Globally Convergent Autocalibration Using Interval Analysis

M. Farenzena, A. Fusiello, A. Benedetti, and A. Busti. Department of Computer Science
University of Verona, Verona - Italy

Overview

Existing autocalibration techniques use numerical optimization algorithms that are prone to the problem of local minima. 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 is guaranteed to converge to the global solution with mathematical certainty and arbitrary accuracy, and the only input information it requires from the user is a set of point correspondences and a search box. The cost function is based on the Huang-Faugeras constraint of the fundamental matrix. A recently proposed interval extension base on Bernstein polynomial forms has been investigated to speed up the search for the solution.

Method

For each pair of images the fundamental matrix is computed;
The cost function, based on Huang and Faugeras constraints, is minimized by an algorithm based on Interval Analysis, that has mathematical garancy of converge;
The final results are the camera intrinsic parameters.

Uncalibrated sequences

Sequence

Interval autocalibration

Other autocalibration

Structure

au	=	[618.28, 619.18]
av	=	[698.99, 700.43]
u0	=	[233.69, 235.28]
v0	=	[372.05, 373.24]

au	=	681.345
av	=	679.285
u0	=	258.802
vu	=	383.188

[Zeller-Faugeras 1996]

au	=	[799.30, 799.84]
av	=	[830.34, 831.19]
u0	=	[405.12, 405.72]
v0	=	[351.21, 352.00]

au	=	837
av	=	837
u0	=	-
vu	=	-

[Ueshiba-Tomita 1998]

au	=	[600.98, 601.53]
av	=	[719.45, 720.00]
u0	=	[409.42, 409.97]
v0	=	[190.49, 191.14]

au	=	604.93
av	=	712.94
u0	=	377.86
vu	=	313.88

[Lourakis-Deriche 1999]

Calibrated sequences

Sequence

Interval autocalibration

Calibration

Structure

au	=	[1485.62, 1486.67]
av	=	[1445.62, 1446.96]
u0	=	[698.00, 699.00]
v0	=	[432.00, 432.75]

au	=	1462.996
av	=	1458.304
u0	=	635.662
vu	=	497.794

au	=	[1436.40, 1436.95]
av	=	[1406.61, 1407.28]
u0	=	[612.00, 613.00]
v0	=	[561.00, 561.75]

au	=	1462.996
av	=	1458.304
u0	=	635.662
vu	=	497.794

au	=	[1439.14, 1439.68]
av	=	[1482.61, 1482.94]
u0	=	[604.50, 605.00]
v0	=	[477.37, 477.75]

au	=	1462.996
av	=	1458.304
u0	=	635.662
vu	=	497.794

Reference Paper

A. Fusiello, A. Benedetti, M. Farenzena, and A. Busti. Globally convergent autocalibration using interval analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(12):1633-1638, December 2004. (PDF)