertex * v2)
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{
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int id0 = v0->id;
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int id1 = v1->id;
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int id2 = v2->id;
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Vector3 p0 = v0->pos;
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Vector3 p1 = v1->pos;
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Vector3 p2 = v2->pos;
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#if 1
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// @@ IC: Wouldn't it be more accurate to return cos and compute 1-cos^2?
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// It does indeed seem to be a little bit more robust.
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// @@ Need to revisit this more carefully!
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float a0, a1, a2;
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triangle_angles(p0, p1, p2, &a0, &a1, &a2);
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float s0 = sinf(a0);
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float s1 = sinf(a1);
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float s2 = sinf(a2);
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/*// Hack for degenerate triangles.
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if (equal(s0, 0) && equal(s1, 0) && equal(s2, 0)) {
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if (equal(a0, 0)) a0 += 0.001f;
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if (equal(a1, 0)) a1 += 0.001f;
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if (equal(a2, 0)) a2 += 0.001f;
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if (equal(a0, PI)) a0 = PI - a1 - a2;
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if (equal(a1, PI)) a1 = PI - a0 - a2;
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if (equal(a2, PI)) a2 = PI - a0 - a1;
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s0 = sinf(a0);
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s1 = sinf(a1);
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s2 = sinf(a2);
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}*/
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if (s1 > s0 && s1 > s2)
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{
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swap(s1, s2);
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swap(s0, s1);
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swap(a1, a2);
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swap(a0, a1);
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swap(id1, id2);
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swap(id0, id1);
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}
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else if (s0 > s1 && s0 > s2)
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{
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swap(s0, s2);
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swap(s0, s1);
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swap(a0, a2);
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swap(a0, a1);
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swap(id0, id2);
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swap(id0, id1);
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}
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float c0 = cosf(a0);
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#else
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float c0, c1, c2;
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triangle_cosines(p0, p1, p2, &c0, &c1, &c2);
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float s0 = 1 - c0*c0;
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float s1 = 1 - c1*c1;
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float s2 = 1 - c2*c2;
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nvDebugCheck(s0 != 0 || s1 != 0 || s2 != 0);
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if (s1 > s0 && s1 > s2)
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{
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swap(s1, s2);
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swap(s0, s1);
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swap(c1, c2);
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swap(c0, c1);
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swap(id1, id2);
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swap(id0, id1);
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}
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else if (s0 > s1 && s0 > s2)
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{
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swap(s0, s2);
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swap(s0, s1);
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swap(c0, c2);
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swap(c0, c1);
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swap(id0, id2);
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swap(id0, id1);
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}
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#endif
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float ratio = (s2 == 0.0f) ? 1.0f: s1/s2;
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float cosine = c0 * ratio;
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float sine = s0 * ratio;
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// Note : 2*id + 0 --> u
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// 2*id + 1 --> v
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int u0_id = 2 * id0 + 0;
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int v0_id = 2 * id0 + 1;
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int u1_id = 2 * id1 + 0;
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int v1_id = 2 * id1 + 1;
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int u2_id = 2 * id2 + 0;
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int v2_id = 2 * id2 + 1;
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// Real part
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A.setCoefficient(u0_id, 2 * row + 0, cosine - 1.0f);
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A.setCoefficient(v0_id, 2 * row + 0, -sine);
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A.setCoefficient(u1_id, 2 * row + 0, -cosine);
|
||
|
|
A.setCoefficient(v1_id, 2 * row + 0, sine);
|
||
|
|
A.setCoefficient(u2_id, 2 * row + 0, 1);
|
||
|
|
|
||
|
|
// Imaginary part
|
||
|
|
A.setCoefficient(u0_id, 2 * row + 1, sine);
|
||
|
|
A.setCoefficient(v0_id, 2 * row + 1, cosine - 1.0f);
|
||
|
|
A.setCoefficient(u1_id, 2 * row + 1, -sine);
|
||
|
|
A.setCoefficient(v1_id, 2 * row + 1, -cosine);
|
||
|
|
A.setCoefficient(v2_id, 2 * row + 1, 1);
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace
|
||
|
|
|
||
|
|
|
||
|
|
bool nv::computeLeastSquaresConformalMap(HalfEdge::Mesh * mesh)
|
||
|
|
{
|
||
|
|
nvDebugCheck(mesh != NULL);
|
||
|
|
|
||
|
|
// For this to work properly, mesh should not have colocals that have the same
|
||
|
|
// attributes, unless you want the vertices to actually have different texcoords.
