Image Processing Reference

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where
λ
j

=−

v
j
/d
N
c

for 1

≤

j

≤

3. For all 1

≤

i<N
c
, we have

a
i

d
N
c

b
i

d
N
c

c
i

d
N
c

a
i
A
λ
1
+

b
i
A
λ
2
+

c
i
A
λ
3

=−

Av
1
−

Av
2
−

Av
3

⎡

⎤

u
i

v
i

1

d
i

d
N
c

⎣

⎦

=−

.

This implies that the last column of the matrix
M
f
of Eq.
3.13
is indeed a linear combination of the

previous ones with coefficients
(λ
1
,λ
2
,λ
3
,
−
d
1
/d
N
c
, ...,
−
d
N
c
−
1
/d
N
c
)
. In the general case, none

of these coefficients is zero. Furthermore, because
A
has full rank and the barycentric coordinates

are independent in general, the first 9 columns of
M
f
are linearly independent. Thus, given the

particular structure of the right half of
M
f
, trying to write any column as a linear combination of all

the others except the last one would yield wrong values on the last three rows, which could only be

corrected by using the last column. This implies that, in general,
M
f
has full rank minus 1.

3.3.2.3 Reconstructing theWhole Mesh

If we now consider a mesh made of
N
v
>
3 vertices with a total of
N
c
correspondences well-spread

over the whole mesh, Eq.
3.13
becomes

⎡

⎣

⎤

⎦

v
1

...

v
N
v

d
1

...

d
N
c

M
m

=

0
,

(3.15)

with

⎡

⎣

⎤

⎦

⎡

⎣

⎤

⎦

u
1

v
1

1

a
1
A
b
1
A
c
1
A

0

...

...

−

0

...

...

...

...

...

...

...

...

...

...

...

...

...

...

⎡

⎣

⎤

⎦

u
j

v
j

1

0

b
j
A
c
j
A
d
j
A

0

...

0

−

0

...

...

M
m
=

.

...

...

...

...

...

...

...

...

...

...

...

⎡

⎣

⎤

⎦

u
l

v
l

1

a
l
A

0

c
l
A

0

e
l
A
...

0

...

−

0

...

...

...

...

...

...

...

...

...

...

...

...

Coefficients similar to those of Eq.
3.14
can be derived to compute
u
N
c
,v
N
c
,
1
T
as a linear

combination of the non-zero columns of the last row. Following a similar reasoning as in the weak

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