The generalized microscopic image reconstruction problem

This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini Nivat (2007) (2002). Consider specimen for inspection, represented as collection points typically organized on grid in plane. Assume each point x has an associated physical value ?x, which we would like to determine. However, it might be that obtaining these values precisely (by what call surgical probe ) is difficult, risky, or impossible. The alternative employ aggregate measuring techniques (such EM, CT, US MRI), whereby measurement taken over larger window, exact at are subsequently extracted computational methods. In this paper, extend MIR framework number ways. First, consider generalized setting where inspected object arbitrary graph G, vector ??Rn assigns ?v node v. A centred v will capture window encompassing its entire neighbourhood N[v], i.e., outcome Pv=?w?N[v]?w. We give criterion graphs extended can solved extracting ? from probes, P={Pv?v?V}. then cases such impossible (namely, G P inconclusive, sense there may more than one yielding P). probes technically available, yet expensive must used sparingly. show cases, still possible achieve based combination ordinary (aggregate) together with suitable set probes. aim identifying minimum necessary unique reconstruction, depending topology. referred Minimum Surgical Probing (MSP). Besides providing solution above problems graphs, also explore range behaviours determining certain specific families, perfect k-ary trees, paths, cycles, grids, tori, tubes hypercubes.