Hi,

I'm researching inverse rendering: extraction of the lighting, materials and geometry out of photos from different viewpoints.
The current algorithm uses a hierarchical refinement procedure, based on a Haar wavelet tree, to guide the optimization process. The process is illustrated below:



As you can see, the estimated lighting is not smooth enough. In game development the most obvious thing to do would be to apply some smoothing filter, but I cannot do it in this case. This would make my estimation much more innaccurate and this is not meant for games but more critical visualisation applications.

So my idea was to replace the Haar wavelets with linear wavelets. Unfortunately, there is far less literature on them.
As we all know, Haar wavelets look like this:



However, I've been scratching my head over what a linear (second order) wavelet would look like. Would it look something like this?


Also, if you know of any good literature on linear wavelets like these, please let me know.

Many thanks,
Jeroen

This is pretty interesting. When using the tree to render the sphere with the estimated lighting, you could use bilinear filtering. This way you could still use your Haar Wavelets and you don't need to smooth them. Just the rendering needs to filter them bilinearly.

PS: I might misunderstand something though

maybe Daubechies D4 wavelet would do the trick?
http://en.wikipedia.org/wiki/Daubechies_wavelet#The_scaling_sequences_of_lowest_approximation_order

Thank you for your replies.

Bilinear filtering is unfortunately out of the question. It smooths the entire environment map, but that is not the intention. Wherever the Haar wavelets can approximate the function adequately, linear wavelets will preserve the quality just as good. However, in difficult areas of big changes in frequency content (e.g. from diffuse to a specular highlight), linear wavelets will provide a smoother result. Bilinear filtering on the other hand, would just blur everything.

Daubechies wavelets are even more complex than linear wavelets. I need to perform a triple product on the wavelets, and doing so with Daubechies wavelets is far from obvious. It has been solved for Haar wavelets though, so I have good hopes that it might work for linear wavelets as well. Performing a triple product on linear wavelets is what I am researching.