Jacobians and Hessians of Mean Value Coordinates for Closed Triangular Meshes

I Background

I-a Mean Value Coordinates for Closed Triangular Meshes

Mean Value Coordinates for closed triangular meshes were introduced in [2]. In this section, we briefly review this work and describe the notations we will use in the rest of this note.

As written in [2], a 3D point $η$ can be expressed as a linear sum of the 3D positions $p_{i}$ of the vertices of a triangular mesh $M$ by: $η = \frac{\sum_{i} w_{i} \cdot p_{i}}{\sum_{i} w_{i}} = \sum_{i} λ_{i} \cdot p_{i}$ .

For a point $x$ onto the surface (a two-dimensional parameter), we note as usual $ϕ_{i} [x]$ the linear function on $M$ that takes value 1 on vertex $i$ and 0 on other vertices, and $p [x]$ its 3D position. The definition of the weights $λ_{i}$ should guarantee linear precision (i.e. $η = \sum_{i} λ_{i} (η) p_{i}$ ).

Since $\int_{B_{η} (M)} \frac{p [x] - η}{| p [x] - η |} d S_{η} (x) = 0$ (the integral of the unit outward normal onto the unit sphere is $0$ ), we have

η = \frac{\int_{B_{η} (M)} \frac{p [x]}{| p [x] - η |} d S_{η} (x)}{\int_{B_{η} (M)} \frac{1}{| p [x] - η |} d S_{η} (x)}

(1)

$B_{η} (M)$ being the projection of the manifold $M$ onto the unit sphere centered in $η$ .

Writing that $\forall x p [x] = \sum_{i} ϕ_{i} [x] p_{i}$ , with $\sum_{i} ϕ_{i} [x] = 1$ , we obtain

η = \frac{\sum_{i} \int_{B_{η} (M)} \frac{ϕ_{i} [x]}{| p [x] - η |} d S_{η} (x) p_{i}}{\int_{B_{η} (M)} \frac{1}{| p [x] - η |} d S_{η} (x)}

(2)

The weights $λ_{i}$ are given by

λ_{i} = \frac{\int_{B_{η} (M)} \frac{ϕ_{i} [x]}{| p [x] - η |} d S_{η} (x)}{\int_{B_{η} (M)} \frac{1}{| p [x] - η |} d S_{η} (x)}

(3)

And the weights $w_{i}$ such that $λ_{i} = \frac{w_{i}}{\sum_{j} w_{j}}$ are given by

w_{i} = \int_{B_{η} (M)} \frac{ϕ_{i} [x]}{| p [x] - η |} d S_{η} (x)

(4)

This definition guarantees linear precision; it gives a linear interpolation of the function onto the triangles of the cage; and it extends it in a regular way to the entire 3D space.

Computing the weights $w_{i}$

The support of the function $ϕ_{i} [x]$ is only composed of the adjacent triangles to the vertex $i$ . Then, we can rewrite Eq. 4 as $w_{i} = \sum_{T \in N 1 (i)} w_{i}^{T}$ , with

w_{i}^{T} = \int_{B_{η} (T)} \frac{ϕ_{i} [x]}{| p [x] - η |} \cdot d ¯ ¯¯ ¯ T

(5)

Given a triangle $T$ with vertices $t_{1}, t_{2}, t_{3}$ , we see that

\begin{matrix} \sum j w_{t_{j}}^{T} \cdot (p_{t_{j}} - η) = & \int_{B_{η} (T)} \frac{\sum_{j} ϕ_{t_{j}} [x] \cdot (p_{t_{j}} - η)}{| p [x] - η |} \cdot d ¯ ¯¯ ¯ T = & \int_{B_{η} (T)} \frac{p [x] - η}{| p [x] - η |} \cdot d ¯ ¯¯ ¯ T ≜ m^{T} \end{matrix}

(6)

This last integral is simply the integral of the unit outward normal on the spherical triangle $¯ ¯¯ ¯ T$ .

By noting $n_{i}^{T} = \frac{N_{i}^{T}}{| N_{i}^{T} |}$ , with $N_{i}^{T} ≜ (p_{t_{i + 1}} - η) \land (p_{t_{i + 2}} - η)$ (see Fig.1), it can be easily expressed as

m^{T} = \sum i \frac{1}{2} θ_{i}^{T} n_{i}^{T}

(7)

This comes from the fact that the integral of the unit outward normal on a closed surface is always 0.

Fig. 1: Triangle T projected on the spherical triangle T̄.

