Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting.
Methods for performing orthogonalization include: When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram–Schmidt process since it is more numerically stable, i.e. rounding errors tend to have less serious effects.
[1] To compensate for the loss of useful signal in traditional noise attenuation approaches because of incorrect parameter selection or inadequacy of denoising assumptions, a weighting operator can be applied on the initially denoised section for the retrieval of useful signal from the initial noise section.
The new denoising process is referred to as the local orthogonalization of signal and noise.
[2] It has a wide range of applications in many signals processing and seismic exploration fields.