|
||
|
|
|
||
|
|
const uint vertexCount = mesh->vertexCount();
|
||
|
|
const uint D = 2 * vertexCount;
|
||
|
|
const uint N = 2 * countMeshTriangles(mesh);
|
||
|
|
|
||
|
|
// N is the number of equations (one per triangle)
|
||
|
|
// D is the number of variables (one per vertex; there are 2 pinned vertices).
|
||
|
|
if (N < D - 4) {
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
SparseMatrix A(D, N);
|
||
|
|
FullVector b(N);
|
||
|
|
FullVector x(D);
|
||
|
|
|
||
|
|
// Fill b:
|
||
|
|
b.fill(0.0f);
|
||
|
|
|
||
|
|
// Fill x:
|
||
|
|
HalfEdge::Vertex * v0;
|
||
|
|
HalfEdge::Vertex * v1;
|
||
|
|
if (!findApproximateDiameterVertices(mesh, &v0, &v1))
|
||
|
|
{
|
||
|
|
// Mesh has no boundaries.
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
if (v0->tex == v1->tex)
|
||
|
|
{
|
||
|
|
// LSCM expects an existing parameterization.
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
for (uint v = 0; v < vertexCount; v++)
|
||
|
|
{
|
||
|
|
HalfEdge::Vertex * vertex = mesh->vertexAt(v);
|
||
|
|
nvDebugCheck(vertex != NULL);
|
||
|
|
|
||
|
|
// Initial solution.
|
||
|
|
x[2 * v + 0] = vertex->tex.x;
|
||
|
|
x[2 * v + 1] = vertex->tex.y;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Fill A:
|
||
|
|
const uint faceCount = mesh->faceCount();
|
||
|
|
for (uint f = 0, t = 0; f < faceCount; f++)
|
||
|
|
{
|
||
|
|
const HalfEdge::Face * face = mesh->faceAt(f);
|
||
|
|
nvDebugCheck(face != NULL);
|
||
|
|
nvDebugCheck(face->edgeCount() == 3);
|
||
|
|
|
||
|
|
const HalfEdge::Vertex * vertex0 = NULL;
|
||
|
|
|
||
|
|
for (HalfEdge::Face::ConstEdgeIterator it(face->edges()); !it.isDone(); it.advance())
|
||
|
|
{
|
||
|
|
const HalfEdge::Edge * edge = it.current();
|
||
|
|
nvCheck(edge != NULL);
|
||
|
|
|
||
|
|
if (vertex0 == NULL)
|
||
|
|
{
|
||
|
|
vertex0 = edge->vertex;
|
||
|
|
}
|
||
|
|
else if (edge->next->vertex != vertex0)
|
||
|
|
{
|
||
|
|
const HalfEdge::Vertex * vertex1 = edge->from();
|
||
|
|
const HalfEdge::Vertex * vertex2 = edge->to();
|
||
|
|
|
||
|
|
setup_abf_relations(A, t, vertex0, vertex1, vertex2);
|
||
|
|
//setup_conformal_map_relations(A, t, vertex0, vertex1, vertex2);
|
||
|
|
|
||
|
|
t++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
const uint lockedParameters[] =
|
||
|
|
{
|
||
|
|
2 * v0->id + 0,
|
||
|
|
2 * v0->id + 1,
|
||
|
|
2 * v1->id + 0,
|
||
|
|
2 * v1->id + 1
|
||
|
|
};
|
||
|
|
|
||
|
|
// Solve
|
||
|
|
LeastSquaresSolver(A, b, x, lockedParameters, 4, 0.000001f);
|
||
|
|
|
||
|
|
// Map x back to texcoords:
|
||
|
|
for (uint v = 0; v < vertexCount; v++)
|
||
|
|
{
|
||
|
|
HalfEdge::Vertex * vertex = mesh->vertexAt(v);
|
||
|
|
nvDebugCheck(vertex != NULL);
|
||
|
|
|
||
|
|
vertex->tex = Vector2(x[2 * v + 0], x[2 * v + 1]);
|
||
|
|
}
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|