Draft:Vanishing moment

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 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. The higher the number of vanishing moments, the more low-frequency components are filtered out during the inner product operation.

Definition

Mathematically, a function $\psi (t)$ is said to have $n$ vanishing moments if:

$\int _{-\infty }^{\infty }t^{k}\psi (t)\,dt=0\quad {\text{for }}k=0,1,\ldots ,n-1$

In simpler terms, a wavelet has vanishing moments up to order $n$ if it is orthogonal to all polynomials of degree $n-1$ or lower.

The vanishing moments of a wavelet determine its ability to capture specific signal characteristics, such as discontinuities or trends.

Calculations

The $k_{th}$ moment $m_{k}$ is defined as:

$m_{k}=\int t^{k}\psi (t)dt$

Calculating this directly can be challenging, especially for large values of $k$. Fourier transform properties can simplify the computation.

Calculating the zeroth noment

To compute $m_{0}$:

1. Calculate the Fourier transform of $g(t)$.

2. Extract the DC component ($f=0$).

Calculating the k-th noment

Using a property of the Fourier transform: differentiating $k$-times in the frequency domain is equivalent to multiplying by $t^{k}$ in the time domain.

${\frac {1}{(-j2\pi )^{k}}}G^{(k)}(f)=\,\int t^{k}g(t)e^{-j2\pi ft}dt$

When $f=0$, this simplifies to:

${\frac {1}{(-j2\pi )^{k}}}G^{(k)}(0)=\,\int t^{k}g(t)dt$

Thus, the $k$-th moment $m_{k}$ can be computed as:

$m_{k}={\frac {1}{(-j2\pi )^{k}}}G^{(k)}(0)$

Common functions and their vanishing moments

Vanishing moments are a critical property for analyzing wavelets, affecting their ability to capture specific signal characteristics. Here are some commonly used functions categorized into continuous functions and discrete coefficients of continuous functions:

Continuous functions

Haar Basis

Expression:

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

Since $\psi (t)$ is an odd function:

$m_{0}=\int \psi (t)dt=0$

However, $t\psi (t)$ is an even function:

$m_{1}=\int t\psi (t)dt\neq 0$

• Vanishing moments: 1
• Simple and compactly supported, often used as a basic wavelet for signal approximation.

Mexican Hat function

Expression:

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

Since $\psi (t)$ is the second derivative of Gaussian function, its Fourier transform is:

$F.T.\{\psi (t)\}=\,F.T.\left\{C{\frac {d^{2}}{dt^{2}}}e^{\pi t^{2}}\right\}=-C4\pi ^{2}f^{2}e^{-\pi f^{2}},\,C={\text{constants}}.$

Using the moment formula:

${\frac {1}{(-j2\pi )^{k}}}G^{(k)}(0)=\,\int t^{k}g(t)dt$

we find:

$m_{0}=m_{1}=0,m_{2}\neq 0$

• Vanishing moments: 2

p-th derivative of Gaussian function

Fourier transform:

$F.T.\{\psi (t)\}=F.T.\left\{{\frac {d^{p}}{dt^{p}}}e^{-\pi t^{2}}\right\}\,=(j2\pi f)^{p}e^{-\pi f^{2}}$

• Vanishing moments: $p$

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.

• For a $2n$-point Daubechies wavelet, vanishing moment $=n$.

Symlet and Coiflets

• Symlets are designed for improved symmetry, and Coiflets ensure both wavelet and scaling functions have vanishing moments.
• For a $2n$-point Symlet, vanishing moment $=n$.
• For a $6n$-point Coiflet, vanishing moment $=n$.

Applications

1. Signal processing

Vanishing moments are critical in applications like signal denoising, compression, and feature extraction. For instance, wavelets with high vanishing moments are effective in compressing images with smooth regions.

2. Data compression

The JPEG 2000 standard uses wavelets with vanishing moments to achieve high compression ratios with minimal loss of detail.

3. Image analysis

Wavelets with vanishing moments are used to detect edges, textures, and singularities in images.

4. Numerical analysis

Vanishing moments are employed in quadrature rules and solving differential equations by decomposing functions into components orthogonal to polynomial bases.

References

• Grossmann, A. (1984). "Decomposition of Hardy functions into square integrable wavelets of constant shape". SIAM Journal on Mathematical Analysis. 15 (4): 723–736. doi:10.1137/S0036144500371907.
• Daubechies, I. (1988). "Orthonormal bases of compactly supported wavelets". Communications on Pure and Applied Mathematics. 41 (7): 909–996. doi:10.1002/cpa.3160410705.
• Strang, G. (1996). Wavelets and Filter Banks. Wellesley-Cambridge Press.
• Mallat, S. (1999). A Wavelet Tour of Signal Processing. Academic Press.
• Percival, D.B. (2000). Wavelet Methods for Time Series Analysis. Cambridge University Press. doi:10.1017/CBO9780511804289.
• Gonzalez, R.C. (2009). Digital Image Processing. Pearson Education India.
• Debnath, L. (2012). Wavelet Transforms and Time-Frequency Signal Analysis. Springer Science & Business Media.
• Addison, P.S. (2017). The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance. CRC Press.