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Variation reduction and <i>LULU</i>-Smoothing
Abstract
In the space of absolutely summable
sequences the total variation of a sequence becomes a natural norm, and is a
measure of smoothness. This norm is preserved by the so-called LULU-smoothers, in that the variation of
the image plus the variation of the difference is equal to the variation of the
original sequence. This surprising property is useful for strategies for the
removal of impulsive and other random noise. Strong shape preserving properties
of these smoothers are derived in the process of proving this. The results shed
light on similar properties of the popular median smoothers, as well as their
sometimes enigmatic behaviour.
Mathematics Subject
Classification (2000):
41A99, 41A46, 65D10.
Key words: Total variation; non-linear;
smoothing; variation reduction; variation preservation.
Quaestiones
Mathematicae 25 (2002), 163-176
sequences the total variation of a sequence becomes a natural norm, and is a
measure of smoothness. This norm is preserved by the so-called LULU-smoothers, in that the variation of
the image plus the variation of the difference is equal to the variation of the
original sequence. This surprising property is useful for strategies for the
removal of impulsive and other random noise. Strong shape preserving properties
of these smoothers are derived in the process of proving this. The results shed
light on similar properties of the popular median smoothers, as well as their
sometimes enigmatic behaviour.
Mathematics Subject
Classification (2000):
41A99, 41A46, 65D10.
Key words: Total variation; non-linear;
smoothing; variation reduction; variation preservation.
Quaestiones
Mathematicae 25 (2002), 163-176