Finally, we obtain

\sum j w_{t_{j}}^{T} \cdot (p_{t_{j}} - η) = m^{T}

(8)

This point was discussed in [2]. As the authors pointed out, by noting $A^{T}$ the 3 by 3 matrix ${p_{t_{1}} - η, p_{t_{2}} - η, p_{t_{3}} - η}$ , we can derive the weights $w_{t_{j}}^{T}$ by

{w_{t_{1}}^{T}, w_{t_{2}}^{T}, w_{t_{3}}^{T}}^{t} = {A^{T}}^{- 1} \cdot m^{T}

(9)

Since ${N_{i}^{T}}^{t} \cdot (p_{t_{j}} - η) = 0 \forall i \neq j$ , we have from Eq. 8 that

w_{t_{i}}^{T} = \frac{{N_{i}^{T}}^{t} \cdot m^{T}}{{N_{i}^{T}}^{t} \cdot (p_{t_{i}} - η)} = \frac{{N_{i}^{T}}^{t} \cdot m^{T}}{det (A^{T})} \forall η \notin S u p p o r t (T)

(10)

Ii MVC Derivatives

We now present the derivatives of the Mean Value Coordinates. Deforming the cage mesh with $f (p_{i}) =_{i}$ induces a deformation of the 3D space by $f = \sum_{i} λ_{i} \cdot_{i}$ . In the rest of the document, for any function $h : E \to F$ , we note $\partial_{x} h, \partial_{y} h, \partial_{z} h$ its derivative by $x, y,$ and $z$ , $\to ▽ h$ its gradient, $J h$ its jacobian, and $H h$ its hessian.

The deformation function $f$ as defined acts now on $R^{3}$ entirely. The derivatives of $f$ can be expressed as a linear sum of positions $_{i} = {_{i},_{i},_{i}}$ :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} J f & = \sum_{i}_{i} \cdot \to ▽ {λ_{i}}^{t} H (f_{x}) & = \sum_{i}_{i} \cdot H λ_{i} H (f_{y}) & = \sum_{i}_{i} \cdot H λ_{i} H (f_{z}) & = \sum_{i}_{i} \cdot H λ_{i} \end{matrix}

(11)

Consequently, it allows to specify implicit equations on the cage in a linear system by giving specified rotations and scales on 3D locations, or to minimize the norm of the hessian to force rigidity, as done in the case of Green Coordinates in [1].

Since $λ_{i} = \frac{w_{i}}{\sum_{j} w_{j}}$ ,

\to ▽ λ_{i} = \frac{\to ▽ w_{i}}{\sum_{j} w_{j}} - \frac{w_{i} \cdot \sum_{j} \to ▽ w_{j}}{(\sum_{j} w_{j})^{2}}

(12)

We also have $\forall c = x, y, z$

\begin{matrix} \partial_{c} (\to ▽ λ_{i}) = & \frac{\partial_{c} (\to ▽ w_{i})}{\sum_{j} w_{j}} - \frac{\to ▽ w_{i} \cdot \sum_{j} \partial_{c} (w_{j})}{(\sum_{j} w_{j})^{2}} - \frac{\partial_{c} (w_{i}) \cdot \sum_{j} \to ▽ w_{j} + w_{i} \cdot \sum_{j} \partial_{c} (\to ▽ w_{j})}{(\sum_{j} w_{j})^{2}} + \frac{2 w_{i} \cdot (\sum_{j} \to ▽ w_{j}) \cdot (\sum_{k} \partial_{c} (w_{k}))}{(\sum_{j} w_{j})^{3}} \end{matrix}

(13)

\begin{matrix} H λ_{i} = & \frac{H w_{i}}{\sum_{j} w_{j}} - \frac{w_{i} \sum_{j} H w_{j}}{(\sum_{j} w_{j})^{2}} - \frac{\to ▽ w_{i} \cdot \sum_{j} \to ▽ (w_{j})^{t} + \sum_{j} \to ▽ (w_{j}) \cdot \to ▽ {w_{i}}^{t}}{(\sum_{j} w_{j})^{2}} + \frac{2 w_{i} (\sum_{j} \to ▽ w_{j}) \cdot (\sum_{j} \to ▽ w_{j})^{t}}{(\sum_{j} w_{j})^{3}} \end{matrix}

(14)

From these expressions, we see that, in order to get $\to ▽ λ_{i} (η)$ and $H λ_{i} (η)$ , we first need to obtain $\to ▽ w_{i} (η)$ and $H w_{i} (η)$ for each vertex $i$ of the cage.

Special case: $η$ lies on the surface of the cage

Mean Value Coordinates define an interpolation process. The function represented onto the vertices of the cage (in our case, a space transformation) is extended to the interior of the triangles with linear interpolation on each triangle. Then it is extended to the space by means of a surfacic integration of the function (see Eq. 1).

