Convolutional neural networks on tetrahedral meshes
View Patent ↗Disclosed herein are methods for graphing convolutional neural networks in tetrahedral meshes. In some embodiments, the methods include computing a volumetric Laplace Beltrami Operator. In some embodiments, the methods include feeding the LBO into a neural network. In some embodiments, the methods include down-sampling a tetrahedral mesh.
1 . A method of using a tetrahedral mesh on a neural network, comprising:
computing a volumetric Laplace Beltrami operator (LBO) for the tetrahedral mesh;
feeding into the neural network the LBO and a set of signals for each vertex of the neural network; and
down-sampling the tetrahedral mesh.
2 . The method of claim 1 , wherein computing the volumetric LBO is performed based on the
following function:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N(i) includes the adjacent vertices of vertex v i , d i is total tetrahedral volume of adjacent tetrahedra to vertex v i , k i,j is the string constant, ƒ(v i ) and ƒ(v j ) are the scalar function values at vertices v i and v j respectively, and Δƒ(v i ) is the Laplacian operator applied to ƒ at vertex v i .
3 . The method of claim 1 , wherein the set of signals comprises at least one input signal and at least one output signal.
4 . The method of claim 3 , wherein the at least one input signal comprises the three-dimensional coordinates of each vertex.
5 . The method of claim 4 , wherein the at least one output signal comprises the output of the convolved signal with filter g.
6 . The method of claim 5 , wherein a convolution is defined as:
𝔵
o
u
t
=
g
*
T𝔵
i
n
=
Φ
(
(
Φ
T
g
)
⊙
(
Φ
τ
𝔵
in
)
)
=
Φ
f
(
Λ
)
Φ
τ
𝔵
in
,
in which ⊙ is the element-wise product, ƒ(Λ) is a general function based on the eigen-value matrix Λ, Φ is the eigen-vector matrix, x in and x out are the input and output graph signals respectively, and Φ T G and Φ T x in are the spectral representations of the filter and input signal obtained via the graph Fourier transform.
7 . The method of claim 5 , wherein a convolution is defined as:
𝔵
o
u
t
=
∑
m
=
0
K
Φ
m
T
m
(
L
t
e
t
)
𝔵
i
n
where θ m are a set of learnable model parameters denoting the coefficients of the polynomials, and Tm∈R n×n is the Chebyshev polynomial of order k, where x in and x out are the input and output graph signals respectively, Φ m denotes the m-th set of learnable parameters, and L Tet is the Laplacian matrix of the tetrahedral mesh.
8 . The method of claim 1 , wherein a lumped discrete LBO on Tis defined as:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N (i) includes the adjacent vertices of vertex v i , and d i is the total tetrahedral volume of adjacent tetrahedral to vertex v i , and k i,j is the string constant, ƒ(v i ) and ƒ(v j ) are the scalar function values at vertices v i and v j respectively, and Δƒ(v i ) is the Laplacian operator applied to ƒ at vertex v i .
9 . The method of claim 8 , wherein the tetrahedral mesh is decimated by an order of two using the function:
d
(
v
i
,
v
j
)
=
-
A
i
,
j
(
1
D
ii
+
1
D
j
j
)
where d(v i ,v j ) is the LBO-based edge weight between vertices v i and v j used in the Graclus pooling objective, A ij is the cotangent weight between vertices v i and v j derived from the stiffness matrix of the tetrahedral mesh, and D ii and D jj are the diagonal entries of the lumped mass matrix D corresponding to the total tetrahedral volume associated with vertices v i and v j respectively.
10 . A method, comprising:
computing a volumetric Laplace Beltrami operator (LBO) for a tetrahedral mesh;
feeding the volumetric LBO into a neural network; and
feeding a set of signals for each vertex of the neural network, wherein computing the volumetric LBO is performed based on the following function:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N(i) includes the adjacent vertices of vertex v i , and d i is total tetrahedral volume of adjacent tetrahedra to vertex v i , and k i,j is the string constant, ƒ(v i ) and ƒ(v j ) are the scalar function values at vertices v i and v j respectively, and Δƒ(v i ) is the Laplacian operator applied to ƒ at vertex v i .
11 . The method of claim 10 , wherein the set of signals comprises at least one input signal and at least one output signal.
12 . The method of claim 11 , wherein the at least one input signal comprises the three-dimensional coordinates of each vertex.