Seeking closed form expression

[Nicholas James (nicholas.james@sympatico.ca)]

I need a wavelet which satifies the following criteria:
- has a closed form expression
- is differentiable
- can be used in a DWT

I think my only option is a biorthogonal spline wavelet, but I'm having great difficulty finding a closed form expression for one. If I do use a biorthogonal spline wavelet, I only need the reconstruction wavelet to be differentiable and expressible in closed form; I don't care what the decomposition wavelet looks like.

Just to be clear, when I say a "closed form expression," what I mean is that there is a way to write (t) in terms of t alone, rather than indirectly defining via its Fourier transform or scaling equations.

...Thanks *very* much in advance!

[Gabriel Peyré (gpeyre@altern.org)]

> Just to be clear, when I say a "closed form expression," what I mean is that there is a way to write (t) in terms of t alone,
> rather than indirectly defining via its Fourier transform or scaling equations.

The scaling spline function has a well known expression (piecewise polynomial).
The wavelet function can be computed using an infinite sum.

Everything is explained in Unser's publications,
e.g. this one.
You should also check his survey paper.

G.

[Nicholas James (nicholas.james@sympatico.ca)]

Thank you very much! I still haven't found exactly what I'm looking for in those papers, but I think you put me on the right track.