Factorization with Uncertainty

  • Michal Irani
  • P. Anandan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1842)


Factorization using Singular Value Decomposition (SVD) is often used for recovering 3D shape and motion from feature correspondences across multiple views. SVD is powerful at finding the global solution to the associated least-square-error minimization problem. However, this is the correct error to minimize only when the x and y positional errors in the features are uncorrelated and identically distributed. But this is rarely the case in real data. Uncertainty in feature position depends on the underlying spatial intensity structure in the image, which has strong directionality to it. Hence, the proper measure to minimize is covariance-weighted squared-error (or the Mahalanobis distance). In this paper, we describe a new approach to covariance-weighted factorization, which can factor noisy feature correspondences with high degree of directional uncertainty into structure and motion. Our approach is based on transforming the raw-data into a covariance-weighted data space, where the components of noise in the different directions are uncorrelated and identically distributed. Applying SVD to the transformed data now minimizes a meaningful objective function. We empirically show that our new algorithm gives good results for varying degrees of directional uncertainty. In particular, we show that unlike other SVD-based factorization algorithms, our method does not degrade with increase in directionality of uncertainty, even in the extreme when only normal-flow data is available. It thus provides a unified approach for treating corner-like points together with points along linear structures in the image.


Singular Value Decomposition Mahalanobis Distance Feature Position Factorization Algorithm Singular Value Decomposition Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Michal Irani
    • 1
  • P. Anandan
    • 2
  1. 1.Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Microsoft CorporationRedmondUSA

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