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Algorithms to perform point-based motion estimation under orthographic and scaled orthographic projection abound in the literature. A key limitation of many existing algorithms is that they operate on the minimum amount of data required, often requiring the selection of a suitable minimal set from the available data to serve as a "local coordinate frame". Such approaches are extremely sensitive to errors and noise in the minimal set, and forfeit the advantages of using the full data set. Furthermore, attention is seldom paid to the statistical performance of the algorithms. We present a new framework that allows all available features to be used in the motion computations, without the need to select a frame explicitly. This theory is derived in the context of the affine camera, which preserves parallelism and generalises the orthographic, scaled orthographic and para-perspective models. We define the affine epipolar geometry for two such cameras, giving the fundamental matrix in this case. The noise resistant computation of the epipolar geometry is discussed, and a statistical noise model constructed so that confidence in the results can be assessed. The rigid motion parameters are then determined directly from the epipolar geometry, using the novel rotation representation of Koenderink and van Doorn (1991). The two-view partial motion solution comprises the scale factor between views, the projection of the 3D axis of rotation and the cyclotorsion angle, while the addition of a third view allows the true 3D rotation axis to be computed (up to a Necker reversal). The computed uncertainties in these parameters permit optimal estimates to be obtained over time by means of a linear Kalman filter. Our theory extends work by Huang and Lee (1989), Harris (1990), and Koenderink and van Doorn (1991), and results are given on both simulated and real data. © 1995 Kluwer Academic Publishers.

Original publication




Journal article


International Journal of Computer Vision

Publication Date





147 - 182