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Generalizations of the projective reconstruction theorem

Nasihatkon, Behrooz

Description

We present generalizations of the classic theorem of projective reconstruction as a tool for the design and analysis of the projective reconstruction algorithms. Our main focus is algorithms such as bundle adjustment and factorization-based techniques, which try to solve the projective equations directly for the structure points and projection matrices, rather than the so called tensor-based approaches. First, we consider the classic case of 3D to 2D projections. Our new theorem shows that...[Show more]

dc.contributor.authorNasihatkon, Behrooz
dc.date.accessioned2018-11-22T00:09:45Z
dc.date.available2018-11-22T00:09:45Z
dc.date.copyright2014
dc.date.created2014
dc.identifier.otherb3579048
dc.identifier.urihttp://hdl.handle.net/1885/151688
dc.description.abstractWe present generalizations of the classic theorem of projective reconstruction as a tool for the design and analysis of the projective reconstruction algorithms. Our main focus is algorithms such as bundle adjustment and factorization-based techniques, which try to solve the projective equations directly for the structure points and projection matrices, rather than the so called tensor-based approaches. First, we consider the classic case of 3D to 2D projections. Our new theorem shows that projective reconstruction is possible under a much weaker restriction than requiring, a priori, that all estimated projective depths are nonzero. By completely specifying possible forms of wrong configurations when some of the projective depths are allowed to be zero, the theory enables us to present a class of depth constraints under which any reconstruction of cameras and points projecting into given image points is projectively equivalent to the true camera-point configuration. This is very useful for the design and analysis of different factorization-based algorithms. Here, we analyse several constraints used in the literature using our theory, and also demonstrate how our theory can be used for the design of new constraints with desirable properties. The next part of the thesis is devoted to projective reconstruction in arbitrary dimensions, which is important due to its applications in the analysis of dynamical scenes. The current theory, due to Hartley and Schaffalitzky, is based on the Grassmann tensor, generalizing the notions of Fundamental matrix, trifocal tensor and quardifocal tensor used for 3D to 2D projections. We extend their work by giving a theory whose point of departure is the projective equations rather than the Grassmann tensor. First, we prove the uniqueness of the Grassmann tensor corresponding to each set of image points, a question that remained open in the work of Hartley and Schaffalitzky. Then, we show that projective equivalence follows from the set of projective equations, provided that the depths are all nonzero. Finally, we classify possible wrong solutions to the projective factorization problem, where not all the projective depths are restricted to be nonzero. We test our theory experimentally by running the factorization based algorithms for rigid structure and motion in the case of 3D to 2D projections. We further run simulations for projections from higher dimensions. In each case, we present examples demonstrating how the algorithm can converge to the degenerate solutions introduced in the earlier chapters. We also show how the use of proper constraints can result in a better performance in terms of finding a correct solution.
dc.format.extentxvi, 137 leaves.
dc.language.isoen_AU
dc.rightsAuthor retains copyright
dc.subject.lcshGeometry, projective
dc.subject.lcshComputer vision
dc.subject.lcshImage reconstruction
dc.subject.lcshImage processing Mathematics
dc.titleGeneralizations of the projective reconstruction theorem
dc.typeThesis (PhD)
local.contributor.supervisorHartley, Richard
local.description.notesThesis (Ph.D.)--Australian National University
local.type.statusAccepted Version
local.contributor.affiliationAustralian National University. Research School of Engineering
local.identifier.doi10.25911/5d5152776452d
dc.date.updated2018-11-21T12:12:27Z
dcterms.accessRightsOpen Access
local.mintdoimint
CollectionsOpen Access Theses

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