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 \(\ell \in \mathbb {R}^n\) that assigns a value \(\ell _i\) 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 \(\mathcal{P}_i = \sum _{j \in N(i) \cup \{i\}} \ell _j\) where N(i) is the open neighborhood of vertex i. Bar-Noy et al. [4] gave a criterion whether the vector \(\ell \) can be recovered from the collection of probes \(\mathcal{P}= \{\, \mathcal{P}_v \; | \; v \in V(G)\}\) alone. However, there are graphs where \(\mathcal{P}\) is inconclusive, i.e., there are several vectors \(\ell \) that are consistent with \(\mathcal{P}\). In these cases, we are allowed to use surgical probes. A surgical probe at vertex i returns \(\ell _i\). The objective of MSP is to recover \(\ell \) from \(\mathcal{P}\) and G using as few surgical probes as possible.