Segmentation of Time Series: Parameter Dependence of Blake-Zisserman and Mumford-Shah Functionals and the Transition from Discrete to Continuous

Abstract

The paper deals with variational approaches to the segmentation of time series into smooth pieces, but allowing for sharp breaks. In discrete time, the corresponding functionals are of Blake-Zisserman type. Their natural counterpart in continuous time are the Mumford-Shah functionals. Time series which minimise these functionals are proper estimates or representations of the signals behind recorded data. We focus on consistent behaviour of the functionals and the estimates, as parameters vary or as the sampling rate increases. For each time continuous time series $f\in L^2 (\lbrack 0,1\rbrack)$ we take conditional expectations w.r.t. to $σ$-algebras generated by finer and finer partitions of the time domain into intervals, and thereby construct a sequence $(f_n)_{n\in\N}$ of discrete time series. As $n$ increases this amounts to sampling the continuous time series with more and more accuracy. Our main result is consistent behaviour of segmentations w.r.t. to variation of parameters and increasing sampling rate.