The generalized microscopic image reconstruction problem

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