On Weighting and Choosing Constraints for Optimally Reconstructing the Geometry of Image Triplets

  title={On Weighting and Choosing Constraints for Optimally Reconstructing the Geometry of Image Triplets},
  author={Wolfgang F{\"o}rstner},
Optimally reconstructing the geometry of image triplets from point correspondences requires a proper weighting or selection of the used constraints between observed coordinates and unknown parameters. By analysing the ML-estimation process the paper solves a set of yet unsolved problems: (1) The minimal set of four linearily independent trilinearities (Shashua 1995, Hartley 1995) actually imposes only three constraints onto the geometry of the image triplet. The seeming contradiction between… 

Matching, reconstructing and grouping 3D lines from multiple views using uncertain projective geometry

  • S. HeuelW. Förstner
  • Computer Science
    Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001
  • 2001
A geometric method for matching 2D line segments from multiple oriented images, optimally reconstructing 3Dline segments and grouping 3D lines to corners is presented, which gives reasonable results.

Conformal geometric algebra in stochastic optimization problems of 3D-vision applications

A new versatile parameter estimation technique is proposed, termed GH-method, that reconciles two different mathematical areas, that is algebra and stochastics, under the umbrella of geometry and shows up the advantages inhering with this combination.

Continuous Stereo Self-Calibration by Camera Parameter Tracking

A consistent framework for continuous stereo self-calibration is presented and a method combining two of the above constraints is proposed: reduced order bundle adjustment gives by far the most accurate results (and might suffice on its own in some environments), but the epipolar constraint yields instantaneous calibration that is not affected by independently moving objects in the scene.

On the Fitting of Surfaces to Data with Covariances

This work considers the problem of estimating parameters of a model described by an equation of special form, and generates a Newton-like iterative scheme that has as its theoretical limit the minimizer of the cost function.


Orientation procedures are preceived as the central part of ph togrammetry. During the last decade the problem of deter mining the interior and the exterior orientation of one or mo re cameras has

Optimization without Minimization Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation

This work presents an alternative approach to what it calls the “standard optimization”, which minimizes a cost function by searching a parameter space, and applies it to line fitting and multiview triangulation, producing a new algorithm far more efficient than existing methods.

Geometric Derivations Of Minimal Sets Of Sufficient Multiview Constraints

Geometric interpretations of four of the most common determinant formulations of multiview constraints are given, showing that they all enforce the same geometry and that all of the forms commonly in

Optimal computation of 3-D similarity: Gauss-Newton vs. Gauss-Helmert

Is covariance information useful in estimating vision parameters?

The consequences for parameter estimation of inaccuracies in the characteristiziaton of noise that inevitably arise in practical computation are investigated.

What value covariance information in estimating vision parameters?

The value of covariances to the estimation process when applied to measured image point locations is considered and tests are undertaken to determine the impact of imprecise covariance information upon the quality of parameter estimates.



Optimal Estimation of Matching Constraints

A numerical library for estimating multi-image matching constraints, or more precisely the multi-camera geometry underlying them, designed to be modular and open-ended, so that new feature types or error models, new constraint types or parametrizations, and new numerical resolution methods are relatively easy to add.

A linear method for reconstruction from lines and points

  • R. Hartley
  • Mathematics
    Proceedings of IEEE International Conference on Computer Vision
  • 1995
The trifocal tensor is shown to be essentially identical to a set of coefficients introduced by Shashua (1994) to effect point transfer in the three-view case and to be extended to allow for the computation of the trifoc tensor given any mixture of sufficiently many line and point correspondences.

On the geometry and algebra of the point and line correspondences between N images

The formalism of the Grassmann-Cayley algebra is proposed to use as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed).

Algebraic Functions For Recognition

  • A. Shashua
  • Mathematics
    IEEE Trans. Pattern Anal. Mach. Intell.
  • 1995
The trilinearity result is shown to be of much practical use in visual recognition by alignment-yielding a direct reprojection method that cuts through the computations of camera transformation, scene structure and epipolar geometry.

Robust parameterization and computation of the trifocal tensor

Grassman–Cayley algebra for modelling systems of cameras and the algebraic equations of the manifold of trifocal tensors

  • O. FaugerasT. Papadopoulo
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 1998
We show how to use the Grassmann–Cayley algebra to model systems of one, two and three cameras. We start with a brief introduction of the Grassmann–Cayley or double algebra and proceed to demonstrate

An improved algorithm for algebraic curve and surface fitting

  • G. Taubin
  • Mathematics, Computer Science
    1993 (4th) International Conference on Computer Vision
  • 1993
A more complex, and better, approximation to the Euclidean distance is introduced from a point to an alegbraic curve or surface, and it is shown that this new approximate distance produces results of the same quality as those based on the exact Euclidesan distance, and much better than those obtained using other available methods.

Computer and Robot Vision

This two-volume set is an authoritative, comprehensive, modern work on computer vision that covers all of the different areas of vision with a balanced and unified approach.

Metrischer Trifokaltensor für die Auswertung von Bildfolgen

Der Artikel stellt ein Verfahren zur Auswertung von Bildfolgen vor, welches Trilinearitaten fur die Bestimmung der Kamerabewegung und Pradiktion von Punkten in Folgebilder einsetzt. Eine spezielle

Computer and Robot Vision

Computer and Robot Vision Vol. 1, by R.M. Haralick and Linda G. Shapiro, Addison-Wesley, 1992, ISBN 0-201-10887-1.