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APPENDIX
We evaluate the gradient of (4) on M
s
× M
r
∇
ˆ
f
h
s,r
,K
(x) =
n
∑
i=1
k
0
d
2
s
(x
s
,x
i
s
)
h
2
s
k
d
2
r
(x
r
,x
i
r
)
h
2
r
2
h
2
s
log
x
i
s
(x
s
),0
r
+
n
∑
i=1
k
d
2
s
(x
s
,x
i
s
)
h
2
s
k
0
d
2
r
(x
r
,x
i
r
)
h
2
r
2
h
2
r
0
s
,log
x
i
r
(x
r
)
.
(10)
Through projection onto T
x
s
M
s
we obtain
dπ
s
(x)
∇
ˆ
f
h
s,r
,K
(x)
=
n
∑
i=1
k
0
d
2
s
(x
s
,x
i
s
)
h
2
s
k
d
2
r
(x
r
,x
i
r
)
h
2
r
2
h
2
s
log
x
i
s
(x
s
). (11)
Riemannian Filters for Multi-variate Mesh Signals
235