# Vanishing Moments

I am reading a book titled "Two Dimensional Wavelets and their relatives" by Antoine et al. and it talks about vanishing moments. I have trouble understanding the exact significance of it. Can anybody give an idea on vanishing moments? • Maybe you could tell which of the hundreds of papers on wavelets you are reading? and in what context the phrase "vanishing moment" is used? Oct 26 '12 at 21:27
• I am reading a book titled "Two Dimensional Wavelets and their relatives" by Antoine et al. I have a pic of the exact place where I am referring to. Please find it here dl.dropbox.com/u/28068989/IMAG0746.jpg Oct 26 '12 at 21:35
• In brief, if a wavelet has $n$ vanishing moments, the output of filtering a $n$-th degree polynomial with this wavelet will be 0. Oct 27 '12 at 0:04
• This is an intuitive explanation "for dummies" I don't know about continuous wavelets, but in discrete wavelets, a wavelet with $n$ vanishing moments, produce low coefficients on the parts of the data which can be approached by a polynomial with degree $n$. It facilitates the identification of parts of the data as "polynomial of order $n$" This should be a comment, and not an answer, but I'm not allowed to comment. Apr 24 '18 at 15:54

For a continuous random variable $X$ with probability density function $f(x)$, the $n$-th moment is $$m_n = \int_{-\infty}^\infty x^n f(x)\,\mathrm dx.$$ The zero-th moment is $1$ (the area under the density is $1$), the first moment is called the mean or expected value of the random variable and the second moment the mean square value. Note that since $f(x) \geq 0$, the second moment cannot be zero.
Even more generally, the $n$-th moment of an arbitrary function $f(x)$ can be defined as $$m_n = \int_{-\infty}^\infty x^n f(x)\,\mathrm dx.$$ Now the restriction of zero-th moment being $1$ and second moment being positive is not applicable any more, and the "vanishing moment" is merely a fancy way of saying that $f(x)$ must be such that $m_0 = m_1 = m_2 = \cdots m_N = 0$. In particular, $m_0$ is the DC value of the wavelet and the authors are insisting that the DC value be $0$.