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QMVC.pptx
- 1. Mean value coordinates for quad cages in 3D Jean-Marc Thiery Telecom ParisTech, LTCI – Université Paris-Saclay Pooran Memari Ecole Polytechnique, LIX – CNRS – Université Paris-Saclay Tamy Boubekeur Telecom ParisTech, LTCI – Université Paris-Saclay
- 2. WHY CAGES? • Various control structures with different pros and cons
- 3. WHY CAGES? • Large scale smooth deformations • Anisotropic stretch • Precise volume control
- 4. • MVC [Floater 2003, Ju et al. 2005] • HC [Joshi et al. 2007] • PMVC [Lipman et al. 2007] • Green [Lipman et al. 2008] • SMVC [Langer et al. 2006] • MEC [Hormann et al. 2008] • … LOTS OF EXISTING COORDINATES Desired properties: • Linear precision • Positive • Smooth • “Shape-aware” • Closed-form expression • Interpolation
- 5. WHY QMVC? HC MVC PMVC Green Why quads? • Easy to model by hand: - box modeling - edge extrusion • Anisotropic surface structure • Easy to manipulate: - cross section selection
- 6. WHY QMVC? Why quads? • Easy to model by hand: - box modeling - edge extrusion • Anisotropic surface structure • Easy to manipulate: - cross section selection • Motivation for most recent automatic/assisted cage [Le&Deng 2017] [Calderdon&Boubekeur 2017] [Xian et al. 2011]
- 7. WHY QMVC? : NON-PLANAR QUADS! MVC QMVC
- 8. QUICK RECAP’ ON MVC
- 9. QUICK RECAP’ ON MVC
- 10. QUICK RECAP’ ON MVC Linear functions on the cage : MVC coordinates : 𝜙0(𝜉) 𝜙1(𝜉) 𝑓𝐶(𝜉)
- 11. QUICK RECAP’ ON MVC Properties • Linear precision: (valid • Interpolation: • Smoothness: 𝐶∞ (everywhere but on the cage
- 12. COMPUTATION Computation of Computation of MVC : 𝜙𝑖(𝜉) 𝑣𝑖
- 13. COMPUTATION ON TRIANGLES Computation of [Ju et al. 2005] Mean value coordinates for closed triangular meshes
- 14. COMPUTATION ON TRIANGLES Computation of Full rank matrix expression
- 15. WHEN CONSIDERING QUADS Choice of basis functions: Bilinear coordinates Valid quads: Extend « convexity » to 3D Our validity predicate: « A quad is valid if convex when projected on any of the 4 planes spanned by its vertices »
- 16. COMPUTATION ON QUADS Computation of Full rank matrix expression
- 17. SPECTRAL ANALYSIS • Rank = 2 : • Point is lying on a planar quad basis functions (MVC are an interpolant, see paper) • Rank = 3 : • Point is lying on the (non-planar) bilinear quad basis functions (MVC are an interpolant, see paper) • Otherwise solution known up to a component along the kernel
- 18. ANALYSIS OF THE LAST CASE Solution Least-norm solution to Eq (1) (Eq (1)) Unique vector in kernel of Eq (1) What we seek for Closed-form expressions:
- 19. « APPROPRIATE » VALUES FOR (Eq (1)) Desired properties of λ: • Any choice (even random values) of λ leads to linear precision (the input matter what) • 𝐶∞ coordinates λ must be 𝐶∞ • Interpolation λ is constrained near the cage: lim 𝜂 → 𝜉 𝑤𝑞 𝜂 = 𝑤𝑞 𝜉 𝜂 𝜉
- 20. PROPOSED STRATEGY Ground truth: Weight estimate: 𝜂 λ estimate: norm correction: Sampling used to approximate the integrals . ~ ∑. ( )
- 21. IMPACT OF NORM CORRECTION
- 22. SAMPLING CONSTRAINTS 𝜂 Desired properties of λ: • Interpolation we make the sampling dependent on the evaluation point η • Smooth coordinates we make the sampling a smooth function of the evaluation point η 𝜂 𝜂 𝜂 𝜂
- 23. SAMPLING THROUGH PROJECTIONS 𝜂 Sampling: • We project η on the quad • We construct a sampling centered on this point in the (u,v) space • Regularity of the sampling inferred by the regularity of the projection
- 24. SAMPLING THROUGH PROJECTIONS Two-step projection: • Challenge: project on a non-convex geometry • Step 1: Project inside the convex hull of the 4 points (tetrahedron), using absolute MVC • Step 2: Project the constructed point by intersecting the line directed by the quad’s • Our validity predicate ensures that this projection is valid (see paper for details)
- 25. TO RECAP’: QMVC IN A NUTSHELL Linear precision equation: kernel vector minimal-norm solution 𝜂 𝜂 Quad’s unnormalized coordinates:
- 27. MVC GC QMVC
- 28. MVC GC QMVC
- 29. MVC GC QMVC
- 30. MVC QMVC
- 31. MVC QMVC
- 32. COMPARISON WITH SMVC [Langer et al. 2006] Spherical barycentric coordinates Why SMVC? • MVC-based
- 33. COMPARISON WITH SMVC [Langer et al. 2006] Spherical barycentric coordinates Why SMVC? • MVC-based • Can use any (convex) planar n-gon more general than us on that account sometimes too restrictive for modelling
- 36. COMPARISON WITH HC Why HC? • Only need to discretize cage faces with their basis functions on a 3d grid can use bilinear quads as well
