Weighted Microscopic Image Reconstruction

Abstract

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.

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 $\mathcal{P}_i = \sum _{j \in N(i)} \ell _j + w_i \ell _i$. 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 $\ell$. 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.

This work was supported by US-Israel BSF grant 2018043 and ARL Cooperative Grant ARL Network Science CTA W911NF-09-2-0053.