An ellipsoidal calculus based on propagation and fusion

Presents an ellipsoidal calculus based solely on two basic operations: propagation and fusion. Propagation refers to the problem of obtaining an ellipsoid that must satisfy an affine relation with another ellipsoid, and fusion to that of computing the ellipsoid that tightly bounds the intersection of two given ellipsoids. These two operations supersede the Minkowski sum and difference, affine transformation and intersection tight bounding of ellipsoids on which other ellipsoidal calculi are based. Actually, a Minkowski operation can be seen as a fusion followed by a propagation and an affine transformation as a particular case of propagation. Moreover, the presented formulation is numerically stable in the sense that it is immune to degeneracies of the involved ellipsoids and/or affine relations. Examples arising when manipulating uncertain geometric information in the context of the spatial interpretation of line drawings are extensively used as a testbed for the presented calculus.

[1]  J. Norton,et al.  State bounding with ellipsoidal set description of the uncertainty , 1996 .

[2]  N. Shor,et al.  New algorithms for constructing optimal circumscribed and inscribed ellipsoids , 1992 .

[3]  J. Norton Recursive computation of inner bounds for the parameters of linear models , 1989 .

[4]  Uwe D. Hanebeck,et al.  Fusing Information Simultaneously Corrupted by Uncertainties with Known Bounds and Random Noise with Known Distribution , 2000, Inf. Fusion.

[5]  Eric Walter,et al.  Trace Versus Determinant in Ellipsoidal Outer-Bounding, with Application to State Estimation , 1996 .

[6]  S. M. Veres,et al.  Using MATLAB toolbox “GBT” in identification and control , 1994 .

[7]  M. Milanese,et al.  Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: Linear families of models and estimators , 1982 .

[8]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[9]  Federico Thomas,et al.  Set membership approach to the propagation of uncertain geometric information , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[10]  Federico Thomas,et al.  Numerical analysis of the instantaneous motions of panel-and-hinge frameworks and its application to computer vision , 2001 .

[11]  L. Giralt,et al.  A kinematic-geometric approach to spatial interpretation of line drawings , 2000 .

[12]  E. Walter,et al.  Estimation of parameter bounds from bounded-error data: a survey , 1990 .

[13]  Michael J. Todd,et al.  The Ellipsoid Method: A Survey , 1980 .

[14]  E. Walter,et al.  Minimal volume ellipsoids , 1994 .

[15]  Walter Whiteley,et al.  Weavings, sections and projections of spherical polyhedra , 1991, Discret. Appl. Math..

[16]  Federico Thomas,et al.  Overcoming Superstrictness in Line Drawing Interpretation , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  Y. F. Huang,et al.  On the value of information in system identification - Bounded noise case , 1982, Autom..

[18]  A. B. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Feedback Control , 1997 .

[19]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

[20]  W. Kahan,et al.  Circumscribing an Ellipsoid about the Intersection of Two Ellipsoids , 1968, Canadian Mathematical Bulletin.

[21]  Hermann A. Maurer,et al.  New Results and New Trends in Computer Science , 1991, Lecture Notes in Computer Science.

[22]  G. Belforte,et al.  An Improved Parameter Identification Algorithm for Signals with Unknown-But-Bounded Errors , 1985 .

[23]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[24]  Michael J. Todd,et al.  Feature Article - The Ellipsoid Method: A Survey , 1981, Oper. Res..