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