Since we represent the cage as triangle mesh in the 3D case, the deformation function cannot be anything more than continuous onto the edges of the cage in 3D. Therefore Jacobians and Hessians of the deformation cannot be evaluated everywhere on the surface of the cage, and we do not provide any formula for Jacobians and Hessians of the deformation onto the surface of the cage.

Ii-a Expression of the Jacobians

In the general case where $det (A^{T}) = (p_{e_{i}} - η)^{t} \cdot N_{i}^{T} \neq 0$ , we have

$\to ▽ w_{t_{i}}^{T} = \frac{{B^{T}}^{t} \cdot N_{i}^{T}}{det (A^{T})}$

with

\begin{matrix} B^{T} = & \sum j \frac{e q_{1} (θ_{j}^{T}) N_{j}^{T} \cdot {N_{j}^{T}}^{t} \cdot J N_{j}^{T}}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{3}} - \sum j \frac{N_{j}^{T} \cdot (2 η - p_{t_{j + 1}} - p_{t_{j + 2}})^{t}}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{2}} + \sum j \frac{e q_{2} (θ_{j}^{T}) J N_{j}^{T}}{2 | p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |} + \sum j w_{t_{j}}^{T} \cdot I_{3} \end{matrix}

and $e q_{1} (x) = \frac{cos (x) sin (x) - x}{sin (x)^{3}}$ and $e q_{2} (x) = \frac{x}{sin (x)}$ two well defined functions on $] 0, π [$ that admit well controlled Taylor expansion around $0$ , $J N_{j}^{T} = (p_{t_{j + 2}} - p_{t_{j + 1}})_{[\land]}$ , $k_{[\land]}$

being the skew 3 by 3 matrix (i.e.

{k_{[\land]}}^{t} = - k_{[\land]}

) such that

k_{[\land]} \cdot u = k \land u \forall k, u \in R^{3}

Special case: $η \in S u p p o r t (T), \notin T$

\begin{matrix} - 2 | T | \to ▽ w_{i}^{T} = & \sum j \frac{e q_{2} (θ_{j}^{T}) (p_{t_{i + 2}} - p_{t_{i + 1}})^{t} \cdot (p_{t_{j + 2}} - p_{t_{j + 1}})}{2 | p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |} n_{T} + \sum j \frac{e q_{1} (θ_{j}^{T}) | p_{t_{j + 2}} - p_{t_{j + 1}} |^{2} {N_{i}^{T}}^{t} \cdot N_{j}^{T}}{4 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{3}} n_{T} + \sum j \frac{cos (θ_{j}^{T}) e q_{3} (θ_{j}^{T}) {N_{i}^{T}}^{t} \cdot N_{j}^{T}}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{2}} n_{T} \end{matrix}

(15)

with $e q_{2} (x) = \frac{x}{sin (x)}, e q_{1} (x) = \frac{cos (x) sin (x) - x}{sin (x)^{3}},$ and $e q_{3} (x) = \frac{cos (x) - 1}{sin (x)^{2}}$ being functions well defined on $] 0, π [$ and that admit controlable Taylor expansion around $0$ .

Ii-B Expression of the Hessians

We note $δ^{x} = ⎛ ⎜ ⎝ \begin{matrix} 100 \end{matrix} ⎞ ⎟ ⎠, δ^{y} = ⎛ ⎜ ⎝ \begin{matrix} 010 \end{matrix} ⎞ ⎟ ⎠, δ^{z} = ⎛ ⎜ ⎝ \begin{matrix} 001 \end{matrix} ⎞ ⎟ ⎠$ .

(16)

with

(17)

and $e q_{6} (x) = \frac{d (e q_{1})}{d x} (x) c o s (x) / s i n (x), e q_{7} (x) = \frac{d (e q_{1})}{d x} (x) s i n (x), e q_{8} (x) = \frac{d (e q_{2})}{d x} (x) c o s (x) / s i n (x)$ , and $e q_{9} (x) = \frac{d (e q_{2})}{d x} (x) s i n (x)$ being functions well defined on $] 0, π [$ and that admit controllable Taylor expansion around $0$ .

Special case: $η \in S u p p o r t (T), \notin T$

H (w_{i}^{T}) (η) = \to ▽ d w_{i}^{T} (η) \cdot n_{T}^{t}

(18)

