Nonlinear smoothing and multiresolution analysis (Q556603)

LULU-theory can be seen as a special part of mathematical morphology. The book presents a framework of LULU-smoothers and a comparison of several useful nonlinear smoothers. The concept of ``smoothness'' and ``trend'' is established and linked to the total variation. The considered LULU-smoothers are proved to ensure variation reduction and shape preservation. Further, these smoothers give rise to define a nonlinear multiresolution analysis that decomposes a sequence into ``trend'' and ``pulse'' content. The introduced theory also helps to explain the success of local median transforms like running median procedures. A discrete pulse transform is established in analogy to the discrete Fourier transform, and the question of its consistency is addressed.