- 37. HC QMVC COMPARISON WITH HC
- 38. HC QMVC COMPARISON WITH HC
- 39. HC QMVC COMPARISON WITH HC
- 40. COMBINATION WITH MEC What are MEC? • Input (invalid) masses m (invalid cage coordinates) • Find « closest » valid weights b
- 41. HC QMVC QMVC+MEC QMVC QMVC+MEC COMBINATION WITH MEC
- 42. HC QMVC QMVC+MEC QMVC QMVC+MEC COMBINATION WITH MEC
- 43. HC QMVC+MEC COMBINATION WITH MEC
- 44. CONCLUSION Contributions: • Entension of MVC to cages featuring non-planar quads • Characterization of valid non-planar quads for cage modelling • A smooth projection operator on non-planar quads Perspectives: • Sampling-free λ construction • Better priors for positive QMVC using the MEC approach • Green coordinates for quad cages? (quasi-conformal) • Non-planar n-gons?
- 45. THANK YOU! Source code available at perso.telecom-paristech.fr/boubek/papers/QMVC/ Contains our implementation of: - QMVC - MVC - SMVC - GC - MEC - Demo viewer for comparison
- 46. Additionals
- 47. TIMINGS IN MS ON A LAPTOP (CPU)
Editor's Notes
- Different control structures are used to animate 3D shapes, depending on the type of intended deformation
- In this talk we will focus on cages, which are typically used to perform large scale deformations featuring anisotropic stretch and offer precise volume control.
- They are a lot of coordinates in the literature, all coming with various properties. The first one is mandatory and allows to recover the input shape with the input cage, which means that the encoding is valid. Positivity results in more natural deformations, as translating parts of the cage in one direction does not result in mesh translation in the opposite direction, The coordinates are required to be smooth so that the resulting deformation is smooth, The resulting deformation should be shape-aware, Ideally the coordinates should have a close-form expression and be computed efficiently Optionnaly, coordinates can lead to interpolation on the cage surface, for precise control.
- What motivates our work is a very simple observation: If you pay attention to cages designed by artists, they feature a lot of quads, and that is because most artists model cages with a box-modeling strategy and extrusion of edges result in quads Quads also allow to better conform to the large features of the shape, and that are also easier to deform than triangles.
- And that is also the conclusion reached by recent work in automatic cage design.
- While artists design triquad cages for these reasons, Most coordinates require the cage to be triangulated for their computation, Resulting in asymetrical artifacts. In this work we present coordinates based on mean value coordinates that support non-planar anisotropic quads.
- Let’s have a look at mean value coordinates A cage is provided to the user, with a function living on the cage surface. MVC allow to extrapolate this function to any point in the 3D space through a simple integral.
- Given an evaluation point, we consider the unit sphere centered on it. A spherical averaging of the function is then performed, while enforcing nearby cage vertices to have a strong influence.
- By considering that both the cage geometry and embedded function are linear, The extrapolated function can be expressed as a linear combination of the values defined at the cage vertices, The barycentric weights being the mean value coordinates of the point.
- This definition implies directly linear precision, Interpolation on the cage, And smoothness of the resulting function.
- To compute Mean value coordiantes, One has to compute the spherical integral of each vertex basis function on each cage facet Sum the contributions of all adjacent facets to compute the unnormalized coordinates And finally normalize them to obtain the MVC. In the end, we see that, in order to compute MVC for any type of linear geometry, one only has to know how to compute the weighted integral of the basis function on a spherical facet.
- On triangle cages, Ju and colleagues noted that summing up the unnormalized coordinates against the vectors starting from the evaluation point and ending at the triangle corners resulted in a very simple quantity, called the mean vector of the face, which is simply the integral of the unit normal on the spherical triangle in red.
- By using this expression, the unnormalized weights at the 3 corners of the triangle can be obtained jointly, since the resulting matrix expression is full rank (in that case, we have three equations, and three unknowns).
