Imprecise expectations for imprecise linear filtering

Abstract

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.

Towards interval-based non-additive deconvolution in signal processing

“…It sums up different results presented in [18]. It is based on replacing the usual probability measure by a more general confidence measure called a capacity (see e.g.…”

“…a capacity that is concave and convex) and the imprecise valued expectation coincides with the usual precise valued expectation when the considered capacity is a probability measure: E P = E P . See [18] for more details.…”

“…However, this method is very computationally expensive and the guarantee relies on the ability to predict an appropriate bound of the reconstruction error. In a previous paper [18], we proposed an alternative method that uses a capacity ν to model imprecise knowledge of the impulse response of a sensor. This modeling entails a generalized convolution operator based on the Choquet integral leading to modeling the measurement process by a non-linear equation of the form:…”

Non-additive interval-valued F-transform

“…Similar to the continuous case, a discrete maxitive kernel defines a convex subset of discrete summative kernels, denoted M(π) [28].…”

“…The expectation concept can easily be extended to concave capacities (see [28]). Let ν be a concave capacity and let f : Ω → R be a L 1 bounded function.…”