Weighted Microscopic Image Reconstruction

Assume that we inspect a specimen represented as a collection of points. The points are typically organized on a grid structure in 2D- or 3D-space, and each point has an associated physical value. The goal of the inspection is to determine these values. Yet, measuring these values directly (by surgical probes) may damage the specimen or is simply impossible. The alternative is to employ aggregate measuring techniques (e.g., CT or MRI), whereby measurements are taken over subsets of points, and the exact values at each point are subsequently extracted by computational methods. In the Minimum Surgical Probing problem (MSP) the inspected specimen is represented by a graph G and a vector ℓ∈ Rⁿ that assigns a value ℓ_i to each vertex i. An aggregate measurement (called probe) centered at vertex i captures its entire neighborhood, i.e., the outcome of a probe centered at i is P_i= ∑ _{j∈N(i)∪{i}}ℓ_j where N(i) is the open neighborhood of vertex i. Bar-Noy et al. [4] gave a criterion whether the vector ℓ can be recovered from the collection of probes P={Pv|v∈V(G)} alone. However, there are graphs where P is inconclusive, i.e., there are several vectors ℓ that are consistent with P. In these cases, we are allowed to use surgical probes. A surgical probe at vertex i returns ℓ_i. The objective of MSP is to recover ℓ from P and G using as few surgical probes as possible. In this work, we introduce the Weighted Minimum Surgical Probing (WMSP) problem in which a vertex i may have an aggregation coefficient w_i, namely P_i= ∑ _j∈N(i)ℓ_j+ w_iℓ_i. We show that WMSP can be solved in polynomial time. Moreover, we analyze the number of required surgical probes depending on the weight vector w. For any graph, we give two boundaries outside of which no surgical probes are needed to recover the vector ℓ. The boundaries are connected to the (Signless) Laplacian matrix. In addition, we focus on the special case, where. We explore the range of possible behaviors of WMSP by determining the number of surgical probes necessary in certain graph families, such as trees and various grid graphs. Finally, we analyze higher dimensional grids graphs. For the hypercube, when, we only need surgical probes if the dimension is odd, and when, we only need surgical probes if the dimension is even. The number of surgical probes follows the binomial coefficients.