A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws

A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The nonlocal calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations

Nonlocal PdES on graphs for active contours models with applications to image segmentation and data clustering

International audienceWe propose a transcription on graphs of recent continuous global active contours proposed for image segmentation to address the problem of binary partitioning of data represented by graphs. To do so, using the framework of Partial difference Equations (PdEs), we propose a family of nonlocal regularization functionals that verify the co-area formula on graphs. The gradients of a sub-graph are introduced and their properties studied. Relations, for the case of a sub-graph, between the introduced nonlocal regularization functionals and nonlocal discrete perimeters are exhibited and the co-area formula on graphs is introduced. Finally, nonlocal global minimizers can be considered on graphs with the associated energies. Experiments show the benefits of the approach for nonlocal image segmentation and high dimensional data clustering

Partial differences as tools for filtering data on graphs

International audienc