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{{Short description|Introduction of vanishing moment}}
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'''Vanishing moments''' are a fundamental concept in [[wavelet]] theory, [[signal processing]], and functional analysis. They describe a property of a wavelet or function, wherein certain [[Integral|integrals]] of the function against polynomial terms up to a specific degree vanish. This property is crucial in determining the ability of a wavelet to represent and compress signals effectively. It is used to evaluate whether the [[mother wavelet]] effectively captures high-frequency components of a signal.

== Origins ==

In the [[Continuous wavelet transform|Continuous Wavelet Transform]] (CWT), the [[mother wavelet]] must satisfy five primary constraints:

1. Compact Support

2. Real Function

3. Even or Odd Symmetry

4. High Vanishing Moments<ref name="Mallat1999" />

5. Admissibility Criterion<ref>{{Cite journal |last=Grossmann |first=A. |title=Decomposition of Hardy functions into square integrable wavelets of constant shape |journal=SIAM Journal on Mathematical Analysis |volume=15 |issue=4 |pages=723–736 |year=1984 |doi=10.1137/S0036144500371907}}</ref>

:<math>
C_{\psi}=\int_{0}^{\infty} \frac{|\Psi(f)|^2}{|f|} df < \infty
</math>,

:where <math>\Psi(f)</math> is the Fourier transform of <math>\psi(t)</math>.

== Definition ==

Mathematically, a function <math>\psi(t)</math> is said to have <math>n</math> vanishing moments if
<ref name="Mallat1999">{{Cite book |last=Mallat |first=S. |title=A Wavelet Tour of Signal Processing |publisher=Academic Press |year=1999}}
</ref>:

:<math>\int_{-\infty}^{\infty} t^k \psi(t)dt = 0,\quad \text{for} \ 0 \leq k < p</math>

In simpler terms, a wavelet has vanishing moments up to order <math>n</math> if it is [[Orthogonality|orthogonal]] to all polynomials of degree <math>n-1</math> or lower.

The <math>k</math>-th moment <math>m_k</math> is defined as:

:<math>
m_k = \int t^k \psi(t) dt
</math>

where <math>\psi(t)</math> has <math>p</math> vanishing moments if <math>m_0 = m_1 = m_2 = \ldots = m_{p-1} = 0</math>.

== Calculations ==

=== Calculating the zeroth moment ===

To compute <math>m_0</math>:

1. Calculate the [[Fourier transform]] of <math>g(t)</math>:

:<math>
G(f) = \int g(t)e^{-j2 \pi ft}dt
</math>

2. Extract the DC component (<math>f = 0</math>)

:<math>
m_0 = G(0) = \int g(t)dt
</math>

=== Calculating the k-th moment ===

Using a property of the [[Fourier transform]]: differentiating <math>k</math>-times in the [[Frequency domain|frequency domain]] is equivalent to multiplying by <math>t^k</math> in the [[Time domain|time domain]]

<math>
\frac{1}{(-j 2\pi)^k} G^{(k)} (f) = \,
\int t^k g(t) e^{-j 2\pi ft}dt
</math>.

When <math>f=0</math>, this simplifies to:

:<math>
\frac{1}{(-j 2\pi)^k} G^{(k)} (0) = \int t^k g(t)dt
</math>

Thus, the <math>k</math>-th moment <math>m_k</math> can be computed as:

:<math>
m_k = \frac{1}{(-j 2\pi)^k} G^{(k)} (0)
</math>

== Common functions and their vanishing moments ==

Vanishing moments are one of the properties for analyzing wavelets. Here are some commonly used functions categorized into '''[[Continuous function|continuous functions]]''' and '''discrete coefficients of continuous functions''':

=== Continuous functions ===

==== Haar basis function ====

:Expression<ref name="Daubechies1988">{{Cite journal |last=Daubechies |first=I. |title=Orthonormal bases of compactly supported wavelets |journal=Communications on Pure and Applied Mathematics |volume=41 |issue=7 |pages=909–996 |year=1988 |doi=10.1002/cpa.3160410705}}</ref>:

::<math>
\psi(t) = \,
\begin{cases}
1 \quad & 0 \leq t < 1/2,\\
-1 \quad & 1/2 \leq t < 1,\\
0 \quad & \text{otherwise.}
\end{cases}
</math>

:Since <math>\psi(t)</math> is an [[odd function]]:

::<math>
m_0 = \int \psi(t)dt=0
</math>

:However, <math>t \psi(t)</math> is an [[even function]]:

