15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Cauchy wavelet | 2/2 | https://en.wikipedia.org/wiki/Cauchy_wavelet | reference | science, encyclopedia | 2026-05-05T07:23:27.637906+00:00 | kb-cron |
f
(
t
)
=
∑
n
=
−
∞
∞
L
n
(
f
)
e
j
n
t
{\displaystyle f(t)=\sum _{n=-\infty }^{\infty }L_{n}(f)e^{jnt}}
. and in fact we have Parseval's identity
|
|
f
|
|
2
=
∑
n
=
−
∞
∞
|
L
n
(
f
)
|
2
{\displaystyle ||f||^{2}=\sum _{n=-\infty }^{\infty }|L_{n}(f)|^{2}}
. where
|
|
f
|
|
2
=
1
2
π
∫
−
π
π
|
f
(
t
)
|
2
d
t
{\displaystyle ||f||^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|f(t)|^{2}\,dt}
i.e. the norm defined in
L
2
(
[
−
π
,
π
]
)
{\displaystyle L^{2}([-\pi ,\pi ])}
. Hence, in this example, the index set
I
{\displaystyle I}
is the integer
Z
{\displaystyle \mathbb {Z} }
, the vector space
V
{\displaystyle V}
is
L
2
(
[
−
π
,
π
]
)
{\displaystyle L^{2}([-\pi ,\pi ])}
and the linear form
L
n
{\displaystyle L_{n}}
is the Fourier coefficient. Furthermore, the absolute value of Fourier coefficients
{
|
L
n
(
f
)
|
}
n
∈
Z
{\displaystyle \{|L_{n}(f)|\}_{n\in \mathbb {Z} }}
can only determine the norm of
f
{\displaystyle f}
defined in
L
2
(
[
−
π
,
π
]
)
{\displaystyle L^{2}([-\pi ,\pi ])}
.
=== Unicity Theorem of the reconstruction === Firstly, we define the Cauchy wavelet transform as:
W
ψ
p
[
x
(
t
)
]
(
a
,
b
)
=
1
b
∫
−
∞
∞
x
(
t
)
ψ
p
(
t
−
a
b
)
¯
d
t
{\displaystyle W_{\psi _{p}}[x(t)](a,b)={\frac {1}{b}}\int _{-\infty }^{\infty }x(t){\overline {\psi _{p}({\frac {t-a}{b}})}}\,dt}
. Then, the theorem is:
Theorem. For a fixed
p
>
0
{\displaystyle p>0}
, if exist two different numbers
b
1
,
b
2
>
0
{\displaystyle b_{1},b_{2}>0}
and the Cauchy wavelet transform defined as above. Then, if there are two real-valued functions
f
,
g
∈
L
2
(
R
)
{\displaystyle f,g\in L^{2}(\mathbb {R} )}
satisfied
|
W
ψ
p
[
f
(
t
)
]
(
a
,
b
1
)
|
=
|
W
ψ
p
[
g
(
t
)
]
(
a
,
b
1
)
|
{\displaystyle |W_{\psi _{p}}[f(t)](a,b_{1})|=|W_{\psi _{p}}[g(t)](a,b_{1})|}
,
∀
a
∈
R
{\displaystyle \forall a\in \mathbb {R} }
and
|
W
ψ
p
[
f
(
t
)
]
(
a
,
b
2
)
|
=
|
W
ψ
p
[
g
(
t
)
]
(
a
,
b
2
)
|
{\displaystyle |W_{\psi _{p}}[f(t)](a,b_{2})|=|W_{\psi _{p}}[g(t)](a,b_{2})|}
,
∀
a
∈
R
{\displaystyle \forall a\in \mathbb {R} }
, then there is a
α
∈
R
{\displaystyle \alpha \in \mathbb {R} }
such that
f
+
(
t
)
=
e
j
α
g
+
(
t
)
{\displaystyle f_{+}(t)=e^{j\alpha }g_{+}(t)}
.
f
+
(
t
)
=
e
j
α
g
+
(
t
)
{\displaystyle f_{+}(t)=e^{j\alpha }g_{+}(t)}
implies that
R
e
{
f
+
(
t
)
}
=
R
e
{
e
j
α
g
+
(
t
)
}
⟹
f
(
t
)
=
cos
α
g
(
t
)
−
sin
α
g
H
(
t
)
{\displaystyle Re\{f_{+}(t)\}=Re\{e^{j\alpha }g_{+}(t)\}\implies f(t)=\cos {\alpha }g(t)-\sin {\alpha }g_{H}(t)}
and
I
m
{
f
+
(
t
)
}
=
I
m
{
e
j
α
g
+
(
t
)
}
⟹
f
H
(
t
)
=
sin
α
g
(
t
)
+
cos
α
g
H
(
t
)
{\displaystyle Im\{f_{+}(t)\}=Im\{e^{j\alpha }g_{+}(t)\}\implies f_{H}(t)=\sin {\alpha }g(t)+\cos {\alpha }g_{H}(t)}
. Hence, we get the relation
f
(
t
)
=
(
cos
α
−
sin
α
tan
α
)
g
(
t
)
−
tan
α
f
H
(
t
)
{\displaystyle f(t)=(\cos {\alpha }-\sin {\alpha }\tan {\alpha })g(t)-\tan {\alpha }f_{H}(t)}
and
f
(
t
)
,
g
H
(
t
)
∈
s
p
a
n
{
f
H
(
t
)
,
g
(
t
)
}
=
s
p
a
n
{
f
(
t
)
,
f
H
(
t
)
}
=
s
p
a
n
{
g
(
t
)
,
g
H
(
t
)
}
{\displaystyle f(t),g_{H}(t)\in span\{f_{H}(t),g(t)\}=span\{f(t),f_{H}(t)\}=span\{g(t),g_{H}(t)\}}
.
Back to the phase retrieval problem, in the Cauchy wavelet transform case, the index set
I
{\displaystyle I}
is
R
×
{
b
1
,
b
2
}
{\displaystyle \mathbb {R} \times \{b_{1},b_{2}\}}
with
b
1
≠
b
2
{\displaystyle b_{1}\neq b_{2}}
and
b
1
,
b
2
>
0
{\displaystyle b_{1},b_{2}>0}
, the vector space
V
{\displaystyle V}
is
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
and the linear form
L
(
a
,
b
)
{\displaystyle L_{(a,b)}}
is defined as
L
(
a
,
b
)
(
f
)
=
W
ψ
p
[
f
(
t
)
]
(
a
,
b
)
{\displaystyle L_{(a,b)}(f)=W_{\psi _{p}}[f(t)](a,b)}
. Hence,
{
|
L
(
a
,
b
)
(
f
)
|
}
a
,
b
∈
R
×
{
b
1
,
b
2
}
{\displaystyle \{|L_{(a,b)}(f)|\}_{a,b\in \mathbb {R} \times \{b_{1},b_{2}\}}}
determines the two dimensional subspace
s
p
a
n
{
f
,
f
H
}
{\displaystyle span\{f,f_{H}\}}
in
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
.
== References ==