I'm interested in wavelet framework for polar coordinates.
In the paper of Hou&Qin (2012) was proposed a general method for definition of MH wavelets on a certain manifold. In short, first we need to solve a diffusion problem for singular initial conditions, then to differentiate the solution to get the wavelet.
It's not easy to solve the diffusion problem in the polar frame when the source is offset from the origin (actually this should be a singular ring). Does anybody know the solution for this problem, or just know the look of the MH-wavelet in polar frame?
An alternative approach may be the foolowing. There is a well known way to define a Bessel functions with angular integration of plane wave:
$$J_0(kr) \sim \int_0^{2\pi} d\alpha\,\exp[- i k r \cos\alpha]$$
With this integration we utilize the axial symmetry of some field, thus eliminating the angular coordinate.
Let $\phi(x-a,s)$ be a 1-D wavelet in Cartesian frame, centered in $a$. May I define the polar wavelet by the similar way:
$$\psi(r, a,s) = \int_0^{2\pi} d\alpha\,\phi(r \cos\alpha - a, s)$$
Thanks in advance.
I've decided to follow the recipe of Hou&Qin cited above and express the radial wavelet via solution of the diffusion problem. The last was obtained using Hankel transform of Green's function equation for time-dependent problem, following lection of S. Gustafson, adopted for cylindrical coordinates. I'll bring it shortly.
Let $G = G(r, q; s)$ satisfies the equation \begin{align} & \partial_s G - \Delta_r G = \frac{\delta(r - q)}{r}\,\delta(s) \\ & G = 0 \quad\text{for}\quad s < 0 \end{align} where $\Delta_r$ is the radial part of Laplasian in polar coordinates. Express the solution in terms of cylindrical bessel functions, then solve the resilting ODE, not forgetting the "jump condition" at $s = 0$: \begin{gather} \partial_s \hat{G} + k^2 \hat{G} = J_0(kq)\,\delta(s) \\ \hat{G}(k, q; s) = J_0(kq)\,e^{-k^2 s} \\ G(r, q; s) = \int_0^\infty dk\,k\,J_0(kr)\,J_0(kq)\,e^{-k^2 s} \end{gather} This is the Green's function of the time-dependent problem and this is the solution for the diffusion problem in radial cylindrical coordinates, for singular initial conditions.
Further let's define the wavelet like this: $$ \Psi(r, q; s) \equiv \partial_s G(r, q; s) = - \int_0^\infty dk\,k^3\,J_0(k r)\,J_0(k q)\,e^{- k^2 s} $$ For a given $f = f(r)$ it's transform should be defined as $$ \tilde{f}(q; s) \equiv \int_0^\infty dr\,r\,\Psi(r, q; s)\,f(r) $$ It's easy to show that function $$ \Phi(r, q; s) \equiv - 2 \int_0^\infty dk\,k\,J_0(k r)\,J_0(k q)\,e^{- k^2 s} $$ gives us the inverse transform: $$ f(r) = \int_0^\infty dq\,q \int_0^\infty ds\,\Phi(r, q; s)\,\tilde{f}(q; s) $$ And don't forget about duality: $$ \int_0^\infty dq\,q \int_0^\infty ds\,\Phi(r, q; s)\,\Psi(p, q; s) = \frac{\delta(r - p)}{r} $$
Other nice things, like compatibility condition, $\partial_s \tilde{f} = \Delta_q \tilde{f}$, and local reconstruction formula are satisfied also.
Thanks to all who cared.