The generalized microscopic image reconstruction problem

Amotz Bar-Noy, Toni Böhnlein, Zvi Lotker, David Peleg, Dror Rawitz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish
Title of host publication30th International Symposium on Algorithms and Computation, ISAAC 2019
EditorsPinyan Lu, Guochuan Zhang
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771306
DOIs
StatePublished - 1 Dec 2019
Event30th International Symposium on Algorithms and Computation, ISAAC 2019 - Shanghai, China
Duration: 8 Dec 201911 Dec 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume149
ISSN (Print)1868-8969

Conference

Conference30th International Symposium on Algorithms and Computation, ISAAC 2019
Country/TerritoryChina
CityShanghai
Period8/12/1911/12/19

Keywords

  • Combinatorics
  • Discrete mathematics
  • Graph spectra
  • Grid graphs
  • Image reconstruction
  • Reconstruction algorithm

ASJC Scopus subject areas

  • Software

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