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The transfer matrix appears in the treatment of refinable functions .
For the mask
h
{\displaystyle h}
a
{\displaystyle a}
b
{\displaystyle b}
h
{\displaystyle h}
T
h
{\displaystyle T_{h}}
(
T
h
)
j
,
k
=
h
2
⋅
j
−
k
{\displaystyle (T_{h})_{j,k}=h_{2\cdot j-k}}
More verbosely
T
h
=
(
h
a
h
a
+
2
h
a
+
1
h
a
h
a
+
4
h
a
+
3
h
a
+
2
h
a
+
1
h
a
⋱
⋱
⋱
⋱
⋱
⋱
h
b
h
b
−
1
h
b
−
2
h
b
−
3
h
b
−
4
h
b
h
b
−
1
h
b
−
2
h
b
)
{\displaystyle T_{h}={\begin{pmatrix}h_{a}&&&&&\\h_{a+2}&h_{a+1}&h_{a}&&&\\h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\&h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\&&&h_{b}&h_{b-1}&h_{b-2}\\&&&&&h_{b}\end{pmatrix}}}
Properties
If you drop the first and the last column and move the odd indexed columns to the left and the even indexed columns to the right, then you obtain a transposed Sylvester matrix .
The determinant of a transfer matrix is essentially a resultant. More precisely:
Let
h
e
{\displaystyle h_{\mathrm {e} }}
h
{\displaystyle h}
(
h
e
)
k
=
h
2
⋅
k
{\displaystyle (h_{\mathrm {e} })_{k}=h_{2\cdot k}}
h
o
{\displaystyle h_{\mathrm {o} }}
h
{\displaystyle h}
(
h
o
)
k
=
h
2
⋅
k
+
1
{\displaystyle (h_{\mathrm {o} })_{k}=h_{2\cdot k+1}}
Then
det
T
h
=
(
−
1
)
⌊
b
−
a
+
1
4
⌋
⋅
h
a
⋅
h
b
⋅
r
e
s
(
h
e
,
h
o
)
{\displaystyle \det T_{h}=(-1)^{\lfloor {\frac {b-a+1}{4}}\rfloor }\cdot h_{a}\cdot h_{b}\cdot \mathrm {res} (h_{\mathrm {e} },h_{\mathrm {o} })}
r
e
s
{\displaystyle \mathrm {res} }
resultant .
This connection allows for fast computation using the Euclidean algorithm .
t
r
T
g
∗
h
=
t
r
T
g
⋅
t
r
T
h
{\displaystyle \mathrm {tr} ~T_{g*h}=\mathrm {tr} ~T_{g}\cdot \mathrm {tr} ~T_{h}}
For the determinant of the transfer matrix of convolved mask holds
det
T
g
∗
h
=
det
T
g
⋅
det
T
h
⋅
r
e
s
(
g
−
,
h
)
{\displaystyle \det T_{g*h}=\det T_{g}\cdot \det T_{h}\cdot \mathrm {res} (g_{-},h)}
where
g
−
{\displaystyle g_{-}}
(
g
−
)
k
=
(
−
1
)
k
⋅
g
k
{\displaystyle (g_{-})_{k}=(-1)^{k}\cdot g_{k}}
References
