Imprecise expectations for imprecise linear filtering

In the last 10 years, there has been increasing interest in interval valued data in signal processing. According to the conventional view, an interval value supposedly reflects the variability of the observation process. Generally, the considered variability is associated with either random noise or the uncertainty that underlies the observation process. In most sensor measure based applications, the raw sensor signal has to be processed by an appropriate filter to increase the signal to noise ratio or simply to recover the signal to be measured. In both cases, the output filter is obtained by convoluting the sensor signal with a supposedly known appropriate impulse response. However, in many real life situations, this impulse response cannot be precisely specified. The filtered value can thus be considered as biased by this arbitrary choice of one impulse response among all possible impulse responses considered in this specific context. In this paper, we propose a new approach to perform filtering that aims at computing an interval-valued signal that contains all outputs of filtering processes involving a coherent family of conventional linear filters. This approach is based on a very straightforward extension of the expectation operator involving appropriate concave capacities.