::<math>
m_1 = \int t \psi(t)dt \neq 0
</math>
   
:* Vanishing moments: 1

==== [[Ricker wavelet]] ====

:Expression<ref>{{Cite journal |last=Farge |first=M. |title=Wavelet Transforms and Their Applications to Turbulence |journal=Annual Review of Fluid Mechanics |volume=24 |issue=1 |pages=395–458 |year=1992 |doi=10.1146/annurev.fl.24.010192.002143|bibcode=1992AnRFM..24..395F }}</ref><ref>{{Cite journal |last1=Torrence |first1=C. |last2=Compo |first2=G. P. |title=A Practical Guide to Wavelet Analysis |journal=Bulletin of the American Meteorological Society |volume=79 |issue=1 |pages=61–78 |year=1998 |doi=10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2|bibcode=1998BAMS...79...61T }}</ref><ref>{{Cite conference |last=Stepanov |first=A. B. |title=Construction of Activation Functions for Wavelet Neural Networks |book-title=2017 XX IEEE International Conference on Soft Computing and Measurements (SCM) |pages=397–399 |publisher=IEEE |date=May 2017 |doi=10.1109/SCM.2017.7970688}}</ref>:

::<math>
\psi (t) = \frac{2^{5/4}}{\sqrt{3}} \,
(1-2 \pi t^2)e^{- \pi t^2}
</math>

:Since <math>\psi (t)</math> is the second derivative of [[Gaussian function]], its [[Fourier transform]] is:

::<math>
F.T.\{\psi (t)\} = \,
F.T. \left\{C \frac{d^2}{dt^2}e^{\pi t^2} \right\} = -C 4\pi^2 f^2 e^{-\pi f^2}, \,
C = \text{constants}.
</math>

:Using the moment formula:

::<math>
\frac{1}{(-j 2\pi)^k} G^{(k)} (0) = \,
\int t^k g(t)dt
</math>

:we find:

::<math>
m_0 = m_1 = 0, m_2 \neq 0
</math>

:* Vanishing moments: 2

==== p-th derivative of Gaussian function<ref>{{Cite journal |last1=Mallat |first1=S. |last2=Hwang |first2=W. L. |title=Singularity detection and processing with wavelets |journal=IEEE Transactions on Information Theory |volume=38 |issue=2 |pages=617–643 |year=1992 |doi=10.1109/18.119727}}</ref> ====

:<math>
F.T.\{ \psi(t)\} = F.T.\left\{ \frac{d^p}{dt^p} e^{- \pi t^2} \right\} \,
= (j 2\pi f)^p e^{-\pi f^2}
</math>

:* Vanishing moments: <math>p</math>

=== Discrete coefficients of continuous functions ===

==== Daubechies wavelet ====

* [[Daubechies wavelet]], widely used in practice, has varying numbers of vanishing moments, offering a balance between localization in time and frequency domains<ref name="Mallat1999" /><ref name="Daubechies1988" /><ref name="Daubechies1992">{{Cite book |last=Daubechies |first=Ingrid |title=Ten Lectures on Wavelets |publisher=SIAM |year=1992 |isbn=978-0898712742}}</ref>.

* For a <math>2n</math>-point Daubechies wavelet, vanishing moment <math>=n</math>.

==== Symlet and Coiflets ====

* [[Symlet|Symlets]] are designed for improved symmetry, and [[Coiflet|Coiflets]] ensure both wavelet and scaling functions have vanishing moments<ref>{{Cite book |last=Percival |first=D.B. |title=Wavelet Methods for Time Series Analysis |publisher=Cambridge University Press |year=2000 |doi=10.1017/CBO9780511804289|doi-broken-date=23 December 2024 }}</ref><ref>{{Cite book |last=Ford |first=M. S. |title=The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science |year=2003 |publisher=CRC Press |isbn=978-0367574000}}</ref>.
* For a <math>2n</math>-point Symlet, vanishing moment <math>=n</math>.
* For a <math>6n</math>-point Coiflet, vanishing moment <math>=n</math>.

== Significance of Vanishing Moments for Functions ==

Vanishing moments serve as an indicator of how a function decreases in magnitude. For example, consider the function:

:<math>f(t) = \frac{\sin(t)}{t^2}</math>

As the input value <math>t</math> increases toward infinity, this function decreases at a rate proportional to <math>\frac{1}{t^2}</math>. The decay rate of such a function can be evaluated using the momentum integral defined as <math>\int_{-\infty}^{\infty} t^k f(t) dt</math>.

In this example:

When <math>k = 0</math>:
The numerator <math>\sin(t)</math> oscillates between <math>[-1,1]</math>, causing the function to oscillate within <math>\left [-\frac{1}{t^2}, \frac{1}{t^2} \right ]</math>.
This oscillatory behavior ensures that:

:<math>
\int_{-\infty}^{\infty} t^k \frac{\sin(t)}{t^2} dt \to 0
</math>

Thus, the integral converges to 0, representing the zeroth momentum <math>m_0 = 0</math>.

When <math>k = 1</math>:

:<math>
\int_{-\infty}^{\infty} t^k \frac{\sin(t)}{t} dt = \pi
</math>

Here, the first momentum is nonzero, <math>m_1 = \pi \neq 0</math>.

For <math>k > 1</math>:
The momentum integral diverges as <math>t \to \infty</math>.

From this example, the maximum value of <math>k</math> for which the momentum integral converges to zero determines the decay rate of the function. This maximum <math>k</math> value is defined as the vanishing moment of the function.

