Michela Farenzena

3D Reconstruction based on interval analysis
Date: Mon 28th June
Location: Law 207 – Note: LAW DEPARTMENT
Speaker: Michela Farenzena, University of Verona, Italy

Interval analysis is an arithmetic defined on intervals, rather than on real numbers. It is a powerful tool that we
used to face the problem of 3D reconstruction in both calibrated and uncalibrated case. In particular our aim is
obtaining rigorous bounds to the position of 3-D points when data are affected by uncertainty. By “rigorous bounds”
we mean that the true unknown 3-D points are guaranteed to lie within the given intervals. For the calibrated case we
consider that both the camera matrix and the image points are affected by uncertainty. Established techniques are
based on statistical analysis of error propagation: given an input error distribution, a characterization of the
output error distribution is produced. We used a different approach: data are represented by intervals containing the
real value, and the width of the interval represent a bound to the error. Arithmetic operations are then performed on
these intervals, with the guarantee that the resulting interval contains the exact result.

As to the uncalibrated case, we address the problem of autocalibration of a moving camera with unknown constant
intrinsic parameters. Existing autocalibration techniques use numerical optimization algorithms whose convergence to
the correct result cannot be guaranteed, in general. To address this problem, we have developed a method where an
interval branch-and-bound method is employed for numerical minimization. Thanks to the properties of Interval Analysis
this method converges to the global solution with mathematical certainty and arbitrary accuracy, and the only input
information it requires from the user are a set of point correspondences and a search interval.