- Before going through the derivation for quads, we need to go over a few differences that exist with the simpler triangle case. The first thing is that there exist an infinity of choices for linear basis functions on a quad, and the choice of basis functions will impact the behavior of the resulting coordinates. In our work, we consider bilinear coordinates, which are smooth and are commonly used in Graphics. The second point is, that it is not always possible to consider any quad for cage modelling. Indeed, for example, consider this planar non-convex quad in red. If we encode MVC with a cage featuring this quad, and we deform the cage to match the convex quad in blue, we face a contradiction. As we want to obtain interpolation on the cage facets, this point should actually be deformed to these two different points when deforming the encoding non-convex quad, which is impossible. In our work, we extend this notion of convexity to 3D, by considering that a quad is valid if convex when projected on any of the 4 planes spanned by its vertices.
- Now going back to the computation, We can still compute the mean vector of the quad, And relate the 4 unnormalized weights through the same matrix expression, But unfortunately we cannot obtain directly the 4 unnormalized weights, as the matrix has a rank that is at most 3.
- We can actually characterize all situations depending on the rank of the matrix A_q If the rank equals 2, we know that the point lies on a planar quad, and we can simply return the quad’s basis functions. When the rank is 3, The point might be still be lying on a non-planar bilinear quad, which is rather simple to test. Otherwise, we know the solution, up to a component along the kernel of A_q.
- More precisely, The solution is given by this expression, Which involves the least-norm solution to Equation 1, And the unique vector in the kernel of A_q. All we are missing is the component along this 4-dimensional vector, component that we call here the lambda coordinate. Note that we expressed these quantities using the Singular Value Decomposition of the matrix A_q, but that it is not actually needed, and these quantities have simple geometric closed-form expressions.
- Let’s see what important properties should be verified by the lambda coordinate. First, any choice will lead to linear precision, because it is the component along the kernel of the linear precision equation. Which means that we can give absolutely any value to this lambda coordinate, and we will obtain valid coordinates. Second, the resulting coordinates should be smooth, so the lambda coordinate should be smooth as well. Finally, we wish to obtain interpolation on the cage facets, which constrains the lambda coordinate near the cage.
- We make the simple observation that, If we knew the ground truth weight vector w_q, We could obtain the ground truth lambda coordinate using a simple projection along the kernel vector. We are going to compute a smooth approximation of lambda, by first Computing a approximate weight vector omega_q, using a Riemann summation in place of the continuous integrals, And we will project this approximate weight vector along the kernel vector. Note that, we added this correction term, That corresponds to scaling the resulting approximation so that it matches the component along the least-norm solution vector, for which we know the ground truth. The underlying assumption is, that the error we make when approximating the weights, are somehow distributed similarly along all components of the spectrum of A_q.
- This experiment validates our assumption. We encoded a straight box within a straight box-cage, and we twisted the box-cage to observe the resulting deformations. On the left, are indicated the sampling size that were used for our approximation. When no norm correction is used, we obtain coordinates that are still smooth, but far from what we want. If we increase the sampling, we obtain better results And eventually it converges to the ground truth solution with a lot of samples When we compare to the results obtained with the norm correction, as you can see, even with a sampling of 3 times 3 per large quad, we obtain a solution that matches almost exactly the solution obtained with more than 10 thousand samples per quad.
- We can not use any sampling for our purpose, And if we look back at the properties that we want to obtain for our coordinates, We want to obtain interpolation on the cage. This forces us to make the sampling dependent on the evaluation point eta, so that the sampling results in a Dirac distribution any time eta belongs to the bilinear quad. As we want to obtain smooth coordinates, we further need to make this sampling a smooth function of the evaluation point eta.
- We proceed as follows: We first project eta on the bilinear quad And we then construct a sampling centered on this point. The regularity of the sampling we be directly given by the regularity of the projection operator that we use.
- The main challenge here is therefore to come up with an operator that can project any 3D point on a non-convex object, such as the bilinear quad. We proceed in two steps, We project on the convex hull of the 4 points using the absolute value of the MVC with respect to this tetrahedron, this results in a point that is inside the tetrahedron, in blue. At this point, we are close enough to the bilinear quad to define a proper projection, by computing the quad’s mean vector, and intersecting the resulting line with the bilinear quad. The validity of this projection is ensured by our validity predicate for the quads!
- The main challenge here is therefore to come up with an operator that can project any 3D point on a non-convex object, such as the bilinear quad. We proceed in two steps, We project on the convex hull of the 4 points using the absolute value of the MVC with respect to this tetrahedron, this results in a point that is inside the tetrahedron, in blue. At this point, we are close enough to the bilinear quad to define a proper projection, by computing the quad’s mean vector, and intersecting the resulting line with the bilinear quad. The validity of this projection is ensured by our validity predicate for the quads!