# Canonic Representations for the Geometries of Multiple Project Views

### Q.-T. Luong and T. Vieville

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-93-772

November 1993

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/CSD-93-772.pdf

We show how a special decomposition of a set of two or three general projection matrices, called canonic enables us to build geometric descriptions for a system of cameras which are invariant with respect to a given group of transformations. These representations are minimal and capture completely the properties of each level of description considered: Euclidean (in the context of calibration, and in the context of structure from motion, which we distinguish clearly), affine, and projective, that we also relate to each other. In the last case, a new decomposition of the well-known fundamental matrix is obtained. Dependencies, which appear when three or more views are available, are studied in the context of the canonic decomposition, and new composition formulas are established. The theory is illustrated by examples with real images.

Keywords: 3D vision, perspective projection, invariants, motion, self-calibration

BibTeX citation:

@techreport{Luong:CSD-93-772, Author = {Luong, Q.-T. and Vieville, T.}, Title = {Canonic Representations for the Geometries of Multiple Project Views}, Institution = {EECS Department, University of California, Berkeley}, Year = {1993}, Month = {Nov}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/6296.html}, Number = {UCB/CSD-93-772}, Abstract = {We show how a special decomposition of a set of two or three general projection matrices, called canonic enables us to build geometric descriptions for a system of cameras which are invariant with respect to a given group of transformations. These representations are minimal and capture completely the properties of each level of description considered: Euclidean (in the context of calibration, and in the context of structure from motion, which we distinguish clearly), affine, and projective, that we also relate to each other. In the last case, a new decomposition of the well-known fundamental matrix is obtained. Dependencies, which appear when three or more views are available, are studied in the context of the canonic decomposition, and new composition formulas are established. The theory is illustrated by examples with real images. <p>Keywords: 3D vision, perspective projection, invariants, motion, self-calibration} }

EndNote citation:

%0 Report %A Luong, Q.-T. %A Vieville, T. %T Canonic Representations for the Geometries of Multiple Project Views %I EECS Department, University of California, Berkeley %D 1993 %@ UCB/CSD-93-772 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/6296.html %F Luong:CSD-93-772