Cauchy wavelet

The Cauchy wavelet of order $p$ is defined as:

$\backslash psi\_p(t)\; =\; \backslash frac\{\backslash Gamma(p+1)\}\{2\backslash pi\}\backslash left\; (\; \backslash frac\{j\}\{t\; +\; j\}\; \backslash right\; )\; ^\{p+1\}$

where $p\; >\; 0$ and $j\; =\; \backslash sqrt\{-1\}$

therefore, its Fourier transform is defined as

$\backslash hat\{\backslash psi\_p\}(\backslash xi)\; =\; \backslash xi^\{p\}e^\{-\backslash xi\}I\_\{[\backslash xi\; \backslash geq\; 0]\}$.

Sometimes it is defined as a function with its Fourier transform{{Cite journal|title=Phase retrieval for the Cauchy wavelet transform|journal=Journal of Fourier Analysis and Applications|arxiv=1404.1183|last1=Mallat|first1=Stéphane|volume=21|pages=1251–1309|last2=Waldspurger|first2=Irène|issue=6|year=2015 |doi=10.1007/s00041-015-9403-4 |doi-access=free|bibcode=2015JFAA...21.1251M }}

$\backslash hat\{\backslash psi\_p\}(\backslash xi)\; =\; \backslash rho(\backslash xi)\backslash xi^\{p\}e^\{-\backslash xi\}I\_\{[\backslash xi\; \backslash geq\; 0]\}$

where $\backslash rho(\backslash xi)\; \backslash in\; L^\{\backslash infty\}(\backslash mathbb\{R\})$ and $\backslash rho(\backslash xi)\; =\; \backslash rho(a\backslash xi)$ for $\backslash xi\; \backslash in\; \backslash mathbb\{R\}$ almost everywhere and $\backslash rho(\backslash xi)\; \backslash neq\; 0$ for all $\backslash xi\; \backslash in\; \backslash mathbb\{R\}$.

Also, it had used to be defined as{{Cite journal|journal=Mechanical Systems and Signal Processing|last1=Argoul|first1=Pierre|volume=17|pages=243–250|last2=Le|first2=Thien-phu|title=Instantaneous Indicators of Structural Behaviour Based on the Continuous Cauchy Wavelet Analysis |issue=1|year=2003|doi=10.1006/mssp.2002.1557 |bibcode=2003MSSP...17..243A |doi-access=free}}

$\backslash psi\_p(t)\; =\; (\backslash frac\{j\}\{t\; +\; j\})^\{p+1\}$

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

$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash hat\{\backslash psi\_p\}(\backslash xi)\; \backslash ,d\backslash xi\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{2\backslash pi\}\{\backslash Gamma(p+1)\}\; \backslash xi^\{p\}e^\{-\backslash xi\}\; \backslash ,d\backslash xi\; =\; 2\backslash pi$

Moreover, we can see that the maximum of the Fourier transform of the Cauchy wavelet of order $p$ is happened at $\backslash xi\; =\; p$ and the Fourier transform of the Cauchy wavelet is positive only in $\backslash xi\; >\; 0$, 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.

$\backslash overline\{FT\backslash \{x\backslash \}(-\backslash xi)\}\; =\; FT\backslash \{x\backslash \}(\backslash xi)$

where $x(t)$ is a real signal ($x(t)\; \backslash in\; \backslash mathbb\{R\}$, for all $t\; \backslash in\; \backslash mathbb\{R\}$)

And the bijection between analytic signal and real signal is that

$x\_\{+\}(t)\; =\; x(t)\; +\; jx\_H(t)$

$x(t)\; =\; Re\backslash \{x\_\{+\}(t)\backslash \}$

where $x\_\{+\}(t)$ is the corresponded analytic signal of the real signal $x(t)$, and $x\_H(t)$ is Hilbert transform of $x(t)$.

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 $\backslash mathbb\{C\}$ and $\backslash \{L\_i\backslash \}\_\{i\; \backslash in\; I\}$ a set of linear forms from $V$ to $\backslash mathbb\{C\}$. We are given the set of all $\backslash \{|L\_i(f)|\backslash \}\_\{i\; \backslash in\; I\}$, for some unknown $f\; \backslash in\; V$ and we want to determine $f$.

This problem can be studied under three different viewpoints:{{Cite journal|title=Phase retrieval for the Cauchy wavelet transform|journal=Journal of Fourier Analysis and Applications|arxiv=1404.1183|last1=Mallat|first1=Stéphane|volume=21|pages=1251–1309|last2=Waldspurger|first2=Irène|issue=6|year=2015 |doi=10.1007/s00041-015-9403-4 |doi-access=free|bibcode=2015JFAA...21.1251M }}

(1) Is $f$ uniquely determined by $\backslash \{|L\_i(f)|\backslash \}\_\{i\; \backslash in\; I\}$ (up to a global phase)?

(2) If the answer to the previous question is positive, is the inverse application $\backslash \{|L\_i(f)|\backslash \}\_\{i\; \backslash in\; I\}\; \backslash implies\; f$ is “stable”? For example, is it continuous? Uniformly Lipschitz?

(3) In practice, is there an efficient algorithm which recovers $f$ from $\backslash \{|L\_i(f)|\backslash \}\_\{i\; \backslash in\; I\}$?