In continuous wavelet transforms, one of the conditions for designing a wavelet mother function is that its support must be finite. The rate at which the wavelet mother function decays within this finite support is characterized by its vanishing moments.

== Equivalent Statements of Vanishing Moments ==

According to the definition, the condition for a wavelet mother function <math>\psi(t)</math> to have <math>p</math> vanishing moments is<ref name="Mallat1999" />:

:<math>\int_{-\infty}^{\infty} t^k \psi(t)dt = 0, \quad \text{for} \ 0 \leq k < p</math>

However, since this definition involves an infinite-range continuous integral, it is not practical for designing wavelet mother functions.

If the scaling function in the wavelet transform is defined as <math>\varphi(t)</math>, and the following relationship between the wavelet mother function and the scaling function holds<ref name="Mallat1999" /><ref>{{cite web|url=http://www.polyvalens.com/blog/?page_id=15#7.+The+scaling+function+%5B7%5D|title=A Really Friendly Guide To Wavelets – PolyValens|website=www.polyvalens.com}}</ref>:

:<math>\left| \psi(t) \right| = O((1+t^2)^{-p/2-1})</math>

:<math>\left| \varphi(t) \right| = O((1+t^2)^{-p/2-1})</math>

Then, the following four statements are equivalent<ref name="Mallat1999" />:

1. The wavelet mother function <math>\psi(t)</math> has <math>p</math> vanishing moments.

2. The Fourier transforms of <math>\psi(t)</math> and <math>\varphi(t)</math>, along with their derivatives up to <math>(p - 1)</math>th order, are zero at <math>\omega = 0</math>.

3. The Fourier transforms of <math>h_\varphi(t)</math> and <math>H_\varphi(e^{j\omega})</math>, along with their derivatives up to <math>p - 1</math>th order, are zero at <math>\omega = 0</math>.

4. For any <math>k</math> in the interval <math>0 \leq k < p</math>,

:<math>q_k(t) = \sum_{n=-\infty}^\infty n^k \varphi(t-n)</math>

:is a polynomial function of degree <math>k</math>.

== Vanishing Moments and Wavelet Function Design ==

When the Fourier transform of a filter satisfies the following condition:

:<math> {\left| H_\varphi (\omega) \right|}^2 + {\left| H_\varphi (\omega + \pi) \right|}^2 = 2 </math>

the filter meets the condition of a conjugate mirror filter<ref name="Mallat2009">{{Cite book |last=Mallat |first=S. |title=A Wavelet Tour of Signal Processing |chapter=Wavelet Bases |pages=263–376 |publisher=Academic Press |year=2009 |isbn=978-0123743701}}</ref>. Here, <math>H_\varphi(\omega)</math> represents the Fourier transform of the discrete low-pass filter <math>h_\varphi[n]</math>.

By combining the conjugate mirror filter condition with the third equivalent statement of vanishing moments, the low-pass filter can be expressed as:

:<math> H_\varphi (e^{j\omega}) = \sqrt{2} \left( \frac{1+e^{j\omega}}{2} \right)^p L(e^{j\omega}) </math>

where <math>L(x)</math> is a polynomial function.

Using the above condition and the equivalent statements of vanishing moments, the design process for wavelet functions can be simplified.

=== Vanishing Moments and Filter Length ===

In wavelet transforms, the scaling function and wavelet mother function can be defined using discrete filters<ref name="Mallat2009" />:

:<math>\varphi(t) = \sum_{n}^{} h_\varphi[n] \sqrt{2}\varphi(2t-n)</math>

:<math>\psi(t) = \sum_{n}^{} h_\psi[n] \sqrt{2}\varphi(2t-n)</math>

Here, <math>h_\varphi[n]</math> is the discrete low-pass filter, and <math>h_\psi[n]</math> is the discrete high-pass filter. The filter length is usually expressed in terms of the size of support.

From the expression <math> H_\varphi (e^{j\omega}) = \sqrt{2} \left( \frac{1+e^{j\omega}}{2} \right)^p L(e^{j\omega}) </math>, we can observe that:

Choosing a higher number of vanishing moments <math>p</math> results in <math>H_\varphi(e^{j\omega})</math> being a polynomial function with higher powers of <math>e^{j\omega}</math>. Consequently, the corresponding <math>h_\varphi[n]</math> will have a longer filter length.

In general, there is a trade-off between having a higher number of vanishing moments and a shorter filter length; both cannot be achieved simultaneously.

Thus, when designing the wavelet mother function for continuous wavelet transforms, considerations should include not only the number of vanishing moments but also the corresponding filter length.

== References ==
<!-- Inline citations added to your article will automatically display here. See en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations. -->
{{reflist}}

* {{Cite book |last=Strang |first=G. |title=Wavelets and Filter Banks |publisher=Wellesley-Cambridge Press |year=1996}}
* {{Cite book |last=Debnath |first=L. |title=Wavelet Transforms and Time-Frequency Signal Analysis |publisher=Springer Science & Business Media |year=2012}}
* {{Cite book |last=Addison |first=P.S. |title=The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance |publisher=CRC Press |year=2017}}