In mathematics, Cauchy wavelets are a family of continuous wavelets, used in the continuous wavelet transform.
The Cauchy wavelet of order
p
\psip(t)=
\Gamma(p+1) | |
2\pi |
\left(
j | |
t+j |
\right)p+1
where
p>0
j=\sqrt{-1}
\hat{\psip}(\xi)=\xipe-\xiI[\xi
Sometimes it is defined as a function with its Fourier transform[1]
\hat{\psip}(\xi)=\rho(\xi)\xipe-\xiI[\xi
where
\rho(\xi)\inLinfty(R)
\rho(\xi)=\rho(a\xi)
\xi\inR
\rho(\xi) ≠ 0
\xi\inR
Also, it had used to be defined as[2]
\psip(t)=(
j | |
t+j |
)p+1
in previous research of Cauchy wavelet. If we defined Cauchy wavelet in this way, we can observe that the Fourier transform of the Cauchy wavelet
infty | |
\int | |
-infty |
\hat{\psip}(\xi)d\xi=
infty | |
\int | |
0 |
2\pi | |
\Gamma(p+1) |
\xipe-\xid\xi=2\pi
Moreover, we can see that the maximum of the Fourier transform of the Cauchy wavelet of order
p
\xi=p
\xi>0
p
p
Since the analytic signal is bijective to the real signal and there is only positive frequency in the analytic signal (the real signal has conjugated frequency between positive and negative) i.e.
\overline{FT\{x\}(-\xi)}=FT\{x\}(\xi)
where
x(t)
x(t)\inR
t\inR
x+(t)=x(t)+jxH(t)
x(t)=Re\{x+(t)\}
where
x+(t)
x(t)
xH(t)
x(t)
A phase retrieval problem consists in reconstructing an unknown complex function
f
V
C
\{Li\}i
V
C
\{|Li(f)|\}i
f\inV
f
f
\{|Li(f)|\}i
\{|Li(f)|\}i\impliesf
f
\{|Li(f)|\}i
The most well-known example of a phase retrieval problem is the case where the
Li
Ln(f)=
1 | |
2\pi |
\pi | |
\int | |
-\pi |
f(t)e-jntdt
n\inZ
where
f
[-\pi,\pi]
f
Ln(f)
f(t)=
infty | |
\sum | |
n=-infty |
jnt | |
L | |
n(f)e |
and in fact we have Parseval's identity
||f||2=
infty | |
\sum | |
n=-infty |
2 | |
|L | |
n(f)| |
where
||f||2=
1 | |
2\pi |
\pi | |
\int | |
-\pi |
|f(t)|2dt
L2([-\pi,\pi])
I
Z
V
L2([-\pi,\pi])
Ln
\{|Ln(f)|\}n
f
L2([-\pi,\pi])
Firstly, we define the Cauchy wavelet transform as:
W | |
\psip |
[x(t)](a,b)=
1 | |
b |
infty | |
\int | |
-infty |
x(t)
\overline{\psi | ||||
|
)}dt
Then, the theorem is as followed
Theorem.[4] For a fixed
p>0
b1,b2>0
f,g\inL2(R)
|W | |
\psip |
[f(t)](a,b1)|=
|W | |
\psip |
[g(t)](a,b1)|
\foralla\inR
|W | |
\psip |
[f(t)](a,b2)|=
|W | |
\psip |
[g(t)](a,b2)|
\foralla\inR
then there is a
\alpha\inR
f+(t)=ej\alphag+(t)
f+(t)=ej\alphag+(t)
Re\{f+(t)\}=Re\{ej\alphag+(t)\}\impliesf(t)=\cos{\alpha}g(t)-\sin{\alpha}gH(t)
Im\{f+(t)\}=Im\{ej\alphag+(t)\}\impliesfH(t)=\sin{\alpha}g(t)+\cos{\alpha}gH(t)
Hence, we get the relation
f(t)=(\cos{\alpha}-\sin{\alpha}\tan{\alpha})g(t)-\tan{\alpha}fH(t)
and
f(t),gH(t)\inspan\{fH(t),g(t)\}=span\{f(t),fH(t)\}=span\{g(t),gH(t)\}
I
R x \{b1,b2\}
b1 ≠ b2
b1,b2>0
V
L2(R)
L(a,
L(a,(f)=
W | |
\psip |
[f(t)](a,b)
\{|L(a,
(f)|\} | |
a,b\inR x \{b1,b2\ |
span\{f,fH\}
L2(R)