The most well-known example of a phase retrieval problem is the case where the $L\_i$ represent the Fourier coefficients:

for example:

$L\_n(f)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash pi\}^\{\backslash pi\}\; f(t)e^\{-jnt\}\; \backslash ,dt$, for $n\; \backslash in\; \backslash mathbb\{Z\}$,

where $f$ is complex-valued function on $[-\backslash pi,\; \backslash pi]$

Then, $f$ can be reconstruct by $L\_n(f)$ as

$f(t)\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; L\_n(f)e^\{jnt\}$.

and in fact we have Parseval's identity

$||f||^2\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; |L\_n(f)|^2$.

where $||f||^2\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash pi\}^\{\backslash pi\}\; |f(t)|^2\; \backslash ,dt$ i.e. the norm defined in $L^2([-\backslash pi,\; \backslash pi])$.

Hence, in this example, the index set $I$ is the integer $\backslash mathbb\{Z\}$, the vector space $V$ is $L^2([-\backslash pi,\; \backslash pi])$ and the linear form $L\_n$ is the Fourier coefficient. Furthermore, the absolute value of Fourier coefficients $\backslash \{|L\_n(f)|\backslash \}\_\{n\; \backslash in\; \backslash mathbb\{Z\}\}$ can only determine the norm of $f$ defined in $L^2([-\backslash pi,\; \backslash pi])$.

Firstly, we define the Cauchy wavelet transform as:

$W\_\{\backslash psi\_p\}[x(t)](a,\; b)\; =\; \backslash frac\{1\}\{b\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x(t)\; \backslash overline\{\backslash psi\_p(\backslash frac\{t-a\}\{b\})\}\; \backslash ,dt$.

Then, the theorem is:

Theorem.{{Cite journal|title=Phase retrieval for the Cauchy wavelet transform|journal=Journal of Fourier Analysis and Applications|arxiv=1404.1183|last1=Mallat|first1=Stéphane|volume=21|pages=1251–1309|last2=Waldspurger|first2=Irène|issue=6|year=2015 |doi=10.1007/s00041-015-9403-4 |doi-access=free|bibcode=2015JFAA...21.1251M }} For a fixed $p\; >\; 0$, if exist two different numbers $b\_1,\; b\_2\; >\; 0$ and the Cauchy wavelet transform defined as above. Then, if there are two real-valued functions $f,\; g\; \backslash in\; L^2(\backslash mathbb\{R\})$ satisfied

$|W\_\{\backslash psi\_p\}[f(t)](a,\; b\_1)|\; =\; |W\_\{\backslash psi\_p\}[g(t)](a,\; b\_1)|$, $\backslash forall\; a\; \backslash in\; \backslash mathbb\{R\}$ and

$|W\_\{\backslash psi\_p\}[f(t)](a,\; b\_2)|\; =\; |W\_\{\backslash psi\_p\}[g(t)](a,\; b\_2)|$, $\backslash forall\; a\; \backslash in\; \backslash mathbb\{R\}$,

then there is a $\backslash alpha\; \backslash in\; \backslash mathbb\{R\}$ such that $f\_\{+\}(t)\; =\; e^\{j\backslash alpha\}g\_\{+\}(t)$.

$f\_\{+\}(t)\; =\; e^\{j\backslash alpha\}g\_\{+\}(t)$ implies that

$Re\backslash \{f\_\{+\}(t)\backslash \}\; =\; Re\backslash \{e^\{j\backslash alpha\}g\_\{+\}(t)\backslash \}\; \backslash implies\; f(t)\; =\; \backslash cos\{\backslash alpha\}\; g(t)\; -\; \backslash sin\{\backslash alpha\}\; g\_H(t)$ and

$Im\backslash \{f\_\{+\}(t)\backslash \}\; =\; Im\backslash \{e^\{j\backslash alpha\}g\_\{+\}(t)\backslash \}\; \backslash implies\; f\_H(t)\; =\; \backslash sin\{\backslash alpha\}\; g(t)\; +\; \backslash cos\{\backslash alpha\}\; g\_H(t)$.

Hence, we get the relation

$f(t)\; =\; (\backslash cos\{\backslash alpha\}-\backslash sin\{\backslash alpha\}\backslash tan\{\backslash alpha\})\; g(t)\; -\; \backslash tan\{\backslash alpha\}\; f\_H(t)$

and $f(t),\; g\_H(t)\; \backslash in\; span\backslash \{f\_H(t),\; g(t)\backslash \}\; =\; span\backslash \{f(t),\; f\_H(t)\backslash \}\; =\; span\backslash \{g(t),\; g\_H(t)\backslash \}$.

Back to the phase retrieval problem, in the Cauchy wavelet transform case, the index set $I$ is $\backslash mathbb\{R\}\; \backslash times\; \backslash \{b\_1,\; b\_2\backslash \}$ with $b\_1\; \backslash neq\; b\_2$ and $b\_1,\; b\_2\; >\; 0$, the vector space $V$ is $L^2(\backslash mathbb\{R\})$ and the linear form $L\_\{(a,\; b)\}$ is defined as $L\_\{(a,\; b)\}(f)\; =\; W\_\{\backslash psi\_p\}[f(t)](a,\; b)$. Hence, $\backslash \{|L\_\{(a,\; b)\}(f)|\backslash \}\_\{a,\; b\; \backslash in\; \backslash mathbb\{R\}\; \backslash times\; \backslash \{b\_1,\; b\_2\backslash \}\}$ determines the two dimensional subspace $span\backslash \{f,f\_H\backslash \}$ in $L^2(\backslash mathbb\{R\})$.

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Category:Mathematical terminology