TY - GEN
T1 - The generalized microscopic image reconstruction problem
AU - Bar-Noy, Amotz
AU - Böhnlein, Toni
AU - Lotker, Zvi
AU - Peleg, David
AU - Rawitz, Dror
N1 - Funding Information: This work was supported by US-Israel BSF grant 2018043. Amotz Bar-Noy: ARL Cooperative Grant, ARL Network Science CTA, W911NF-09-2-0053 Funding Information: Funding This work was supported by US-Israel BSF grant 2018043. Amotz Bar-Noy: ARL Cooperative Grant, ARL Network Science CTA, W911NF-09-2-0053 Dror Rawitz: ISF grant no. 497/14 Publisher Copyright: © Amotz Bar-Noy, Toni Böhnlein, Zvi Lotker, David Peleg, and Dror Rawitz; licensed under Creative Commons License CC-BY
PY - 2019/12/1
Y1 - 2019/12/1
N2 - This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini and Nivat [7, 12]. 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 `x, which we would like to determine. However, it might be that obtaining these values precisely (by 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 ` ∈ Rn assigns a value `v to each node v. A probe centered at a vertex v will capture a window encompassing its entire neighborhood N[v], i.e., the outcome of a probe centered at v is Pv = Pw∈N[v] `w. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector ` from the collection of probes, P¯ = {Pv | 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 ` yielding P). Let us assume that surgical probes (whose outcome at vertex v is the exact value of `v) are technically available to us (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 standard 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 behaviors 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 and tubes.
AB - This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini and Nivat [7, 12]. 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 `x, which we would like to determine. However, it might be that obtaining these values precisely (by 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 ` ∈ Rn assigns a value `v to each node v. A probe centered at a vertex v will capture a window encompassing its entire neighborhood N[v], i.e., the outcome of a probe centered at v is Pv = Pw∈N[v] `w. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector ` from the collection of probes, P¯ = {Pv | 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 ` yielding P). Let us assume that surgical probes (whose outcome at vertex v is the exact value of `v) are technically available to us (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 standard 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 behaviors 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 and tubes.
KW - Combinatorics
KW - Discrete mathematics
KW - Graph spectra
KW - Grid graphs
KW - Image reconstruction
KW - Reconstruction algorithm
UR - http://www.scopus.com/inward/record.url?scp=85076336707&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.ISAAC.2019.42
DO - https://doi.org/10.4230/LIPIcs.ISAAC.2019.42
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 30th International Symposium on Algorithms and Computation, ISAAC 2019
A2 - Lu, Pinyan
A2 - Zhang, Guochuan
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 30th International Symposium on Algorithms and Computation, ISAAC 2019
Y2 - 8 December 2019 through 11 December 2019
ER -