non-normalized tangent vectors

[HelliceSaouli]

Hello sir, can you please explain to me how you exactly computed non-normalized tangent vectors u and v

[sebastianlipponer]

Are you referring to the computation of u and v for triangles?

surface_splatting/surface_splatting/main.cpp

Lines 109 to 167 in c376d6c

	void
	steiner_circumellipse(float const* v0_ptr, float const* v1_ptr,
	float const* v2_ptr, float* p0_ptr, float* t1_ptr, float* t2_ptr)
	{
	Matrix2f Q;
	Vector3f d0, d1, d2;
	{
	Map<const Vector3f> v[3] = { v0_ptr, v1_ptr, v2_ptr };
	d0 = v[1] - v[0];
	d0.normalize();
	d1 = v[2] - v[0];
	d1 = d1 - d0 * d0.dot(d1);
	d1.normalize();
	d2 = (1.0f / 3.0f) * (v[0] + v[1] + v[2]);
	Vector2f p[3];
	for (unsigned int j(0); j < 3; ++j)
	{
	p[j] = Vector2f(
	d0.dot(v[j] - d2),
	d1.dot(v[j] - d2)
	);
	}
	Matrix3f A;
	for (unsigned int j(0); j < 3; ++j)
	{
	A.row(j) = Vector3f(
	p[j].x() * p[j].x(),
	2.0f * p[j].x() * p[j].y(),
	p[j].y() * p[j].y()
	);
	}
	FullPivLU<Matrix3f> lu(A);
	Vector3f res = lu.solve(Vector3f::Ones());
	Q(0, 0) = res(0);
	Q(1, 1) = res(2);
	Q(0, 1) = Q(1, 0) = res(1);
	}
	Map<Vector3f> p0(p0_ptr), t1(t1_ptr), t2(t2_ptr);
	{
	SelfAdjointEigenSolver<Matrix2f> es;
	es.compute(Q);
	Vector2f const& l = es.eigenvalues();
	Vector2f const& e0 = es.eigenvectors().col(0);
	Vector2f const& e1 = es.eigenvectors().col(1);
	p0 = d2;
	t1 = (1.0f / std::sqrt(l.x())) * (d0 * e0.x() + d1 * e0.y());
	t2 = (1.0f / std::sqrt(l.y())) * (d0 * e1.x() + d1 * e1.y());
	}
	}

[HelliceSaouli]

yes the t1, and t2 ? and is it possible to compute it if we are not loading a triangular mesh. ?

[sebastianlipponer]

Whats the actual representation of your input geometry?

The code above converts a triangle mesh to a point based representation by computing the circumellipse for each triangle of the triangle mesh. The vectors t1 and t2 are the ellipses semi-major and semi-minor axis.

While the code allows you to render a triangle mesh using surface splatting in meaningful way, the resulting point based representation is certainly not an optimal reconstruction of the given input geometry.

You may want to use a full-blown point-based surface reconstruction algorithm. For example the technique described by Wu and Kobbelt [1].

[1] Optimized Sub-Sampling of Point Sets for Surface Splatting