Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices

Miranda Krekovic, Ivan Dokmanic, Martin Vetterli

Research output: Contribution to journalArticlepeer-review


We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the 'structure from sound' approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate all cases of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D.

Original languageEnglish (US)
Article number9044768
Pages (from-to)2480-2498
Number of pages19
JournalIEEE Transactions on Signal Processing
StatePublished - 2020


  • Point-to-plane distance matrix
  • collocated source and receiver
  • indoor localization and mapping
  • inverse problem in the Euclidean space
  • uniqueness of the reconstruction

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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