Generalizations of the projective reconstruction theorem
Download (20.29 MB)

Altmetric Citations
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 factorizationbased techniques, which try to solve the projective equations directly for the structure points and projection matrices, rather than the so called tensorbased approaches. First, we consider the classic case of 3D to 2D projections. Our new theorem shows that...[Show more]
dc.contributor.author  Nasihatkon, Behrooz  

dc.date.accessioned  20181122T00:09:45Z  
dc.date.available  20181122T00:09:45Z  
dc.date.copyright  2014  
dc.date.created  2014  
dc.identifier.other  b3579048  
dc.identifier.uri  http://hdl.handle.net/1885/151688  
dc.description.abstract  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 factorizationbased techniques, which try to solve the projective equations directly for the structure points and projection matrices, rather than the so called tensorbased 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 camerapoint configuration. This is very useful for the design and analysis of different factorizationbased 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.extent  xvi, 137 leaves.  
dc.language.iso  en_AU  
dc.rights  Author retains copyright  
dc.subject.lcsh  Geometry, projective  
dc.subject.lcsh  Computer vision  
dc.subject.lcsh  Image reconstruction  
dc.subject.lcsh  Image processing Mathematics  
dc.title  Generalizations of the projective reconstruction theorem  
dc.type  Thesis (PhD)  
local.contributor.supervisor  Hartley, Richard  
local.description.notes  Thesis (Ph.D.)Australian National University  
local.type.status  Accepted Version  
local.contributor.affiliation  Australian National University. Research School of Engineering  
local.identifier.doi  10.25911/5d5152776452d  
dc.date.updated  20181121T12:12:27Z  
dcterms.accessRights  Open Access  
local.mintdoi  mint  
Collections  Open Access Theses 
Download
File  Description  Size  Format  Image 

b35790489_Nasihatkon_B.pdf  20.29 MB  Adobe PDF 
Items in Open Research are protected by copyright, with all rights reserved, unless otherwise indicated.
Updated: 22 January 2019/ Responsible Officer: University Librarian/ Page Contact: Library Systems & Web Coordinator