A helicopter view of the self-consistency framework for wavelets and other signal extraction methods in the presence of missing and irregularly spaced data
Xiao‐Li Meng · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2007
A common frustration in signal processing and, more generally, information recovery is the presence of irregularities in the data. At best, the standard software or methods will no longer be directly applicable when data are missing, incomplete or irregularly spaced (e.g., as with wavelets). Self-consistency is a very general and powerful statistical principle for dealing with such problems. Conceptually it is extremely appealing, for it is essentially a mathematical formalization of iterating common-sense "trial-and-error" methods until no more improvement is possible. Mathematically it is elegant, with one fixed-point equation to solve and a general projection theorem to establish optimality. Practically it is straightforward to program because it directly uses the regular/complete-data method for iteration. Its major disadvantage is that it can be computationally intensive. However, increasingly efficient (approximate) implementations are being discovered, such as for wavelet de-noising with hard and soft thresholding. This brief overview summarizes the author's keynote presentation on those points, based on joint work with Thomas Lee on wavelet applications and with Zhan Li on the theoretical properties of the self-consistent estimators.