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Crises or large-scale emergencies such as earthquakes and hurricanes cause massive damage to lives and property. Crisis response is an essential task to mitigate the impact of a crisis. An effective response to a crisis necessitates information gathering and analysis. Traditionally, this process has been restricted to the information collected by first responders on the ground in the affected region or by official agencies such as local governments involved in the response.

Crises and situations of mass emergency such as earthquakes and hurricanes cause massive damage to lives and property. Crisis response is an essential task to mitigate the impact of a crisis. An effective response to a crisis necessitates information gathering and analysis. Traditionally this process has been restricted to the information collected by first responders on the ground in the affected region or official agencies such as local governments involved in the response.

In [doi{10.1109/ICMEW.2014.6890711}], a~graph-based filtering of noisy images is performed by directly computing a projection of the image to be filtered onto a lower dimensional Krylov subspace of the graph Laplacian, constructed using non-negative graph weights determined by distances between image data corresponding to image pixels. We extend the construction of the graph Laplacian to the case, where some graph weights can be negative.

We propose signal reconstruction algorithms which utilize a guiding subspace that represents desired properties of reconstructed signals. Optimal reconstructed signals are shown to belong to a convex bounded set, called the ``reconstruction'' set. Iterative reconstruction algorithms, based on conjugate gradient methods, are developed to approximate optimal reconstructions with low memory and computational costs. Effectiveness of the proposed method is demonstrated with an application to image magnification.

With the increasing ubiquity and power of mobile devices, as well as the prevalence of social systems, more and more