How does the non-orthogonality of an expansion manifest in the resulting signal coefficients? i.e.: what are the differences between the signal coefficients resulting from an orthogonal decomposition scheme (precisely an orthogonal filter bank) and those resulting from a non-orthogonal one? Please attach your answers with an explicit example if it is possible and thank you in advance.
Asalam Alicom Abdelhak,
If you use orthogonal eignfunctions, (Psi_n(t)), for expansion then the resulting coefficients can be used directly with the orthogonal functions to reconstruct the signal back. f(t)= Sum_(n=1)^N ( Psi_n(t)), where the coefficients are given by c_n=.
If you use non-orthogonal expansion, then the above relations does not hold. You can not use the relations above. There are multiple possibilities. First, the non-orthogonal basis might be redundant, so you have redundant coefficients. Second, the the non-orthogonal basis are not redundant, but since they are not orthogonal then they imply components that are parallel to each other, and this implies redundancy in some directions and lack of information in others. The signal can be retrieved back from the non-orthogonal expansion if the Bi-orthogonal set is found or exist. If not, then the reconstruction is not guaranteed. Some cases can be retrieved others can not. You can see my article "New wavelet family based on second order LTI systems" where I show reconstruction of non-orthogonal expansion. However, this is a special case.
Tariq
You can also use an overcomplete basis such as multiscale Gaussian pyramids, and this has certain advantages (e.g. no aliasing in subbands). Look at papers by Eero Simoncelli's group for image statistical descriptors using such non orthogonal expansions. (multiscale directional overcomplete filter banks)
wa aleikoum assalam sir Tarik.
Thank you very much for your explanation. But I want exactly to know if the application on a non-orthogonal filter bank to a signal can lead in coefficients of well-known characteristics or no?
Sincerely, Abdelhak.
I am not sure what you mean by well-known characteristics. They are non-minimal, which means that they are not the smallest set to fully describe the signal, in other words they contain redundancy. Redundancy means also that the information contained in the non-orthogonal coefficients can be obtained from applying a different set of filters.
Dear Abdelhak Bouhali
You can see the problem like this explanation -
Decomposition depends on basic question why we are doing it (depend on application). It is because we want to segregate signal in desired and undesired features. Later we can removed undesired signal part and reconstruct by desired pattern. If pattern are orthogonal then elimination of undesired will not effect your desired signal. But if they are non-orthogonal then it will eliminate desired signal part with undesired. It is because our desired and and undesired are mixed with each other in non-orthogonal decomposition.
orthogonal expansions are unique... non-orthogonal expansions are not unique... (unique means there is only one answer)... Another issue is "completeness"...
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