1. The affine group behind the wavelet transform

Continuing my previous post on the uncertainty principle for the windowed Fourier transform, we now come to another integral transform: The wavelet transform.

In contrast to the windowed Fourier transform (which analyzes a function with respect to position and frequency) the wavelet transform analyzes a function with respect to position and scale. For a given analyzing function ${\psi}$ and a signal ${f}$, the wavelet transform is (for ${a\neq 0}$, ${b\in{\mathbb R}}$): $\displaystyle W_\psi f(a,b) = \int_{\mathbb R} f(x) \tfrac{1}{\sqrt{|a|}}\psi(\tfrac{x-b}{a})dx.$

In the same way, the windowed Fourier transform could be written as inner products of ${f}$ with a translated and modulated window function, the wavelet transform can be written as inner products of ${f}$ with translated and scaled functions ${\psi}$. And again, these modifications which happen to the analyzing function come from a group.

Definition 1 The affine group ${G_{\text{aff}}}$ is the set ${({\mathbb R}\setminus\{0\})\times {\mathbb R}}$ endowed with the operation $\displaystyle (a,b)(a',b') = (a\,a',a\,b' + b).$

Indeed this is group (with identity ${(1,0)}$ and inverse ${(a,b)^{-1} = (a^{-1},a^{-1}b)}$). The name affine group stems from the fact that the group operation behaves like the composition of one dimensional affine linear functions: For ${f(x) = ax+b}$ and ${g(x) = a'\,x+b'}$ we have ${f(g(x)) = (a\,a')\, x + a\,b' + b}$.

The affine group admits a representation on the space of unitary operators on ${L^2({\mathbb R})}$: $\displaystyle \Pi(a,b)f(x) = \tfrac{1}{\sqrt{|a|}}\psi(\tfrac{x-b}{a})$

(note the normalizing factor ${1/\sqrt{|a|}}$).

2. The affine uncertainty principle

I am not sure who has to credited for the group theoretical background behind wavelets however, the two-part paper “Transforms associated to square integrable group representations” by Grossmann, Morlet and Paul has been influential (and can be found, e.g. in the compilation “Fundamental papers in wavelet theory” by Heil and Walnut.

As done by Stephan Dahlke and Peter Maass in “The Affine Uncertainty Principle in One and Two Dimensions” and can proceed in analogy to the windowed Fourier transform and the corresponding Weyl-Heisenberg group and compute the infinitesimal operators: Take the derivative of the representation with the respect to the group parameters and evaluate at the identity: $\displaystyle T_a f(x) := \frac{d}{da}[\Pi(a,b)f(x)|_{(a,b) = (1,0)} = -\tfrac{1}{2}f(x) - xf'(x)$

and $\displaystyle T_b f(x) := \frac{d}{db}[\Pi(a,b)f(x)|_{(a,b) = (1,0)} = -f'(x).$

Again, these operators are skew adjoint and hence, multiplying by ${\mathrm{i}}$ gives self-adjoint operators.

These operators ${\mathrm{i} T_b}$ and ${\mathrm{i} T_a}$ do not commute and hence, applying Robertson’s uncertainty principle gives an inequality. The commutator of ${\mathrm{i} T_a}$ and ${\mathrm{i} T_b}$ is $\displaystyle [\mathrm{i} T_a,\mathrm{i} T_b] f(x) = f'(x).$ $\displaystyle \tfrac{1}{2}|\langle f',f\rangle| \leq \|(\mathrm{i} T_a - \mu_1 I)f\|\,\| (\mathrm{i} T_b - \mu_2 I)f\| \ \ \ \ \ (1)$

and with some manipulation this turn to (for ${\|f\|=1}$) $\displaystyle \tfrac{1}{4}|\langle f',f\rangle|\leq (\|f'\|^2 - \mu_2^2)(\|xf'\|^2 -\tfrac{1}{4} - \mu_1^2). \ \ \ \ \ (2)$

Again, one can derive the functions for which equality in attained and these are the functions of the form $\displaystyle f(x) = c(x-\mathrm{i} \lambda) ^{-1/2 + \mathrm{i}\lambda\mu_2+\mathrm{i}\mu_1}$

for real ${\lambda}$. (By the way, these functions are indeed wavelets and sometimes called Cauchy-wavelets because of their analogy with the Cauchy kernel from complex analysis.)

By the way: These functions are necessarily complex valued. If one restricts oneself to real valued functions there is a simpler inequality, which one may call “real valued affine uncertainty”. First, observe that ${\langle f',f\rangle = 0}$ for real valued ${f}$, and hence, the left hand side in (1) is zero (which make the inequality a bit pointless). Using that for real valued ${f}$ we have ${\langle T_a f,f\rangle = 0}$, and that ${\|T_a f\|^2 = \|xf'\| - \tfrac{1}{4}\|f\|^2}$ together with ${\|(\mathrm{i} T_b -\mu_2)f\|^2\neq 0}$ for ${f\neq 0}$ we obtain (with ${\mu_1=0}$) from (1) $\displaystyle \tfrac{1}{2} \|f\| \leq \|xf'\|.$

Since we know that equality is only attained for the Cauchy wavelets (which are not real valued we can state:

Corollary 2 (Real valued affine uncertainty) For any real valued function which is in the domain of ${[T_a,T_b]}$ it holds that $\displaystyle \tfrac{1}{2} \|f\| < \|xf'\|.$

As some strange curiosity, one can derive this “real valued affine uncertainty principle” by formal integration by parts and Cauchy-Schwarz inequality totally similar to the Heisenberg uncertainty principle (as I’ve done in my previous post): $\displaystyle \begin{array}{rcl} \|g\|_2^2 & =& \int_{\mathbb R} 1\cdot|g(x)|^2dx\\ & = &-\int_{\mathbb R} x\tfrac{d}{dx}|g(x)|^2dx\\ & = &-2\int_{\mathbb R} xg(x)g'(x)dx\\ & \leq &2\int_{\mathbb R} |xg'(x)|\,|g(x)|dx\\ & \leq &2\Big(\int_{\mathbb R} |xg'(x)|^2dx\Big)^{1/2}\Big(\int_{\mathbb R} |g(x)|^2dx\Big)^{1/2}. \end{array}$

Dividing by ${2\|g\|}$ gives the “real valued affine uncertainty” (but only in the non-strict way).