Uncertainty Modeling for Optimal Structure from Motion
The parameters estimated by Structure from Motion (SFM) contain inherent indeterminacies which we call gauge freedoms. Under a perspective camera, shape and motion parameters are only recovered up to an unknown similarity transformation. In this paper we investigate how covariance-based uncertainty can be represented under these gauge freedoms. Past work on uncertainty modeling has implicitly imposed gauge constraints on the solution before considering covariance estimation. Here we examine the effect of selecting a particular gauge on the uncertainty of parameters. We show potentially dramatic effects of gauge choice on parameter uncertainties. However the inherent geometric uncertainty remains the same irrespective of gauge choice. We derive a Geometric Equivalence Relationship with which covariances under different parametrizations and gauges can be compared, based on their true geometric uncertainty. We show that the uncertainty of gauge invariants exactly captures the geometric uncertainty of the solution, and hence provides useful measures for evaluating the uncertainty of the solution. Finally we propose a fast method for covariance estimation and show its correctness using the Geometric Equivalence Relationship.
KeywordsUncertainty Modeling Motion Parameter Tangent Plane Generalize Inverse Normal Covariance
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- 1.K. B. Atkinson, Close Range Photogrammetry and Machine Vision, Whittles Publ., Caithness, Scotland, (1996).Google Scholar
- 2.R. Hartley, Euclidean reconstruction from uncalibrated views, Proc. DARPAESPRIT Workshop App. Invariants in Comp. Vis., Portugal, 1993 187–202.Google Scholar
- 3.A. Heyden & K. Astrom, Euclidean reconstruction from image sequences with varying and unknown focal length and principal point, Proc. Comp. Vision Patt. Recog., Peurto Rico, 1997 438–443.Google Scholar
- 4.T. Kanade & D. D. Morris, Factorization methods for structure from motion, Phil. Trans. R. Soc. Long. A, 1998 1153–1173.Google Scholar
- 7.K. Kanatani & D. D. Morris, Gauges and gauge transformations in 3-D reconstruction from a sequence of images, To appear in ACCV 2000.Google Scholar
- 8.L. L. Kontsevich, M. L. Kontsevich & A. K. Shen, Two algorithms for reconstructing shapes, Avtometriya (1987), (5), 76–81, (Transl. Optoelec., Instrum. Data Proc., no. 5. 76-81, 1987).Google Scholar
- 9.D. D. Morris & T. Kanade, A unified factorization algorithm for points, line segments and planes with uncertainty models, Proc. Sixth Int. Conf. Comp. Vision, Bombay, India, 1998 696–702.Google Scholar
- 10.C. Poelman & T. Kanade, A paraperspective factorization method for shape and motion recovery, IEEE Trans. Patt. Anal. Mach. Intell. (Mar. 1997), 19(3), 206–18.Google Scholar
- 12.P. Sturm & B. Triggs, A factorization based algorithm for multi-image projective structure and motion, Proc. Euro. Conf. Comp. Vision, Cambridge, UK, 1996 709–720.Google Scholar
- 13.R. Szeliski & S. B. Kang, Recovering 3D shape and motion from image streams using non-linear least squares, J. of Visual Comm. and Image Rep. (Mar. 1994), 5(1), 10–28.Google Scholar
- 14.R. Szeliski & S. B. Kang, Shape ambiguities in structure from motion, IEEE Trans. Patt. Anal. Mach. Intell. (May 1997), 19(5), 506–512.Google Scholar
- 15.J. I. Thomas, A. Hanson & J. Oliensis, Refining 3D reconstructions: A theoretical and experimental study of the effect of cross-correlations, CVGIP (Nov. 1994), 60(3), 359–370.Google Scholar