In mathematics, Cauchy wavelets are a family of continuous wavelets, used in the continuous wavelet transform.
The Cauchy wavelet of order
therefore, its Fourier transform is defined as
Sometimes it is defined as a function with its Fourier transform[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
Moreover, we can see that the maximum of the Fourier transform of the Cauchy wavelet of order
and the Fourier transform of the Cauchy wavelet is positive only in
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.
) And the bijection between analytic signal and real signal is that
is the corresponded analytic signal of the real signal
A phase retrieval problem consists in reconstructing an unknown complex function
from a set of phaseless linear measurements.
be a vector space, whose vectors are complex functions, on
a set of linear forms from
This problem can be studied under three different viewpoints:[1]
(2) If the answer to the previous question is positive, is the inverse application
(3) In practice, is there an efficient algorithm which recovers
The most well-known example of a phase retrieval problem is the case where the
represent the Fourier coefficients: for example:
and in fact we have Parseval's identity
Furthermore, the absolute value of Fourier coefficients
Firstly, we define the Cauchy wavelet transform as:
and the Cauchy wavelet transform defined as above.
Back to the phase retrieval problem, in the Cauchy wavelet transform case, the index set
determines the two dimensional subspace