On a fast bilateral filtering formulation using functional rearrangements

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.Comment: 29 pages, Journal of Mathematical Imaging and Vision, 2015. arXiv admin note: substantial text overlap with arXiv:1406.712

Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.Comment: Final version. To appear in Nolinear Analysis Real World Applications (2016

Are Historical Biogeographical Events Able to Promote Biological Diversification?

One of the goals of evolutionary biology is understanding how biological diversification change across spatial and temporal scales. Theoretically, it has been established that external (i.e., dispersals) and internal (i.e., origin of a key innovations) factors can modulate shifts in rates of species diversification. However, the role of historical events as trigger of species diversification rates have not been well understood in empirical studies. I reviewed the literature linking historical biogeographic events and species diversification in many groups. Many of studies conclude that dispersals can be associated with exceptional changes in species diversification rates in insular and mainland areas. I discuss the limitations of some approaches used to discover the link between historical biogeography and macroevolution. I propose some predictions under biogeographic scenarios to gain understanding in how historical events promote biological diversification. I suggest that future studies linking biogeography and macroevolution should incorporate ecologically-relevant traits to discern the mechanisms underlying these historical associations. Although new developments in phylogenetic comparative methods have been done, still is necessary more traditional field-based ecological and evolutionary research. The link between biogeography and diversification still remains narrative and a comprehensive approach is necessary to establish how diversification was triggered by historical events