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Minimal Representations for Testing and Estimation in Projective Spaces

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Abstract:

Testing and estimation using homogeneous coordinates and matrices has to cope with obstacles such as singularities of covariance matrices and redundant parametrisations. The paper proposes a representation of the uncertainty of all types of geometric entities which (1) only requires the minimum number of parameters, (2) is free of singularities, (3) enables to exploit the simplicity of homogeneous coordinates to represent geometric constraints and (4) allows to handle geometric entities which are at infinity or at least very far away. We develop the concept, discuss its usefulness for bundle adjustment and demonstrate its applicability for determining 3D lines from observed image line segments in a multi view setup.

German
Beim Testen und Schätzen mit homogenen Koordinaten und Matrizen treten wegen der Redundanz der Repräsentationen und der daraus folgenden Singularität der Kovarianzmatrizen Schwierigkeiten auf. Der Beitrag schlägt eine Repräsentation für die Unsicherheit geometrischer Elemente vor, die (1) eine minimale Zahl von Parametern benötigt, (2) frei von Singularitäten ist, (3) die Einfachheit homogenener Koordinaten bei der Formulierung geometrischer Bedingungen belässt und (4) uneigentliche Elemente, d.h. Elemente im Unendlichen bzw. sehr weit entfernte Elemente behandeln kann. Wir stellen das Konzept vor, diskutieren seine Nützlichkeit bei der Bündelausgleichung und zeigen seine Anwendbarkeit für die Schätzung von 3D Geraden aus mehreren Bildern.

Keywords: ESTIMATION; MINIMAL REPRESENTATION; PROJECTIVE SPACE

Document Type: Research Article

DOI: https://doi.org/10.1127/1432-8364/2012/0112

Publication date: 2012-06-01

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  • Photogrammetrie - Fernerkundung - Geoinformation (PFG) is an international scholarly journal covering the progress and application of photogrammetric methods, remote sensing technology and the intricately connected field of geoinformation processing.

    Papers published in PFG highlight new developments and applications of these technologies in practice. The journal hence addresses both researchers and student of these disciplines at academic institutions and universities and the downstream users in both the private sector and public administration.

    PFG places special editorial emphasis on the communication of new methodologies in data acquisition, new approaches to optimized processing and interpretation of all types of data which were acquired by photogrammetric methods, remote sensing, image processing and the computer-aided interpretation of such data in general.

    PFG is the official journal of the German Society of Photogrammetry and Remote Sensing.
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