Smoothing for signals with discontinuities using higher order
  Mumford-Shah models

There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

Abstract

Minimizing the Mumford-Shah functional is frequently used for smoothing signals or time series with discontinuities. A significant limitation of the standard Mumford-Shah model is that linear trends -- and in general polynomial trends -- in the data are not well preserved. This can be improved by building on splines of higher order which leads to higher order Mumford-Shah models. In this work, we study these models in the univariate situation: we discuss important differences to the first order Mumford-Shah model, and we obtain uniqueness results for their solutions. As a main contribution, we derive fast minimization algorithms for Mumford-Shah models of arbitrary orders. We show that the worst case complexity of all proposed schemes is quadratic in the length of the signal. Remarkably, they thus achieve the worst case complexity of the fastest solver for the piecewise constant Mumford-Shah model (which is the simplest model of the class). Further, we obtain stability results for the proposed algorithms. We complement these results with a numerical study. Our reference implementation processes signals with more than 10,000 elements in less than one second.

Related collections

Most cited references 22

Record: found
Abstract: found
Article: found

Is Open Access

Natural Scales in Geographical Patterns

Telmo Menezes, Camille Roth (2017)

Human mobility is known to be distributed across several orders of magnitude of physical distances , which makes it generally difficult to endogenously find or define typical and meaningful scales. Relevant analyses, from movements to geographical partitions, seem to be relative to some ad-hoc scale, or no scale at all. Relying on geotagged data collected from photo-sharing social media, we apply community detection to movement networks constrained by increasing percentiles of the distance distribution. Using a simple parameter-free discontinuity detection algorithm, we discover clear phase transitions in the community partition space. The detection of these phases constitutes the first objective method of characterising endogenous, natural scales of human movement. Our study covers nine regions, ranging from cities to countries of various sizes and a transnational area. For all regions, the number of natural scales is remarkably low (2 or 3). Further, our results hint at scale-related behaviours rather than scale-related users. The partitions of the natural scales allow us to draw discrete multi-scale geographical boundaries, potentially capable of providing key insights in fields such as epidemiology or cultural contagion where the introduction of spatial boundaries is pivotal.