Cauchy wavelet explained

In mathematics, Cauchy wavelets are a family of continuous wavelets, used in the continuous wavelet transform.

Definition

The Cauchy wavelet of order

p

is defined as:

\psip(t)=

\Gamma(p+1)
2\pi

\left(

j
t+j

\right)p+1


where

p>0

and

j=\sqrt{-1}


therefore, its Fourier transform is defined as

\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)

and

\rho(\xi)=\rho(a\xi)

for

\xi\inR

almost everywhere and

\rho(\xi)0

for all

\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

is happened at

\xi=p

and the Fourier transform of the Cauchy wavelet is positive only in

\xi>0

, it means that:
(1) when

p

is low then the convolution of Cauchy wavelet is a low pass filter, and when

p

is high the convolution of Cauchy wavelet is a high pass filter.
Since the wavelet transform equals to the convolution to the mother wavelet and the convolution to the mother wavelet equals to the multiplication between the Fourier transform of the mother wavelet and the function by the convolution theorem.
And,
(2) the design of the Cauchy wavelet transform is considered with analysis of the analytic signal.

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)

is a real signal (

x(t)\inR

, for all

t\inR

)
And the bijection between analytic signal and real signal is that

x+(t)=x(t)+jxH(t)


x(t)=Re\{x+(t)\}


where

x+(t)

is the corresponded analytic signal of the real signal

x(t)

, and

xH(t)

is Hilbert transform of

x(t)

.

Unicity of the reconstruction

Phase retrieval problem

A phase retrieval problem consists in reconstructing an unknown complex function

f

from a set of phaseless linear measurements. More precisely, let

V

be a vector space, whose vectors are complex functions, on

C

and

\{Li\}i

a set of linear forms from

V

to

C

. We are given the set of all

\{|Li(f)|\}i

, for some unknown

f\inV

and we want to determine

f

.
This problem can be studied under three different viewpoints:[3]
(1) Is

f

uniquely determined by

\{|Li(f)|\}i

(up to a global phase)?
(2) If the answer to the previous question is positive, is the inverse application

\{|Li(f)|\}i\impliesf

is “stable”? For example, is it continuous? Uniformly Lipschitz?
(3) In practice, is there an efficient algorithm which recovers

f

from

\{|Li(f)|\}i

?

The most well-known example of a phase retrieval problem is the case where the

Li

represent the Fourier coefficients:
for example:

Ln(f)=

1
2\pi
\pi
\int
-\pi

f(t)e-jntdt

, for

n\inZ

,

where

f

is complex-valued function on

[-\pi,\pi]


Then,

f

can be reconstruct by

Ln(f)

as

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

i.e. the norm defined in

L2([-\pi,\pi])

.
Hence, in this example, the index set

I

is the integer

Z

, the vector space

V

is

L2([-\pi,\pi])

and the linear form

Ln

is the Fourier coefficient. Furthermore, the absolute value of Fourier coefficients

\{|Ln(f)|\}n

can only determine the norm of

f

defined in

L2([-\pi,\pi])

.

Unicity Theorem of the reconstruction

Firstly, we define the Cauchy wavelet transform as:

W
\psip

[x(t)](a,b)=

1
b
infty
\int
-infty

x(t)

\overline{\psi
p(t-a
b

)}dt

.

Then, the theorem is as followed

Theorem.[4] For a fixed

p>0

, if exist two different numbers

b1,b2>0

and the Cauchy wavelet transform defined as above. Then, if there are two real-valued functions

f,g\inL2(R)

satisfied
|W
\psip

[f(t)](a,b1)|=

|W
\psip

[g(t)](a,b1)|

,

\foralla\inR

and
|W
\psip

[f(t)](a,b2)|=

|W
\psip

[g(t)](a,b2)|

,

\foralla\inR

,

then there is a

\alpha\inR

such that

f+(t)=ej\alphag+(t)

.

f+(t)=ej\alphag+(t)

implies that

Re\{f+(t)\}=Re\{ej\alphag+(t)\}\impliesf(t)=\cos{\alpha}g(t)-\sin{\alpha}gH(t)

and

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)\}

.

Back to the phase retrieval problem, in the Cauchy wavelet transform case, the index set

I

is

R x \{b1,b2\}

with

b1b2

and

b1,b2>0

, the vector space

V

is

L2(R)

and the linear form

L(a,

is defined as

L(a,(f)=

W
\psip

[f(t)](a,b)

. Hence,

\{|L(a,

(f)|\}
a,b\inR x \{b1,b2\
} determines the two dimensional subspace

span\{f,fH\}

in

L2(R)

.

Notes and References

  1. Phase retrieval for the Cauchy wavelet transform. Journal of Fourier Analysis and Applications. 1404.1183. Mallat. Stéphane. 21. 1251–1309. Waldspurger. Irène. 6. 2015 . 10.1007/s00041-015-9403-4 . free. 2015JFAA...21.1251M .
  2. Mechanical Systems and Signal Processing. Argoul. Pierre. 17. 243–250. Le. Thien-phu. Instantaneous Indicators of Structural Behaviour Based on the Continuous Cauchy Wavelet Analysis . 1. 2003. 10.1006/mssp.2002.1557 . 2003MSSP...17..243A . free.
  3. Phase retrieval for the Cauchy wavelet transform. Journal of Fourier Analysis and Applications. 1404.1183. Mallat. Stéphane. 21. 1251–1309. Waldspurger. Irène. 6. 2015 . 10.1007/s00041-015-9403-4 . free. 2015JFAA...21.1251M .
  4. Phase retrieval for the Cauchy wavelet transform. Journal of Fourier Analysis and Applications. 1404.1183. Mallat. Stéphane. 21. 1251–1309. Waldspurger. Irène. 6. 2015 . 10.1007/s00041-015-9403-4 . free. 2015JFAA...21.1251M .