Continuous wavelets
In mathematics, Cauchy wavelets are a family of continuous wavelets, used in the continuous wavelet transform.
The Cauchy wavelet of order
is defined as:
![{\displaystyle \psi _{p}(t)={\frac {\Gamma (p+1)}{2\pi }}\left({\frac {j}{t+j}}\right)^{p+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a73e2e67b9bdc25ce35ca45be633f846027f11)
where
and ![{\displaystyle j={\sqrt {-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fec2c043c1bedcc4aba77e29a310a0adcb96a929)
therefore, its Fourier transform is defined as
.
Sometimes it is defined as a function with its Fourier transform[1]
![{\displaystyle {\hat {\psi _{p}}}(\xi )=\rho (\xi )\xi ^{p}e^{-\xi }I_{[\xi \geq 0]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fa60f544e42d8f6a75bef025f182f166edc1be)
where
and
for
almost everywhere and
for all
.
Also, it had used to be defined as[2]
![{\displaystyle \psi _{p}(t)=({\frac {j}{t+j}})^{p+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2259a08776649f7f922cddbb7c88bc44697d3a)
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
Moreover, we can see that the maximum of the Fourier transform of the Cauchy wavelet of order
is happened at
and the Fourier transform of the Cauchy wavelet is positive only in
, it means that:
(1) when
is low then the convolution of Cauchy wavelet is a low pass filter, and when
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.
![{\displaystyle {\overline {FT\{x\}(-\xi )}}=FT\{x\}(\xi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d6390e661d2a34616a18216fd27714788f4b46)
where
is a real signal (
, for all
)
And the bijection between analytic signal and real signal is that
![{\displaystyle x_{+}(t)=x(t)+jx_{H}(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a357f17bb3ee76ec124ebc354355bc01dca70465)
![{\displaystyle x(t)=Re\{x_{+}(t)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23fc5bf6c27dd7a82023d3c7f7c44fb3098870fe)
where
is the corresponded analytic signal of the real signal
, and
is Hilbert transform of
.
Unicity of the reconstruction
[edit]
Phase retrieval problem
[edit]
A phase retrieval problem consists in reconstructing an unknown complex function
from a set of phaseless linear measurements. More precisely, let
be a vector space, whose vectors are complex functions, on
and
a set of linear forms from
to
. We are given the set of all
, for some unknown
and we want to determine
.
This problem can be studied under three different viewpoints:[1]
(1) Is
uniquely determined by
(up to a global phase)?
(2) If the answer to the previous question is positive, is the inverse application
is “stable”? For example, is it continuous? Uniformly Lipschitz?
(3) In practice, is there an efficient algorithm which recovers
from
?
The most well-known example of a phase retrieval problem is the case where the
represent the Fourier coefficients:
for example:
, for
,
where
is complex-valued function on
Then,
can be reconstruct by
as
.
and in fact we have Parseval's identity
.
where
i.e. the norm defined in
.
Hence, in this example, the index set
is the integer
, the vector space
is
and the linear form
is the Fourier coefficient. Furthermore, the absolute value of Fourier coefficients
can only determine the norm of
defined in
.
Unicity Theorem of the reconstruction
[edit]
Firstly, we define the Cauchy wavelet transform as:
.
Then, the theorem is as followed
Theorem.[1] For a fixed
, if exist two different numbers
and the Cauchy wavelet transform defined as above. Then, if there are two real-valued functions
satisfied
,
and
,
,
then there is a
such that
.
implies that
and
.
Hence, we get the relation
and
.
Back to the phase retrieval problem, in the Cauchy wavelet transform case, the index set
is
with
and
, the vector space
is
and the linear form
is defined as
. Hence,
determines the two dimensional subspace
in
.