# Tutorial on 2nd generation wavelets (with lifting)?

For some denoising and deconvolution experiments, I'd like to apply a 2nd generation wavelet transform (using lifting steps) to images.

I know that there are several implementations available, but most of them use matlab, while I want to work in C++ with OpenCV. Since there is no built-in wavelet transform implementation in OpenCV 2.x, I plan to implement it myself (plus, it will make a good exercise for me). After some research, I've been able to find the original articles about the 2nd generation transform, but I'm still a bit confused about the exact way the algorithm works.

Taking for main reference the paper [1] by Sweldens: The Lifting scheme: a construction of second generation wavelets, I'm still confused by the definition of the index sets $\mathcal{K}(j)$ : what is their size ? how are they built ? ...

Hence my question : does anyone know about some resources about the 2nd generation wavelet transform (papers, tutorials, slides...) that are either in a tutorial-like form, or that provide a more algorithmic view (rather than a mathematical one), which would help me designing my own implementation ?

References

My main reference is:

[1] Sweldens, W. (1998). The lifting scheme: A construction of second generation wavelets. SIAM Journal on Mathematical Analysis, 29(2), 511.

And I am also learning from:

[2] Daubechies, I., & Sweldens, W. (1998). Factoring wavelet transforms into lifting steps. Journal of Fourier analysis and applications, 4(3), 247–269.

[3] Kovacevic, J., & Sweldens, W. (2000). Wavelet families of increasing order in arbitrary dimensions. Image Processing, 9(3), 480–496. doi:10.1109/83.826784

• It would probably help, if you link to the original papers and explain, why you are confused by them. Also, you say, that there are many matlab (scripting language) implementations, which you could read to get an idea, how the algorithm works. – bjoernz May 24 '12 at 8:38
• There are already C++ wavelet libraries. If you're going to code something for exercise, why don't you pick one of the newer multiscale transforms like beamlets, ridgelets, or curvelets so the community can benefit? – Emre May 24 '12 at 11:14
• @Emre: As said before, OpenCV doesn't include a wavelet transform, and I don't wand to add dependencies. I'll check Blitzwave code anyway to see how things are done. For now, I only need 2nd gen. wavelets, but more recent tools (starting with the curvelets) are an option for later work. – sansuiso May 24 '12 at 15:21
• @bjoernz: I added a precise question about a small part of Sweldens's paper that confuses me. – sansuiso May 24 '12 at 15:26
• Could you please reference your articles/books that you are learning this from? – Spacey May 24 '12 at 16:33