with

\begin{matrix} - 2 | T | \to ▽ d w_{i}^{T} = - \sum j \frac{((p_{t_{i + 2}} - p_{t_{i + 1}})^{t} \cdot (p_{t_{j + 2}} - p_{t_{j + 1}})) (2 η - p_{t_{j + 2}} - p_{t_{j + 1}})}{(| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{2}} + \sum j \frac{e q_{1} (θ_{j}^{T}) ((p_{t_{i + 2}} - p_{t_{i + 1}})^{t} \cdot (p_{t_{j + 2}} - p_{t_{j + 1}})) {J N_{j}^{T}}^{t} \cdot N_{j}^{T}}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{3}} + \sum j \frac{| p_{t_{j + 2}} - p_{t_{j + 1}} |^{2} ({N_{i}^{T}}^{t} \cdot N_{j}^{T}) (2 η - p_{t_{j + 2}} - p_{t_{j + 1}})}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{4}} + \sum j \frac{e q_{1} (θ_{j}^{T}) | p_{t_{j + 2}} - p_{t_{j + 1}} |^{2} ({J N_{j}^{T}}^{t} \cdot N_{i}^{T} + {J N_{i}^{T}}^{t} \cdot N_{j}^{T})}{4 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{3}} - \sum j \frac{e q_{4} (θ_{j}^{T}) | p_{t_{j + 2}} - p_{t_{j + 1}} |^{2} ({N_{i}^{T}}^{t} \cdot N_{j}^{T}) {J N_{j}^{T}}^{t} \cdot N_{j}^{T}}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{5}} + \sum j \frac{cos (θ_{j}^{T}) e q_{3} (θ_{j}^{T}) ({J N_{j}^{T}}^{t} \cdot N_{i}^{T} + {J N_{i}^{T}}^{t} \cdot N_{j}^{T})}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{2}} - \sum j \frac{(1 - 2 cos (θ_{j}^{T})) ({N_{i}^{T}}^{t} \cdot N_{j}^{T}) (2 η - p_{t_{j + 2}} - p_{t_{j + 1}})}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{3}} + \sum j \frac{e q_{5} (θ_{j}^{T}) ({N_{i}^{T}}^{t} \cdot N_{j}^{T}) {J N_{j}^{T}}^{t} \cdot N_{j}^{T}}{2 (| p_{t_{j + 2}} - η | | p_{t_{j + 1}} - η |)^{4}} \end{matrix}

(19)

and $e q_{1} (x) = \frac{cos (x) sin (x) - x}{sin (x)^{3}}, e q_{4} (x) = \frac{2 cos (x) sin (x)^{3} + 3 (sin (x) cos (x) - x)}{sin (x)^{5}}, e q_{3} (x) = \frac{cos (x) - 1}{sin (x)^{2}},$ and $e q_{5} (x) = \frac{cos (x) sin (x)^{2} (1 - 2 cos (x)) - 2 cos (x)^{2} + 2 cos (x)}{sin (x)^{4}}$ being functions well defined on $] 0, π [$ and that admit controllable Taylor expansion formula around $0$ .

References

[1] M. Ben-Chen, O. Weber, and C. Gotsman, Variational harmonic maps for space deformation, ACM Transactions on Graphics (Proc. of ACM SIGGRAPH) (2009), 1–11.
[2] T. Ju, S. Schaefer, and J. Warren, Mean value coordinates for closed triangular meshes, ACM Transactions on Graphics (Proc. of ACM SIGGRAPH) 24 (2005), no. 3, 561–566.

Jacobians and Hessians of Mean Value Coordinates for Closed Triangular Meshes

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One-parameter tetrahedral mesh generation for spheroids

Dihedral Rigidity and Deformation

Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing

Plücker Correction Problem: Analysis and Improvements in Efficiency

I Background

I-a Mean Value Coordinates for Closed Triangular Meshes

Computing the weights $w_{i}$

Ii MVC Derivatives

Special case: $η$ lies on the surface of the cage

Ii-a Expression of the Jacobians

Special case: $η \in S u p p o r t (T), \notin T$

Ii-B Expression of the Hessians

Special case: $η \in S u p p o r t (T), \notin T$

References

Jacobians and Hessians of Mean Value Coordinates for Closed Triangular Meshes

Authors

3D Texture Coordinates on Polygon Mesh Sequences

Modelling Developable Ribbons Using Ruling Bending Coordinates

Fat Pad Cages for Facial Posing

One-parameter tetrahedral mesh generation for spheroids

Dihedral Rigidity and Deformation

Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing

Plücker Correction Problem: Analysis and Improvements in Efficiency

This week in AI

I Background

I-a Mean Value Coordinates for Closed Triangular Meshes

Computing the weights wi

Ii MVC Derivatives

Special case: η lies on the surface of the cage

Ii-a Expression of the Jacobians

Special case: η∈Support(T),∉T

Ii-B Expression of the Hessians

Special case: η∈Support(T),∉T

References

Plücker Correction Problem: Analysis and Improvements in Efficiency

Computing the weights $w_{i}$

Special case: $η$ lies on the surface of the cage

Special case: $η \in S u p p o r t (T), \notin T$

Special case: $η \in S u p p o r t (T), \notin T$